state standards curriculum companion · standards descriptors algebra 1 geometry algebra 2...

Ron Larson

Laurie Boswell

Timothy D. Kanold

Lee Stiff

State StandardsCurriculum Companion

Teacher’s Edition

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Teacher’s Edition

Correlation to Common Core State Standards . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4-Year Scope and Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Essential Course of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Pacing for 50-Minute Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Pacing for 90-Minute Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Course Planner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Skills Readiness Pre-Course Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Additional Content

Investigating Geometry Activity 4.2A Rigid Motions in the Plane . . . . . . . . . . . . . . . . . . . CC1

Lesson 4.2B Relate Transformations and Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC3

Construction 4.5A Rigid Motions and Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC10

Investigating Geometry Activity 6.3A Explore Properties of Dilations. . . . . . . . . . . . . . . . CC12

Lesson 6.3B Relate Transformations and Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC13

Investigating Geometry Activity 6.4A Dilations and AA Similarity . . . . . . . . . . . . . . . . . . CC20

Extension 6.7A Partition Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC22

Construction 10.4A Tangent Lines and Inscribed Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . CC24

Extension 11.4A Measure Angles in Radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC27

Extension 12.4A Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC30

Extension 12.5A Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC32

Contents

Larson GeometryCommon Core State StandardsCurriculum Companion

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Standards Descriptors Algebra 1 Geometry Algebra 2

Standards for Mathematical Content(1 5 advanced; * 5 also a Modeling Standard)

Number and Quantity

CC.9-12.N.RN.1 Explain how the defi nition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we defi ne 51/3 to be the cube root of 5 because we want (51/3)3 5 5(1/3)3 to hold, so (51/3)3 must equal 5.

SE: 509–510 SE: 414–419, 459, 466, 469, 833, 1015

CC.9-12.N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

SE: 509–510 SE: 139, 423, 451, 457–459

SE: 420–427, 467, 469, 474, 505, 1015

CC.9-12.N.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

CCCC: CC8–CC9

CC.9-12.N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.*

Found throughout the text. See for example:SE: 17–18, 19–20, 27,

37, 42, 44–45, 47, 48, 137, 140–141, 227–228, 230–232, 429, 432–433, 519, 609, 612–613, 614, 665, 667–668, 886, 887–892, 893, 894

SE: 50–56, 63, 68, 74, 78, 97, 160, 197, 317, 529, 705, 722, 723–725, 731–735, 739–742, 745, 747–752, 755–761, 763, 767–768, 777, 778, 780, 782–784, 801, 803–809, 810–817, 818, 820–825, 827, 829–836, 839–845, 850–854, 855, 858–861, 863–865, 866–867, 878, 888–889, 897, 916–917

Found throughout the text. See for example:SE: 5, 7, 9, 20, 24, 27,

30–31, 32, 34–36, 42, 46–47, 63, 74, 100, 103–104, 134, 137, 239, 242–243, 264, 345, 356, 358, 610, 616, 618–619, 624–625, 631, 987, 991–993, 995

SE 5 Student Edition CCCC 5 Common Core Curriculum Companion

Standards for Mathematical Content Correlation for Holt McDougal Larson Algebra 1, Geometry, and Algebra 2

2 Correlation to Standards for Mathematical Content

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CC.9-12.N.Q.2 Defi ne appropriate quantities for the purpose of descriptive modeling.*

SE: 230, 337, 342, 888, 891, 893

Found throughout the text. See for example:SE: 13, 19, 20, 29, 34,

35, 36, 42, 54, 63, 66, 100, 101, 134, 155, 162, 181, 239, 254, 261, 262, 356, 373, 389, 560, 829

CC.9-12.N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.*

CCCC: CC1–CC6 SE: 482, 727–728, 763, 765–768

CC.9-12.N.CN.1 Know there is a complex number i such that i 2 5 21, and every complex number has the form a 1 bi with a and b real.

SE: 275–276

CC.9-12.N.CN.2 Use the relation i 2 5 21 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

SE: 276–278, 279–281, 291, 320–321, 323, 335, 1013

CC.9-12.N.CN.3 (1) Find the conjugate of a complex number; use conjugates to fi nd moduli and quotients of complex numbers.

SE: 278–280, 291, 321, 323, 1013

CC.9-12.N.CN.4 (1) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

SE: 278–280

CC.9-12.N.CN.5 (1) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 – Ï

}

3 i)3 5 8 because (1 – Ï

}

3 i) has modulus 2 and argument 120°.

SE: 281, 282

CC.9-12.N.CN.6 (1) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

SE: 281, 282

CC.9-12.N.CN.7 Solve quadratic equations with real coeffi cients that have complex solutions.

SE: 275, 279, 291, 323, 327, 1013


<… Correlation to Standards for Mathematical Content

Correlation to Standards for Mathematical Content 3



CC.9-12.N.CN.8 (1) Extend polynomial identities to the complex numbers. For example, rewrite x2 1 4 as (x 1 2i )(x 2 2i ).

SE: 380–382, 384, 407

CC.9-12.N.CN.9 (1) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

SE: 379–385, 405, 407

CC.9-12.N.VM.1 (1) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v ).

SE: 574–577, 587, A5, A8

SE: A7, A9–A11

Number and Quantity

CC.9-12.N.VM.2 (1) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

SE: A7, A9 SE: A8, A10–A11

CC.9-12.N.VM.3 (1) Solve problems involving velocity and other quantities that can be represented by vectors.

SE: A9 SE: A11

CC.9-12.N.VM.4 (1) Add and subtract vectors.

a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

c. Understand vector subtraction v 2 w as v 1 (2w ), where 2w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

SE: A5–A9 SE: A8, A10–A11





CC.9-12.N.VM.5 (1) Multiply a vector by a scalar.

a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) 5 (cvx, cvy).

b. Compute the magnitude of a scalar multiple cv using ||cv || 5 |c |v. Compute the direction of cv knowing that when |c |v Þ 0, the direction of cv is either along v (for c . 0) or against v (for c , 0).

SE: A7, A9 SE: A8, A10

CC.9-12.N.VM.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

SE: 94 SE: 583, 586–587 SE: 189, 192, 193

CC.9-12.N.VM.7 (1) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

SE: 95 SE: 627–631, 639, 913 SE: 188, 191, 194, 224–225, 1012

CC.9-12.N.VM.8 (1) Add, subtract, and multiply matrices of appropriate dimensions.

SE: 95 SE: 581, 584–585, 587, 912

SE: 187–188, 190–191, 194, 195–202, 209, 224, 1012

CC.9-12.N.VM.9 (1) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfi es the associative and distributive properties.

SE: 582–583, 586, 587, 912

SE: 188

CC.9-12.N.VM.10 (1) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

SE: 210–212, 214, 1012

CC.9-12.N.VM.11 (1) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

SE: 582, 592–594, 600, 603

SE: A11

CC.9-12.N.VM.12 (1) Work with 2 3 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

SE: 592–594, 600, 603 SE: 202






Algebra

CC.9-12.A.SSE.1 Interpret expressions that represent a quantity in terms of its context.*

a. Interpret parts of an expression, such as terms, factors, and coeffi cients.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 1 r )n as the product of P and a factor not depending on P.

Found throughout the text. See for example:SE: 96, 97–98, 99, 115,

121, 126–127, 244–245, 247–249, 253, 255, 256

Found throughout the text. See, for example:

SE: 49–52, 52–56, 433–435, 437, 439, 659–660, 699–700, 720–722, 730–732, 737–739, 747, 749, 755–757, 763, 779, 803–806, 810–813, 820–822


66, 90, 239, 254, 261, 262, 337, 347, 356, 373, 389, 431, 829

CC.9-12.A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 2 y4 as (x2)2 2 (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 2 y2)(x2 1 y2).

SE: 96–98, 99–101, 105, 106, 120, 123–124, 125, 555–556, 561, 562–563, 569–570, 582, 583–584, 586–588, 592, 593–594, 596–597, 600–601, 603–604, 606–608, 610

SE: 106, 713, 804–806, 810–813, 819–822, 829–831, 872, 873

SE: 12–13, 14, 16, 24, 62, 65, 252–253, 255–256, 259–260, 263–264, 265, 319–320, 323, 346–347, 353–355, 356–357

CC.9-12.A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor a quadratic expression to reveal the zeros of the function it defi nes.

b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defi nes.

c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t

ø 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

SE: 524, 536, 593, 594, 595, 597, 598, 601, 602, 603, 604, 607, 609, 612, 641–642, 647, 669–670

SE: 245–246, 248, 249, 255–256, 261–262, 287, 289–290, 490–491, 496

CCCC: CC2–CC3






CC.9-12.A.SSE.4 Derive the formula for the sum of a fi nite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

SE: 812–813, 815–817, 818, 820, 839, 841, 843, 847, 848, 872, 1021

CC.9-12.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

SE: 554–556, 557–559, 561, 562–565, 565–568, 569–571, 572–574, 580, 581, 589, 605, 615, 616–617, 621, 624

SE: 346–348, 349–352, 368, 369, 403, 407, 427, 474

CCCC: CC7–CC8

CC.9-12.A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x 2 a is p (a ), so p (a ) 5 0 if and only if (x 2 a ) is a factor of p (x ).

SE: 363–365, 366–367, 371–373, 374–375, 404, 407, 411, 451, 1014

CC.9-12.A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defi ned by the polynomial.

SE: 607, 641–642 SE: 353–356, 356–359, 362–365, 366–368, 369, 370–373, 374–377, 380, 382, 384, 387, 390–391, 399, 401, 404–405, 407, 419, 451

CC.9-12.A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 1 y2)2 5 (x2 2 y2)2 1 (2xy )2 can be used to generate Pythagorean triples.

SE: 569–571, 572–574, 600–602, 603–605, 741

SE: 347–348, 349–350, 353–355, 356–359

CC.9-12.A.APR.5 (1) Know and apply the Binomial Theorem for the expansion of (x 1 y )n in powers of x and y for a positive integer n, where x and y are any numbers, with coeffi cients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)

SE: 693–694, 695, 697, 723, 735, 737, 741, 1019





CC.9-12.A.APR.6 Rewrite simple rational expressions in different forms; write a(x )/b(x ) in the form q(x ) 1 r (x )/b(x ), where a(x ), b(x ), q(x ), and r (x ) are polynomials with the degree of r (x ) less than the degree of b (x ), using inspection, long division, or, for the more complicated examples, a computer algebra system.

SE: 783, 784–787, 788–791, 794–797, 797–800, 810–811, 832, 835, 839, 949

SE: 362–364, 366, 377, 404, 407, 474, 1014

CCCC: CC5–CC6

CC.9-12.A.APR.7 (1) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

SE: 802–805, 806–809, 812–815, 816–819, 826, 829, 830, 833–834, 835, 906, 949

SE: 573–577, 577–580, 581, 582–585, 586–588, 595, 602, 605, 607, 625, 678, 1017

CCCC: CC7–CC8

CC.9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*

Found throughout the text. See for example:SE: 137, 138–140, 143,

145–146, 150, 152–153, 155, 158–159, 358, 360–361, 365, 367–368, 371, 372–374, 380–381, 383, 385–386

Found throughout the text. See, for exampleSE: 13, 16, 19, 26, 30,

36, 39, 41, 45, 64, 65, 69, 309, 311, 313, 325, 330, 332, 339, 345, 348, 689–691, 692–693

SE: 19–20, 23–24, 42, 44, 46–47, 54, 57–58, 59, 64, 269, 270–271, 290, 295, 306, 356, 373, 376, 516, 594–595, 600, 937

CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.*

Found throughout the text. See, for example:SE: 37, 39–40, 218,

219–221, 226–228, 229–232, 245, 247–249, 254–255, 257–259, 263, 265, 267–268, 283–285, 286–289, 292–295, 296–299, 303–305, 306–308, 313, 315–316

SE: 173–174, 175–177, 180–183, 184–187

Found throughout the text. See, for example:SE: 89–92, 93–96,

98–101, 101–104, 105, 106, 107, 109–111, 115–117, 118–119, 124–125, 127, 153–155, 157–158, 162, 166, 174–175, 176, 181, 184–185, 206, 209, 213, 216, 239, 242–243






CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.*


39–40, 81, 83–84, 90, 92–93, 98, 100–101, 150, 152–153, 285, 288–289, 408, 410–411, 437, 438, 440–441, 453, 456–457, 468, 471–472, 473


101, 103–104, 105, 134, 139–138, 139, 162, 165–166, 174–175, 176, 181, 185, 186, 213, 239, 242–243

CC.9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V 5 IR to highlight resistance R.*

SE: 184–186, 187–189, 190, 191, 196, 197, 199, 212, 940

SE: 483, 486–487, 843, 877

SE: 26–29, 30–32, 40, 58, 63, 65, 69, 88, 1010

CC.9-12.A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Found throughout the text. See for example:SE: 134–137, 137–138,

141–143, 144, 148–149, 150, 154–156, 168–169, 176–178, 184–186, 191, 192–196

CCCC: CC11–CC12

Found throughout the text. See for example:SE: 104, 105–106,

108–109, 111, 119, 136, 138, 178, 212, 899

Found throughout the text. See for example:SE: 18–20, 26–29

CC.9-12.A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

SE: 729–731, 732–734, 735, 755, 757, 758–759, 760–761, 772, 820–822, 823–826, 830, 834, 835, 906, 948–949

SE: 452–455, 456–459, 460–461, 462–463, 464, 465, 468, 469, 473, 474, 498, 513, 557, 589–592, 592–595, 596–597, 598–599, 600, 601, 602, 606, 607, 619, 678





CC.9-12.A.REI.3 Solve linear equations and inequalities in one variable, including equations with coeffi cients represented by letters.

SE: 132–133, 134–137, 137–140, 141–143, 144–146, 148–150, 150–153, 154–156, 157–159, 160, 161, 163–164, 165–167, 173, 177–178, 179–181, 184–186, 187–189, 190, 191, 192–194, 196, 197, 354, 356–358, 359–361, 362, 363–365, 366–368, 369–371, 372–374, 377–378, 380–383, 384–387, 388, 390–392, 393–395


54, 84, 89, 91, 155, 158, 161, 186, 229, 266, 268, 303, 311, 323, 330, 339, 357, 358, 363, 385

SE: 18–21, 21–24, 25, 26–29, 30–32, 33, 34–36, 37–40, 41–44, 44–47, 51–55, 55–58, 59, 62–64, 65, 66–67, 68–69

CC.9-12.A.REI.4 Solve quadratic equations in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x 2 p)2 5 q that has the same solutions. Derive the quadratic formula from this form.

b. Solve quadratic equations by inspection (e.g., for x2 5 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a 6 bi for real numbers a and b.

SE: 585, 586, 589, 595–596, 597, 599, 602, 603, 605, 613, 618–619, 621, 622–623, 652–655, 655–658, 659, 661, 664–665, 666–668, 671–673, 674–676, 677, 678–680, 681–683, 695, 698–699, 701, 702–703, 707, 727

SE: 499, 882–883 SE: 252–255, 255–258, 259–262, 263–265, 266–269, 269–271, 272–273, 274, 282, 284–286, 288–291, 292–295, 296–299, 315, 319–321, 323

CC.9-12.A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

CCCC: CC18–CC19






CC.9-12.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

SE: 426, 427–430, 430–433, 434, 435–438, 439–441, 442, 443, 444–447, 447–450, 451–454, 454–457, 458, 459–462, 462–465, 472, 473, 474, 475–478, 479, 480–481, 482–483, 485, 508

SE: 183, 186, 880 SE: 152, 153–155, 156–158, 159, 160–163, 164–167, 177, 178–181, 182–185, 186, 193, 202, 203–207, 207–209, 210–213, 214–217, 218–219, 220, 221, 222–224, 226, 227

CC.9-12.A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, fi nd the points of intersection between the line y 5 23x and the circle x 2 1 y 2 5 3.

CCCC: CC21–CC27 SE: 658–661, 661–664, 667, 672, 673, 674–675, 677, 1018

CC.9-12.A.REI.8 (1) Represent a system of linear equations as a single matrix equation in a vector variable.

SE: 212–213, 214–217, 219, 226, 227, 1012

CC.9-12.A.REI.9 (1) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 3 or greater).

SE: 210–213, 214–217, 218–219, 226, 227, 1012

CC.9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

SE: 215 SE: 74

CC.9-12.A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y 5 f(x) and y 5 g(x) intersect are the solutions of the equation f(x) 5 g(x); fi nd the solutions approximately, e.g., using technology to graph the functions, make tables of values, or fi nd successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*

SE: 251–252, 643–646, 647–649, 651, 654, 713

CCCC: CC13–CC14

SE: 272–273, 360–361, 372, 374–375, 382–383, 387, 455, 460–461, 518, 523–525, 526–257, 931, 934, 938–939





CC.9-12.A.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

SE: 404, 405–408, 409–412, 413, 418, 419, 422–423, 465, 466–468, 469–472, 473, 474, 478, 479, 494, 559, 568, 580

SE: 207, 881 SE: 132–135, 135–138, 139, 140, 144, 145, 148, 150, 167, 168–170, 171–173, 174–175, 176, 186, 193, 209, 217, 221, 223, 227, 230, 232, 291, 299

Functions

CC.9-12.F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

SE: 35–36, 38, 43–45, 48, 49–50, 52, 56, 57, 167, 263, 264, 266–268

SE: 73–74, 77–78, 96, 141, 145, 148, 232, 1011

CC.9-12.F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

SE: 262–265, 265–268, 269, 274, 275, 279, 330, 396–397, 941

SE: 75–76, 78, 81, 120, 127, 130–131, 141, 145, 149, 209, 258, 265, 291, 307, 379–383, 383–385, 388–389, 390–391, 393, 397, 399, 419, 428–431, 432–434, 435, 437, 439–441, 443, 445

CC.9-12.F.IF.3 Recognize that sequences are functions, some-times defi ned recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defi ned recursively by f(0) 5 f(1) 5 1, f(n 1 1) 5 f(n) 1 f(n 2 1) for n $ 1.

SE: 309–310, 539–540, A3–A4, A5

SE: 78 SE: 794, 826, 827–830, 830–833, 835, 838, 839, 842, 843, 844, 846–847, 848, 1021






CC.9-12.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

SE: 227–228, 230–232, 233, 238, 241–242, 267, 313, 315, 335–337, 339–341, 631, 633–634, 637, 639–640, 646, 648–649

CCCC: CC28–CC34

Found throughout the text. See, for example: SE: 91, 94–95, 106,

119, 125, 128–129, 130–131, 239, 241–243, 246–247, 250–251, 308, 311, 314, 336, 339, 387–389, 390–392, 396, 398, 908–911, 912–914

CCCC: CC2–CC3,CC9–CC16

CC.9-12.F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

SE: 44–45, 46, 51, 56, 57, 217–218, 219–221, 228, 232, 233, 263, 267 313, 315, 526, 631, 633, 781

SE: 72, 76, 78–79, 94, 96, 233, 251, 344, 391, 446–449, 49–451, 479, 482, 484, 487–488, 489, 491, 493–494, 496, 498, 503, 504, 559–561, 561–563, 565, 911

CC.9-12.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specifi ed interval. Estimate the rate of change from a graph.*

SE: 237–238, 240–242, 269, 294–295, 299, 301, 304–305, 307, 326, 327–330

CCCC: CC35

SE: 85, 86–88, 104, 106, 115, 117, 118–119, 139, 143, 145, 146–147, 148

CCCC: CC9–CC16





CC.9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph square root, cube root, and piecewise-defi ned functions, including step functions and absolute value functions.

c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

d. (1) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

SE: 216–218, 219–220, 222, 225–228, 229–231, 244–246, 247–250, 251–252, 254, 257, 259, 262–265, 265–268, 269, 272–274, 275, 303, 306, 313, 315, 396–397, 521, 524–525, 532–533, 535–536, 560, 628–631, 632–634, 35–636, 638, 641–642, 643–646, 647, 650–651, 669–670, 692–693, 710–713, 714–716, 717, 766–767, 771, 773–774, 775–778, 779–781, 786–787, 792–793, A1–A2

SE: 182, 185–186, 499, 882–883

SE: 75–76, 77–78, 89–92, 93–96, 97, 121–122, 123–126, 127–129, 130–131, 236–239, 240–243, 245—248, 249–251, 336, 339–341, 342–344, 345, 387–389, 390–392, 446–449, 449–451, 478–480, 482–484, 486–488, 489–491, 493–494, 496–497, 502–503, 504–505, 558–561, 561–563564, 565–567, 568–571, 908–912, 912–914, 915–919, 919–922

CC.9-12.F.IF.8 Write a function defi ned by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y 5 (1.02)t, y 5 (0.97)t, y 5 (1.01)12t, y 5 (1.2)t/10, and classify them as representing exponential growth or decay.

SE: 225–228, 229–230, 244–246, 247–250, 283–285, 286–289, 292–295, 296–299, 302–305, 305–308, 311–313, 314–316, 344, 522, 523, 534, 535, 635–636, 638–640, 641–642, 669–670

SE: 236–239, 240–243, 244, 245–248, 249–251, 265, 478–481, 482–483, 486, 489

CCCC: CC2–CC3

CC.9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

SE: 396–397, 521, 532, 628–630, 776

CCCC: CC28–CC34

CCCC: CC9–CC16






CC.9-12.F.BF.1 Write a function that describes a relationship between two quantities.*

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

c. (+) Compose functions. For example, if T(y ) is the temperature in the atmosphere as a function of height, and h(t ) is the height of a weather balloon as a function of time, then T(h(t )) is the temperature at the location of the weather balloon as a function of time.

SE: 285, 288–289, 294–295, 298–299, 304–305, 307–308, 313, 315–316, 326–327, 327–330, 331–332, 334, 335–338, 338–341, 342, 343, 348, 349, 352, 353, 520, 522–523, 524–525, 530, 531, 533, 535, 537, 686–687, 701, 778, 781, 787, 789–790, 799, 805, 808–809, 815, 817–819

SE: 112, 115–117, 117–120, 143, 145, 146, 148, 308, 311, 314, 316, 322, 323, 327, 393–396, 397–399, 400, 406, 407, 410–411, 428–431, 432–434, 435, 528, 529–533, 533–536, 542, 543, 547, 774, 775–777, 778–780, 781, 782, 786, 787, 791, 826, 827–830, 830–833, 941–943, 944–947

CC.9-12.F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*

SE: 309–310, 539–540, A3–A4, A5

SE: 798, 802–804, 806–808, 810–812, 814–816, 826, 827–830, 830–833, 838, 839, 841–842, 843, 844–845, 846

CCCC: CC44–CC45

CC.9-12.F.BF.3 Identify the effect on the graph of replacing f(x ) by f(x ) 1 k, kf(x ), f (kx ), and f (x 1 k) for specifi c values of k (both positive and negative); fi nd the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

SE: 263–265, 265–268, 269, 274, 290–291, 396–397, 521, 524, 532, 535–536, 669–670, 710–712, 713–714, 773–774, 775–777, 779

SE: 121–122, 123–126, 127–129, 139, 144, 145, 236–237, 240, 245, 249, 446–448, 449–450, 479, 482, 487, 489, 493, 496, 503, 504, 58–559, 561–562, 909–912, 913, 915–919, 919–922, 941–943, 944–947

CCCC: CC2–CC3, CC9–CC16





CC.9-12.F.BF.4 Find inverse functions.

a. Solve an equation of the form f(x ) 5 c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x ) 5 2x 3 for x . 0 or f (x ) 5 (x 1 1)/(x 2 1) for x Þ 1.

b. (1) Verify by composition that one function is the inverse of another.

c. (1) Read values of an inverse function from a graph or a table, given that the function has an inverse.

d. (1) Produce an invertible function from a non-invertible function by restricting the domain.

SE: 483, 485, 486–488 SE: 437, 438–442, 442–445, 453, 458, 474, 499, 501–502, 506, 516, 519, 522, 874, 875–877, 878–880, 931–934, 935–937

CC.9-12.F.BF.5 (1) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

SE: 499–502, 503–505, 506, 511, 513, 516–519, 520–522, 530–532, 538, 541–542, 543, 545, 546, 678, 1016

CC.9-12.F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.*

a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

SE: 520–523, 523–527, 531, 535, 539–540, 684–687, 688–691, 692–693

CCCC: CC28–CC34

SE: 393–394, 774, 775–777, 778–780, 809






CC.9-12.F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).*

SE: 44–45, 46–47, 283–285, 286–289, 292–295, 296–299, 302–305, 305–308, 309–310, 311–313, 314–316, 317, 318, 320, 321–322, 326, 327–330, 35–338, 338–341, 342, 520–523, 523–527, 530, 531–534, 535–538, 539–540, A3–A4, A5

SE: 98–101, 101–104, 105, 106, 108, 109, 112, 115–117, 118–119, 480–481, 483–485, 488, 489–491, 495, 496–497, 528, 529–530, 533–535, 798, 802–804, 806–808, 810–812, 814–816, 826, 827–830, 830–833, 838, 839, 841–842, 843, 844–845, 846

CC.9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.*

CCCC: CC28–CC34 SE: 547

CCCC: CC9–CC16

CC.9-12.F.LE.4 For exponential models, express as a logarithm the solution to abct 5 d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.*

SE: 516, 520–522, 531, 536, 537, 538, 542, 543

CC.9-12.F.LE.5. Interpret the parameters in a linear or exponential function in terms of a context.*

SE: 285, 294–295, 299, 304, 327, 329–330, 522–523, 527, 533–534, 537

SE: 91, 94, 106, 480–481, 482, 486, 489, 494

CC.9-12.F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

SE: 860–862, 863–865

CC.9-12.F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

SE: 859–861, 866–870, 870–872, 899





CC.9-12.F.TF.3 (1) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, p + x, and 2π – x in terms of their values for x, where x is any real number.

