concept map geometry

PUHSD 2009 of 31 Curriculum Division

Mathematics Concept Map Geometry 1-2 (Regular and Honors) Phoenix Union High School District

April 2009

Copyright 2009 Phoenix Union High School District. All rights reserved. No part of this document may be reproduced without the express prior written permission of the Phoenix Union High School District.

PHOENIX UNION HIGH SCHOOL DISTRICT NO. 210

4502 North Central Avenue Phoenix, AZ 85012


Phoenix Union High School District Mathematics Concept Map

Table of Contents

Section Page #

Definition of Concept Map 3

Geometry 1-2 Concept Maps 4

Geometry 1-2 Scope and Sequence 17

Geometry 1-2 Honors Concept Maps 18

Geometry 1-2 Honors Scope and Sequence 31

Acknowledgements The Mathematics Curriculum Guide and Concept Maps were developed through the hard work of the 2008 Curriculum Revision Team

Algebra Team Kathy Long North Fernando Galvez Maryvale Melanie Dalager Central

Geometry Team Oscar Ramirez Cesar Chavez Eddie Fylling Bostrom Lydia Horstman Fairfax Joseph Prevost Maryvale

Algebra 3-4 Team Nicole Fernandez North Diane Welch Metro Tech Juli Schexnayder Maryvale Sheryl Filliater Metro Tech

Facilitator

Mona Toncheff Math Content Specialist CES


Concept Map Definition Page

Concept Unit

Enduring Understanding:

Essential Questions:

Power Standards Key Vocabulary

Examples:

Topic: The organization of performance objectives into a common theme that promotes student engagement and focuses student inquiry.

Essential Question: The mental questions that help students form a conceptual understanding of the concept or concepts. They point toward key ideas and issues and suggest meaningful and provocative inquiry into content.

Enduring Understanding: A central and organizing notion that gives meaning and connection to facts. It has lasting value and can transfer to other inquiries and requires “uncoverage”.

Examples: Examples illustrate level of how students can solve problems within each concept unit or how concepts can be assessed.

Key Vocabulary: The vocabulary that is important for students to know in order to demonstrate an understanding of a topic.

Power Standards: The ideas that connect the PO’s to the overarching topic.

Resources

Resources: Lists important lessons, activities, assessments, or links for the instructional .unit.

Technology Standards


Concept Unit Map Grade Level: Geometry 1

Using Points, Lines, Planes, and Angles

Enduring Understanding: Recognizing the relationship of points, lines, and planes and the geometric figures they form is the foundation for applying mathematics to the real world

Identify, name, and draw points, lines, rays, segments, planes and angles.

Solve problems related to complementary, supplementary, congruent angles, bisectors, angles or segment concepts.

Solve problems created by parallel lines cut by a transversal.

Essential Question(s): What is the difference between a postulate and a theorem? How can slope be used to determine the relationship between two lines? What is the distance and midpoint between two points? What is the relationship between points, lines and planes?

Key Vocabulary collinear/Linear coplanar/plane skew point segment ray line angle segment perpendicular bisector linear pair complementary/supplementary vertical angles transversal parallel/perpendicular adjacent angles midpoint postulate theorem

Resources Algebra Help http://www.algebrahelp.com/index.jsp Textbook resources http://classzone.com Segment addition postulate tutorial http://www.hstutorials.net/math/geometry/theoPosts/seg_add.htm

Examples Determine congruence of segments.

Determine the distance and midpoint between 2 points in a coordinate system.

Given: M is the midpoint of AB . Find AB. 2x+10 4x-6

B M A

Identify a line segment, ray, plane, line, a pair of skewed lines, perpendicular segments, three coplanar points, two collinear points.

F

E

D

C B

A

H

G

Key Concepts

Cabri Jr. Discovery activities www.timath.com • Angle bisector • Midpoint of a line



Perpendicular and Parallel lines

Enduring Understandings: Parallel and perpendicular lines provide the foundation for most physical structures.

Identify parallel lines, perpendicular lines, skew lines, parallel planes, and a line perpendicular to a plane.

Identify and solve problems using angles formed by perpendicular lines.

Identify congruent angles and supplementary angles formed by parallel lines and a transversal.

Determine if lines are parallel or perpendicular based on slope or angle relationships.