SE: 853–854, 868, 874

CC.9-12.F.TF.4 (1) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

The opportunity to address this standard can be found on the following pages: SE: 866–870, 908–912,

924

CC.9-12.F.TF.5 Choose trigonometric functions to model periodic phenomena with specifi ed amplitude, frequency, and midline.*

SE: 910–911, 913–914, 916, 921–922, 940, 941–943, 944–947, 948, 963, 967, 969, 972

CC.9-12.F.TF.6 (1) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

SE: 875, 897, 899

CC.9-12.F.TF.7 (1) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.*

SE: 931–934, 935–937, 938–939, 940, 947, 954, 964, 967, 969, 973, 1023

CC.9-12.F.TF.8 Prove the Pythagorean identity sin2(u) 1 cos2(u) 5 1 and use it to calculate trigonometric ratios.

SE: Ex. 32, p. 478 SE: 924–927, 928–929, 934, 947, 954, 958, 966, 969, 973, 1023

CC.9-12.F.TF.9 (1) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

SE: 949–951, 952–954, 962, 964, 968, 969, 973, 1023

Geometry

CC.9-12.G.CO.1 Know precise defi nitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefi ned notions of point, line, distance along a line, and distance around a circular arc.

SE: 246, 247, 318–319, 321

SE: 2–3, 24–25, 81, 82, 147, 651, 746–747, 749

SE: 84, 859, 861






CC.9-12.G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

SE: 213–214 SE: 271, 273–275, 276–279, 285, 286, 289, 291, 408, 409–411, 412–415, 416, 421, 422, 427, 572–574, 576–579, 581, 585, 588, 589–590, 592, 593–594, 596, 597, 599–600, 602–603, 605, 606, 607, 608–609, 612–613, 615, 628, 630–631, 633


SE: 121–122, 123–126, 127–129, 988–989

CC.9-12.G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and refl ections that carry it onto itself.

SE: 214 SE: 619–621, 621–624, 639, 640, A10–A11

CC.9-12.G.CO.4 Develop defi nitions of rotations, refl ections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

SE: 572–574, 576–579, 588, 589–592, 593–596, 598–601, 602–605, 607, 608–611, 611–615

CC.9-12.G.CO.5 Given a geometric fi gure and a rotation, refl ection, or translation, draw the transformed fi gure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given fi gure onto another.

SE: 213–214, 920–921 SE: 271, 273–275, 276, 278, 279, 280, 285, 286, 291, 572, 574, 576–578, 587, 588, 589–590, 593–594, 597, 598–599, 602–603, 606, 607, 608–611, 611–614, 616–618





CC.9-12.G.CO.6 Use geometric descriptions of rigid motions to transform fi gures and to predict the effect of a given rigid motion on a given fi gure; given two fi gures, use the defi nition of congruence in terms of rigid motions to decide if they are congruent.

SE: 272–275, 276–279, 280, 285, 286, 289, 290–291, 572–575, 576–579, 581, 584–585, 587, 588, 589–592, 593–596, 597, 598–601, 602–605, 606, 607, 608–611, 611–615, 616–618, 634, 635, 636–638, 640


CC.9-12.G.CO.7 Use the defi nition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.


CC.9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the defi nition of congruence in terms of rigid motions.

CCCC: CC10–CC11

CC.9-12.G.CO.9 Prove geometric theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

SE: 113–114, 118, 124–126, 129–130, 137, 153, 155–156, 159–160, 162–163, 168, 177, 190–192, 196, 303, 308

CC.9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

SE: 216, 218–219, 224, 264–265, 269, 294, 295, 297, 300–301, 303–305, 308, 310, 312, 315–316, 318, 319–321, 323–324, 326–327, 328–329, 30, 334, 335, 338, 340–341, 932–936






CC.9-12.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

SE: 514 515–517, 518–521, 522–525, 526–529, 530–531, 533–536, 537–540, 552–553, 554–557, 559, 561–563, 564

CC.9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, refl ective devices, paper folding, dynamic geometry software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

SE: 33–34, 169, 198–199, 235, 258, 261–262, 305, 307, 312, 314, 323, 401, 408, 527, 625, 629, 665, 671, 767

CC.9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

SE: 767

CCCC: CC24–CC25

CC.9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

SE: 408, 414, 625, 631, 633

CCCC: CC12, CC13–CC19

CC.9-12.G.SRT.2 Given two fi gures, use the defi nition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding angles and the proportionality of all corresponding pairs of sides.

CCCC: CC13–CC19

CC.9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

CCCC: CC20





CC.9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

SE: 388–390, 394–395, 396, 397–398, 402–403, 448, 449, 452, 455–456, 457, 459, 463

CC.9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric fi gures.

SE: 174–175 SE: 225–228, 228–231, 234–236, 236–239, 240–242, 243–246, 248, 249–252, 252–255, 256–258, 259–263, 283–284, 286, 290–291, 300, 372–375, 376–379, 381–383, 384–387, 388–391, 391–395, 397–399, 400–403, 405, 416, 420–421, 422, 424, 426, 449–452, 453–456, 457–460, 461–464

CC.9-12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to defi nitions of trigonometric ratios for acute angles.

SE: 466–467, 469, 473, 477

SE: 852–853

CC.9-12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

SE: 480 SE: 924, 927–928, 966

CC.9-12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

SE: 738–739, 741–742, 746, 752, 757, 760

SE: 434, 437–439, 443, 445–446, 465, 468, 471–472, 474–476, 479–480, 482, 484–485, 487–488, 492, 496, 498, 500, 503

SE: 855, 857–858, 865, 877, 879–880, 896, 899, 901, 902–903, 914

CC.9-12.G.SRT.9 (1) Derive the formula A 5 1 } 2 ab sin(C)

for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

SE: 479 SE: 885, 887






CC.9-12.G.SRT.10 (1) Prove the Laws of Sines and Cosines and use them to solve problems.

SE: 490–491 SE: 881, 882–884, 886–888, 889–891, 892–894, 896, 897, 900, 901, 1022

CC.9-12.G.SRT.11 (1) Understand and apply the Law of Sines and the Law of Cosines to fi nd unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

SE: 490–491 SE: 881, 882–884, 886–888, 889–891, 892–894, 896, 897, 900, 901, 1022

CC.9-12.G.C.1 Prove that all circles are similar. CCCC: CC13–CC19

CC.9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

SE: 650, 653–654, 656–658, 659–661, 661–663, 664–666, 667–670, 671, 672–675, 676–679, 705, 709–710, 712

CC.9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

SE: 306, 307, 312, 314, 675, 678

CC.9-12.G.C.4 (1) Construct a tangent line from a point outside a given circle to the circle.

CCCC: CC24–CC25

CC.9-12.G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and defi ne the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

SE: 747, 749, 756, 758

CCCC: CC27–CC28

SE: 860–861

CC.9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to fi nd the center and radius of a circle given by an equation.

SE: 699, 703 SE: 626, 656, 664, 672, 673, 678, 1018

CC.9-12.G.GPE.2 Derive the equation of a parabola given a focus and directrix.

SE: 620–622

CC.9-12.G.GPE.3 (1) Derive the equations of ellipses and hyperbolas given foci and directrices.

SE: 638, 646

CC.9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a fi gure defi ned by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, Î

}

3 ) lies on the circle centered at the origin and containing the point (0, 2).

SE: 294, 296–297, 298–301, 302, 309, 316, 320–321, 322, 344, 350–351, 17, 518–519, 525, 526–527, 531, 532, 538, 542, 546–547, 549, 555

SE: 614, 617, 619





CC.9-12.G.GPE.5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., fi nd the equation of line parallel or perpendicular to a given line that passes through a given point).

SE: 318–320, 321–323, 330, 343, 347, 349, 353

SE: 172–173, 175–176, 179, 180–181, 185–186, 193, 195, 197, 201, 204–205, 206, 209, 210–211

SE: 84–85, 86, 99, 102–103, 106, 145, 149

CC.9-12.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

CCCC: CC22–CC23

CC.9-12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.*

SE: 750 SE: 22, 50–51, 53, 58, 63, 724, 732

CC.9-12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

SE: 761, 769, 819–820, 828, 829

SE: 857

CC.9-12.G.GMD.2 (1) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid fi gures.

SE: 821–822, 824, 827, 832, 836, 859, 919

CC.9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

SE: 819–822, 822–825, 826–827, 829–831, 832–836, 837, 840–841, 843–845, 854, 855, 856, 859–860, 861, 862–863, 864–865, 866–867, 919

SE: 332, 334–335, 350–351, 356, 357–359, 360–361, 367, 369, 373, 386, 389, 392, 400, 407, 408, 410, 567, 569–571, 574, 579–580, 601, 610

CC.9-12.G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

SE: 797, 799–801, 818, 821, 825, 839, 864

CCCC: CC32–CC33

SE: 649, 657, 667






CC.9-12.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*

SE: 188–189, 494, 525, 791

Found throughout the text. See, for example: SE: 508, 510, 512, 517,

519–520, 523–524, 526, 528, 531, 532, 537, 539, 545, 657, 663, 665, 669, 674, 679, 682, 685, 717, 722, 725, 731, 735, 747, 751, 755, 760, 767, 796, 800, 805, 807–808, 813, 814

CC.9-12.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*

SE: 30, 153, 183, 412, 516–517, 878

CCCC: CC30–CC31

CC.9-12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*

Found throughout the text. See, for example: SE: 7, 23, 31, 54, 58,

68, 107, 132, 151, 159, 170, 189, 213, 217, 223, 226, 230, 236, 238, 242, 269, 274, 278, 291, 295, 300, 317, 329, 342, 362, 390, 416, 455, 616–618, 677, 679, 722, 725, 738, 742

Statistics and Probability

CC.9-12.S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).*

SE: 882–885, 886, 887–892, 893, 894, 900, 901, 904–905, 950

CCCC: CC42

SE: 888–889 SE: 724–730, 731, 1008–1009

CC.9-12.S.ID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.*

SE: 874, 875–878, 879–880, 883, 885, 887, 891–892, 893, 901, 918

CCCC: CC44–CC45

SE: 887 SE: 744–745, 749, 751, 787, 791





CC.9-12.S.ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).*

SE: 878, 880, 885, 888, 891, 892, 893, 894, 901

SE: 746–748, 756

CC.9-12.S.ID.4 Use the mean and standard deviation of a data set to fi t it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.*

SE: 758–762, 785, 787, 1020

CCCC: CC31, CC33

CC.9-12.S.ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.*

SE: 844, 847, 848, 870

CCCC: CC37–CC41

SE: 722, 1008

CC.9-12.SID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.*

a. Fit a function to the data; use functions fi tted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

b. Informally assess the fi t of a function by plotting and analyzing residuals.

c. Fit a linear function for a scatter plot that suggests a linear association.

SE: 26–330, 331–332, 335–341, 342–343, 348, 349, 352–353, 942

CCCC: CC15–CC16

SE: 115–117, 119, 120, 139, 143, 145, 148, 233, 271, 311, 314, 323, 327, 396, 398–399, 400, 530, 532–535, 537, 543, 547, 774, 775–780, 786, 787, 791, 943, 946, 947, 969, 1011, 1014, 1020

CC.9-12.S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.*

SE: 237–238, 240–241, 245, 248–249, 304–305, 338, 340, 341

SE: 173, 177–178, 182, 186

SE: 87, 91, 94

CC.9-12.S.ID.8 Compute (using technology) and interpret the correlation coeffi cient of a linear fi t.*

SE: 332, 333 SE: 114, 117–118

CC.9-12.S.ID.9 Distinguish between correlation and causation.*

SE: 333 SE: 120

CC.9-12.S.IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.*

SE: 871, 874

CCCC: CC36

SE: 369 SE: 770–771

CCCC: CC34–CC35






CC.9-12.S.IC.2 Decide if a specifi ed model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?*

SE: 849–850, 868–869 SE: 714, 722

CCCC: CC28–CC29

CC.9-12.S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.*

SE: 871, 873–874 SE: 766–767, 769, 773, 782, 1020

CCCC: CC36–CC41

CC.9-12.S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.*

CCCC: CC36 SE: 768–771, 780, 782, 787, 1020

CCCC: CC34–CC35

CC.9-12.S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are signifi cant.*

SE: 850, 869, 874

CCCC: CC37–CC41

CCCC: CC42–CC43

CC.9-12.S.IC.6 Evaluate reports based on data.* SE: 874 SE: 369, 770 SE: 771

CCCC: CC36–CC41

CC.9-12.S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).*

SE: 843, 846, 861, 865–867, 870, 930–931

SE: 698, 706, 707–713, 716, 732, 1019

CC.9-12.S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.*

SE: 862–865, 898, 901, 907, 950

SE: 777, 893 SE: 717–723

CCCC: CC17–CC24

CC.9-12.S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.*

SE: 863 SE: 893 SE: 722

CCCC: CC17–CC24





CC.9-12.S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classifi ed. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.*

SE: 844, 847–848 SE: 19, 722

CCCC: CC17–CC24

CC.9-12.S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.*

SE: 717, 719–720, 722

CCCC: CC17–CC24

CC.9-12.S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.*

SE: 863, 865 SE: 719–721, 722–723

CCCC: CC17–CC24

CC.9-12.S.CP.7 Apply the Addition Rule, P(A or B ) 5 P (A ) 1 P (B ) 2 P (A and B ), and interpret the answer in terms of the model.*

SE: 862, 864, 865, 898 SE: 707–708, 710–711, 713, 736, 737

CC.9-12.S.CP.8 (1) Apply the general Multiplication Rule in a uniform probability model, P(A and B) 5 P (A)P (B|A ) 5 P (B)P (A|B ), and interpret the answer in terms of the model.*

SE: 862–864 SE: 718–722, 736, 737

CCCC: CC17–CC24

CC.9-12.S.CP.9 (1) Use permutations and combinations to compute probabilities of compound events and solve problems.*

SE: 853, 855, 857, 859, 861–867

SE: 699, 702, 712

CC.9-12.S.MD.1 (1) Defi ne a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.*

SE: 724–725, 727–728






CC.9-12.S.MD.2 (1) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.*

SE: 750

CC.9-12.S.MD.3 (1) Develop a probability distribution for a random variable defi ned for a sample space in which theoretical probabilities can be calculated; fi nd the expected value. For example, fi nd the theoretical probability distribution for the number of correct answers obtained by guessing on all fi ve questions of a multiple-choice test where each question has four choices, and fi nd the expected grade under various grading schemes.*

SE: 724–726, 727–730, 731, 732, 736–737

CC.9-12.S.MD.4 (1) Develop a probability distribution for a random variable defi ned for a sample space in which probabilities are assigned empirically; fi nd the expected value. For example, fi nd a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to fi nd in 100 randomly selected households?*

SE: 726–729

CC.9-12.S.MD.5 (1) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and fi nding expected values.*

a. Find the expected payoff for a game of chance. For example, fi nd the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

SE: 750

CC.9-12.S.MD.6 (1) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).*

SE: 173, 872 SE: 766–767CCCC: CC25–CC26

CC.9-12.S.MD.7 (1) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).*

SE: 847 SE: 723

CCCC: CC25–CC26




Pre-Algebra Algebra 1 Geometry Algebra 2

Problem Solving

Skills

Problem Solving Strategies R R R R

Identify relationships R R R R

Choose an operation R R R R

Choose a method of computation T R R R

Make generalizations R R R R

Use a formula T R R R

Estimate or give an exact answer R R R R

Prioritize and sequence information R R R R

Identify too much or too little information R R R R

Write an equation T TR R R

Write the problem in your own words/Restate the question R R R R

Eliminate answer choices R R R R

Check that your answer is reasonable R R R R

Write algebraic expressions R R R R

Analyze units R R R R

Use a simulation T R R R

Interpret unfamiliar words/Understand the words in the problem R R R R

Identify important details in the problem R R R R

Choose a problem-solving strategy R R R R

Check that the question is answered R R R R

Break into simpler parts R R R R

Translate between words and math R R R R

Identify missing information R R R R

Reasoning

Make and test predictions R R R R

Explain and justify answers R R R R

Evaluate evidence and conclusions T R R R

Interpret charts, tables, and graphs T T R T

Classify and sort R R R R

Identify spatial relationships R R R R

4-Year Scope and SequenceHolt McDougal Larson Geometry

I (Introduce) T (Teach and Test) R (Reinforce and Maintain)

30 4-Year Scope and Sequence

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<…4-Year Scope and Sequence


Use visual representations to solve problems R R R R

Solve nonroutine problems R R R R

Compare and contrast R R R R

Draw conclusions R R R R

Inductive and deductive reasoning I T T R

Number and Quantity

Read and write numbers

Evaluate exponents R R R R

Negative exponents T T R

Evaluate rational exponents I T

Properties of Exponents I T R

Scientifi c notation T R R

Properties of Real Numbers I T R

Integers R R R R

Square roots T R R R

Absolute value T R R R

Quantities

Choose and interpret units I T R R

Precision and accuracy IT R

Complex Numbers

Operations with complex numbers IT

Complex numbers in the complex plane IT

Ratio

Cross products T R R R

Proportion

Indirect measurement T T R R

Solve proportions T R R R

Scale factor T R R R

Scale drawings TR R R

Similar fi gures TR R T R

Percent

Percents greater than 100% and less than 1% T TR R

Percent of a number TR TR R R

Percent one number is of another T TR R R

Percent change (increase and decrease) T TR R R


4-Year Scope and Sequence 31



Find number when percent is known T TR R R

Circle graph T R R R

Simple interest T TR

Compound interest IT T R

Operations

Order of operations

Order of operations T TR R R

Addition and Subtraction

Decimals TR R R R

Fractions R R R R

Mixed numbers TR R R R

Integers TR TR R R

Multiplication and Division

Of exponential expressions IT TR R R

Decimal by a whole number R R R R

Decimal by a decimal TR R R R

Fraction by a whole number R R R R

Fraction by a fraction TR R R R

Mixed numbers TR R R R

Integers TR TR R R

Algebra and Functions

Equations and Expressions

Formulas R R R R

Variables R R R R

Write and evaluate algebraic expressions R R R R

Identify and combine like terms R R R R

Monomials: simplify, operations IT T R

Polynomials IT T R R

Binomials and trinomials, defi nition IT T R

Degree IT R

Simplify polynomial expressions IT T R R

Add and subtract polynomials IT T R R

Multiply binomials IT T R R

FOIL method IT T R






Difference of squares IT R

Perfect-square trinomial IT R

Multiply polynomials by monomials IT T R R

Divide polynomials by monomials T R R

Divide polynomials by polynomials IT R

Properties of polynomial and rational expressions IT

Factor binomials IT R R

Factor trinomials IT R R

Factor difference of squares IT R

Factor perfect-square trinomials IT R

Factor Theorem IT

Binomial expansion IT

Binomial Theorem IT

Rational expressions: simplify, graph IT R TR

Radical expressions: simplify, evaluate IT R TR

Simplify expressions with complex numbers IT

Write linear equations T TR R R

Solve equations

1-step equations T TR R R

2-step equations T TR R R

Multistep equations T TR R R

Equations with variables on both sides T TR R R

Relate graphs and equations T R R R

Solve equations by factoring IT R R

Linear equations T T R R

Systems of equations IT T R TR

Absolute-value equations IT R TR

Rational equations IT R TR

Quadratic equations IT T R TR

Polynomial equations IT TR

Exponential equations IT IT R TR

Logarithmic equations IT

Radical equations IT R TR





Inequalities

Compare numbers R R R R

Algebraic inequality T R R R

Write an inequality for a problem situation T R R R

Solve inequalities

1-step inequalities T TR R R

2-step inequalities T T R R

Graph inequalities T T R R

Graph compound inequalities I T R R

Graph inequalities in two variables IT R R

Absolute-value inequalities IT TR

Rational inequalities IT

Radical inequalities IT

Coordinate plane

Ordered pairs R R R R

Origin R R R R

Axes R R R R

Graph in four quadrants T R R R

Find area by coordinates R

Relations R R

Functions T TR R R

Transformations T TR R R

Linear equations T TR R R

Nonlinear equations T T R R

Systems of equations T T R R

Inequalities T T R R

Systems of inequalities IT R R

Quadratic equations T T R R

Conics

Conic sections IT

Parabolas T T R TR

Circles T R R

Ellipses IT

Hyperbolas IT






General equation for conics IT

Identify conic from equation IT

Transformations of conics IT

Vectors

Magnitude, direction I I

Vector addition I I

Patterns

Arithmetic sequences T R R

Arithmetic series IT

Geometric sequences I T

Geometric series IT

Infi nite sequence IT

Infi nite geometric series IT

Sigma notation IT

Fibonacci sequence IT R R R

Pascal’s triangle T

Fractals IT I T

Binomial expansion IT

Recursion I T

Functions and relations

Evaluate functions T TR R

Operations with functions T

Composite functions IT

Relations IT T

Inverse of function or relation IT

Linear functions T T R R

Rational functions IT T

Quadratic functions IT IT T

Exponential functions IT T T

Logarithmic functions IT

Polynomial functions IT T

Radical functions IT T

Trigonometric functions IT





Modeling

Linear models IT R

Exponential models IT R

Quadratic models I TR

Matrices

Matrix operations IT IT T

Determinants IT

Identity and inverse matrices IT

Solve systems of equations IT

Transformation matrices IT

Probability

Probability as ratio, proportion, decimal, percent T R R R

Making predictions T R R R

Finding outcomes

Tree diagrams T R R

Combinations T T R

Permutations T T R

Fundamental Counting Principle T R R

Factorial IT T R

Theoretical probability

Mutually exclusive T T R R

Complementary events T T R R

Independent/dependent events T T R R

Conditional Probability IT TR

Experimental probability

Simulations T T T

Random numbers I I T

Odds

Odds T T T

Data Analysis and Statistics

Organizing and Displaying Data

Frequency table/chart R R

Stem-and-leaf plot T R R R

Two-way tables IT R






Dot plot IT TR R R

Venn diagram T R R R

Histogram T R R R

Box-and-whisker plot T R R R

Scatter plot R R R R

Analyzing data

Surveys, experiments, and observational studies I TR

Identify correlation T T

Quartiles T T R

Interquartile range T T

Line of best fi t T T R

Make predictions R R R

Mean, median, mode T R R R

Determine best measure of central tendency T R R

Standard deviation I T

Variance I T

Frequency distribution IT

Normal distribution (bell curve) IT

Binomial distribution IT

Shape of distribution I TR

Standard normal curve IT

Geometry

Points, lines, planes R R R

Angles

Ray R R

Vertex R R

Classify R R

Vertical, adjacent, complementary, supplementary R R R

Congruent R R

Relationships of angles formed by parallel lines and a transversal IT R

Angle relationship theorems R

Sum of angle measures R R R

Identify unknown angle measures R R





Lines and line segments

Properties of intersecting lines and segments T TR

Properties of parallel lines and segments T R TR

Properties of perpendicular lines and segments T R TR

Triangles

Classify T R

Sum of the measures of the angles is 180 degrees T R R R

Right triangle relationships T R R R

Pythagorean Theorem T R R R

Prove triangles congruent IT

Isosceles triangle properties and proofs IT

Triangle inequality IT

Similar triangles, identify T T

Exterior Angle Theorem IT

Quadrilaterals

Classify T R

Angles T T

Sum of the measures of the angles is 360 degrees T R R

Congruent quadrilaterals T

Diagonals T T

Circles

Meaning of π R R R R

Radius R R R R

Diameter R R R R

Chord IT

Arc IT R

Central angle IT R

Inscribed angles and arcs IT

Chords, secants and tangents IT

Area of sector IT R

Area R R R R

Circumference R R R R

Equation of a circle IT T






Other plane fi gures

Classify R R

Polygons R R R

Similar fi gures

Similarity R R R

Corresponding parts T R R

Transformations

Translations, refl ections T R R R

Rotations T R R

Dilations R R

Isometry IT

Transformation, defi nition T R R

Mapping, image, preimage IT R

Transformation matrices IT

Congruence and transformations IT

Similarity and transformations IT

Tessellation T

Symmetry T R R

Perimeter

Perimeter R R R R

Area

Regular polygons T

Composite fi gures T T

Parallelograms and triangles T R R R

Squares T R R R

Trapezoids T R R R

Circles T R R R

Solid fi gures

Vertices, edges, faces R R

Hemisphere, great circle T

Sphere I T

Pyramid, cube, prism T R R

Cone, cylinder T R R

Polyhedron T

Solids of revolution IT





Surface area

Prism T R R

Pyramid T R R

Cylinder T R R

Cone T R R

Sphere I T R

Volume

Prism T R R R

Pyramid T R R R

Cylinder T R R R

Cone T R R R

Sphere I I T R

Coordinate geometry

Transformations in the coordinate plane T R R R

Distance in the coordinate plane I T R

Coordinates in space I

Reasoning and Proof

Logical reasoning in problem solving IT R

Theorem and postulate IT

Inductive reasoning I T T R

Conjecturing I T T

If-then statements I T R

Venn diagrams T R R R

Truth tables IT

Deductive reasoning I T T R

Proofs

Line segment proofs IT

Angle relationship proofs IT

Parallel lines proofs IT

Triangle Sum Theorem proof IT

Prove triangles congruent IT

Isosceles triangle proofs IT

Segments in triangles proofs IT

Right triangle proofs IT






Parallelogram proofs IT

Rhombus proofs IT

Trapezoid proofs IT

Similar triangle proofs IT

Prove lines parallel IT

Pythagorean Theorem proof IT

Circle Theorem proofs IT

Tangent proofs IT

Trigonometry

Trigonometric ratios T T R

Inverse trigonometric ratios I T R

Applications of right triangle trigonometry IT R

Law of sines IT IT

Law of cosines IT IT

Area of triangles IT T

Solving right triangles IT T

Special right triangles IT R

Unit circle IT

Radian measure IT TR

Trigonometric functions, general angles IT

Trigonometric functions, special angles IT

Period IT

Graphs of trigonometric functions IT

Trigonometric equations IT




Essential Course of Study Holt McDougal Larson Geometry

42 Essential Course of Study

ChapterPacing(Days)

LessonContentStandards

Chapter 1 – Essentials of Geometry

1 1.1 Identify Points, Lines, and Planes CC.9-12.G.CO.11 1.2 Use Segments and Congruence CC.9-12.G.CO.1, CC.9-12.G.CO.7

1 1 } 2 1.3 Use Midpoint and Distance Formulas CC.9-12.G.GPE.7

1 1 } 2 1.4 Measure and Classify Angles CC.9-12.G.CO.1, CC.9-12.G.CO.7

1 } 2 1.4 Construction: Copy and Bisect Segments and

AnglesCC.9-12.G.CO.12

1 1 } 2 1.5 Describe Angle Pair Relationships CC.9-12.G.CO.1

1 1 } 2 1.6 Classify Polygons CC.9-12.G.MG.1, Prepare for CC.9-12.G.GMD.4

Chapter 2 – Reasoning and Proof

1 2.1 Use Inductive Reasoning Prepare for CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11

2 2.2 Analyze Conditional Statements Prepare for CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11

2 2.3 Apply Deductive Reasoning CC.9-12.G.CO.1, Prepare for CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11

1 2.4 Use Postulates and Diagrams Prepare for CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11

1 } 2 2.5 Activity: Justify a Number Trick CC.9-12.A.REI.1

1 2.5 Reason Using Properties from Algebra CC.9-12.A.REI.1, Prepare for CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11

2 2.6 Prove Statements about Segments and Angles Prepare for CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11

1 } 2 2.7 Activity: Angles and Intersecting Lines CC.9-12.G.CO.9

1 1 } 2 2.7 Prove Angle Pair Relationships CC.9-12.G.CO.9

See pp. 2–29 for the full text of the Common Core Mathematics Content Standards for High School.

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ChapterPacing(Days)


Chapter 3 – Parallel and Perpendicular Lines

1 } 2 3.1 Activity: Draw and Interpret Lines CC.9-12.G.CO.1

1 3.1 Identify Pairs of Lines and Angles CC.9-12.G.CO.1, CC.9-12.G.CO.9

1 } 2 3.2 Activity: Parallel Lines and Angles CC.9-12.G.CO.9

1 3.2 Use Parallel Lines and Transversals CC.9-12.G.CO.91 3.3 Prove Lines are Parallel CC.9-12.G.CO.9

1 1 } 2 3.4 Find and Use Slopes of Lines CC.9-12.G.GPE.5

1 } 2 3.4 Activity: Investigate Slopes CC.9-12.G.GPE.5

1 1 } 2 3.5 Write and Graph Equations of Lines CC.9-12.G.GPE.5, CC.9-12.A.CED.2

1 1 } 2 3.6 Prove Theorems about Perpendicular

LinesCC.9-12.G.CO.9

1 } 2 3.6 Construction: Parallel and Perpendicular

LinesCC.9-12.G.CO.12

Chapter 4 – Congruent Triangles

1 } 2 4.1 Activity: Angle Sums in Triangles CC.9-12.G.CO.10

1 4.1 Apply Triangle Sum Properties CC.9-12.G.CO.10

1 1 } 2 4.2 Apply Congruence and Triangles CC.9-12.G.CO.7

1 } 2 4.2A Activity: Rigid Motions in the Plane CC.9-12.G.CO.2, CC.9-12.G.CO.7

1 4.2B Relate Transformations and Congruence CC.9-12.G.CO.2, CC.9-12.G.CO.6, CC.9-12.G.CO.7

1 } 2 4.3 Activity: Investigate Congruent Figures CC.9-12.G.CO.6, CC.9-12.G.CO.8

1 4.3 Prove Triangles Congruent by SSS CC.9-12.G.CO.8

1 1 } 2 4.4 Prove Triangles Congruent by SAS and HL CC.9-12.G.CO.8

1 } 2 4.4 Activity: Investigate Triangles and

CongruenceCC.9-12.G.CO.7, CC.9-12.G.CO.8

1 1 } 2 4.5 Prove Triangles Congruent by ASA and

AASCC.9-12.G.CO.8

1 } 2 4.5A Construction: Rigid Motions and

CongruenceCC.9-12.G.CO.8

1 4.6 Use Congruent Triangles CC.9-12.G.CO.121 4.7 Use Isosceles and Equilateral Triangles CC.9-12.G.CO.10

1 } 2 4.8 Activity: Investigate Slides and Flips CC.9-12.G.CO.2, CC.9-12.G.CO.5

1 4.8 Perform Congruence Transformations CC.9-12.G.CO.2, CC.9-12.G.CO.6

<… Essential Course of Study

Essential Course of Study 43



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ChapterPacing(Days)


Chapter 5 – Relationships within Triangles

1 } 2 5.1 Activity: Investigate Segments in Triangles CC.9-12.G.CO.10, CC.9-12.G.GPE.4

1 1 } 2 5.1 Midsegment Theorem and

Coordinate ProofCC.9-12.G.CO.10, CC.9-12.G.GPE.4

2 5.2 Use Perpendicular Bisectors CC.9-12.G.CO.9, CC.9-12.G.CO.12, CC.9-12.G.C.31 5.3 Use Angle Bisectors of Triangles CC.9-12.G.C.3

1 } 2 5.4 Activity: Intersecting Medians CC.9-12.G.CO.12

1 5.4 Use Medians and Altitudes CC.9-12.G.CO.10, CC.9-12.G.CO.12

1 } 2 5.4 Activity: Investigate Points of Concurrency CC.9-12.G.CO.12

1 5.5 Use Inequalities in a Triangle Extend CC.9-12.G.CO.7

1 1 } 2 5.6 Inequalities in Two Triangles and

Indirect ProofExtend CC.9-12.G.CO.7

Chapter 6 – Similarity

1 } 2 6.3 Activity: Similar Polygons CC.9-12.G.SRT.5

1 1 } 2 6.3 Use Similar Polygons CC.9-12.G.SRT.5

1 } 2 6.3A Activity: Explore Properties of Dilations CC.9-12.G.SRT.1

1 6.3B Relate Transformations and Similarity CC.9-12.G.SRT.1, CC.9-12.G.SRT.2, CC.9-12.G.C.1

1 } 2 6.4A Activity: Dilations and AA Similarity CC.9-12.G.SRT.3

1 6.4 Prove Triangles Similar by AA CC.9-12.G.SRT.3

1 1 } 2 6.5 Prove Triangles Similar by SSS and SAS CC.9-12.G.SRT.4

1 } 2 6.6 Activity: Investigate Proportionality CC.9-12.G.SRT.4

1 1 } 2 6.6 Use Proportionality Theorems CC.9-12.G.SRT.4

1 } 2 6.6 Extension: Fractals CC.9-12.G.SRT.5, CC.9-12.G.MG.3

1 } 2 6.7 Activity: Dilations CC.9-12.G.CO.2, CC.9-12.G.SRT.1b

1 1 } 2 6.7 Perform Similarity Transformations CC.9-12.G.CO.2, CC.9-12.G.SRT.1, CC.9-12.G.GPE.4

1 } 2 6.7A Extension: Partition Segments CC.9-12.G.GPE.6



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ChapterPacing(Days)


Chapter 7 – Right Triangles and Trigonometry

2 7.1 Apply the Pythagorean Theorem CC.8.G.6, CC.8.G.7, CC.9-12.G.SRT.4, CC.9-12.G.GPE.7, CC.9-12.G.SRT.8

1 7.2 Use the Converse of the Pythagorean Theorem

CC.8.G.6, CC.8.G.7, CC.9-12.G.SRT.8

1 } 2 7.3 Activity: Similar Right Triangles CC.9-12.G.SRT.4, CC.9-12.G.SRT.5

1 1 } 2 7.3 Use Similar Right Triangles CC.9-12.G.SRT.4, CC.9-12.G.SRT.5, CC.9-12.G.MG.1

1 1 } 2 7.4 Special Right Triangles CC.9-12.G.SRT.6

1 7.5 Apply the Tangent Ratio CC.9-12.G.SRT.6, CC.9-12.G.SRT.82 7.6 Apply the Sine and Cosine Ratios CC.9-12.G.SRT.6, CC.9-12.G.SRT.8, CC.9-12.G.SRT.9

1 1 } 2 7.7 Solve Right Triangles CC.9-12.G.SRT.8

1 } 2 7.7 Extension: Law of Sines and Law of

CosinesCC.9-12.G.SRT.10, CC.9-12.G.SRT.11

Chapter 8 – Quadrilaterals

2 8.1 Find Angle Measures in Polygons CC.8.G.5, CC.9-12.G.CO.11

1 } 2 8.2 Activity: Investigate Parallelograms CC.9-12.G.CO.11

1 8.2 Use Properties of Parallelograms CC.9-12.G.CO.11, CC.9-12.G.SRT.5

1 1 } 2 8.3 Show that a Quadrilateral is a

ParallelogramCC.9-12.G.CO.11, CC.9-12.G.SRT.5

1 1 } 2 8.4 Properties of Rhombuses, Rectangles,

and SquaresCC.9-12.G.CO.11, CC.9-12.G.GPE.7, CC.9-12.G.SRT.5

1 } 2 8.5 Activity: Midsegment of a Trapezoid CC.9-12.G.SRT.5

1 1 } 2 8.5 Use Properties of Trapezoids and Kites CC.9-12.G.SRT.5, CC.9-12.G.GPE.4

1 8.6 Identify Special Quadrilaterals CC.9-12.G.CO.11


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ChapterPacing(Days)


Chapter 9 – Properties of Transformations

1 9.1 Translate Figures and Use Vectors CC.9-12.G.CO.42 9.2 Use Properties of Matrices CC.9-12.G.CO.4, CC.9-12.N.VM.6, CC.9-12.N.VM.12

1 } 2 9.3 Activity: Refl ections in the Plane CC.9-12.G.CO.2, CC.9-12.G.CO.5

1 1 } 2 9.3 Perform Refl ections CC.9-12.G.CO.4

1 1 } 2 9.4 Perform Rotations CC.9-12.G.CO.4

1 } 2 9.5 Activity: Double Refl ections CC.9-12.G.CO.2, CC.9-12.G.CO.5

1 1 } 2 9.5 Apply Compositions of Transformations CC.9-12.G.CO.5

1 9.5 Extension: Tessellations CC.9-12.G.CO.51 9.6 Identify Symmetry CC.9-12.G.CO.3

1 } 2 9.7 Activity: Investigate Dilations CC.9-12.G.CO.2, CC.9-12.G.CO.5

1 9.7 Identify and Perform Dilations CC.9-12.G.SRT.1

1 } 2 9.7 Activity: Compositions with Dilations Extend CC.9-12.G.SRT.2

Chapter 10 – Properties of Circles

1 } 2 10.1 Activity: Explore Tangent Segments CC.9-12.G.C.4

2 10.1 Use Properties of Tangents CC.9-12.G.CO.1, CC.9-12.G.C.21 10.2 Find Arc Measures CC.9-12.G.CO.11 10.3 Apply Properties of Chords CC.9-12.G.CO.12, CC.9-12.G.C.3

1 } 2 10.4 Activity: Explore Inscribed Angles CC.9-12.G.C.2

2 10.4 Use Inscribed Angles and Polygons CC.9-12.G.C.2, CC.9-12.G.C.3, CC.9-12.G.C.4, CC.9-12.G.C.5

1 10.4A Construction: Tangent Lines and Inscribed Squares

CC.9-12.G.CO.13, CC.9-12.G.C.4

1 10.5 Apply Other Angle Relationships in Circles CC.9-12.G.C.2, CC.9-12.G.C.5

1 } 2 10.6 Activity: Investigate Segment Lengths CC.9-12.G.C.2

1 10.6 Find Segment Lengths in Circles CC.9-12.G.C.22 10.7 Write and Graph Equations of Circles CC.9-12.G.GPE.1



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ChapterPacing(Days)


Chapter 11 – Measuring Length and Area

2 11.4 Circumference and Arc Length CC.9-12.G.C.5, CC.9-12.G.GMD.11 11.4A Extension: Measure Angles in Radians CC.9-12.G.C.51 11.5 Areas of Circles and Sectors CC.9-12.G.C.51 11.6 Areas of Regular Polygons CC.9-12.G.CO.13, CC.9-12.G.SRT.81 11.7 Use Geometric Probability CC.9-12.G.MG.1, CC.9-12.S.CP.1

Chapter 12 – Surface Area and Volume of Solids

1 } 2 12.1 Activity: Investigate Solids CC.9-12.G.GMD.4

1 12.1 Explore Solids CC.9-12.G.GMD.4

1 1 } 2 12.4 Volume of Prisms and Cylinders CC.9-12.G.GMD.1, CC.9-12.G.GMD.3, CC.9-12.G.GMD.4

1 } 2 12.4A Extension: Density CC.9-12.G.MG.2

1 } 2 12.5 Activity: Investigate the Volume of a

PyramidCC.9-12.G.GMD.1

1 12.5 Volume of Pyramids and Cones CC.9-12.G.GMD.1, CC.9-12.G.GMD.3, CC.9-12.G.MG.3

1 } 2 12.5 Activity: Minimize Surface Area CC.9-12.G.MG.3

1 } 2 12.5A Extension: Solids of Revolution CC.9-12.G.GMD.4

1 12.6 Surface Area and Volume of Spheres CC.9-12.G.GMD.3

1 } 2 12.7 Activity: Investigate Similar Solids CC.9-12.G.GMD.3

1 12.7 Explore Similar Solids CC.9-12.G.GMD.3

1 } 2 12.7 Extension: Symmetries of Solids CC.9-12.G.CO.3





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48 Pacing Guide for 50-Minute Classes

Pacing Guide for 50-Minute ClassesHolt McDougal Larson Geometry

This sequence was created as a guide to assist you in covering the Common Core State Standards for Geometry. This 170-day schedule includes time for review and assessment. The schedule leaves room for you to customize the pacing to your students’ needs.

Chapter 1 DAY 1 DAY 2 DAY 3 DAY 4 DAY 5

Lesson 1.1 Lesson 1.2 Lesson 1.3 Lesson 1.3 (cont.)Mixed Review of Problem Solving

Lesson 1.4

DAY 6 DAY 7 DAY 8 DAY 9 DAY 10

Lesson 1.4 (cont.)Investigating Geometry Construction 1.4

Lesson 1.5 Lesson 1.5 (cont.)Lesson 1.6

Lesson 1.6 (cont.)Mixed Review of Problem SolvingChapter 1 Review

Chapter 1 Test

DAY 11

Standardized Test Preparation Standardized Test Practice


Lesson 2.1 Lesson 2.2 Lesson 2.2 (cont.) Lesson 2.3 Lesson 2.3 (cont.)


Lesson 2.4 Mixed Review of Problem SolvingInvestigating Geometry Activity 2.5

Lesson 2.5 Lesson 2.6 Lesson 2.6 (cont.)


Investigating Geometry Activity 2.7Lesson 2.7

Lesson 2.7 (cont.) Mixed Review of Problem Solving Chapter 2 Review

Chapter 2 Test Standardized Test PreparationStandardized Test Practice

GE_CCCC_TE_618258_P50.indd 48GE_CCCC_TE_618258_P50.indd 48 12/17/10 3:19:50 AM12/17/10 3:19:50 AM

<… Pacing Guide for 50-Minute Classes

Pacing Guide for 50-Minute Classes 49



Lesson 3.1 (cont.)Investigating Geometry Activity 3.2

Lesson 3.2 Lesson 3.3 Mixed Review of Problem SolvingLesson 3.4


Lesson 3.4 (cont.) Technology Activity 3.4Lesson 3.5

Lesson 3.5 (cont.) Lesson 3.6 Lesson 3.6 (cont.)Investigating Geometry Construction 3.6

DAY 11 DAY 12 DAY 13 DAY 14

Mixed Review of Problem SolvingChapter 3 Review


Cumulative Review



Lesson 4.1 (cont.)Lesson 4.2

Lesson 4.2 (cont.) Investigating Geometry Activity 4.2ALesson 4.2B

Lesson 4.2B (cont.)Investigating Geometry Activity 4.3


Lesson 4.3 Lesson 4.4 Lesson 4.4 (cont.)Technology Activity 4.4

Mixed Review of Problem SolvingLesson 4.5

Lesson 4.5 (cont.)


Investigating Geometry Activity 4.5ALesson 4.6



Lesson 4.8 Mixed Review of Problem SolvingChapter 4 Review

DAY 16 DAY 17






Lesson 5.1 (cont.) Lesson 5.2 Lesson 5.2 (cont.) Lesson 5.3


Mixed Review of Problem SolvingInvestigating Geometry Activity 5.4

Lesson 5.4 Technology Activity 5.4Lesson 5.5


Lesson 5.6 (cont.)

DAY 11 DAY 12 DAY 13





Lesson 6.3 (cont.) Investigating Geometry Activity 6.3ALesson 6.3B

Lesson 6.3B (cont.)Mixed Review of Problem Solving

Investigating Geometry Activity 6.4ALesson 6.4



Lesson 6.5 (cont.) Investigating Geometry Activity 6.6Lesson 6.6

Lesson 6.6 (cont.) Extension 6.6Investigating Geometry Activity 6.7


Lesson 6.7 Lesson 6.7 (cont.)Extension 6.7A



DAY 16

Cumulative Review





Lesson 7.1 Lesson 7.1 (cont.) Lesson 7.2 Investigating Geometry Activity 7.3Lesson 7.3

Lesson 7.3 (cont.)


Lesson 7.4 Lesson 7.4 (cont.)Mixed Review of Problem Solving

Lesson 7.5 Lesson 7.6 Lesson 7.6 (cont.)


Lesson 7.7 Lesson 7.7 (cont.)Extension 7.7




Lesson 8.1 Lesson 8.1 (cont.) Investigating Geometry Activity 8.2Lesson 8.2


Lesson 8.3 (cont.)




Lesson 8.5 (cont.) Lesson 8.6

DAY 11 DAY 12 DAY 13






Lesson 9.1 Lesson 9.2 Lesson 9.2 (cont.) Investigating Geometry Activity 9.3Lesson 9.3

Lesson 9.3 (cont.)




Lesson 9.5 (cont.) Extension 9.5


Lesson 9.6 Investigating Geometry Activity 9.7Lesson 9.7

Lesson 9.7 (cont.)Technology Activity 9.7


Chapter 9 Test

DAY 16

Standardized Test PreparationStandardized Test Practice



Lesson 10.1 (cont.) Lesson 10.1 (cont.)Lesson 10.2




Lesson 10.4 Lesson 10.4 (cont.) Investigating GeometryConstruction 10.4A

Lesson 10.5 Mixed Review of Problem SolvingInvestigating Geometry Activity 10.6


Lesson 10.6 Lesson 10.7 Lesson 10.7 (cont.) Mixed Review of Problem SolvingChapter 10 Review

Chapter 10 Test

DAY 16 DAY 17

Standardized Test PreparationStandardized Test Practice

Cumulative Review





Lesson 11.4 Lesson 11.4 (cont.) Extension 11.4A Lesson 11.5 Lesson 11.6






Lesson 12.1 (cont.) Mixed Review of Problem SolvingLesson 12.4

Lesson 12.4 (cont.) Extension 12.4AInvestigating GeometryActivity 12.5


Lesson 12.5 Spreadsheet Activity 12.5Extension 12.5A

Lesson 12.6 Investigating Geometry Activity 12.7Lesson 12.7

Lesson 12.7 (cont.)Extension 12.7




Cumulative Review



Pacing Guide for 90-Minute ClassesHolt McDougal Larson Geometry

This sequence was created as a guide to assist you in covering the Common Core State Standards for Geometry. This 85-day schedule includes time for review and assessment. The schedule leaves room for you to customize the pacing to your students’ needs.


Lesson 1.1Lesson 1.2

Lesson 1.3 Mixed Review of Problem Solving

Lesson 1.4Investigating Geometry Construction 1.4

Lesson 1.5Lesson 1.6`


DAY 6

Chapter 1 TestStandardized TestPreparation and Practice





Mixed Review of Problem SolvingInvestigating GeometryActivity 2.5Lesson 2.5

Lesson 2.6

DAY 6 DAY 7 Day 8

Investigating GeometryActivity 2.7Lesson 2.7


Chapter 2 TestStandardized Testreparation and Practice


Investigating GeometryActivity 3.1Lesson 3.1Investigating GeometryActivity 3.2



TechnologyActivity 3.4Lesson 3.5

Lesson 3.6Investigating GeometryConstruction 3.6

DAY 6 DAY 7 DAY 8



Cumulative Review





Investigating GeometryActivity 4.1Lesson 4.1Lesson 4.2

Lesson 4.2 (cont.)Investigating GeometryActivity 4.2ALesson 4.2B

Lesson 4.2B (cont.)Investigating GeometryActivity 4.3Lesson 4.3

Lesson 4.4TechnologyActivity 4.4



Investigating GeometryActivity 4.5ALesson 4.6Lesson 4.7

Lesson 4.7 (cont.)Investigating GeometryActivity 4.8





Lesson 5.2 Lesson 5.3Mixed Review of Problem SolvingInvestigating GeometryActivity 5.4

Lesson 5.4TechnologyActivity 5.4Lesson 5.5


DAY 6 DAY 7





Investigating GeometryActivity 6.3ALesson 6.3BMixed Review of Problem Solving

Investigating GeometryActivity 6.4ALesson 6.4 Lesson 6.5

Lesson 6.5 (cont.)Investigating GeometryActivity 6.6Lesson 6.6

Lesson 6.6 (cont.)Extension 6.6Investigating GeometryActivity 6.7


Lesson 6.7Extension 6.7A



Cumulative Review




Lesson 8.1 Investigating GeometryActivity 8.2Lesson 8.2Lesson 8.3

Lesson 8.3 (cont.)Mixed Review of Problem SolvingLesson 8.4



DAY 6 DAY 7


Chapter 8 Test Standardized TestPreparation and Practice

Chapter 7DAY 1 DAY 2 DAY 3 DAY 4 DAY 5

Lesson 7.1 Lesson 7.2Investigating GeometryActivity 7.3Lesson 7.3



Lesson 7.6

DAY 6 DAY 7 DAY 8

Lesson 7.7Extension 7.7








Lesson 9.5 (cont.)Extension 9.5


Lesson 9.6Investigating GeometryActivity 9.7Lesson 9.7

Lesson 9.7 (cont.)TechnologyActivity 9.7








Lesson 10.1 (cont.)Lesson 10.2Lesson 10.3


Lesson 10.4 (cont.)Investigating GeometryConstruction 10.4A

Lesson 10.5Mixed Review of Problem SolvingInvestigating GeometryActivity 10.6





Cumulative Review


Lesson 11.4 Extension 11.4ALesson 11.5







Extension 12.4AInvestigating GeometryActivity 12.5Lesson 12.5

SpreadsheetActivity 12.5Extension 12.5ALesson 12.6

Investigating GeometryActivity 12.7Lesson 12.7 Extension 12.7

DAY 6 DAY 7 DAY 8



Cumulative Review


58 Chapter Prerequisites

Chapter Key Skills and Concepts Content Standards Prerequisites

Chapter 1 – Essentials of Geometry

Name and sketch geometric fi gures. Use postulates to identify congruent segments in the coordinate plane. Find lengths of segments in the coordinate plane. Find the midpoint of a segment. Name, measure, and classify angles.

CC.9-12.G.CO.1, CC.9-12.G.CO.7, CC.9-12.G.CO.12, CC.9-12.G.GPE.7, CC.9-12.G.MG.1

Chapter 2 – Reasoning and Proof

Use inductive reasoning to make and test conjectures. Analyze conditional statements and write the converse, inverse, and contrapositive of a conditional statement. Use deductive reasoning, the Law of Detachment, and the Law of Syllogism to develop simple logical arguments. Use properties of equality and the laws of logic to prove basic theorems.

CC.9-12.A.REI.1, CC.9-12.G.CO.1, CC.9-12.G.CO.9, CC.9-12.G.CO.10, CC.9-12.G.CO.11

Lessons – 1.1, 1.2, 1.4, 1.5

Chapter 3 – Parallel and Perpendicular Lines

Classify angle pairs formed by three intersecting lines. Study angle pairs formed by a line that intersects two parallel lines. Use angle relationships to prove lines parallel. Investigate slopes of lines and study the relationship between slopes of parallel and perpendicular lines. Find equations of lines. Prove theorems about perpendicular lines. Find the distance between parallel lines on the coordinate plane.

CC.9-12.A.CED.2, CC.9-12.G.CO.1, CC.9-12.G.CO.9, CC.9-12.G.CO.12, CC.9-12.G.GPE.5

Lessons – 1.1, 1.4, 1.5, 2.3, 2.5, 2.6

Chapter 4 – Congruent Triangles

Classify triangles. Find measures of angles in triangles. Identify congruent fi gures. Prove triangles congruent. Use theorems about isosceles and equilateral triangles. Perform transformations.

CC.9-12.G.CO.2, CC.9-12.G.CO.5, CC.9-12.G.CO.6, CC.9-12.G.CO.7, CC.9-12.G.CO.8, CC.9-12.G.CO.10, CC.9-12.G.CO.12

Lessons – 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 2.1, 2.6, 3.1, 3.2

Chapter PrerequisitesHolt McDougal Larson Geometry

GE_CCCC_TE_618258_PRE.indd 58GE_CCCC_TE_618258_PRE.indd 58 12/17/10 3:20:33 AM12/17/10 3:20:33 AM

Chapter Prerequisites 59


Chapter 5 – Relationships within Triangles

Use properties of midsegments to fi nd lengths of segments in triangles. Write a coordinate proof. Explore perpendicular bisectors and use the concurrency of perpendicular bisectors of a triangle to solve problems. Use angle bisectors to fi nd distance relationships and explore the concurrency of angle bisectors of a triangle. Use medians of a triangle to fi nd the centroid and to fi nd segment lengths. Use altitudes of a triangle to fi nd and explore the orthocenter. Relate side lengths and angle measures of a triangle. Find possible side lengths for the third side of a triangle. Use inequalities to make comparisons in two triangles. Use the Hinge Theorem and its converse to solve multi-step problems. Write indirect proofs.