Essential Questions: What angle relationships are formed by parallel lines cut by a transversal? What angle relationships are formed by perpendicular lines? How does slope determine the relationship between two lines? How can angle relationships be used to determine measures of other angles?

Key Vocabulary parallel lines perpendicular lines skew lines parallel planes line perpendicular to a plane complementary angles supplementary angles transversal corresponding angles alternate interior angles alternate exterior angles same-side interior angles congruent converse theorem postulate

Resources Algebra Help http://www.algebrahelp.com/index.jsp Textbook resources http://classzone.com Geometry lesson on parallel and perp. http://www.homeschoolmath.net/teaching/g/parallel_and_perpendicular.php

Examples

Key Concepts

Find the value of x.

110°

( )2 10x − °( )5x + °

( )2 25x + °

Find the value of x. Apply properties, theorems, and constructions about parallel lines, perpendicular lines, and angles to prove theorems.

Cabri Jr. Discovery activities www.timath.com • Vertical and

adjacent angles • Transversals



Logical Reasoning

Enduring Understandings: Logical reasoning is essential to synthesize information, put the world in perspective, and apply knowledge.

Construct a simple formal deductive proof.

Prove a conjecture false by finding a counterexample.

Draw valid conclusions from conditional statements using law of detachment and syllogism.

Determine triangle congruency using SSS, SAS, ASA, HL and AAS congruence postulates.

Essential Questions: What conjecture can be made based on an observation? How can you use a counterexample to prove a conjecture false? How can a logical conclusion be made using law of detachment and syllogism? How can you use triangle congruence theorems to prove two triangles are congruent? How do you write a paragraph proof?

Key Vocabulary conjecture inductive reasoning counterexample patterns deductive reasoning conditional statement hypothesis conclusion law of detachment syllogism inverse converse contrapositive if-then statement reflexive property symmetric property transitive property proofs writing proofs paragraph proofs two-column proofs writing a formal proof valid invalid

Resources Algebra Help http://www.algebrahelp.com/index.jsp Textbook resources http://classzone.com

Examples State the inverse, converse, and contrapositive of a given if-then statement and determine if it is true or false.

Critique inductive and deductive arguments concerning geometric ideas and relationships.

Using the Law of Syllogism, which statement follows from the pair of true statements? If I do not get paid this week, then I cannot pay my bills. If I cannot pay my bills, then the bill collectors will call.

What is the next number you expect? 45, 90, 135, 180……..

Key Concepts

List related if…then statements in logical order.

Identify a valid conjecture using inductive reasoning.

Cabri Jr. Discovery activities www.timath.com • Conditional statements • ASA,SAS,SSS Triangle

congruence



Attributes and Properties of Triangles

Enduring Understandings: Triangles are the support system for most objects that we use on a daily basis. Are you sitting on a triangle now?

Classify triangles according to lengths of sides and angle measures.

Solve problems using triangle sum and exterior angles theorems.

Solve problems related to base angles, equilateral, and equiangular triangle theorems.

Use the Pythagorean Theorem and Distance formula to find unknown lengths.

Essential Questions: How are triangles classified? How do triangle attributes help solve problems related to angles measures and side lengths? How are the Pythagorean Theorem and Distance formula related? What is the relationship between segments formed by the intersection of the medians of a triangle?

Key Vocabulary triangle equilateral triangle isosceles triangle scalene triangle right triangle obtuse triangle acute triangle equiangular triangle vertex length measure corollary interior angles exterior angles base legs base angles hypothesis Pythagorean theorem square square root distance classify median of a triangle centroid triangle inequality Decide whether a

triangle could have the given angle measures. Explain your reasoning. 18°, 34°, 128° Resources

Algebra Help http://www.algebrahelp.com/index.jsp Textbook resources http://classzone.com Geometry Interactive activities http://www.shodor.org/interactivate/activities/

Examples

Key Concepts

Determine and solve problems using angle and side length relationships.

Solve problems using median and centroids of a triangle.

Solve problems using the triangle inequality theorem.

Classify the triangle with the most specific name.

8 9

10Find the value of x.

51°x°

102°Cabri Jr. Discovery activities www.timath.com • Triangle Inequalities • Ratio of Right Triangles • Distance in the coordinate

plane


Concept Unit Map

Grade Level: Geometry 1

Similarity

Enduring Understandings: Properties of similar shapes can be used to solve problems and create and critique inductive and deductive arguments

Solve applied problems using the attributes of similar triangles.