CC.9-12.G.CO.9, CC.9-12.G.CO.10, CC.9-12.G.CO.12, CC.9-12.G.C.3, CC.9-12.G.GPE.4

Lessons – 1.3, 1.4, 2.2, 3.1, 4.1

Chapter 6 – Similarity

Use ratios, proportions, and geometric means to solve geometry problems. Use ratios to fi nd the scale of a drawing and then use the scale to fi nd actual distances. Use proportions to identify similar polygons and fi nd the scale factor between two polygons. Use the AA Similarity Postulate, the SSS Similarity Theorem, or the SAS Similarity Theorem to determine whether two triangles are similar. Use proportions and the Triangle Proportionality Theorem or its converse to fi nd the lengths of segments related to triangles or parallel lines. Perform dilations that are reductions or enlargements and verify that a fi gure is similar to its dilation.

CC.9-12.G.CO.2, CC.9-12.G.SRT.1, CC.9-12.G.SRT.2, CC.9-12.G.SRT.3, CC.9-12.G.SRT.4, CC.9-12.G.SRT.5, CC.9-12.G.C.1, CC.9-12.G.GPE.4, CC.9-12.G.GPE.6, CC.9-12.G.MG.3

Lessons – 1.4, 3.1, 3.4, 4.1

<… Chapter Prerequisites


60 Chapter Prerequisites


Chapter 7 – Right Triangles and Trigonometry

Investigate side lengths and angles in triangles. Use the Pythagorean theorem to fi nd the length of the third side in a right triangle. Use the Converse of the Pythagorean Theorem, and other theorems, to decide if three given side lengths form an acute, right, or obtuse triangle. Explore ratios and lengths formed by an altitude to the hypotenuse of a right triangle. Use the ratios of side lengths for 458-458-908 and 308-608-908 triangles.triangle. Apply trigonometric ratios to fi nd side lengths and angle measures in triangles.

CC.8.G.6, CC.8.G.7, CC.9-12.G.SRT.4, CC.9-12.G.SRT.5, CC.9-12.G.SRT.6, CC.9-12.G.SRT.8, CC.9-12.G.SRT.9, CC.9-12.G.SRT.10, CC.9-12.G.SRT.11, CC.9-12.G.GPE.7, CC.9-12.G.MG.1

Lessons – 4.1, 4.7, 5.1, 5.5

Chapter 8 – Quadrilaterals

Find angle measures in polygons. Investigate properties of parallelograms and learn what information is needed to conclude that a quadrilateral is a parallelogram. Study special quadrilaterals such as rhombuses, rectangles, squares, trapezoids, and kites.

CC.8.G.5, CC.9-12.G.CO.11, CC.9-12.G.SRT.5, CC.9-12.G.GPE.4, CC.9-12.G.GPE.7

Lessons – 1.1, 1.2, 1.3, 1.5, 2.5, 2.6, 3.1, 3.3, 3.4, 4.1, 4.3, 4.5, 5.1

Chapter 9 – Properties of Transformations

Perform translations with vectors, algebra, and matrices. Refl ect fi gures in a given line. Rotate fi gures about a point. Identify line and rotational symmetry. Perform transformations using drawing tools and matrices.

CC.9-12.G.CO.2, CC.9-12.G.CO.3, CC.9-12.G.CO.4, CC.9-12.G.CO.5, CC.9-12.G.SRT.1, CC.9-12.G.SRT.2

Lessons – 1.3, 1.6, 3.1, 4.4, 4.8, 5.2, 6.7, 8.2

Chapter 10 – Properties of Circles

Draw tangents to circles and see how a tangent to a circle is related to the radius at the point of tangency. Use intercepted arcs of circles to measure angles formed by chords in a circle. Measure angles formed by secants and tangents to a circle. Explore relationships between segment lengths of chords that intersect in a circle. Investigate relationships between segment lengths of secants and tangents to a circle. Use the standard equation of a circle to graph and describe circles in a coordinate plane.

CC.9-12.G.CO.1, CC.9-12.G.CO.12, CC.9-12.G.CO.13, CC.9-12.G.C.2, CC.9-12.G.C.3, CC.9-12.G.C.4, CC.9-12.G.C.5, CC.9-12.G.GPE.1

Lessons – 1.1, 1.3, 1.5, 3.3, 5.2, 7.1


Chapter Prerequisites 61


Chapter 11 – Measuring Length and Area

Develop and use formulas for the area of triangles, parallelograms, trapezoids, and other polygons. Use ratios to fi nd areas of similar polygons. Use ratios of areas to fi nd missing lengths in similar fi gures. Explore circles, relating arc length and circumference to areas of sectors. Develop and use a formula for the area of a regular polygon. Use lengths of segments and areas of regions to calculate probabilities.

CC.9-12.N.Q.3, CC.9-12.G.CO.13, CC.9-12.G.SRT.8, CC.9-12.G.C.5, CC.9-12.G.GPE.7, CC.9-12.G.GMD.1

Lessons – 1.2, 5.1, 7.1, 7.4, 7.5, 7.6, 8.2, 8.4, 8.5, 10.2

Chapter 12 – Surface Area and Volume of Solids

Identify and name solids. Use Euler’s Theorem to relate the number of faces, vertices, and edges of solids. Describe cross sections of solids. Find the surface areas and lateral areas of prisms and cylinders. Find the surface area and volume of prisms, cylinders, cones, pyramids, spheres, and composite solids. Use scale factors in similar solids to compare the ratios of the surface areas and the ratios of the volumes of the solids.

CC.9-12.7.G.6, CC.9-12.G.GMD.1, CC.9-12.G.GMD.3, CC.9-12.G.GMD.4, CC.9-12.G.MG.1, CC.9-12.G.MG.3

Lessons – 1.6, 6.3, 7.1, 7.5, 11.5, 11.6

<… Chapter Prerequisites


Course Planner for Differentiated InstructionChapter 1 – Essentials of Geometry


On-Level Learners[RTI Tier 1]

Special Needs Learners[RTI Tier 2]

1.1 Identify Points, Lines, and Planes(1 day)

CC.9-12.G.CO.1 ❏ Practice B 1.1, CR❏ Notetaking Guide 1.1❏ Key Questions to Ask, TE❏ Study Guide 1.1, CR

❏ Inclusion Notes 1.1, DIR❏ Differentiated Instruction, TE❏ Remediation Book❏ Practice A 1.1, CR

1.2 Use Segments and Congruence(1 day)

CC.9-12.G.CO.1, CC.9-12.G.CO.7 ❏ Practice B 1.2, CR❏ Notetaking Guide 1.2❏ Key Questions to Ask, TE❏ Study Guide 1.2, CR


1.3 Use Midpoint and Distance Formulas(1 day)

CC.9-12.G.GPE.7 ❏ Practice B 1.3, CR❏ Notetaking Guide 1.3❏ Key Questions to Ask, TE❏ Study Guide 1.3, CR


Assessment Options ❏ Quiz for 1.1 to 1.3, SE❏ Online Quiz❏ Quiz 1, AR❏ Test and Practice Generator

❏ Quiz for 1.1 to 1.3, SE❏ Online Quiz❏ Quiz 1, AR❏ Test and Practice Generator

1.4 Measure and Classify Angles(1 day)



See pp. 2–29 for the full text of the Common Core Mathematics Standards for High School.

62 Course Planner

GE_CCCCETE618258-CP.indd 62GE_CCCCETE618258-CP.indd 62 12/18/10 3:58:26 AM12/18/10 3:58:26 AM

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Red Type Minimum Course of Study CC Curriculum Companion

CR Chapter Resources DIR Differentiated Instruction Resources TE Teacher’s Edition

AR Assessment Resources SE Student Edition PAP Pre-AP Resources

KEY

Developing Learners[RTI Tiers 1 and 2]

Advanced Learners English Language Learners

❏ Practice A 1.1, CR❏ Notetaking Guide 1.1❏ Key Questions to Ask, TE❏ Differentiated Instruction, DIR❏ Study Guide 1.1, CR

❏ Practice C 1.1, CR ❏ Challenge 1.1, CR❏ Pre-AP Best Practices 1.1, PAP

❏ Spanish Study Guide, 1.1❏ Student Resources in Spanish, 1.1❏ English Learner Notes 1.1, DIR❏ Multi-Language Visual Glossary









❏ Quiz 1.1 to 1.3, Student Resources in Spanish❏ Online Quiz❏ Quiz 1, Spanish AR❏ Test and Practice Generator




Course Planner 63


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Course Planner for Differentiated InstructionChapter 1 – Essentials of Geometry




Construction: Copy and Bisect Segments and Angles (1/2 day)

CC.9-12.G.CO.12

1.5 Describe Angle Pair Relationships(1 day)





1.6 Classify Polygons(1 day)

CC.9-12.G.MG.1 ❏ Practice B 1.6, CR❏ Notetaking Guide 1.6❏ Key Questions to Ask, TE❏ Study Guide 1.6, CR


1.7 Find Perimeter, Circumference, and Area

❏ Practice B 1.7, CR❏ Notetaking Guide 1.7❏ Key Questions to Ask, TE❏ Study Guide 1.7, CR


Assessment Options ❏ Quiz for 1.6 to 1.7, SE❏ Online Quiz❏ Quiz 3, AR❏ Chapter Test, SE❏ Chapter Test B, AR❏ Standardized Test, AR❏ SAT/ACT Test, AR❏ Alternative Assessment, AR❏ Test and Practice Generator

❏ Quiz for 1.6 to 1.7, SE❏ Online Quiz❏ Quiz 3, AR❏ Chapter Test, SE❏ Chapter Test A, AR❏ Chapter Test, Benchmark Tests❏ Test and Practice Generator



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❏ Practice C 1.5, CR ❏ Challenge 1.5, CR❏ Pre-AP Best Practices 1.5, PAP❏ Pre-AP Copymaster 1.5, PAP












❏ Quiz for 1.6 to 1.7, SE❏ Online Quiz❏ Quiz 3, AR❏ Chapter Test, SE❏ Chapter Test C, AR❏ Standardized Test, AR❏ SAT/ACT Test, AR❏ Alternative Assessment, AR❏ Test and Practice Generator

❏ Quiz 1.6 to 1.7, Student Resources in Spanish❏ Online Quiz❏ Quiz 3, AR❏ Chapter Test, Student Resources in Spanish❏ Chapter Test B, Spanish AR❏ Standardized Test, Spanish AR❏ SAT/ACT Test, Spanish AR❏ Alternative Assessment, Spanish AR❏ Test and Practice Generator


66 Course Planner




2.1 Use Inductive Reasoning(1 day)

Prepare for CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11



2.2 Analyze Conditional Statements(1 1/2 days)




Activity: Logic Puzzles

2.3 Apply Deductive Reasoning (1 1/2 days)

CC.9-12.G.CO.1, Prepare for CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11



Extension: Symbolic Notation and Truth Tables



2.4 Use Postulates and Diagrams(1 day)





Course Planner for Differentiated InstructionChapter 2 – Reasoning and Proof


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68 Course Planner

Course Planner for Differentiated InstructionChapter 2 – Reasoning and Proof




Activity: Justify a Number Trick (1/2 day)

CC.9-12.A.REI.1

2.5 Reason Using Properties from Algebra(1 day)

CC.9-12.A.REI.1, Prepare for CC.9-12.G.CO.9, CC.9-12.G.CO.10, and CC.9-12.G.CO.11





2.6 Prove Statements about Segments and Angles(2 days)




Activity: Angles and Intersecting Lines(1/2 day)

CC.9-12.G.CO.9

2.7 Prove Angle Pair Relationships(1 1/2 days)







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70 Course Planner

Course Planner for Differentiated InstructionChapter 3 – Parallel and Perpendicular Lines




Activity: Draw and Interpret Lines (1/2 day)

CC.9-12.G.CO.1

3.1 Identify Pairs of Lines and Angles(1 day)



Activity: Parallel Lines and Angles (1/2 day)

CC.9-12.G.CO.9

3.2 Use Parallel Lines and Transversals(1 day)





3.3 Prove Lines are Parallel(1 day)



3.4 Find and Use Slopes of Lines(1 1/2 days)





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Activity: Investigate Slopes(1/2 day)

CC.9-12.G.GPE.5



3.5 Write and Graph Equations of Lines(1 1/2 days)

CC.9-12.G.GPE.5, CC.9-12.A.CED.2



3.6 Prove Theorems about Perpendicular Lines(1 1/2 days)



Construction: Parallel and Perpendicular Lines(1/2 day)

CC.9-12.G.CO.12




Course Planner for Differentiated InstructionChapter 3 – Parallel and Perpendicular Lines


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Activity: Angle Sums in Triangles(1/2 day)

CC.9-12.G.CO.10

4.1 Apply Triangle Sum Properties(1 day)



4.2 Apply Congruence and Triangles(1 1/2 days)



Activity: Rigid Motions in the Plane (CC) (1/2 day)

CC.9-12.G.CO.2, CC.9-12.G.CO.7

4.2B Relate Transformations and Congruence (CC) (1 day)

CC.9-12.G.CO.2, CC.9-12.G.CO.6, CC.9-12.G.CO.7

❏ Practice B 4.2B, CR❏ Key Questions to Ask, TE❏ Study Guide 4.2B, CR

❏ Differentiated Instruction, TE❏ Remediation Book

Activity: Investigate Congruent Figures (1/2 day)

CC.9-12.G.CO.6, CC.9-12.G.CO.8

4.3 Prove Triangles Congruent by SSS (1 day)





4.4 Prove Triangles Congruent by SAS and HL(1 1/2 days)



Activity: Investigate Triangles and Congruence (1/2 day)

CC.9-12.G.CO.6, CC.9-12.G.CO.8


Course Planner for Differentiated InstructionChapter 4 – Congruent Triangles


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❏ Key Questions to Ask, TE❏ Differentiated Instruction, DIR❏ Study Guide 4.2B, CR

❏ Challenge 4.2B, CR❏ Pre-AP Best Practices 4.2B, PAP

❏ Multi-Language Visual Glossary











76 Course Planner




4.5 Prove Triangles Congruent by ASA and AAS(1 1/2 days)



Activity: Rigid Motions and Congruence (CC) (1/2 day)

CC.9-12.G.CO.8

4.6 Use Congruent Triangles(1 day)





4.7 Use Isosceles and Equilateral Triangles(1 day)



Activity: Investigate Slides and Flips (1/2 day)

CC.9-12.G.CO.2, CC.9-12.G.CO.5

4.8 Perform Congruence Transformations(1 day)






Course Planner for Differentiated InstructionChapter 4 – Congruent Triangles


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78 Course Planner




Activity: Investigate Segments in Triangles (1/2 day)

CC.9-12.G.CO.10, CC.9-12.G.GPE.4

5.1 Midsegment Theorem and Coordinate Proof(1 1/2 days)

CC.9-12.G.CO.10, CC.9-12.G.GPE.4 ❏ Practice B 5.1, CR❏ Notetaking Guide 5.1❏ Key Questions to Ask, TE❏ Study Guide 5.1, CR


5.2 Use Perpendicular Bisectors(2 days)

CC.9-12.G.CO.9, CC.9-12.G.CO.12, CC.9-12.G.C.3





5.3 Use Angle Bisectors of Triangles(1 day)

CC.9-12.G.C.3 ❏ Practice B 5.3, CR❏ Notetaking Guide 5.3❏ Key Questions to Ask, TE❏ Study Guide 5.3, CR


Activity: Intersecting Medians (1/2 day)

CC.9-12.G.CO.12

5.4 Use Medians and Altitudes(1 day)




Course Planner for Differentiated InstructionChapter 5 – Relationships within Triangles


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Activity: Investigate Points of Concurrency (1/2 day)

CC.9-12.G.CO.12



5.5 Use Inequalities in a Triange(1 day)

Extend CC.9-12.G.CO.7 ❏ Practice B 5.5, CR❏ Notetaking Guide 5.5❏ Key Questions to Ask, TE❏ Study Guide 5.5, CR


5.6 Inequalities in Two Triangles and Indirect Proof(1 1/2 days)

Extend CC.9-12.G.CO.7 ❏ Practice B 5.6, CR❏ Notetaking Guide 5.6❏ Key Questions to Ask, TE❏ Study Guide 5.6, CR





Course Planner for Differentiated InstructionChapter 5 – Relationships within Triangles


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82 Course Planner




6.1 Ratios, Proportions, and the Geometric Mean



6.2 Use Proportions to Solve Geometry Problems





Activity: Similar Polygons (1/2 day)

CC.9-12.G.SRT.5

6.3 Use Similar Polygons(1 1/2 days)

CC.9-12.G.SRT.5 ❏ Practice B 6.3, CR❏ Notetaking Guide 6.3❏ Key Questions to Ask, TE❏ Study Guide 6.3, CR


Activity: Explore Properties of Dilations (CC) (1/2 day)

CC.9-12.G.SRT.1

6.3B Relate Transformations and Similarity (CC)(1 day)

CC.9-12.G.SRT.1, CC.9-12.G.SRT.2, CC.9-12.G.C.1

❏ Practice B 6.3B, CR❏ Key Questions to Ask, TE❏ Study Guide 6.3B, CR

❏ Differentiated Instruction, TE❏ Remediation Book

Activity: Dilations and AA Similarity (CC)(1/2 day)

CC.9-12.G.SRT.3

6.4 Prove Triangles Similar by AA(1 day)




Course Planner for Differentiated InstructionChapter 6 – Similarity


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KEY















❏ Key Questions to Ask, TE❏ Differentiated Instruction, DIR❏ Study Guide 6.3B, CR

❏ Challenge 6.3B, CR❏ Pre-AP Best Practices 6.3B, PAP

❏ Multi-Language Visual Glossary





84 Course Planner

Course Planner for Differentiated InstructionChapter 6 – Similarity




6.5 Prove Triangles Similar by SSS and SAS(1 1/2 days)





Activity: Investigate Proportionality (1/2 day)

CC.9-12.G.SRT.4

6.6 Use Proportionality Theorems(1 1/2 days)



Extension: Fractals (1/2 day)

CC.9-12.G.SRT.5, CC.9-12.G.MG.3

Activity: Dilations (1/2 day)

CC.9-12.G.CO.2, CC.9-12.G.SRT.1b

6.7 Perform Similarity Transformations(1 1/2 days)

CC.9-12.G.CO.2, CC.9-12.G.SRT.1, CC.9-12.G.GPE.4



Extension: Partition Segments (CC) (1/2 day)

CC.9-12.G.GPE.6





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86 Course Planner

Course Planner for Differentiated InstructionChapter 7 – Right Triangles and Trigonometry




7.1 Apply the Pythagorean Theorem(2 days)

CC.8.G.6, CC.8.G.7, CC.9-12.G.SRT.4, CC.9-12.G.SRT.8, CC.9-12.G.GPE.7



7.2 Use the Converse of the Pythagorean Theorem(1 day)

CC.8.G.6, CC.8.G.7, CC.9-12.G.SRT.8





Activity: Similar Right Triangles (1 1/2 days)

CC.9-12.G.SRT.4, CC.9-12.G.SRT.5

7.3 Use Similar Right Triangles(1/2 day)

CC.9-12.G.SRT.4, CC.9-12.G.SRT.5, CC.9-12.G.MG.1



7.4 Special Right Triangles(1 1/2 days)







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88 Course Planner

Course Planner for Differentiated InstructionChapter 7 – Right Triangles and Trigonometry




7.5 Apply the Tangent Ratio(1 day)

CC.9-12.G.SRT.6, CC.9-12.G.SRT.8 ❏ Practice B 7.5, CR❏ Notetaking Guide 7.5❏ Key Questions to Ask, TE❏ Study Guide 7.5, CR


7.6 Apply the Sine and Cosine Ratios(2 days)

CC.9-12.G.SRT.6, CC.9-12.G.SRT.8, CC.9-12.G.SRT.9



7.7 Solve Right Triangles(1 1/2 days)



Extension: Law of Sines and Law of Cosines (1/2 day)

CC.9-12.G.SRT.10, CC.9-12.G.SRT.11





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90 Course Planner




8.1 Find Angle Measures in Polygons(2 days)

CC.8.G.5, CC.9-12.G.CO.11 ❏ Practice B 8.1, CR❏ Notetaking Guide 8.1❏ Key Questions to Ask, TE❏ Study Guide 8.1, CR


Activity: Investigate Parallelograms (1/2 day)

CC.9-12.G.CO.11

8.2 Use Properties of Parallelo-grams (1 day)

CC.9-12.G.CO.11, CC.9-12.G.SRT.5 ❏ Practice B 8.2, CR❏ Notetaking Guide 8.2❏ Key Questions to Ask, TE❏ Study Guide 8.2, CR




8.3 Show that a Quadrilateral is a Parallelogram(1 1/2 days)



8.4 Properties of Rhombuses, Rectangles, and Squares(1 1/2 days)

CC.9-12.G.CO.11, CC.9-12.G.GPE.7, CC.9-12.G.SRT.5






Course Planner for Differentiated InstructionChapter 8 – Quadrilaterals


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92 Course Planner

Course Planner for Differentiated InstructionChapter 8 – Quadrilaterals




Activity: Midsegment of a Trapezoid (1/2 day)

CC.9-12.G.SRT.5

8.5 Use Properties of Trapezoids and Kites(1 1/2 days)

CC.9-12.G.SRT.5, CC.9-12.G.GPE.4 ❏ Practice B 8.5, CR❏ Notetaking Guide 8.5❏ Key Questions to Ask, TE❏ Study Guide 8.5, CR


8.6 Identify Special Quadrilaterals(1 day)







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9.1 Translate Figures and Use Vectors (1 day)



9.2 Use Properties of Matrices(2 days)

CC.9-12.G.CO.4, CC.9-12.N.VM.6, CC.9-12.N.VM.12





Activity: Refl ections in the Plane (1/2 day)

CC.9-12.G.CO.2, CC.9-12.G.CO.5

9.3 Perform Refl ections(1 1/2 days)



9.4 Perform Rotations(1 1/2 days)



9.5 Activity: Double Refl ections (1/2 day)

CC.9-12.G.CO.2, CC.9-12.G.CO.5

9.5 Apply Composition of Transformations (1 1/2 days)




Course Planner for Differentiated InstructionChapter 9 – Properties of Transformations


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Course Planner for Differentiated InstructionChapter 9 – Properties of Transformations




Extension: Tessellations (1 day)

CC.9-12.G.CO.5



9.6 Identify Symmetry (1 day)



Activity: Investigate Dilations (1/2 day)

CC.9-12.G.CO.2, CC.9-12.G.CO.5

9.7 Identify and Perform Dilations(1 day)



Activity: Compositions with Dilations (1/2 day)

Extend CC.9-12.G.SRT.2





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❏ Practice C 9.7, CR ❏ Challenge 9.7, CR❏ Pre-AP Best Practices 9.7, PAP❏ Pre-AP Copymaster 9.7, PA






98 Course Planner

Course Planner for Differentiated InstructionChapter 10 – Properties of Circles




Activity: Explore Tangent Segments (1/2 day)

CC.9-12.G.C.4

10.1 Use Properties of Tangents (2 days)

CC.9-12.G.CO.1, CC.9-12.G.C.2 ❏ Practice B 10.1, CR❏ Notetaking Guide 10.1❏ Key Questions to Ask, TE❏ Study Guide 10.1, CR


10.2 Find Arc Measures(1 day)



10.3 Apply Properties of Chords (1 day)

CC.9-12.G.CO.12, CC.9-12.G.C.3 ❏ Practice B 10.3, CR❏ Notetaking Guide 10.3❏ Key Questions to Ask, TE❏ Study Guide 10.3, CR




Activity: Explore Inscribed Angles (1/2 day)

CC.9-12.G.C.2

10.4 Use Inscribed Angles and Polygons(2 days)

CC.9-12.G.C.2, CC.9-12.G.C.3, CC.9-12.G.C.4, CC.9-12.G.C.5





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100 Course Planner

Course Planner for Differentiated InstructionChapter 10 – Properties of Circles




Construction: Tangent Lines and Inscribed Squares (CC) (1 day)

CC.9-12.G.CO.13, CC.9-12.G.C.4

10.5 Apply Other Angle Relationships in Circles(1 day)

CC.9-12.G.C.2, CC.9-12.G.C.5 ❏ Practice B 10.5, CR❏ Notetaking Guide 10.5❏ Key Questions to Ask, TE❏ Study Guide 10.5, CR




Activity: Investigate Segment Lengths (1/2 day)

CC.9-12.G.C.2

10.6 Find Segment Lengths in Circles(1 day)



10.7 Write and Graph Equations of Circles(2 days)




❏ Quiz for 10.6 to 10.7, SE❏ Online Quiz❏ Quiz 3, AR❏ Chapter Test, SE❏ Chapter Test B, AR❏ Standardized Test, AR❏ SAT/ACT Test, AR❏ Alternative Assessment, AR❏ Test and Practice Generator



Course Planner 101

<… Course Planner




KEY



















102 Course Planner




11.1 Areas of Triangles and Parallelograms



11.2 Areas of Trapezoids, Rhombuses, and Kites



11.3 Perimeter and Area of Similar Figures





11.4 Circumference and Arc Length(2 days)

CC.9-12.G.C.5, CC.9-12.G.GMD.1 ❏ Practice B 11.4, CR❏ Notetaking Guide 11.4❏ Key Questions to Ask, TE❏ Study Guide 11.4, CR


Extension: Measure Angles in Radians (CC) (1 day)

CC.9-12.G.C.5


Course Planner for Differentiated InstructionChapter 11 – Measuring Length and Area


Course Planner 103

<… Course Planner




KEY












❏ Quiz 11.1 to 11.3, SE❏ Online Quiz❏ Quiz 1, Spanish AR❏ Test and Practice Generator







104 Course Planner

Course Planner for Differentiated InstructionChapter 11 – Measuring Length and Area




11.5 Areas of Circles and Sectors(1 day)





11.6 Areas of Regular Polygons(1 day)



11.7 Use Geometric Probability(1 1/2 days)

CC.9-12.G.MG.1, CC.9-12.S.CP.1 ❏ Practice B 11.7, CR❏ Notetaking Guide 11.7❏ Key Questions to Ask, TE❏ Study Guide 11.7, CR


Activity: Investigate Geometric Probability





Course Planner 105

<… Course Planner




KEY



















106 Course Planner




Activity: Measuring Angles in Radians(1/2 day)

12.1 Explore Solids(1 day)

CC.9-12.G.GMD.4 ❏ Practice B 12.1, CR❏ Notetaking Guide 12.1❏ Key Questions to Ask, TE❏ Study Guide 12.1, CR


12.2 Surface Area of Prisms and Cylinders



12.3 Surface Area of Pyramids and Cones





12.4 Volume of Prisms and Cylinders(1 1/2 days)

CC.9-12.G.GMD.1, CC.9-12.G.GMD.3, CC.9-12.G.GMD.4



12.4A Extension: Density (CC) (1/2 day)

CC.9-12.G.MG.2

Activity: Investigate the Volume of a Pyramid (1/2 day)

CC.9-12.G.GMD.1

12.5 Volume of Pyramids and Cones(1 day)

CC.9-12.G.GMD.1, CC.9-12.G.GMD.3, CC.9-12.G.MG.3




Course Planner for Differentiated InstructionChapter 12 – Surface Area and Volume of Solids


Course Planner 107

<… Course Planner




KEY















❏ Practice A 12.4, CR ❏ Notetaking Guide 12.4❏ Key Questions to Ask, TE❏ Differentiated Instruction, DIR❏ Study Guide 12.5, CR







108 Course Planner




Activity: Minimize Surface Area (1/2 day)

CC.9-12.G.MG.3

Extension: Solids of Revolution (CC) (1/2 day)

CC.9-12.G.GMD.4



12.6 Surface Area an Volume of Spheres(1 day)



Activity: Investigate Similar Solids (1/2 day)

CC.9-12.G.GMD.3

12.7 Explore Similar Solids(1 day)



Extension: Symmetries of Solids (1/2 day)

CC.9-12.G.CO.3




Course Planner for Differentiated InstructionChapter 12 – Surface Area and Volume of Solids


Course Planner 109

<… Course Planner




KEY
















Pre-Course TestSkills Readiness

EVALUATE POWERSFind the value of the expression.

1. 72 + 32 2. 82 + 62

ROUNDING AND ESTIMATIONRound the decimal to the indicated place value.

3. 9.53; tenth 4. 3.076; hundredth

SIMPLIFY FRACTIONSWrite the fraction in simplest form.

5. 24 _ 40

6. 6 _ 33

RATIOSUse the table to write each ratio in simplest form.

Rental Cars

Gray 15

White 9

Blue 6

7. gray cars to white cars

8. blue cars to total cars

MEASURE WITH CUSTOMARY AND METRIC UNITSMeasure the segment to the nearest eighth of an inch and to the nearest half of a centimeter.

9.

10.

NAME AND CLASSIFY ANGLESName and classify the angle.

11.

S

R

P 12. F

G

H

MEASURE ANGLESUse a protractor to measure the angle.

13. 14.

ANGLE RELATIONSHIPSUse the diagram to give an example of the angle pair.

15. complementary angles

16. adjacent angles

JK

L

M

NP

O

CLASSIFY TRIANGLESTell whether the triangle is acute, right, or obtuse.

17.

298

618

18.

358 1208

GE_MTNAESE476896_PT.indd TN68 30/08/2010 12:57:56GE_MTNAESE476896_PT.indd TN69 30/08/2010 12:58:13

40. 1y

x21(1, 21)

(0, 23)

41. y

x22

(2, 1)

(0, 0)

42. 1

y

x1

(0, 23)

Answers:1. 582. 1003. 9.54. 3.08

5. 3 _ 5

6. 2 __ 11

7. 5 to 3, 5 : 3, or 5 _ 3

8. 1 to 5, 1 : 5, or 1 _ 5

9. 1 5 _ 8 in.; 4.0 cm

10. 1 in.; 3.0 cm11. /PRS ; obtuse12. /FGH ; straight13. 40°14. 135°15. Sample answer: /JOK and

/KOL16. Sample answer: /KOM and

/MON17. right18. obtuse19. 5 Ï

}

2 20. 4 Ï

}

3 21. 35 in.22. 8 ft23. 28 cm24. 81 in.2

25. 10.5 cm2

26. 15 ft2

27. 6π cm; 9π cm2 28. 10π ft, 25π ft2 29. 72

30. 3 _ 5

31. 3532. 22733. 134. 4035. p 5 636. q 5 1937. e 5 21538. d 5 2939.

1

y

x1

(1, 4)(0, 3)

110 Skills Readiness

GE_CCCCETE618258-PCT.indd 110GE_CCCCETE618258-PCT.indd 110 12/18/10 4:07:50 AM12/18/10 4:07:50 AM

<… Answers to Countdown

<… Skills ReadinessPre-Course Test

GE_MTNAESE476896_PT.indd TN68 30/08/2010 12:57:56

PYTHAGOREAN THEOREMFind x in the right triangle. If the length is not a whole number, give the answer in simplest radical form.

19.

5

5 x

20.

8

4x

FIND PERIMETERFind the perimeter of the figure.

21. equilateral pentagon with side length 7 in.

22. square with side length 2 ft

23. rectangle with length 8 cm and width 6 cm

AREA OF POLYGONSFind the area of the figure.

24. square with side length 9 in.

25. rectangle with length 3.5 cm and width 3 cm

26. triangle with base 10 ft and height 3 ft

CIRCUMFERENCE AND AREA OF CIRCLESFind the circumference and area of the circle. Give your answers in terms of p.

27.

3 cm

28.

10 ft

SIMPLIFY RADICAL EXPRESSIONSSimplify the expression.

29. Ï}

64 · Ï}

81 30. Ï}

36 _ 100

EVALUATE EXPRESSIONSEvaluate the expression for the given value of the variable.

31. 4t + 7 for t = 7

32. 5w -7 for w = -4

33. g

_ 3

+ (-2) for g = 9

34. (h + 9)(h - 9) for h = 11

SOLVE MULTI-STEP EQUATIONSSolve.

35. 7p + 5 = 47 36. 2q - 7 = 31

37. e _ 5

- 6 = -9 38. d _ 3

+ 4 = 1

GRAPH LINEAR FUNCTIONSGraph the function.

39. y = x + 3 40. y = 2x - 3

41. y = 1 _ 2

x 42. y = -3

SOLVE PROPORTIONSSolve the proportion.

43. 4 _ 5

= h _ 35

44. 7 _ m = 21 _ 30

45. n _ 2

= 5 _ 4

46. 2 _ 7

= 5 _ p

ORDERED PAIRSGraph the point.

47. A(2, 2) 48. B(-3, 3)

49. C(-3, -3) 50. D(3, -3)

GE_MTNAESE476896_PT.indd TN69 30/08/2010 12:58:13

Items Skill

1–2 8

3–4 9

5–6 10

7–8 12

9–10 20

11–12 23

13–14 24

15–16 25

17–18 29

19–20 31

21–23 36

24–26 37

27–28 39

29–30 53

31–34 60

35–38 69

39–42 75

43–46 77

47–50 79

43. h 5 2844. m 5 1045. n 5 2.546. p 5 17.5

47–50.

1

x

y

DC

BA

21

InterventionSkills Readiness, available on the Easy Planner, provides review and practice for the items on the Pre-Course Test.

Pre-Course TestPre-Course TestSkills Readiness

Skills Readiness 111

GE_CCCCETE618258-PCT.indd 111GE_CCCCETE618258-PCT.indd 111 12/18/10 4:08:05 AM12/18/10 4:08:05 AM

112 Additional Content

Additional Content Geometry

Investigating Geometry Activity 4.2A Rigid Motions in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . CC1

Lesson 4.2B Relate Transformations and Congruence . . . . . CC3

Construction 4.5A Rigid Motions and Congruence . . . . . . CC10

Investigating Geometry Activity 6.3A Explore Properties of Dilations . . . . . . . . . . . . . . . . . . . . . . CC12

Lesson 6.3B Relate Transformations and Similarity . . . . . . CC13

Investigating Geometry Activity 6.4A Dilations and AA Similarity . . . . . . . . . . . . . . . . . . . . . . . . . CC20

Extension 6.7A Partition Segments . . . . . . . . . . . . . . . . . . . CC22

Construction 10.4A Tangent Lines and Inscribed Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC24

Extension 11.4A Measure Angles in Radians. . . . . . . . . . . . CC27

Extension 12.4A Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . CC30

Extension 12.5A Solids of Revolution . . . . . . . . . . . . . . . . . CC32

Contents

GE_CCCCETE618258-AC.indd 112GE_CCCCETE618258-AC.indd 112 12/18/10 3:58:06 AM12/18/10 3:58:06 AM

CC1

4.2B Relate Transformations and Congruence CC1

4.2A Rigid Motions in the PlaneMATERIALS • graph paper • ruler • protractor

Q U E S T I O N Which transformations are rigid motions?

Transformations are functions that map points onto points. The result of transforming a fi gure is called its image. The original fi gure is called the preimage.

You can use function rules expressed with coordinate notation to describe some transformations.

E X P L O R E 1 Use function rules for transformations

STEP 1 Graph a triangle

Graph the triangle whose vertices have the coordinates (1, 4), (3, 22), and (21, 1).

y

xO 3

3

(3, –2)

(–1, 1)

(1, 4)

STEP 2 Transform the triangle

Transform each vertex of the triangle using this function rule:

(x, y) → (x 2 2, y 2 1).

y

xO 3

3

(3, –2)

(–1, 1)

(1, 4)(–1, 3)

(1, –3)

(–3, 0)

STEP 3 Describe the transformation

The transformation is a slide, or translation, 2 units left and 1 unit down.

The image is congruent to the preimage.

STEP 4 Repeat Steps 1–3 for different transformations

For each function rule, draw a triangle and its image. Describe the transformation. Then tell whether the image is congruent to the preimage.

a. (x, y) → (2x, y)

b. (x, y) → (2x, 2y)

c. (x, y) → (2y, x)

d. (x, y) → (x, 2y)

he Plane

Use before Lesson 4.2BUse bUse beACTIVITYCTIVITYInvestigating GeometryIIIIIIIIIIII iiiii iiiiiInvestigatingIIIIIIIIIIIIInnnnnnnnnnnnnvvvvvvvvvvvvveeeeeeeeeeessssssssssssttttttttttttiiiiiiiiiiggggggggggggaaaaaaaaaatttttttttttttiiiiiiiiiinnnnnnnnnnnnngggggggggggg GeometryGGGGGGGGGGGGGeeeeeeeeeeoooooooooommmmmmmmmmmmmeeeeeeeeeeettttttttttttrrrrrrrrrrryyyyyyyyyyyy

g ggggggggg ggggggg

See margin.

GE_CCESE621852_0402A_ACT.indd Sec1:1 12/11/10 3:49:11 AM

Explore 1, Step 4. Sample answers are given.4a. The transformation is a flip, or reflection, in the y-axis. The image is congruent to the preimage.4b. The transformation is an enlargement, or dilation. The image is not congruent to the preimage.4c. The transformation is a 908 turn, or rotation, counterclockwise about the origin. The image is congruent to the preimage.4d. The transformation is a stretch, or shear, in a vertical direction. The image is not congruent to the preimage.

1 PLAN AND PREPARE

Explore the Concept• Students will investigate trans-

formations in the coordinate plane, looking at lengths and angle measures.

• Students will decide whether a simple transformation is a rigid motion.

• This activity leads into further study of rigid motions and their relation to congruence.

MaterialsEach student will need:• graph paper• ruler• protractor

Recommended TimeWork activity: 15 minDiscuss results: 5 min

GroupingStudents should work individually.

2 TEACHTips for SuccessIn Step 4 of Explore 1, recommend that students use two colors when drawing their triangles on a coordi-nate grid. This will help them describe each transformation.

Key Question• What must be true of a

transformation for it to be a rigid motion? The transformation must preserve length and angle measure.

LGE_CCETE621999_04-2A_LS.indd CC1LGE_CCETE621999_04-2A_LS.indd CC1 12/22/10 4:10:40 AM12/22/10 4:10:40 AM

CC2

CC2 Chapter 4 Congruent Triangles

D R A W C O N C L U S I O N S Use your observations to complete these exercises

1. Use coordinate notation to write a function rule that describes reflection in the x-axis. Is this transformation a rigid motion?

2. Use coordinate notation to write a function rule that describes a rotation of 908 clockwise about the origin. Is this transformation a rigid motion?

3. Use coordinate notation to write a function rule that describes a rotation of 1808 about the origin. Is this transformation a rigid motion?

4. A triangle has vertices (21, 2), (1, 3), and (2, 0). What are the vertices of the image after a transformation described by the function rule (x, y) → (22x, 3y)? Is this transformation a rigid motion? Explain.

5. Which transformations in Explore 2 preserve length?

6. Which transformations in Explore 2 preserve angle measure?

7. Which transformations in Explore 2 are rigid motions?

8. If a transformation preserves area, is it always a rigid motion? If so, explain. If not, give a counterexample.

E X P L O R E 2 Determine if a transformation is a rigid motion

STEP 1 Draw transformation

Draw a triangle and its image after a reflection. You may want to fold your paper along the line of reflection to trace the triangle.

m

STEP 2 Measure sides and angles

Use a protractor to measure the angles and a ruler to measure the side lengths of the preimage and image.

m

66

44

55

838838568568

418418

STEP 3 Tell whether it is a rigid motion

The reflection preserves lengths and angle measures.

So, it is a rigid motion.

STEP 4 Repeat Steps 1–3 for different transformations

Draw a triangle and its image after an example of the transformation. Tell whether the transformation is a rigid motion. a. translation (slide) yes b. rotation (turn) yes c. dilation (enlargement)no

A rigid motion is a transformation that preserves length and angle measure.

(x, y) → (x, 2y); yes

(x, y) → (y, 2x); yes

(x, y) → (2x, 2y); yes

translation, refl ection, rotation

translation, refl ection, rotation

translation, refl ection, rotation, dilation

(2, 6), (22, 9), (24, 0); no; neitherlengths nor angles are preserved,

so it is not a rigid motion.

No; a transformation such as (x, y) → (2x, 0.5y) preserves area but does not preserve length or angle measure, so it is not a rigid motion.

GE_CCESE621852_0402A_ACT.indd Sec1:2 12/11/10 12:33:57 AM

Alternative StrategyExplore 2 can be done on a coordi-nate plane if it is easier for stu-dents. Lengths can be confirmed using the distance formula if the endpoints have integer coordi-nates. However, only rotations of 908, 1808, and 2708 can be easily graphed. Reflections may need to be confined to reflections across an axis, across the line y 5 x, and across the line y 5 2x.

Key DiscoverySome transformations, but not all, preserve length and angle measure.

3 ASSESS AND RETEACH

Write a function rule using coordinate notation for the transformation. Sample answers are given.1. A transformation that preserves

both length and angle measure (x, y) → (x 1 2, y 2 3)2. A transformation that preserves

angle measure but not length (x, y) → (2x, 2y)3. A transformation that does not

preserve length and does not preserve angle measure

(x, y) → (x 1 2, 2y)



4.2BRelate Transformations and Congruence

Transformations in the plane move or change a figure to produce a new figure. A rigid motion is a transformation that preserves length, angle measure, and area. A rigid motion is also called an isometry. Translations, reflections, and rotations are examples of rigid motions.

Recall that two figures are congruent if and only if the corresponding sides and the corresponding angles are congruent. Two geometric figures are congruent if and only if there is a rigid motion or a combination of rigid motions that move one of the figures onto the other.

E X A M P L E 1 Describe rigid motions to show congruence

Describe the transformation(s) you can use to move the blue figure onto the red figure.

a.

P

b.

Solution

a. rotation about P b. translation and then reflection

READ VOCABULARY

Translations are also known as slides, reflections are also known as flips, and rotations are also known as turns.

Key Vocabulary• rigid motion

KEY CONCEPT For Your NotebookCongruent Figures and Transformations

Two figures are congruent if and only if one or more rigid motions can be used to move one figure onto the other. If any combination of translations, reflections, and rotations can be used to move one shape onto the other, the figures are congruent.

Before You identified congruent figures.

Now You will use transformations to show congruence.

Why So you can complete an architect’s drawing, as in Ex. 28.

reflectiontranslation rotation

© R

ich

ard K

lun

e/C

orb

is

GE_CCESE621852_0402B_EXPO.indd Sec1:3 12/10/10 4:38:11 AM

1 PLAN AND PREPARE

Warm-Up Exercises1. In the diagram, nABC > nDEF.

Name the pairs of corresponding angles and corresponding sides.

C B E F

A D

∠ A and ∠ D, ∠ B and ∠ E, ∠ C and ∠ F, } AB and } DE , } BC and } EF , } CA and } FD Find the image of the points after the transformation whose rule is given.2. (24, 3), (1, 4), (2, 25), (x, y ) → (x 2 2, y 1 1) (26, 4), (21, 5), (0, 24)3. (22, 1), (3, 5), (21, 24), (x, y ) → (2y, x ) (21, 22), (25, 3), (4, 21)4. (2, 24), (23, 4), (4, 22), (x, y ) → (2x, y ) (22, 24), (3, 4), (24, 22)

PacingBasic: 1 dayAverage: 1 dayAdvanced: 1 dayBlock: 0.5 block

2 FOCUS AND MOTIVATE

Essential QuestionBig Idea 2, p. 215How do you identify a rigid motion in the plane? Tell students they will learn how to answer this question by using transformations in the plane that preserve length and angle measure.

Chapter Resources• Practice level B

• Study Guide

• Challenge

• Pre-AP notes

Teaching Options• Activity Generator provides editable

activities for all ability levels

Interactive Technology• Activity Generator

• Animated Algebra

• Test Generator

• eEdition

See also the Differentiated Instruction

Resources for more strategies for

meeting individual needs.

Ch t R T h

Resource Planning Guide

CC3


CC4


✓ GUIDED PRACTICE for Example 1

Describe the transformation(s) you can use to move the blue figure onto the red figure.

1.

P

2.

E X A M P L E 2 Show figures are not congruent

Explain why figure A and figure B are not congruent using transformations.

Solution

Translate and then reflect figure A to compare it to figure B. Although some of the corresponding sides are congruent, the corresponding angles are not all congruent. So, the figures are not congruent.

A

B

A

B

E X A M P L E 3 Move one figure in a pattern onto another

QUILTING The quilt shown is made using a repeating pattern.

a. Describe a transformation that moves figure A onto figure B.

b. Explain why figures C and D are congruent by using the marked angles and sides.

Solution

a. A translation or a rotation will move figure A onto figure B.

b. Figures C and D are congruent because all pairs of corresponding sides and all pairs of corresponding angles are congruent.

© D

. Hu

rst/Alam

y

translation and then rotation translation and then refl ection

GE_CCESE621852_0402B_EXPO.indd Sec1:4 12/10/10 4:38:31 AM

3 TEACHExtra Example 1Describe the transformation(s) you can use to move figure A onto figure B.

A

BP

rotation about P and then reflection

Extra Example 2Explain why figure A and figure B are not congruent using transformations.

A

B

P

Rotate figure A about point P and then reflect that image to compare it to figure B. Although one pair of corresponding sides is congruent, none of the corresponding angles are congruent. So, the figures are not congruent.

Differentiated Instruction

English Language Learners Rigid motion and isometry are terms that can be used interchangeably. Ask students to keep a journal listing of all the terms related to transformations, including diagrams of each term.See also the Differentiated Instruction Resources for more strategies.

Motivating the LessonAsk students to draw examples of flips, turns, and slides as they remember them from previous instruction. Have them identify some real-life examples of flips, turns, and slides.


CC5


EXAMPLE 1

on p. CC3for Exs. 3–8

1. VOCABULARY Examples of transformations that are rigid motions are ? , ? , and ? .

2. ★ WRITING Explain why a transformation that maps one figure onto a congruent figure is a rigid motion.

IDENTIFYING TRANSFORMATIONS Identify the transformation you can use to move the blue figure onto the red figure.

3. 4. 5.

★ OPEN-ENDED MATH Copy the figure. Draw an example of the effect of the given transformation on the figure.

6. translation 7. reflection 8. rotation

HOMEWORKKEY

5 WORKED-OUT SOLUTIONSfor Exs. 11, 17, and 21

★ 5 STANDARDIZED TEST PRACTICEExs. 2, 6–9, 21, 25, 28

4.2B EXERCISES

SKILL PRACTICE

✓ GUIDED PRACTICE for Examples 2 and 3

Tell whether the two figures in the quilt are congruent or not congruent. If they are congruent, describe the transformation(s) you can use to move figure A onto figure B.

3. 4.

(l) a

nd (r

) © A

mer

ica

/Ala

my

not congruent congruent; refl ection

translations, refl ections, rotations

A transformation that maps one fi gure onto a congruent fi gure preserves lengths and angle measures, so it is a rigid motion.

rotation

translation refl ection

6–8. Check students’ drawings.

A

GE_CCESE621852_0402B_EXE.indd 5 12/11/10 12:35:01 AM

Extra Example 3The quilt shown is made using a repeating pattern.

A

B

C

D

a. Describe a transformation that moves fi gure A onto fi gure B.A reflection or rotation will move figure A onto figure B.

b. Explain why fi gures C and D are congruent by using the marked angles and sides. Figures C and D are congruent because all pairs of corresponding sides and all pairs of corresponding angles are congruent.

Closing the LessonHave students summarize the major points of the lesson and answer the Essential Question: How do you identify a rigid motion in the plane?• Rigid motions preserve length

and angle measure.• Two figures are congruent if and

only if one or more rigid motions can be used to move one figure onto the other.

• Translations, reflections, and rotations are rigid motions.

If the transformation of a figure results in an image that has corre-sponding angles and correspond-ing sides congruent to those in the preimage, then the transformation is a rigid motion.


Kinesthetic Learners You might wish to help your kinesthetic learners understand congruence in terms of rigid motions by providing them with large cardboard triangles they can use to perform translations, refl ections, and rotations.See also the Differentiated Instruction Resources for more strategies.


CC6


9. ★ MULTIPLE CHOICE Which is not an example of a rigid motion?

A B

C D

EXAMPLES 2 AND 3

on p. CC4 for Exs. 9–17

SHOWING FIGURES CONGRUENT Tell whether a rigid motion can move the blue figure onto the red figure. If so, describe the transformation(s) that you can use. If not, explain why the figures are not congruent.

10. y

xO

1

1

11. y

xO

1

1

12. y

xO

1

1

13. y

xO

1

3

14. y

xO

1

1

15. y

xO

1

1

16. Jerome describes a transformation in the coordinate plane using the notation (x, y) → (x � 3, y � 1). Explain why this is a rigid motion.

17. Jen describes a transformation in the coordinate plane using the notation (x, y) → (x � 1, 2y). Explain why this is not a rigid motion.

CHALLENGE Determine whether a rigid motion can move one triangle onto the other. Justify your answer.

18. 19.

10–15. See margin.

See margin.

See margin.

Yes; check students’ drawings; a translation followed by a refl ection, or a rotation followed by a refl ection

No; check students’ drawings.

B

C

C


13. yes; rotation 908 counterclockwise about the origin14. yes; translation 3 units right and 5 units up15. No; a rotation does not map one figure onto the other, because corresponding sides lengths are not congruent.16. Sample answer: The function rule describes a translation 3 units to the right and 1 unit down. A translation is a rigid motion.

17. Sample answer: The function rule moves points 1 unit to the left and then stretches points vertically away from the x-axis. The transformation is not a rigid motion, because lengths and angles are not pre-served. The triangle with vertices (0, 0), (1, 0), and (1, 1) is transformed to a taller triangle with vertices (21, 0), (0, 0), and (0, 2).

4 PRACTICEAND APPLY

Assignment GuideBasic:Day 1: pp. CC5–CC8Exs. 1–9, 11–17 odd, 20–22, 30–38Average: Day 1: pp. CC5–CC8Exs. 1–9, 11, 13, 15–17, 20–23, 26–28, 30–38Advanced: Day 1: pp. CC5–CC8Exs. 3–9, 10–16 even, 17–19, 21, 23–26, 28–38Block:pp. CC5–CC8Exs. 1–9, 11, 13, 15–17, 20–23, 26–28, 30–38

Differentiated InstructionSee Differentiated Instruction Resources for suggestions on addressing the needs of a diverse classroom.

Homework CheckFor a quick check of student under-standing of key concepts, go over the following exercises:Basic: 3, 6, 9, 13, 20Average: 4, 7, 11, 21, 23Advanced: 5, 8, 16, 24, 26

Extra Practice• Practice B in Chapter Resources

10. No; a reflection maps one side to a congruent side, but other sides are not congruent.11. yes; reflection in the line y 5 x12. yes; translation 3 units right and 2 units down


CC7


20. GAME SOFTWARE In a game, the goal is to move shapes into congruent spaces where they will fit so that completed rows can be eliminated. Describe a combination of transformations that can be used to move game piece A into congruent space B.

21. ★ SHORT RESPONSE Describe a way in which nABC can be moved onto nDCB using just one transformation. Then describe a way in which nABC can be moved onto nDCB using exactly two transformations.

FLOORING Designs for floor tiles are shown below. Describe a rigid motion or combination of rigid motions that can be used to move the blue figure onto the red figure.

22. 23. 24.

25. ★ OPEN-ENDED MATH Create a design using a combination of translations, reflections, or rotations.

26. PENROSE TILES The mathematician Roger Penrose investigated the patterns that can be made with tiles like the ones shown below.

a. Choose two tiles of the same color in the pattern and show how to map one tile onto the other using a rotation.

b. Describe the rotation angle and center.

c. Explain how you calculated the rotation angle.

11.4

15.7

6

18.5

B

A

DB

A C

EXAMPLE 3


1808 rotation around the midpoint of } BC ; refl ection across } BC followed by refl ection across the perpendicular bisector of } BC

908 rotation (either way), followed by translation across and down

1808 rotation 908 rotation counterclockwise 1208 rotation clockwise, followed by translation

Check students’ designs.

a–c. See margin.

B

PROBLEM SOLVING

A

GE_CCESE621852_0402B_EXE.indd Sec2:7 12/11/10 12:35:13 AM

26a. Check students’ rotations.26b. Sample answer: Yellow tiles that share an edge can be rotated either 728 around the vertex of the smaller angle or 1088 around the vertex of the larger angle. Red tiles that share an edge can be rotated 368 around the vertex of the smaller angle or 1448 around the vertex of the larger angle. 26c. Sample answer: You can calculate the angles of the tiles by observing how many of each type meet at various vertices in the design.