Show that two triangles are similar using AA, SSS, and SAS Theorems

Use the triangle proportionality and midsegment theorem.

Essential Questions: How do I apply my knowledge of ratios and proportions to similiarity? What makes polygon similiar? Can I show two triangles are similar using AA, SSS, or SAS? Can I use the Triangle Proportionality Theorem?

Key Vocabulary ratio proportion cross product property similar polygons scale factor midsegment of a triangle similarity Angle-Angle Similarity (AA) Side-Side-Side Similarity (SSS) Side-Angle-Side Similarity (SAS) triangle proportionality theorem midsegment theorem

Resources Textbook resources http://classzone.com similarity lesson http://www.gogeometry.com/geometry/similarity_ratio_proportion_index.html

Examples

Key Concepts

Identify similar polygons.

Solve problems using ratio and proportions.

Cabri Jr. Discovery activities www.timath.com • Constructing

similar triangles

A). 14 B). 7 C). 127

D). 1412

Solve the proportion 5 72x x=

−

Write the ratio of the values of 4 nickels to the values of 10 dimes?

A). 15

B). 25

C). 410

D). 104

28 34

40

What value of x will make the two triangles similar? x

51

42



Quadrilaterals

Enduring Understandings: The building blocks of society are based on Polygons, especially four sided objects (Quadrilaterals).

Identify and classify polygons.

Identify properties of parallelograms.

Prove that a given quadrilateral is a parallelogram.

Solve problems using properties of trapezoids.

Essential Questions: Can I identify and classify polygons? Can I find measures of quadrilaterals? Can I apply properties of parallelograms to solve problems? Can I prove that a quadrilateral is a parallelogram?

Key Vocabulary polygon side of a polygon vertex of a polygon diagonal of a polygon parallelogram rhombus rectangle square trapezoid bases, legs, and base angles of a trapezoid isosceles trapezoid midsegment of a trapezoid

Resources Textbook resources http://classzone.com

Examples

Solve problems using properties of special quadrilaterals including rhombuses, rectangles, and squares.

Classify the figure in as many way as possible Find the value of x and y.

• y x 14

9

Find the angle measures of quadrilaterals.

Key Concepts

Cabri Jr. Discovery activities www.timath.com • Properties of

parallelograms • Angles in a

quadrilateral Trapezoid JKLM

(11x)° 89°

(4y + 3)°

M L

K J 103°

x = y =

JKLM is a rhombus. Find the value of x.

50°

5x°

J

K

L

M



Probability

The radii of the concentric circles on a dart board are 2 in, 4 in, 6 in, and 8 in respectively. What is the probability of getting at least 20 points on the first throw?

A) 516

B) 34

C) 12

D) 916

Enduring Understandings: Understanding theoretical probability makes you better informed when making future decisions.

Determine the theoretical probability of events, estimate probabilities using experiments, and compare the results.

Use concepts and formulas of area to calculate geometric probabilities.

Determine the number of possible outcomes of an event.

Apply appropriate means of computing the number of possible arrangements of items using permutations where order matters, and combinations where order does not matter.

Essential Questions: How do you determine the number of possible outcomes of an event? What is the difference between theoretical and experimental probability? How can you determine the number of arrangements that can be made given an event? How can you determine the number of combinations in an event? How can you use theoretical probability to make predictions?

Which of the following expressions could be used to find the number of four-letter arrangements that can be made from the letters in the word ALGEBRA?

A) 7! B) 7 4P C) 4 7P D) 7!4!

Resources Algebra Help http://www.algebrahelp.com/index.jsp Textbook resources http://classzone.com Combinations and permutations calculator http://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html

Examples

Apply the addition and multiplication principles of counting and represent these principles algebraically using factorial notation.

• Use simulators to generate experimental probability data

• Compute the number of possible arrangement with permutations and combinations

• Computational use

Key Vocabulary outcome theoretical probability experimental probability event counting principle factorial notation arrangements combinations permutations

Key Concepts


Concept Unit Map

Grade Level: Geometry 2

Data Analysis

Enduring Understandings: Data analysis is used to make important decisions in all facets of our lives, including education and corporate America.

Draw inferences about data sets from lists, tables, matrices, and plots.

Organize collected data into an appropriate graphical representation.

Display data, make predictions and observations.

Make inferences by comparing data sets.