Teaching StrategyExercises 10–15 Some students may have difficulty recognizing a rigid motion in the plane using coordinates. Suggest they use the distance formula to check that the corresponding sides of the preimage and image are congruent.

Study StrategyExercises 16–17 You may wish to have students provide examples of transformations in the plane in their explanations. Suggest that they save their examples for study-ing purposes.

Reading StrategyExercises 22–24 Ask students to focus on the color of the figures they are describing. The blue figure is the preimage figure throughout these exercises. The red figure is the resulting image figure.


CC8


27. CLOTHING DESIGN A clothing manufacturer needs two panels cut from cloth that are reflections of each other to create part of a dress. Explain why folding the fabric in half and cutting both pieces together will produce the two panels.

28. ★ EXTENDED RESPONSE The diagram shows an architect’s preliminary design for an A-frame house.

a. Interpret Use rigid motions to explain how the architect knows nRTX > nVTX.

b. Reason Explain how the architect can use rigid motions to conclude that SW 5 UW.

c. Calculate The architect knows that the proportion TS }

SW 5 TR }

RX is true. Find the lengths SW and TW.

29. CHALLENGE Draw examples of two congruent triangles that can be mapped onto each other by the given transformation(s). Show the lines(s) of reflection you use.

a. exactly one reflection b. exactly two reflections c. exactly three reflections

T

USW

R X V

14 ft

16 ft

14 ft

15 ft15 ft

16 ft

?

?

For Exercises 30–32, identify the property of congruent segments that justifies the statement. (Lesson 2.6)

30. If } AB > } CD , then } CD > } AB .

31. For any segment AB, } AB > } AB .

32. If } AB > } CD and then } CD > } EF , then } AB > } EF .

33. A top view of the path between two buildings is shown. Use the Alternate Interior Angles Theorem to prove that ∠ 1 > ∠ 4. (Lessons 2.6 and 3.2)

Find the slope of the line that passes through the points. (Lesson 3.4)

34. (0, 0), (3, 12) 35. (21, 4), (0, 3) 36. (3, 22), (5, 22)

37. Write an equation of the line passing through the point (21, 22) that is parallel to the line with the equation y 5 2x 1 1. (Lesson 3.5)

38. Write an equation of the line passing through the point (2, 3) that is perpendicular to the line with the equation y 5 x 2 4. (Lesson 3.5)

2

1

3

4

PREVIEW

Prepare for Lesson 4.3in Exs. 30–32

MIXED REVIEW

Symmetric Property

See margin.

See margin.

Refl exive Property

Transitive Property

See margin.

4 21 0

y 5 2x 2 3

y 5 2x 1 5

C

Cutting both pieces together will make all corresponding sides and angles of the pattern congruent.

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GE_CCESE621852_0402B_EXE.indd 8 12/14/10 10:10:08 PM

29a–c. Sample answer:

m1

m1

m2

m2

m3

m1


Daily Homework Quiz1. Identify the transformation(s)

you can use to move fi gure A onto fi gure B.

BA

translation, then reflection2. Tell whether a rigid motion can

move fi gure A onto fi gure B. Explain.

y

xO

1

4A B

No; a reflection maps one side to a congruent side, but the other sides are not congruent.

Diagnosis/Remediation• Practice B in Chapter Resources• Study Guide in Chapter Resources

ChallengeAdditional challenge is available in the Chapter Resources.

28a. The rigid motion of reflection across a vertical line maps nRTX onto nV T X, so nR T X > nV T X.28b. The same rigid motion that maps nRTX onto nVTX also maps nSTW onto nUTW, so nSTW > nUTW. Because the triangles are congruent, the corresponding sides are congruent, so SW 5 UW.28c. SW 5 8 ft; by the Pythagorean Theorem, TW ø 13.9 ft

33. From the markings of the diagram, ∠ 1 > ∠ 2 and ∠ 3 > ∠ 4. By the Alternate Interior Angles Theorem, ∠ 2 > ∠ 3. Therefore, ∠ 1 > ∠ 3 by the Transitive Property of Angle Congruence. Finally, ∠ 1 > ∠ 4 by the Transitive Property of Angle Congruence again.


Com

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Mastering the StandardsThe topics described in the Standards for Mathematical

Content will vary from year to year. However, the way in

which you learn, study, and think about mathematics will

not. The Standards for Mathematical Practice describe skills

that you will use in all of your math courses.

for Mathematical PracticeMathematical Practices1. Make sense of problems and

persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

In your bookVerbal Models and the Problem Solving Plan help you translate the information in a problem into a model and then analyze your solution.

Mathematically proficient students start by explaining to themselves the meaning of a problem... They analyze givens, constraints, relationships, and goals. They make conjectures about the form... of the solution and plan a solution pathway...

Make sense of problems and persevere in solving them.

MM

MM1

Tennessee Grade Six Mathematics Standards

form... of the solution and plan a aaa solution pathway...

You have two summer jobs at a youth center. You earn $8 per hour teaching basketball and $10 per hour teaching swimming. Let x represent the amount of time (in hours) you teach basketball each week, and let y represent the amount of time (in hours) you teach swimming each week. Your goal is to earn at least $200 per week.

of x and y.

allow you to meet your goal.

1

8 x 1 10 y 200

x 1 10y 200.

x 1 10y 5 200 , so use

a solid line.

Next, test (5, 5) in 8x 1 10y 200:

8(5) 1 10(5) 200

90 200

Finally, shade the part of Quadrant I that does not contain (5, 5),

three points on the graph, such as (13, 12), (14, 10), and (16, 9). The table shows the total earnings for each combination of hours.

8. In Example 6, suppose that next summer you earn $9 per hour

combinations of hours that will help you meet your goal.

Basketball (hours)

Swim

min

g (h

ours

)

10 20 30

20

10

00 x

y

(5, 5)

8x 1 10y ≥ 200

28 Chapter 1 Expressions, Equations, and Functions

0.1 mi

0.15 mi

Use a Problem Solving Plan

Before You used problem solving strategies.

Now You will use a problem solving plan to solve problems.

Why? So you can determine a route, as in Example 1.

1.5

Key Vocabulary• formula

RUNNING You run in a city. Short blocks are north-south and are 0.1 mile long. Long blocks are east-west and are 0.15 mile long. You will run 2 long blocks east, a number of short blocks south, 2 long blocks west, and back to your start. You want to run 2 miles at a rate of 7 miles per hour. How many short blocks must you run?

Solution

STEP 1 Read and Understand

What do you know?

You know the length of each size block, the number of long blocks you will run, and the total distance you want to run.

You can conclude that you must run an even number of short blocks because you run the same number of short blocks in each direction.

What do you want to find out?

You want to find out the number of short blocks you should run so that, along with the 4 long blocks, you run 2 miles.

STEP 2 Make a Plan Use what you know to write a verbal model that represents what you want to find out. Then write an equation and solve it, as in Example 2.

KEY CONCEPT For Your NotebookA Problem Solving Plan

STEP 1 Read and Understand Read the problem carefully. Identify what you know and what you want to find out.

STEP 2 Make a Plan Decide on an approach to solving the problem.

STEP 3 Solve the Problem Carry out your plan. Try a new approach if the first one isn’t successful.

STEP 4 Look Back Once you obtain an answer, check that it is reasonable.

E X A M P L E 1 Read a problem and make a plan

ANOTHER WAY

For an alternative method for solving the problem in Example 1, turn to page 34 for the Problem Solving Workshop.

laa111se_0105.indd 28 10/20/10 12:50:34 AM

LA1_CCESE612355_filler_page.indd T25 12/10/10 12:10:06 AM


CC10


4.5 Rigid Motions and CongruenceMATERIALS • compass • straightedge

Q U E S T I O N How does triangle congruence follow from rigid motions?

In the following constructions, rigid motions will be used to explain the criteria for triangle congruence.

E X P L O R E 1 Construct triangles from three segments (SSS)

Given three segments, like the ones at the right, use these steps to construct triangles with those three side lengths.

STEP 1 Copy a segment Copy one of the line segments. In the diagram, the longest segment is copied. Label the endpoints A and B. BA

STEP 2 Draw circles At both endpoints of the segment, draw circles using the other two given side lengths as radii. BA

STEP 3 Find vertices Identify the four places where circles of different radii intersect. Label the points C, D, E, and F.

B

C D

E F

A

STEP 4 Draw triangles Draw nABC, nABE, nBAD, and nBAF.

BA

C D

E F

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Use after Lesson 4.5

GE_CCESE621852_0405A_ACT.indd CC10 12/10/10 4:40:09 AM

1 PLAN ANDPREPARE

Explore the Concept• Students will construct congruent

triangles given three segments, or given two segments and an included angle.

• Students will discover that their work is completely symmetric, and this may help them see that reflection is a rigid motion for mapping the resulting triangles to each other.

MaterialsEach student will need:• compass• straightedge



2 TEACHTips for SuccessBefore beginning Explore 1, make sure students are familiar with the construction for copying a segment. Also, point out that it is easiest if students use three different lengths. Stress that if two of the segments are the same length, then only two congruent (isosceles) triangles will be created. Similarly, if all three segments are the same length, only two congruent (equilateral) triangles result.

Alternative StrategyAsk students to try Explore 1 using the shortest segment as their first segment. The resulting drawings will look different than those shown, but four congruent triangles will still be created.


CC11

4.5 Prove Triangles Congruent by ASA and AAS CC11

E X P L O R E 2 Construct triangles from two segments and an angle (SAS)

Given two segments and one angle, like the ones at the right, use these steps to construct triangles with those side lengths and included angle.

STEP 1 Copy a segment Copy one of the line segments. Label the endpoints A and B.

STEP 2 Copy angles Use the copy an angle construction to make two copies of the angle at point A, one above and one below the segment. Then do the same thing at point B.

A B

STEP 3 Copy segments Using the other given side length as radius, draw arcs with centers A and B. Label where the arcs intersect the lines from Step 2 as points C, D, E, and F.

C D

E

A B

F

STEP 4 Draw triangles Draw nABC, nABE, nBAD, and nBAF. C D

E

A B

F


1. Refer to your work in Explore 1 and 2. Describe a rigid motion you can use to show that nABC > nABE and nBAD > nBAF.

2. Refer to your work in Explore 1 and 2. Describe a rigid motion you can use to show that nABC > nBAD and nABE > nBAF.

3. Is there a rigid motion you can use to show that nABC > nBAF and nABD > nBAE? If so, describe it.

4. What can you conclude about the 4 triangles in each Explore?

5. Follow steps like those in Explore 1 and 2 to construct triangles given two angles and an included side (ASA). Can you use rigid motions to show these triangles are all congruent? Explain.

DC

F

BA

E

refl ection in the line that contains } AB

yes; rotation of 1808 around the midpoint of } AB

See margin.

See margin.

refl ection in the perpendicular bisector of } AB

GE_CCESE621852_0405A_ACT.indd CC11 12/10/10 4:40:22 AM

Tips for SuccessBefore beginning Explore 2, make sure students are familiar with the construction for copying an angle. You may wish to review the method, which involves creating two arcs with a compass, each with a different center.

Key DiscoveryTriangle criteria like three side lengths (SSS) and two side lengths and the included angle (SAS) are sufficient to generate unique trian-gles. Any triangles created using those criteria can be shown to be congruent to each other by finding rigid motions that map one triangle onto another.


1. You have shown that a triangle built from three segments or from two segments and an included angle is unique. Explain how this is related to congruent triangles.

Sample answer: For any three given segment lengths, all tri-angles built using those lengths will be congruent. Likewise, all triangles built using two given side lengths and the angle between them will be congruent.

4. Rigid motions can be used to transform the triangles onto each other, so the triangles are all congruent.

5. Yes; the same rigid motions of reflection can be used to show that the triangles are all congruent.


CC12

CC12 Chapter 6 Similarity

6.3A Explore Properties of DilationsMATERIALS • graph paper • ruler • protractor

Q U E S T I O N How do dilations affect lines, segments, and angles?

Dilations are non-rigid transformations that map points to points in the coordinate plane. A function rule in coordinate notation for a dilation with center at the origin is (x, y) → (kx, ky).

The ratio of the length of an image segment to its corresponding preimage segment is the scale factor k of the dilation.

E X P L O R E Draw a dilation of a triangle in the plane

STEP 1

x

y

1

1A

B

C

Graph a triangle Graph a triangle in a coordinate plane one of whose vertices is (0, 0).

STEP 2

x

y

1

1A

B

D

EC

Dilate the triangle Graph a dilation of the triangle with scale factor 2 and centered at the origin.


1. Measure the sides and angles of n ABC and n ADE. Tell whether the dilation preserves lengths or angle measures.

2. Compare the ratios DE } BC

, DA } BA

, and EA } CA

. What do you notice?

3. What is the effect of the dilation on the center of dilation? Explain.

4. How is ‹ ] › AB related to

‹ ] › AD ? How does a dilation affect a line that passes

through the center of dilation? Explain.

5. How is ‹ ] › BC related to

‹ ] › DE ? How does a dilation affect a line that does not

pass through the center of dilation? Explain.

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s of Dilations

Use before Lesson 6.3BUse beUse beACTIVITYACCTIVITYAC

1–5. See margin.

GE_CCESE621852_0603A_ACT.indd 12 12/22/10 9:46:14 PM

1 PLAN AND PREPARE

Explore the Concept• Students will discover the

properties of dilations by using graph paper to dilate a figure.

MaterialsEach student will need:• graph paper• ruler• protractor



2 TEACHKey Questions• In the Explore, to what segments

are }AB , }AC , and }BC transformed?}AD , }AE , and }DE

• Which segment and its image appear to be parallel? }BC and }DE

Key DiscoveryA dilation takes a line not passing through the center of dilation to a parallel line, and lines passing through the center are unchanged. The lengths of segments are changed by a ratio called the scale factor. All dilations preserve angle measures.

3 ASSESS ANDRETEACH

1. A dilation with scale factor 1.5 and center (0, 0) is applied to a triangle with sides of lengths 1, 1, and Ï

}

2 . Describe the longest side of the image. Its length is 1.5 Ï

}

2 and it is parallel to the longest side of the preimage.

1. AB 5 Ï}

10 , AC 5 Ï}

10 , BC 5 2 Ï}

2 , AD 5 2 Ï}

10 , AE 5 2 Ï

}

10 , DE 5 4 Ï}

2 ; m ∠ A ø 538, m ∠ B 5 m ∠ D ø 63.58, m ∠ C 5 m ∠ E ø 63.58; the dilation does not preserve lengths but it does preserve angle measures.

2. The ratios are all equal to the scale factor, 2.

3. The center of dilation (0, 0) is mapped to (k ? 0, k ? 0) 5 (2 ? 0, 2 ? 0) 5 (0, 0). The center of dilation is mapped to itself.

4. The lines are the same; the image of a line that passes through the center of dilation is the same as the preimage line; the center of dilation is mapped to itself, and any line that contains the origin is mapped to a line that contains the origin.

5. The lines are parallel; the image of a line that does not pass through the center of dila-tion is a line parallel to the preimage line; because ∠ C > ∠ E, the lines are parallel by the Corresponding Angles Converse.

1–5. Sample answers are given.

LGE_CCETE621999_6-3A_ACT.indd CC12LGE_CCETE621999_6-3A_ACT.indd CC12 12/22/10 9:48:17 PM12/22/10 9:48:17 PM

6.3B Relate Transformations and Similarity CC13

6.3B Before You identified rigid motions in the plane.

Now You will identify similarity transformations called dilations.

Why? So you can find the dimensions of a scale drawing, as in Ex. 30.

Key Vocabulary• dilation• scale factor

A dilation is a transformation that preserves angle measures and results in an image with lengths proportional to the preimage lengths.

The ratio of the lengths of the corresponding sides of the image and the preimage is called the scale factor of the dilation. Dilations can also be called similarity transformations.

Relate Transformationsand Similarity

KEY CONCEPT For Your NotebookDilations and SimilarityIf a dilation can be used to move one figure onto another, the two figures are similar.

nABC , nXYZ

E X A M P L E 1 Describe a dilation

nFEG is similar to nFDH. Describe the dilation that moves nFEG onto nFDH.

Solution

The figure shows a dilation with center F.

The scale factor is 2 because the ratio of FH to FG is 20 : 10, or 2 : 1.

E

DH

F

G10

10

KEY CONCEPT For Your NotebookCombining Dilations and Rigid MotionsIf a dilation followed by any combination of rigid motions can be used to move one figure onto the other, the two figures are similar.

nABC , nDEF and nDEF > nGHJ, so nABC , nGHJ.

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A

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GE_CCESE621852_0603B_EXPO.indd CC13 12/11/10 4:39:59 AMChapter Resources• Practice level B

• Study Guide

• Challenge

• Pre-AP notes

Teaching Options• Activity Generator provides editable

activities for all ability levels

Interactive Technology• Activity Generator

• Animated Algebra

• Test Generator

• eEdition

See also the Differentiated Instruction

Resources for more strategies for

meeting individual needs.

Ch t R T hi

Resource Planning Guide

1 PLAN AND PREPARE

Warm-Up Exercises1. Given nRST , nXYZ with

RS } XY

5 ST } YZ

5 3 } 2 . Find RT }

XZ . 3 }

2

2. Given nEFG , nMNP with

EF } MN

5 2 } 1 . If FG 5 4.5, fi nd NP.

2.25

PacingBasic: 1 dayAverage: 1 dayAdvanced: 1 dayBlock: 0.5 block

2 FOCUS ANDMOTIVATE

Essential QuestionBig Idea 2, p. 355How do you identify a similarity transformation in the plane? Tell students they will learn how to answer this question by examining dilations, transformations in the plane that preserve angle measure while keeping the lengths of corresponding sides proportional.

Motivating the LessonAsk students to draw examples of figures that are similar based on their recollection from previous work. Have them point out any examples of similar figures they see in the classroom.

CC13

LGE_CCETE621999_6-3B_LS.indd CC13LGE_CCETE621999_6-3B_LS.indd CC13 12/22/10 4:27:34 AM12/22/10 4:27:34 AM

CC14


E X A M P L E 2 Describe a combination of transformations

nABC is similar to nFGE. Describe a combination of transformations that moves nABC onto nFGE.

Solution

A dilation with center B and scale factor 2 } 3

moves nABC onto nDBE. Then a rotation of nDBE with center E moves nDBE onto nFGE. The angle of rotation is equal to the measure of ∠C.

F

G

BA

C

D

E


The two figures are similar. Describe the transformation(s) that move the blue figure onto the red figure.

1. D

E 8 6 BC

A

2.

69A D X

YZB C

E X A M P L E 3 Use transformations to show figures are not similar

Use transformations to explain why ABCDE and KLQRP are not similar.

Solution

Corresponding sides in the pentagons are proportional with a scale factor of 2 }

3 .

However, this does not necessarily mean the pentagons are similar.

A dilation with center A and scale factor 2 }

3 moves ABCDE onto AFGHJ.

Then a reflection moves AFGHJ onto KLMNP.

KLMNP does not exactly coincide with KLQRP, because not all of the corresponding angles are congruent. (Only ∠ A and ∠K are congruent.) Since angle measure is not preserved, the two pentagons are not similar.

AB

C

DE

9

69

12

12

K L

Q

RP

8

8

64

6

AB F

G

H

M

NJ

C

DE

96 8

46

69

84

4

K L

QRP

8

8

64

6

dilation with center B and scale factor 7 }

3

dilation with scale factor 2 } 3

and refl ection

GE_CCESE621852_0603B_EXPO.indd CC14 12/11/10 4:40:10 AM

Corresponding sides in the pentagons are proportional with a scale factor of 2. However, a dilation with center F moves EFGHD onto IFLKJ. Then a reflection moves IFLKJ onto RSWXV. But RSWXV does not exactly coincide with RSTUV, since not all of the corresponding angles are congruent. Since angle measure is not preserved, the two pentagons are not similar.

3 TEACHExtra Example 1nABC is similar to nAFG. Describe the dilation that moves nABC onto nAFG.

A

B

C10 40G

F

dilation with center A and scale factor 5 }

4

Extra Example 2nABC is similar to nFGE. Describe a combination of transformations that moves nABC onto nFGE.

A

B

D

EF

G

C 2030

A dilation with center A and scale factor 5 }

3 moves nABC onto nADE.

Then a reflection moves nADE onto nFGE.

Extra Example 3Use transformations to explain why pentagons EFGHD and RSTUV are not similar.

F R S

T

U

V

G

H

E

D

14

15

32

28

30

28

20

16

10

14

F R S

TW

UX

V

G

L

K

J

I

H

E

D

1414

15

15

2832 32

32

2828

3030

28

2020

16

10

14


CC15


E X A M P L E 4 Use similar figures

GRAPHIC DESIGN A design for a party mask is made using all equilateral

triangles and a scale factor of 1 } 2

.

A

BCD

a. Describe transformations that move triangle A onto triangle B.

b. Describe why triangles C and D are similar by using the given infomation.

Solution

a. The figure shows a dilation with scale factor 1 } 2

followed by a clockwise rotation of 608.

b. Triangles C and D are similar because all pairs of corresponding sides are proportional with a ratio of 1 }

2 and all pairs of corresponding angles of equilateral

triangles have the same measure.

A

B


Refer to the floor tile designs shown below. In each design, the red shape is a regular hexagon.

3. Tile design 1 is made using two hexagons. Explain why the red and blue hexagons are not similar.

4. Tile design 2 is made using two similar geometric shapes. Describe the transformations that move the blue hexagon to the red hexagon.

5. Tile design 3 shows congruent angles and sides. Explain why the red and blue hexagons are similar, using the given information.

6. If the lengths of all the sides of one polygon are proportional to the lengthsof all the corresponding sides of another polygon, must the polygons be similar? Explain.

Tile design 1 Tile design 2 Tile design 3

3–6. See margin.

GE_CCESE621852_0603B_EXPO.indd CC15 12/11/10 4:40:17 AM

Extra Example 4The design for a stained glass win-dow uses two sizes of similar isos-celes right triangles with a scale factor of Ï

}

2 .

AB

C

D

a. Describe the transformations

that move triangle A onto triangle B. A dilation with scale factor Ï

}

2 centered at the vertex of the right angle in triangle A followed by a clockwise rotation of 1358 about that same point will move triangle A onto triangle B.

b. Describe why triangles C and D are similar using the given information. Triangles C and D are similar because all pairs of corresponding sides are propor-tional with a ratio of Ï

}

2 to 1 and corresponding angles of isosce-les right triangles have the same measure (458 or 908).

Closing the LessonHave students summarize the major points of the lesson and answer the Essential Question: How do you identify a similarity transformation in the plane?• A dilation is a transformation that

preserves angle measures and results in an image with lengths proportional to the preimage lengths.

• Dilations are similarity transfor-mations.

If a dilation or a dilation followed by any combination of rigid motions can move one figure onto another, then the two figures are similar.

3. The red hexagon has all sides congruent, but the blue hexagon has 3 shorter sides and 3 longer sides, so ratios of corre-sponding side lengths are not constant.

4. dilation followed by a rotation of 308 about the center of the figures

5. All angles are congruent, so angle mea-sure is preserved, and all side lengths are congruent in each hexagon, so the ratio of any two corresponding side lengths is constant.

6. No; even though corresponding sides might be proportional, if corresponding angles are not congruent, the polygons are not similar.

3–6. Sample answers are given.


CC16


EXAMPLE 1


EXAMPLE 2


EXAMPLE 3

on p. CC14for Ex. 10–12

1. VOCABULARY Draw an example of a dilation.

2. ★ WRITING Describe the results of a similarity transformation.

DESCRIBING DILATIONS Describe the dilation that moves the blue figure onto the red figure.

3.

2 3

4.

10

5

5.

6

8

6. ★ WRITING Jenny described a transformation in the coordinate plane using the notation (x, y) → (3x, 3y). Explain why her transformation is a dilation with center at the origin.

DRAWING SIMILARITY TRANSFORMATIONS Copy the figure. Draw an example of the given similarity transformation of the figure with center O.

7. O 8.

O

9. O

dilation dilation then refl ection dilation then rotation

10. ★ MULTIPLE CHOICE Which of the following transformations does not involve dilation?

A B

C D

HOMEWORKKEY

5 WORKED-OUT SOLUTIONSfor Exs. 5, 15, 19, and 27

★ 5 STANDARDIZED TEST PRACTICEExs. 2, 6, 10, 17, 26, and 30

6.3B EXERCISES

SKILL PRACTICE

dilation with scale factor 14 }

8 5 7 }

4 and center at

intersection of black lines


C

Check students’ drawings.See margin.

See margin.


and center at intersection of black lines

dilation with scale factor 1 }

2 and

center at intersection of black lines

A

B


4 PRACTICEAND APPLY

Assignment GuideBasic:Day 1: pp. CC16–CC19Exs. 1, 3, 7–11, 13, 15, 18, 25–27, 33–37 oddAverage: Day 1: pp. CC16–CC19Exs. 1–10, 11–25 odd, 26, 28–30, 32–38Advanced: Day 1: pp. CC16–CC19Exs. 4–9, 12–16 even, 20–31, 32–38 evenBlock:pp. CC16–CC19Exs. 1–10, 11–25 odd, 26, 28–30, 32–38

Differentiated InstructionSee Differentiated Instruction Resources for suggestions on addressing the needs of a diverse classroom.

Homework CheckFor a quick check of student under-standing of key concepts, go over the following exercises:Basic: 3, 8, 13, 18, 25Average: 5, 9, 15, 19, 25Advanced: 5, 8, 16, 20, 26

Extra Practice• Practice B in Chapter Resources

2. Sample answer: A similarity transformation maps one figure onto a similar figure. The corre-sponding sides have lengths that are proportional and the corre-sponding angles have the same measures.

6. The function notation is for a dilation with scale factor 3. Corresponding sides will be proportional with a ratio of 3 to 1. Choosing a sample figure and drawing its image will show that the corresponding angles of the figures have the same measure.


CC17


IDENTIFYING DILATIONS The red figure is the image of the blue figure under a transformation. Tell whether the transformation involves a dilation. If so, give the scale factor of the dilation.