Essential Questions: Can I draw inferences about data sets from lists, tables, matrices and

plots? Can I organize collected data into an appropriate graphical

representation? Can I make inferences by comparing data sets? Can I identify misrepresentations and distortions in displays of data?

Key Vocabulary measures of center mean, median, mode, range collect data data analysis frequency tables matrices lists make predictions and observations make inferences interval sample population misrepresentation and distortions of data display network graphs adjacency matrices Vertex, edges

Resources Textbook resources http://classzone.com Comparing data activity http://www.northcanton.sparcc.org/~technology/excel/files/comparing_data.html Path puzzles http://www.mathmaniacs.org/lessons/12-euler/PencilPuzzles.html

Examples

Key Concepts

Determine the most appropriate measure of center for a given situation.

Identify misrepresentations and distortions in displays of data.

Danesha is choosing the number of events to have this year for her community service club. Members of the club e-mail Danesha with the number of events they prefer. If Danesha wants to choose the most popular number of events that were e-mailed to her, which measure of centreer should she use to make her choice? A) Mean B)Median C)Mode D) Range

• Enter data sets into lists to create graphical representations, then use 1-VAR Stats under LISTS to compute the measures of central tendency


Solve network problems using graphs and matrices

Determine the shortest route for the recycling trucks that start and end at A?

4

6

11

5

4

23

7

2



Transformations

Enduring Understandings: A clear understanding of transformations allows students to visualize the movement of objectives in three dimensional space and on a plane.

Know that after any of transformation (rotation, reflection, and translation), the shape still has the same size, area, angles and side lengths.

Know that if one shape can become another using roation, reflection, and translation, then the two shapes are called congruent.

Understand that during transformation called a dilation, (enlargement or reduction), the shape becomes bigger or smaller.

Dilation does not result in congruent shapes, the shapes will be similar. Same shapre, yet different sizes are similar shapes.

Essential Questions: What does the word transformation means? What is the relationship of Geometry transformations and symmetry? Can you use more than one transformation to describe the relationship?

Key Vocabulary transformation translation image pre-image reflection line of symmetry dilation glide reduction enlargement rotation center of rotation angle of rotation rotational symmetry tessellation isometry plane figure scale factor

Resources Cabri Jr. Java Applet Sketchpad TI Calculator Textbook Resources http://classzone.com

Key Concepts

Examples

RRoottaattiioonn 4455°°

RReefflleeccttiioonn

TTrraannssllaattiioonn

DDiillaattiioonn

TRANSFORMATIONS

Discover the line of reflection, the center of rotation, and the center of dilation.

Cabri Jr. Discovery activities www.timath.com • Students can use the

transformation tool on Cabri Jr. to explore translation, reflections, rotation or dilations



Right Triangle Trigonometry

Enduring Understandings: Demonstrate understanding of the following equation types to solve problems involving right triangles and use them in real life situations.

• Pythagorean Theorem • Definition of Sine • Definition of Cosine • Definition of Tangent • Sum of Angles

Understand that the term “solving the triangle” means that if we start with a right triangle and know any two sides, we can find or solve for the unknown side.

Investigate the fundamental concepts behind trigonometry: three basic trig functions and how to determine which trig function to use.

Use SOHCAHTOA to memorize the three main trigonometric functions.

SOH → oppositehypotenuse

sin x =

CAH → adjacenthypotenuse

cos x =

TOA → oppositeadjacent

tan x =

Know that recognizing special right triangles in geometry can help you to problem solve. A special right triangle is a right triangle whose sides are in a particular ratio. ( 30 60 90, , ) & ( 45 45 90, , )

Essential Questions: Could we use Pythagorean Theorem with triangles that are not right triangles? How do you determine if a triangle is a right triangle?

Key Vocabulary radical radicand 45°-45°-90° triangle 30°-60°-90° triangle trigonometric ratios leg opposite an angle leg adjacent to an angle tangent sine cosine special right triangle

If we know the two legs of a right triangle, we can solve for the hypotenuse using the formula:

where a and b are the lengths of the two legs of the triangle, and h is the hypotenuse

Examples

Key Concepts

Relative to angle A, this is how the sides of a right triangle would be labeled

Cabri Jr. Discovery activities www.timath.com • Investigating special

triangles • Use the calculator to find

sine, cosine, or tangents

2a

2b

2c

AC

BPythagorean Theorem

2 2 2a b c+ =

Resources Cabri Jr. Java Applet Sketchpad Computer TI Calculator Power Point Textbook Resources http://classzone.com



Circles

Enduring Understandings: Circular objects are abundant in our Universe formed by nature, specifically by the force of gravity.