11.

2030

40

40

12.

3

3

33

3

3

COMBINING TRANSFORMATIONS Coordinates of the vertices of a preimage and image figure are given. Describe the transformations that move the first figure onto the second.

13. O(0, 0), B(0, 3), C(2, 0); O(0, 0), D(0, 26), E(4, 0)

14. O(0, 0), P(0, 8), Q(6, 0); O(0, 0), R(4, 0), S(0, 23)

15. J(24, 0), K(0, 4), L(6, 0); L(6, 0), M(0, 6), N(29, 0)

16. A(26, 0), B(0, 6), C(9, 0), D(0, 215); W(0, 22), X(22, 0), Y(0, 3), Z(5, 0)

17. ★ SHORT RESPONSE A fractal called the Sierpinski triangle can be imagined by thinking of an infinite sequence of stages, the first of which are shown below. Calculate the side lengths of the light blue triangles in Stages 1, 2, and 3. Then describe the 3 dilations you can apply to any stage to generate the next stage.

Stage 0 Stage 1 Stage 2 Stage 3

16

16 16

SIMILARITY OF CIRCLES Prove the circles are similar by finding a center and scale factor of a dilation that moves the blue circle onto the red circle.

18. concentric 19. intersecting

2

4

53

20. tangent 21. not intersecting

4 5

2

4

no

yes; dilation with scale factor Î}

2


scale factor 3

scale factor 5 } 4 scale factor 1 }

2

scale factor 3 } 5

13–16. See margin.

See margin.


Teaching StrategyExercise 10 You might wish to ask students to explain the transforma-tion that moves one figure onto the other in each of the answer choices.

Avoiding Common ErrorsExercises 13–16 While some stu-dents may try to visualize the fig-ures mentally, urge all students to graph the points on a coordinate grid so they can visualize the trans-formations that move the first fig-ure onto the second figure.

Study StrategyExercise 17 Suggest that students make a table of the side lengths of the triangle(s ) at each stage. Doing so will help them determine the ratio of the corresponding side lengths.

13. dilation with center O and scale factor 2, then reflection in the x-axis

14. dilation with center O and scale factor 1 }

2 , then rotation 908

clockwise around O15. dilation with center O and

scale factor 3 } 2 , then reflection

in the y-axis16. dilation with center O and

scale factor 1 } 3 , then rotation

908 counterclockwise around O17. Sample answer: The lengths

are 8, 4, and 2; dilate the previ-ous stage 3 times with scale factor 1 }

2 using each corner of

the triangle as a center to generate the next stage.


Below Level For Exercises 13–16, have students use a large sheet of grid paper to graph each pair of fi gures. Have them measure the sides and angles of the preimage and of the image. Ask students to fi nd the ratio of the corresponding side lengths and to compare the corresponding angle measures.See also the Differentiated Instruction Resources for more strategies.


CC18


EXAMPLE 4


PROBLEM SOLVING

22. CHALLENGE The dilation of a rectangle has a scale factor of 4 to 1.

a. Describe the effect of the dilation on the perimeter of the rectangle.

b. Describe the effect of the dilation on the area of the rectangle.

23. TEXTILES A cloth purse maker uses a pattern for a small bag. The maker wants to start making a similar bag that is exactly twice as big as the small one and has the same design. Explain how you might efficiently create the design for the large bag.

24. GRAPHIC ARTS Describe how using an overhead projector involves dilations when creating a larger image that can be traced onto a poster.

25. CRAFTS A rug design uses similar triangles that become larger on one side while getting smaller on the other side. What transformations are used to move figure A onto figure B?

A

B

26. ★ OPEN-ENDED Create a design using a combination of dilations and other transformations.

27. TOY IDEA A flashlight has a removable cap with a picture on it. When the flashlight is on, the bug can be projected onto a wall. Using the information in the diagram below, find the scale factor of the dilation. Then calculate the height of the bug projected onto the wall.

h 2 cm

2.5 cm

10 cm

28. ANIMATION You are designing an animated lightning bolt on a computer screen. You want the distance of each bolt from P to be 1 }

5 greater than the distance of

the previous bolt. Describe the transformation that moves each bolt to the next larger bolt.

29. CHALLENGE Describe a method for finding the center and scale factor of a dilation that moves a given circleonto another given circle with a different radius.

B

A

P

C

dilation followed by a rotation

See margin.

See margin.

See margin.

See margin.

Check students’ designs.


The scale factor of the dilation is 10 }

25 5 4.

Therefore, the height of the bug on the wall will be 4 times the height of the bug on the fl ashlight cap; 4 p 2 cm 5 8 cm.


Study StrategyExercise 23 Suggest that students draw a diagram to model this situa-tion and to help them formulate their explanation.

Teaching StrategyExercise 26 You may want stu-dents to share and discuss their designs with a partner. Have them describe the transformations they used, trade their designs with their partner, and then have the partner also describe the transformations. They can then finish the discussion by comparing their descriptions.

30a. Sample answer: The floor plan is a model. Each part of the actual house will copy the floor plan and be a dilation of the shape. The final size of every inch on the scale draw-ing will be 2 feet, or 24 inches, so the scale factor is 24 to 1.

30b. Sample answer: Multiply each dimension on the floor plan by 24 to find the actual size in inches; divide by 12 to find the actual size in feet. So, the living room that is 10 in. by 5 in. on the floor plan is 20 ft by 10 ft.

30c. Bedroom 1: 15 ft by 10 ft Storage: 5 ft by 10 ft Den: 8 ft by 10 ft Bedroom 2: 12 ft by 10 ft Bath: 8 ft by 6 ft Kitchen: 16 ft by 8 ft Living: 20 ft by 10 ft

22a. The perimeter of the image is 4 times the perimeter of the preimage.

22b. The area of the image is 16 times the area of the preimage.

23. Sample answer: Because the purses are similar, the designs of the purses should have the same shape but not the same size. Use a copy machine to enlarge the pattern from the smaller purse. For a purse twice as big, use a setting of 200% on the copy machine. Then transfer the pattern to the larger purse.

24. Sample answer: An overhead projector enlarges a figure onto a screen as a function of the distance from the pro-jector to the screen. In place of the screen, affix a poster board. The pattern can be traced onto the board.

29. Sample answer: Draw cor-responding radii in the circles parallel to each other. Draw a line through the endpoints of the radii that lie on the circles. Draw a line through the centers of the circles. The center of dilation is the inter-section of the lines. The scale factor of the dilation is the ratio of the radii of the circles.


CC19


30. ★ EXTENDED RESPONSE The figure below shows an architect’s initial layout of the floor plan for a house. The scale is 1 }

2 inch 5 1 foot.

a. Interpret How is a dilation of the floor plan used in the house construction? What is the scale factor of the dilation?

b. Model Explain how to find the actual size of the living room in the house using the floor plan.

c. Calculate Find the actual size of each room in the house.

31. ★ OPEN-ENDED A transformation in the coordinate plane is described using the notation (x, y) → (2x, 5y). Explain why the transformation is not a similarity transformation by showing how it affects the angles and sides of a polygon.

32. CHALLENGE A dilation of a triangle is shown, in which the center of dilation lies in the interior of the triangle.

a. Use the slope formula to show that corresponding sides are parallel.

b. The black rays in the diagram are transversals that intersect parallel segments. Prove the angles marked are congruent. Explain why the angles of the preimage and image triangles are preserved under the dilation.

PREVIEW

Prepare for Lesson 6.4in Exs. 33–36

C

Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. (Lesson 4.2)

33. S Y

R T Z X

34.

A C F D

B E

Find the values of x and y. (Lesson 4.2)

35.

(2y 1 5)8

(3x)8

458

36.

(2y )8

(6x 1 10)8708

Is it possible to construct a triangle with the given side lengths? If not, explain why not. (Lesson 5.5)

37. 4, 5, 9 38. 2, 8, 9 39. 20, 20, 45

7.5 in. 2.5 in. 4 in. 6 in.

5 in.

5 in. 5 in.

3 in.

4 in.4 in.

8 in.10 in.

LIVING KITCHEN

BATH

DEN

STO

RA

GE

BEDROOM1

BEDROOM2

MIXED REVIEW

See margin.

No; 20 1 20 is not greater than 45.yesNo; 4 1 5 is not

greater than 9.

10; 1015; 20

nRST c nXYZ ; } RS c } XY ; } ST c } YZ ; } TR c } ZX ; ∠ R c ∠ X ; ∠ S c ∠ Y ; ∠ T c ∠ Z

nABC c nDEF ; } AB c } DE ; } BC c } EF ; } CA c } FD ; ∠A c ∠D ; ∠B c ∠E ; ∠C c ∠F

See margin.

See margin.(ka, kb)

(kc, kd )

(ke, kf )

y

x

(a, b)

(c, d)

(e, f )

GE_CCESE621852_0603B_EXE.indd 19 12/13/10 8:47:58 PM

31. Sample answer: An isosceles right triangle with vertices (0, 0), (1, 0), and (1, 1), is mapped to a triangle with vertices (0, 0), (2, 0), and (2, 5). The image triangle is not isosceles; its legs are not the same length and its acute angles do not have the same measure. Image side lengths are not in proportion to preimage side lengths, and angles are not preserved, so the transformation is not a similarity transformation.

32a. The slope of an image side is the same as the slope of a corresponding preimage side. For example:

kd 2 kb } kc 2 ka

5 k(d 2 b) }

k(c 2 a) 5 (d 2 b)

} (c 2 a)

Similar reasoning applies to the other sides of the triangle.


Daily Homework QuizDescribe the dilation that moves the smaller figure onto the larger figure.

1.

8

16

dilation with scale factor 3 and center at the intersection of the thin lines

2.

5 3

dilation with scale factor 8}5 and

center at center of the circles3. The coordinates of the verti-

ces of a nABC and its image nDEF are given. Describe the transformation(s) that move nABC fi gure onto nDEF.A (1, 1), B (22, 2), C (2, 2); D (2, 22), E (24, 24), F (4, 24)

Sample answer: dilation with center (0, 0) and scale factor 2, then reflection in the x-axis

Diagnosis/Remediation• Practice B in Chapter Resources• Study Guide in Chapter Resources

ChallengeAdditional challenge is available in the Chapter Resources.

32b. The black ray that passes through (a, b) and (ka, kb) is a transversal intersecting parallel segments, so the angles marked are congruent. Therefore, their sums are also equal. Similar reasoning applies to the other angles of the triangle. So, angles are preserved under the dilation.


CC20


6.4 Dilations and AA SimilarityMATERIALS • ruler • protractor

Q U E S T I O N How can you use a dilation to map a triangle onto a similar triangle with two pairs of corresponding angles congruent?

Recall that a dilation preserves angle measures but not lengths.

E X P L O R E Build similar triangles given two angles

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milarity

Use before Lesson 6.4Use beUse beACTIVITYACCTIVITYAC

STEP 1 Draw triangles Draw n ABC with any angles. Use a protractor to draw a larger n DEF, with ∠ D > ∠ A and ∠ E > ∠ B. Inside n DEF, draw a segment parallel to } DE the same length as } AB . Label its endpoints G and H.

C

A

F

B G H

D E

STEP 2 Copy angles Copy ∠ D at G and copy ∠ E at H. Extend the sides until they intersect. Label the intersection J. C

A

F

B G H

J

D E

STEP 3 Find center of dilation Draw three rays that intersect:

]

› DG , ]

› EH , and

]

› FJ . Label their point of

intersection O. C

A

F

B GO

H

J

D E


1. Measure ∠ C, ∠ F, and ∠ J. What do you notice?

2. A dilation maps nDEF to nGHJ. Why is ∠ D > ∠ G and ∠ E > ∠ H?

3. Find GO } DO

, HO } EO

, and JO

} FO

. What is the scale factor of the dilation?

4. Prove that nGHJ > n ABC. Justify your reasoning.

5. What combination of transformations maps n DEF to n ABC?

1–5. See margin.

GE_CCESE621852_0604A_ACT.indd 20 12/11/10 4:41:52 AM

1 PLAN AND PREPARE

Explore the Concept• Students will build similar

triangles given two angle measures.

• This activity supplements the study of the Angle-Angle (AA) Similarity Postulate.

MaterialsEach student will need:• ruler• protractor



2 TEACHKey DiscoveryIf two triangles have two pairs of corresponding angles congruent, then a dilation (or a combination of a dilation and a rigid motion) can be used to move one triangle onto the other.

3 ASSESS ANDRETEACH

1. In the Explore, suppose you drew nGHJ inside nDEF so no sides were parallel to sides of nDEF. Would you be able to fi nd a center of dilation? Sample answer: No; preimage and image sides in a dilation must be parallel, so you would not be able to find a center of dilation.

2. In the Explore, suppose you drew nGHJ outside nDEF. Would you be able to fi nd a cen-ter of dilation? Sample answer: Yes; draw lines through cor-responding vertices to find the center of dilation; it would lie outside the triangles.

1. m ∠ C 5 m ∠ F 5 m ∠ J; the third angles of the triangles are congruent because the sum of the angles of a trian-gle is 1808.2. Dilations preserve angle measures, so corresponding angles of nDEF and nGHJ are congruent.

3. Check students’ work. The ratios should be equal to the scale factor of the dilation.

4. ∠ A > ∠ G and ∠ B > ∠ H because dilation preserves angle measures, and } AB > } GH because this was given in Step 1. So, nGHJ > nABC by ASA.5. dilation with scale fac-tor GO }

DO and center O, then

translation a distance GA

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Phot

oDis

c/G

etty

Imag

es
















In your bookApplication exercises and Mixed Reviews of Problem Solving apply mathematics to other disciplines and in real-world scenarios.

Mathematically proficient students can apply...mathematics... to... problems... in everyday life, society, and the workplace...

Model with mathematics.

MM

MM4

58. A diver dives from a cliff when her center of gravity is 46 feet above the surface of the water. Her initial vertical velocity leaving the cliff is 9 feet per second. After how many seconds does her center of gravity enter the water?

59. You plan to make a scrapbook. On the cover, you want to show three pictures with space between them, as shown. Each of the pictures is twice as long as it is wide.

a. Write a polynomial that represents the area of the scrapbook cover.

b. The area of the cover will be 96 square centimeters. Find the length and width of the pictures you will use.

60. You throw a ball into the air with an initial vertical velocity of 31 feet per second. The ball leaves your hand when it is 6 feet above the ground. You catch the ball when it reaches a height of 4 feet. After how many seconds do you catch the ball? Explain how you can use the solutions of an equation to find your answer.

61. The Parthenon in Athens, Greece, is an ancient structure that has a rectangular base. The length of the Parthenon’s base is 8 meters more than twice its width. The area of the base is about 2170 square meters. Find the length and width of the Parthenon’s base.

62. An African cat called a serval leaps from the ground in an attempt to catch a bird. The serval’s initial vertical velocity is 24 feet per second.

a. Write an equation that gives the serval’s height (in feet) as a function of the time (in seconds) since it left the ground.

b. Use the equation from part (a) to make a table that shows the height of the serval for t 5 0, 0.3, 0.6, 0.9, 1.2, and 1.5 seconds.

c. Plot the ordered pairs in the table as points in a coordinate plane. Connect the points with a smooth curve. After how many seconds does the serval reach a height of 9 feet? Justify your answer using the equation from part (a).

5 5 5

2 cm

2 cm

2 cm 2 cm

1 cm 1 cm

2x

4x

1. Flying into the wind, a helicopter takes 15 minutes to travel 15 kilometers. The return flight takes 12 minutes. The wind speed remains constant during the trip.

a. Find the helicopter’s average speed (in kilometers per hour) for each leg of the trip.

b. Write a system of linear equations that represents the situation.

c. What is the helicopter’s average speed in still air? What is the speed of the wind?

2. At a grocery store, a customer pays a total of $9.70 for 1.8 pounds of potato salad and 1.4 pounds of coleslaw. Another customer pays a total of $6.55 for 1 pound of potato salad and 1.2 pounds of coleslaw. How much do 2 pounds of potato salad and 2 pounds of coleslaw cost? Explain.

3. During one day, two computers are sold at a computer store. The two customers each arrange payment plans with the salesperson. The graph shows the amount y of money (in dollars) paid for the computers after x months. After how many months will each customer have paid the same amount?

Months since purchase

Am

ount

pai

d(d

olla

rs)

2 4 6 8 10

400

200

00 x

y

4. Describe a real-world problem that can be modeled by a linear system. Then solve the system and interpret the solution in the context of the problem.

5. A hot air balloon is launched at Kirby Park, and it ascends at a rate of 7200 feet per hour. At the same time, a second hot air balloon is launched at Newman Park, and it ascends at a rate of 4000 feet per hour. Both of the balloons stop ascending after 30 minutes. The diagram shows the altitude of each park. Are the hot air balloons ever at the same height at the same time? Explain.

6. A chemist needs 500 milliliters of a 20% acid and 80% water mix for a chemistry experiment. The chemist combines x milliliters of a 10% acid and 90% water mix and y milliliters of a 30% acid and 70% water mix to make the 20% acid and 80% water mix.

a. Write a linear system that represents the situation.

b. How many milliliters of the 10% acid and 90% water mix and the 30% acid and 70% water mix are combined to make the 20% acid and 80% water mix?

c. The chemist also needs 500 milliliters of a 15% acid and 85% water mix. Does the chemist need more of the 10% acid and 90% water mix than the 30% acid and 70% water mix to make this new mix? Explain.

S E A L E V E L Not drawn to scale

1705 ft 3940 ft


LGE_CCETE621999_6-4A_ACT.indd Sec1:21LGE_CCETE621999_6-4A_ACT.indd Sec1:21 12/22/10 4:28:50 AM12/22/10 4:28:50 AM

CC22


Use after Lesson 6.7

Extension Partition Segments

E X A M P L E 1 Find a point along a directed line segment

Find the coordinates of point P along the directed line segment AB so that the ratio of AP to PB is 3 to 2.

Solution

In order to divide the segment in the ratio 3 to 2, think of dividing, or partitioning, the segment into 3 1 2, or 5 congruent pieces.

Point P is the point that is 3 } 5

of the way from point A to point B.

The diagram shows the rise and run from point A to point B.

slope of } AB 5 10 2 4 } 6 2 3

5 6 } 3

5 rise } run

To find the coordinates of point P, add 3 } 5

of the run to the x-coordinate of A, and

add 3 } 5

of the rise to the y-coordinate of A.

run: 3 } 5

of 3 5 1.8

rise: 3 } 5

of 6 5 3.6

c So, the coordinates of P are (3 1 1.8, 4 1 3.6) 5 (4.8, 7.6). The ratio of AP to PB is 3 to 2.

x

y

1

1O

B(6, 10)

A(3, 4)

x

y

1

1O

B(6, 10)

3.6

A(3, 4) 1.8

P(4.8, 7.6)6

3

Recall that the slope of a nonvertical line is the ratio of rise (the vertical change) to run (the horizontal change) between any two points on the line.

slope 5 rise } run

A directed line segment AB is a segment that represents moving from point A to point B. The following example shows how to use slope to find a point at a specific location on a directed line segment.

AVOID ERRORS

Do not simplify the slope to lowest terms.

GOAL Find the point that partitions a directed line segment in a given ratio.

GE_CCESE621852_0607A_EXT.indd CC22 12/13/10 8:42:04 PM

1 PLAN AND PREPARE

Warm-Up ExercisesFind the rise and the run from point A to point B in the coordinate plane.1. A (24, 3), B (2, 25)

rise: 28; run: 62. A (2, 21), B (23, 24)

rise: 23; run: 253. A (0, 24), B (1, 6)

rise: 10; run: 1


Essential QuestionBig Idea 1, p. 355How do you find the point that partitions a directed line segment in a given ratio? Tell students they will learn how to answer this question by finding the slope of the segment and then using the given ratio.

3 TEACHExtra Example 1Find the coordinates of a point Palong the directed line segment AB with endpoints A (2, 4) and B (9, 6) so that the ratio of AP to PB is 2 to 3.P(4.8, 4.8)

Alternative StrategyStudents may want to think of per-cents when finding coordinates of points that partition segments. In Example 1, for instance, they may

think of the point that is 3}5 of the way

from A to B as being “60% of the way” from A to B. Therefore, they need to add 60% of the run and 60% of the rise to the coordinates of point A in order to obtain the coordi-nates of point P.

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CC23

Extension: Partition Segments CC23

E X A M P L E 2 Construct a point along a directed line segment

Construct the point L on } AB so that the ratio of AL to LB is 3 to 1.

Solution

STEP 1 Draw } AB of any length. Choose any point C not on

‹ ] › AB . Draw

]

› AC .

STEP 2 Place the point of a compass at A and make an arc of any radius intersecting

]

› AC at D. Using the

same compass setting, make

three more arcs on ]

› AC as shown.

Label the points of intersection E, F, and G, and note that AD 5 DE 5 EF 5 FG.

STEP 3 Draw } GB . Use the copy an angle construction to copy ∠ AGB at D, E, and F. The new sides are all parallel, and they intersect } AB at J, K, and L, dividing } AB equally, so that AJ 5 JK 5 KL 5 LB.

c Point L divides directed line segment AB in the ratio 3 to 1.

C

D

A B

EF

G

C

D

A J K L B

EF

G

PARTITIONING Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio.

1. A(1, 3), B(8, 4); 4 to 1 P (6.6, 3.8) 2. A(22, 1), B(4, 5); 3 to 7 P (20.2, 2.2)

3. A(8, 0), B(3, 22); 1 to 4 P (7, 20.4) 4. A(22, 24), B(6, 1); 3 to 2 P (2.8, 21)

CONSTRUCTION Draw a segment with the given length. Construct the point that divides the segment in the given ratio. 5–8. Check students’ constructions.

5. length: 3 in.; ratio: 1 to 4 6. length: 2 in.; ratio: 2 to 3

7. length: 12 cm; ratio: 1 to 3 8. length: 9 cm; ratio: 2 to 5

9. REASONING In Example 2, what theorem helps you to conclude that AJ 5 JK 5 KL 5 LB ? Explain.

10. VISUALIZATION Suppose point P divides } XY so that XP to PY is 3 to 5. Describe the point that divides } YX so that YP to PX is 5 to 3. It is point P.

11. WHAT IF? Make a conjecture about how to find the coordinates of a point that lies beyond point B along

]

› AB . Use an example to support your

conjecture.

PRACTICE EXAMPLE 1


EXAMPLE 2


See margin.

See margin.

GE_CCESE621852_0607A_EXT.indd CC23 12/13/10 8:42:05 PM

9. Sample answer: If parallel lines intersect two transversals, they divide the transversals proportionally. Since AD 5 DE 5 EF 5 FG, any two segments have a ratio of 1. Therefore any two segments of } AB have a ratio of 1, which means AJ 5 JK 5 KL 5 LB.

11. Sample answer: To find a point that lies beyond point B, use a fraction that is greater than 1 along with the rise and run from A to B to find the required coordinates.

Extra Example 2Construct the point L on a directed line segment AB so that the ratio of AL to LB is 3 to 2.

A B

H

GC

F

E

D

J K L M

Closing the LessonHave students summarize the major points of the lesson and answer the Essential Question: How do you find the point that par-titions a directed line segment in a given ratio?• A directed line segment AB is a

segment that represents moving from point A to point B.

First find the rise and the run of the given segment. Then multiply the run by the final ratio of the parti-tioned segments and add that value to the x-coordinate of the first point. Finally, multiply the rise by the final ratio of the partitioned segments and add that value to the y-coordinate of the first point.

4 PRACTICEAND APPLY

Avoiding Common ErrorsExercises 5–8 Some students may construct the wrong number of congruent segments for these exercises. Remind them to add the two numbers given in the ratio to determine the total number of congruent segments that need to be constructed.


CC24

CC24 Chapter 10 Properties of Circles

10.4 Tangent Lines and Inscribed Squares MATERIALS p compass p straightedge

Q U E S T I O N What constructions use right angles inscribed in circles?

Recall that in a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

E X P L O R E 1 Construct a tangent to a circle

STEP 1 Find midpoint Given (C and point A, draw } AC . Construct the bisector of the segment and label the midpoint M.

STEP 2 Draw circle Construct (M with radius MA. Label one of the points where (M intersects (C as point B.

STEP 3 Construct tangent Draw ‹ ] › AB . It is the

tangent to (C that passes through A.

C M A

C MB

A

C MB

A


1. Explain how to construct the other tangent to (C from point A.

2. Add } BC to your construction. Explain how nABC can be used to prove that

‹ ] › AB is a tangent to (C.

3. Can (M be larger than (C in the construction of a tangent? If so, describe under what conditions this can occur.

d Squares

Use after Lesson 10.4Use afUse afCONSTRUCTION ONSTRUCTIONInvestigating GeometryIIIIIIIIIII ttiiiiiiiiiiiii ttiiiiiiiiiiiiiInvestigatingIIIIIIIIIIIIInnnnnnnnnnnnnnvvvvvvvvvvvvvveeeeeeeeeeeesssssssssssssstttttttttttttiiiiiiiiiiiiggggggggggggggaaaaaaaaaaaaaattttttttttttttiiiiiiiiiiiiinnnnnnnnnnnnnngggggggggggggg GeometryGGGGGGGGGGGGGGeeeeeeeeeeeeeooooooooooooommmmmmmmmmmmmmeeeeeeeeeeeeeettttttttttttttrrrrrrrrrrrrrryyyyyyyyyyyyyy

g gggggggggg gggggggg

Draw the line from A through the other point where (M intersects (C.

See margin.

Sample answer: yes; (M is larger than (C if M lies outside (C ; this happens when AC is greater than twice the radius of (C.

GE_CCESE621852_1004A_ACT.indd 24 12/13/10 8:40:19 PM

2. Sample answer: If one side of an inscribed triangle is a diameter of a circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. Because } AC is a diameter of (M, nABC is a right triangle, and

‹ ] › AB is perpen-

dicular to radius } BC at point B. So, ‹ ] › AB is a

tangent to the (C, because in a plane, a line is tangent to a circle if the line is perpendic-ular to a radius at its endpoint on the circle.

1 PLAN AND PREPARE

Explore the Concept• Students will construct a tangent

line to a circle from a point outside the circle, and students will construct a square inscribed in a circle.

• This activity supplements the study of tangents and inscribed angles.

MaterialsEach student will need:• compass• straightedge



2 TEACH

Alternative StrategyHave students work in pairs, with each partner doing one of the con-structions. Then have the students share their results with their partner.

Tips for SuccessBefore starting Explore 1, be sure students recall how to construct the bisector of a segment. Stress that they should not change the compass setting when construct-ing the arcs that are used to find a bisector.

Key Questions• How many tangents to a circle

can be constructed from a point outside the circle? two

• What conjecture can you make about the distances from point A to the two points of tangency on the circle? The distances are equal.


CC25

10.4 Use Inscribed Angles and Polygons CC25

E X P L O R E 2 Construct a square inscribed in a circle

STEP 1 Draw diameter Given (C, draw any diameter. Label the endpoints A and B.

STEP 2 Construct bisector Construct the bisector of the diameter. Label the points where it intersects (C as points D and E.

STEP 3 Form square Connect points A, D, B, and E to form square ADBE.

A

B

C

A

B

C DE

A

B

C DE


4. In Explore 2, Step 2, why can you conclude that ‹ ] › DE '

‹ ] › AB ?

5. Show that ADBE is a rectangle by proving it has four right angles.

6. Show that ADBE is a rhombus by proving it has four congruent sides.

7. How do you know quadrilateral ADBE is a square?

8. Is a quadrilateral that is inscribed in a circle always a square? Explain why or why not.

9. Tell how you might extend the construction in Explore 2 to construct an inscribed regular octagon in (C.

10. Tell how you might extend the construction in Explore 2 to inscribe a circle in the square in (C.

An inscribed polygon is a polygon that has all of its vertices on a circle. So, an inscribed square must have all four of its vertices on the circle.

4–10. See margin.

GE_CCESE621852_1004A_ACT.indd 25 12/11/10 4:43:31 AM

4. The same compass setting was used when constructing the segment bisector, so the points used to draw it are equidistant from points A and B. This makes the bisector a per-pendicular bisector.

5. Because } AB is a diam-eter, ∠ ADB and ∠ BEA are right angles. Also, because } DE is a diameter, ∠ EAD and ∠ DBE are right angles. ADBE has four right angles, so it is a rectangle.

6. Because the diameters are perpendicular, all four minor arcs of the circle measure 908. In a circle, if two minor arcs are congru-ent then their corresponding chords are congruent, so all four chords are congruent. ADBE has four congruent sides, so it is a rhombus.

Key DiscoveryYou can use a theorem about inscribed angles to justify the methods used to construct a tan-gent to a circle from an external point and to inscribe a square.


1. Suppose you construct both tangents to a circle from a point P outside the circle. What is the most specifi c name for the quad-rilateral formed by the segments from point P to the points of tangency and the two radii to the points of tangency? Justify your answer. The quadrilateral is a kite or a square. Sample answer: The quadrilateral has two pairs of congruent sides (the tangents are congruent and the radii are congruent). The angles at the points of tangency are both right angles. If the other two angles are also right angles, the figure is a square.

7. ADBE is a square, because it is both a rectangle (Exercise 5) and a rhombus (Exercise 6).

8. No; it is possible to inscribe other quadrilaterals that are not squares. For example, any two non-perpendicular diam-eters can be used to draw an inscribed rectangle.

9. Bisect the central angles at C. Label the points where the bisectors intersect the circle as F, G, H, and I. Then draw polygon AFDGBHEI, a regular octagon.

10. Bisect one of the central angles at C. Label the intersec-tion of the angle bisector and the side of the square as point L. Use CL as the radius and center C to draw the inscribed circle.


1. When Bill and his brother have the same number of books in their collections, how many books will each of them have?

2. Graph the equations above on the same coordinate plane. What do you notice about the graphs and the solution you found above?

Use a table to solve the linear system.

3. y 5 2x 1 3 4. y 5 2x 1 1 5. y 5 23x 1 1 y 5 23x 1 18 y 5 2x 2 5 y 5 5x 2 31

5 1 5 1

A system of linear equations, or linear system, consists of two or more linear equations in the same variables. A solution of a linear system is an ordered pair that satisfies each equation in the system. You can use a table to find a solution to a linear system.

Bill and his brother collect comic books. Bill currently has 15 books and adds 2 books to his collection every month. His brother currently has 7 books and adds 4 books to his collection every month. Use the equations below to find the number x of months after which Bill and his brother will have the same number y of comic books in their collections.

y 5 2x 1 15

y 5 4x 1 7

Copy and complete the table of values shown.

Find an x-value that gives the same y-value for both equations.

Use your answer to Step 2 to find the number of months after which Bill and his brother have the same number of comic books.

Get

ty Im

ages

/Im

age

Sour

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In your bookProblem Solving Workshops explore alternative methods as tools for problem solving. A variety of Activities use concrete and technological tools to explore mathematical concepts.

Mathematically proficient students consider the available tools when solving a... problem... [and] are... able to use technological tools to explore and deepen their understanding...

Use appropriate tools strategically.

MM

UsU5

You can solve the problem by breaking it into parts.

the number of calories you burn when running.

the calories you burn when swimming.

the calories you burn when doing each activity. You burn a total of 570 calories.

In Example 5 on page 98, you saw how to solve a problem about exercising using a verbal model and an equation. You can also solve the problem by breaking it into parts.

Your daily workout plan involves a total of 50 minutes of running and swimming. You burn 15 calories per minute when running and 9 calories per minute when swimming. Find the number of calories you burn in your 50 minute workout if you run for 20 minutes.

1. Your family is taking a vacation for 10 nights. You will spend some nights at a campground and the rest of the nights at a motel. A campground stay costs $15 per night, and a motel stay costs $60 per night. Find the total cost of lodging if you stay at a campground for 6 nights. Solve this problem using two different methods.

2. In Exercise 1, suppose the vacation lasts 12 days. Find the total cost of lodging if you stay at the campground for 6 nights. Solve this problem using two different methods.

3. During the summer, you work 35 hours per week at a florist shop. You get paid $8 per hour for working at the register and $9.50 per hour for making deliveries. Find the total amount you earn this week if you spend 5 hours making deliveries. Solve this problem using two different methods.

4. Describe and correct the error in solving Exercise 3.

2 5

$8 per hour 5 hours 5 $40

$9.50 per hour 30 hours 5 $285

$40 1 $285 5 $325

15 calories 20 minutes 5 300 calories per minute

9 calories 30 minutes 5 270 calories per minute

300 calories 1 270 calories 5 570 calories



CC27

Extension: Measure Angles in Radians CC27

ExtensionUse after Lesson 11.4

Measure Angles in Radians

Key Vocabulary• radian

In the Activity below, you will explore the relationship between radii and arc length for a central angle of concentric circles.

STEP 1 Find the lengths of the arcs intercepted by a central angle of 60° for four concentric circles with radii 1, 2, 3, and 4.

Radius Arc Length Calculation Arc Length

1 m C AB } 3608

p 2πr 5 608 }

3608 p 2π p 1 5

π } 3

π } 3

2 ? ?

3 ? ?

4 ? ?

STEP 2 Explain how to express the arc length as a direct variation.

STEP 3 Make a conjecture about what happens to the arc length if the radius is doubled.

ACTIVITY CONSTANT OF PROPORTIONALITY

In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 3608, and therefore

arc length of C AB 5 m C AB } 3608

p 2πr

A

BP

r

The form of this equation shows that the arc length associated with a central angle is proportional to the radius of the circle.

The constant of proportionality, m C AB } 3608

p 2π, is defined to be the radianmeasure of the central angle associated with the arc.

The radian measure can be thought of as the length of the arc associated with a given central angle in a circle of radius 1.

GOAL Find the radian measure of an angle.

Since all circles are similar, if the radius is doubled, the arc length doubles.

See margin.

The arc length y is of the form y 5 k p r, with k 5 608 }

3608 p 2p.

GE_CCESE621852_1104A_EXT.indd 27 12/15/10 12:38:11 AM

Activity, Step 1.

Radius Arc Length Calculation

1 m C AB } 3608 p 2pr 5 608 } 3608 p 2p p 1 5 p } 3

2 m C AB } 3608 p 2pr 5 608 } 3608 p 2p p 2 5 2p } 3

3 m C AB } 3608 p 2pr 5 608 } 3608 p 2p p 3 5 p

4 m C AB } 3608 p 2pr 5 608 } 3608 p 2p p 4 5 4p } 3

1 PLAN AND PREPARE

Warm-Up ExercisesFind the measure of the arc.

OB

XYZ

D F

ACE

1208

1. m C AB 908

2. m C CY 1208

3. m C FZ 1508


Essential QuestionBig Idea 3, p. 719How do you find the radian measure of an angle? Tell students they will learn how to answer this question using a circle of radius 1.

3 TEACHActivity NoteBefore beginning Step 1, emphasize that arc length is a distance. Stress that it is a portion of the circumfer-ence, which is a distance. Remind students that arc measure, such as m C AB , is a degree measure. Caution students to distinguish between arc measure and arc length when discussing these topics. In Step 2, you may remind students that a direct variation is a relationship between two variables x and y such that y 5 ax, where a is a non-zero number called the constant of variation. In Step 3, have students compare the arc lengths for the radii 1 and 2, and then the arc lengths for the radii 2 and 4.


CC28

CC28 Chapter 11 Measuring Length and Area

ARC LENGTH Find the length of the arc associated with the given central angle and radius.

1. 1208; radius 4 2. 1358; radius 1.5 3. 2408; radius 3

CONVERSION Convert the degree measure to radian measure.

4. 158 5. 708 6. 3008

CONVERSION Convert the radian measure to degree measure.

7. 4π } 3

radians 8. 11π }

12 radians 9. π }

8 radians

10. Copy and complete the table by giving the equivalent degree or radian measure of the benchmark arcs.

Degrees 308 458 ? ? 1208 ? ? ? 3608

Radians ? ? π } 3 π } 2 ? 3π } 4

π 3π } 2

?

11. The arc length on a circle can be also be found using the formula s 5 r p u, where s is the arc length, r is the radius of the circle, and u is the central angle (measured in radians) associated with the arc. Find the length of an arc in a circle when the radius is 4 inches and the central angle is 3p

} 4

radians.

PRACTICE

E X A M P L E Convert between degree and radian measure

a. Convert 458 to radians. b. Convert 3p }

2 radians to degrees.

Solution

a. 458 p 2π } 3608

5 1 } 4

π or π } 4

b. 3π } 2

p 3608 } 2π 5 2708

So, 458 5 π } 4

radians. So, 3π } 2

radians 5 2708.

CONVERSION The radian measure of a complete circle (360�) is exactly 2π radians, because the circumference of a circle of radius 1 is exactly 2π. You can use this fact to convert from degree measure to radian measure and vice versa.

To convert from degree measure to radian measure, use this relationship:

degree measure p 2π }

3608 5 radian measure

To convert from radian measure to degree measure, use this relationship:

radian measure p 3608 }

2π 5 degree measure

ACTIVITY


EXAMPLE


8p }

3 9p

} 8 4p

p } 12 7p }

18 5p

} 3

2408 1658 22.58

3p in. or about 9.42 in.

See margin.

GE_CCESE621852_1104A_EXT.indd Sec1:28 12/11/10 12:03:55 AM

10.

Degrees 308 458 608 908 1208 1358 1808 2708 3608

Radians p } 6 p } 4 p } 3 p } 2 2p }

3 3p

} 4 p 3p

} 2 2p

Extra Examplea. Convert 1508 to radians. 5p } 6 radiansb. Convert 11π } 6 radians to degrees. 3308

Teaching StrategyYou may tell students that radian measure will be used in Algebra 2 when graphing trigonometric functions.

Closing the LessonHave students summarize the major points of the lesson and answer the Essential Question: How do you find the radian measure of an angle?• In a circle, the arc length of

C AB 5 m C AB } 3608 p 2pr.

• To convert from degree measure to radian measure, use the conversion factor 2p } 3608

.

• To convert from radian measure to degree measure, use the con-version factor 3608 } 2p

.To find the radian measure of an angle, find the length of the arc cut off by that central angle on a circle of radius 1.

4 PRACTICE AND APPLY

Teaching StrategyExercise 10 After completing this exercise, have students add a copy of the completed table to their notebooks for future reference.

Mathematical ReasoningExercise 11 Another way to help students remember how to apply the formula s 5 r • u is to state it in words: “Arc length equals radius times central angle measure in radians.”


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In your bookVerbal Models and the Problem Solving Plan help you translate the information in a problem into a model and then analyze your solution.

Mathematically proficient students start by explaining to themselves the meaning of a problem... They analyze givens, constraints, relationships, and goals. They make conjectures about the form... of the solution and plan a solution pathway...

Make sense of problems and persevere in solving them.

MM

MM1


form... of the solution and plan a aaa solution pathway...

You have two summer jobs at a youth center. You earn $8 per hour teaching basketball and $10 per hour teaching swimming. Let x represent the amount of time (in hours) you teach basketball each week, and let y represent the amount of time (in hours) you teach swimming each week. Your goal is to earn at least $200 per week.

of x and y.

allow you to meet your goal.

1

8 x 1 10 y 200

x 1 10y 200.

x 1 10y 5 200 , so use

a solid line.

Next, test (5, 5) in 8x 1 10y 200:

8(5) 1 10(5) 200

90 200

Finally, shade the part of Quadrant I that does not contain (5, 5),

three points on the graph, such as (13, 12), (14, 10), and (16, 9). The table shows the total earnings for each combination of hours.

8. In Example 6, suppose that next summer you earn $9 per hour

combinations of hours that will help you meet your goal.

Basketball (hours)

Swim

min

g (h

ours

)

10 20 30

20

10

00 x

y

(5, 5)

8x 1 10y ≥ 200

28 Chapter 1 Expressions, Equations, and Functions

0.1 mi

0.15 mi

Use a Problem Solving Plan

Before You used problem solving strategies.

Now You will use a problem solving plan to solve problems.

Why? So you can determine a route, as in Example 1.

1.5

Key Vocabulary• formula

RUNNING You run in a city. Short blocks are north-south and are 0.1 mile long. Long blocks are east-west and are 0.15 mile long. You will run 2 long blocks east, a number of short blocks south, 2 long blocks west, and back to your start. You want to run 2 miles at a rate of 7 miles per hour. How many short blocks must you run?

Solution

STEP 1 Read and Understand

What do you know?

You know the length of each size block, the number of long blocks you will run, and the total distance you want to run.

You can conclude that you must run an even number of short blocks because you run the same number of short blocks in each direction.

What do you want to find out?

You want to find out the number of short blocks you should run so that, along with the 4 long blocks, you run 2 miles.

STEP 2 Make a Plan Use what you know to write a verbal model that represents what you want to find out. Then write an equation and solve it, as in Example 2.

KEY CONCEPT For Your Notebook

A Problem Solving Plan

STEP 1 Read and Understand Read the problem carefully. Identify what you know and what you want to find out.

STEP 2 Make a Plan Decide on an approach to solving the problem.

STEP 3 Solve the Problem Carry out your plan. Try a new approach if the first one isn’t successful.

STEP 4 Look Back Once you obtain an answer, check that it is reasonable.

E X A M P L E 1 Read a problem and make a plan

ANOTHER WAY

For an alternative method for solving the problem in Example 1, turn to page 34 for the Problem Solving Workshop.

laa111se_0105.indd 28 10/20/10 12:50:34 AM



CC30

CC30 Chapter 12 Surface Area and Volume of Solids


Density

Density is the amount of matter that an object has in a given unit of volume. The density of an object is calculated by dividing its mass by its volume.

density 5 mass } volume

Different materials have different densities, so density can be used to distinguish between materials that look similar. For example, table salt and sugar look alike. However, table salt has a density of 2.16 grams per cubic centimeter, while sugar has a density of 1.58 grams per cubic centimeter.

Key Vocabulary• density• population density

E X A M P L E 1 Determine which substance has greater density

A piece of copper with a volume of 8.25 cubic centimeters has a mass of 73.92 grams. A piece of iron with a volume of 5 cubic centimeters has a mass of 39.35 grams. Which metal has the greater density?

Solution

Calculate the density of each metal.

Copper: density 5 mass } volume

5 73.92 g

} 8.25 cm3

5 8.96 g/cm3

Iron: density 5 mass } volume

5 39.35 g

} 5 cm3

5 7.87 g/cm3

c Copper has the greater density.

POPULATION DENSITY Another use of the word density occurs in the term population density. The population density of a city, county, or state is a measure of how many people live within a given area.

population density 5 number of people

}} area of land

Population density is usually given in terms of square miles, but can be expressed using other units such as city blocks.

copper

GOAL Use density to solve problems.

GE_CCESE621852_1204A_EXT.indd CC30 12/11/10 12:06:25 AM

1 PLAN ANDPREPARE

Warm-Up ExercisesIdentify the greater quantity.1. 6.09 cm, 6.90 cm 6.90 cm2. 12.705 mm, 12.75 mm 12.75 mm3. Find the area of a trapezoid

with a height of 3.5 feet and bases 2.5 feet and 5 feet long. 13.125 ft2

4. Find the area of a rectangle with a length of 6.2 meters and a width of 10.4 meters. 64.78 m2


Essential QuestionBig Idea 1, p. 791How do you use density to compare two physical quantities? Tell students they will learn how to answer this question by finding density and population density.

3 TEACH

Extra Example 1A piece of concrete with a volume of 8 cubic centimeters has a mass of 32.4 grams. A rock with a volume of 3 cubic centimeters has a mass of 12.84 grams. Which has the greater density? rock

Reading StrategyPoint out that density is the mass per unit volume, and population density is the number of people per unit area. Be sure to caution students about writing the results of their calculations with the correct units.

LGE_CCETE621999_11-4A_EXT.indd CC30LGE_CCETE621999_11-4A_EXT.indd CC30 12/22/10 9:51:47 PM12/22/10 9:51:47 PM

CC31

Extension: Density CC31

1. METALS Toni found an irregular piece of metal. She dropped it into a container partially filled with water and measured that the water level rose 4.8 centimeters. The square base of the container is 8 centimeters on a side. Toni measures the mass of the metal to be 450 grams. What is the density of the metal? Round to the nearest tenth.

2. POPULATION DENSITY The population of Colorado in 2009 was about 5,024,748. The land area can be approximated by a rectangle with coordinates (0, 0), (369, 0), (369, 281), and (0, 281), with each unit on the coordinate plane being 1 mile. What was the population density of Colorado in 2009?

3. POPULATION DENSITY In 2000, Texas had about 2.74 persons per household, 7,393,354 households, and a land area of about 261,797 square miles. What was the population density of Texas in 2000? If the population in 2009 was about 24,782,302, how did the density in 2009 compare to the density in 2000?

4. COOLING On average during the summer, a 30,000 cubic foot house costs $7 per day to cool, while a 25,000 cubic foot house costs $6.50 per day to cool. Which house costs less per cubic foot to cool? Explain.

5. REASONING If two objects have the same volume, which object has a greater mass, the heavier object or the lighter object? Explain.

PRACTICE EXAMPLE 1

on p. CC30for Ex. 1

EXAMPLE 2


E X A M P L E 2 Find a population density

The population of Vermont in 2009 was 621,760. The state can be modeled by a trapezoid with vertices at (0, 0), (0, 160), (80, 160), and (40, 0), with each unit on the coordinate plane being 1 mile. Find the population density of Vermont.

Solution

STEP 1 Sketch the simplified model of Vermont.

STEP 2 Calculate the area of the trapezoid. The height is 160 miles and the bases have length 80 miles and 40 miles.

A 5 1 } 2

h(b1 1 b2) 5 1 } 2

(160)(80 1 40) 5 9600 mi2

STEP 3 Find the population density.

density 5 number of people

}} area of land

5 621,760 people

}} 9600 mi2

ø 64.8 people/mi2

c In 2009, there were about 65 people per square mile living in Vermont.

80 mi

40 mi

160 miVT

1.5 g/cm3

about 48 people per square mile

See margin.

See margin.

See margin.


3. About 77 people per square mile; in 2009, the population density was about 95 people per square mile, or 18 people per square mile greater than in 2000.4. The 30,000 cubic foot house costs less to cool; $7

} 30,000 ft3

ø $0.00023/ft3, and $6.50 }

25,000 ft3 ø $0.00026/ft3.

5. Sample answer: The heavier object; the weight of an object is the product of its mass and the gravity on Earth. Because the effect of gravity is the same for both objects, the object that is heavier has a greater mass.

Extra Example 2The population of Nevada in 2009 was 2,643,085. Its borders can be modeled by a right trapezoid with vertices at (0, 290), (0, 500), (310, 500), and (310, 0), with each unit on the coordinate plane being 1 mile. Find the population density of Nevada. about 24 people per square mile

Key Question to Ask for Example 2• Why can the 160-mile dimension

of the figure be used as the height of the trapezoid? The diagram marks indicate that this side of the trapezoid is perpendicular to both of the bases.

Closing the LessonHave students summarize the major points of the lesson and answer the Essential Question: How do you use density to solve certain problems?• The density of an object is the

quotient of its mass and its volume.

• The population density of a city, county, state, or country is the quotient of the number of people living there and its land area.

To compare the density of two objects, find the quotient of each object’s mass and volume, and then compare the results. To find the population density of a region, find the area of the region and then divide the number of people living there by this area.

4 PRACTICEAND APPLY

Study StrategyExercise 5 If students have diffi-culty with this question, have them research the difference between weight and mass in their science textbooks or on the Internet.


CC32

CC32 Chapter 12 Surface Area and Volume of Solids


Solids of Revolution

A solid of revolution is a three-dimensional figure that is formed by rotating a two-dimensional shape around an axis. The line around which the shape is rotated is called the axis of revolution.

For example, if you rotate a rectangle around a line that contains one of its sides, the solid of revolution that is produced is a cylinder.

Key Vocabulary• solid of revolution• axis of revolution

E X A M P L E 1 Sketch and describe a solid of revolution

Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid.

a. 4

9

b. 2

5

Solution

a. 4

9

b. 2

5

Cylinder with height 9 Cone with height 5 and base radius 4 and base radius 2

Because the dimensions of the original figure can be used to describe the solid of revolution, you can use these dimensions to calculate the volume of the solid of revolution.

GOAL Sketch and describe solids produced by rotating a two-dimensional figure around an axis in space.


1 PLAN AND PREPARE

Warm-Up ExercisesFind the volume of the solid.1. Cylinder with height 5 and base

radius 2 20p

2. Cone with height 9 and base radius 5 75p


Essential QuestionWhat happens if you rotate atwo-dimensional figure around an axis in space? Tell students they will learn how to answer this question by sketching anddescribing solids of revolution.

3 TEACH

Extra Example 1Sketch the solid produced byrotating a rectangle with length 3 and width 2 around an axis that contains a longer side. Identify and describe the solid.

2

3

The solid is a cylinder with height 3 and base radius 2.

Key Question to Ask for Example 1• What happens when a segment

that is perpendicular to the axis of rotation is rotated?The segment generates a circle, such as the base of a cylinder or cone.

4. Cylinder with height 4 and base radius 4

4

4


25

6. Cone with height 6 and base radius 9

9

6


CC33

Extension: Solids of Revolution CC33

Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid.

1.

8

6

2.

4

3

3.

7

7

Sketch the solid of revolution. Then identify and describe the solid.

4. A square with side length 4 rotated around one side.

5. A rectangle with length 5 and width 2 rotated around its longer side.

6. A right triangle with legs of length 6 and 9 rotated around its shorter leg.

Sketch the solid produced by rotating the figure around the given axis. Then find its volume.

7.

8

8

8.

6

6

9.

2

2

52

10. CHALLENGE A 308-308-1208 isosceles triangle has two legs of length 4 units. If it is rotated around an axis that contains one leg, what is the volume of the solid of revolution?

PRACTICE

EXAMPLE 1


EXAMPLE 2


E X A M P L E 2 Find the volume of a solid of revolution

Sketch the solid produced by rotating the figure around the given axis. Then find its volume.

a.

5

6

b.

33

2

Solution

a.

5

6

b. 2

33

The solid is a cylinder with The solid is made of two cones, each height 6 and base radius 5. with height 3 and base radius 2.

V 5 πr 2h 5 π(52)(6) 5 150π V 5 2 p 1 } 3

πr 2h 5 2 p 1 } 3

p(22)(3) 5 8π

1–3. Check students’ sketches.

See margin.

7–9. See margin.

4–6. See margin.


3. Sphere with radius 7

2. Cone with height 4 and base radius 3


Extra Example 2Sketch the solid produced by rotating the figure in Example 2(b) around a vertical axis of rotation that contains the altitude. Then find its volume.

3

2

The solid is a cone with height 2 and base radius 3.

V 5 1 } 3 p(32)(2) 5 6p

Key Question to Ask for Example 2• How is the segment that lies

along the axis of rotation in Example 2(a) related to the solid of revolution? The segment is the height of the solid.

Closing the LessonHave students summarize the major points of the lesson and answer the Essential Question: What happens if you rotate atwo-dimensional figure around an axis in space?A solid of revolution is created when a two-dimensional figure is rotated around an axis.• A rectangle rotated around an

axis that contains an edgeproduces a cylinder.

• A right triangle rotated around an axis that contains a legproduces a cone.

4 PRACTICEAND APPLY

Mathematical ReasoningExercise 10 Students will need to recall the relationships of lengths in 308-608-908 right triangles to sketch the solid of revolution. A cone with base radius 2 Ï

}

3 and height 2 is carved out of a cone with base radius 2 Ï

}

3 and height 6.

7–9. Check students’ sketches.

7. V 5 p (82)(8) 5 512p

8. V 5 1 } 3 p (62)(6) 5 72p

9. V 5 2 p 1 } 3 p (22)(2) 5 5 1 } 3 p

10. V 5 1 } 3 p (2 Ï}

3 )2 (6) 2 1 } 3 p (2 Ï}

3 )2 (2)

5 24p 2 8p

5 16p

4

42 3

4 3

2


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