Use properties of tangents

Use properties of arcs and central angles; identify and name

Essential Questions: How are segments and lines related to circles? How can properties of tangents, arcs, chords and inscribed angles be used to solve problems? What is the relationship between radii, diameters, chords and tangents?


Key Concepts

Find the circumference and area of circles

Determine measures of central and inscribed angles and their intercepted arcs.

Identify and name parts of a circle

Use the diagram below to find the intercepted arc or inscribed angle

Examples

C

100°

E

A

D

B

70°

37°

146

Solve problems by applying the relationship between radii, diameters, chords, and tangents.

Cabri Jr. Discovery activities www.timath.com • Relationship between

circumference and diameter

• Measure angles and related arc

Key Vocabulary chord secant tangent minor arc major arc arc length inscribed angle intercepted arc diameter radius semi circle circumscribed point of tangency circumference area inscribed polygon locus

The diagram below show special segments and lines of a circle.

J

G

F

E

DB

A

CI

H



Two-Dimensional Figures

Enduring Understandings: Analyze characteristics and properties of two-geometric shapes and develop mathematical arguments about the relationships of geometric figures in the real world.

Compare and contrast polygons two-dimensional figures.

Identify and classify polygons using manipulatives and create two-dimensional figures.

Identify and describe properties of a circle, kite, trapezoid, parallelogram, rectangle, square, and rhombus.

Essential Questions: What attributes define a polygon? What is the relationship between the angles and the number of sides in a polygon? How do you calculate the measure of the interior and exterior angles of a polygon? How do you measure area and perimeter of polygons?

Key Vocabulary equilateral polygon equiangular polygon regular polygon concave/convex apothem area perimeter circumference circle center of the circle radius diameter sector area central angle interior angle exterior angle height of trapezoid midsegment of trapezoid Polygon Names triangle quadrilateral parallelogram rhombus square rectangle trapezoid kite pentagon hexagon heptagon octagon decagon n-gon

Find the area of the parallelogram.

A. 16 m2 C. 36 m2 B. 81 m2 D. 13 m2

Identify and describe polygons (concave, convex, regular, pentagon, hexagon, and n-gonal, circles and sector areas).

Resources Cabri Jr. Sketchpad TI Calculator Power Point Textbook Resources http://classzone.com

Examples

Key Concepts

Identify how many vertices, sides, edges a polygon has.

Use congruent relationships of two-dimensional figures to determine unknown values, such as angles, side lengths, perimeter or circumference and areas.

Apply the interior and exterior angle sum of convex polygons to solve problems.

Find the area of a circle with a radius of 10 millimeters. Round to the nearest tenth. A. 31.4 mm2 C. 314.2 mm2

B. 314.1 mm2 D. 3,142 mm2

Find the area of the polygon.

A. 42 ft2 C. 30 ft2 B. 60 ft2 D. 36 ft2

Cabri Jr. Discovery activities www.timath.com • Volume • Area of the circle



Three-Dimensional Figures

Enduring Understandings: Analyze characteristics and properties of three-dimensional geometric shapes and develop mathematical arguments about geometric relationships in the real world.

Identify how many vertices, sides, edges, and/or faces of a three-dimensional figure.

Apply surface area and volume formulas for prisms, pyramids, cylinders, cones and spheres.

Use manipulatives to create three-dimensional figures

Draw three-dimensional figures with appropriate labels and make a three-dimensional model from a net.

Essential Questions: How do you determine which formula to use with a given solid? How does changing a dimension affect the volume? How do I find lateral areas of prisms and cylinders? How do I find lateral areas of cones? How do I identify and use three-dimensional figures and use nets to draw them?

Key Vocabulary polyhedron base face edge surface area lateral area slant height height prism pyramid cylinder cone sphere hemisphere net

Find the volume of the rectangular prism below.

A. 25 3cm B. 200 3cm C. 392 3cm D. 480 3cm

Resources Cabri Jr. Java Applet Sketchpad Compute TI Calculator Power Point Textbook Resources www.classzone.com

Examples

Key Concepts

Compare and contrast three-dimensional figures.

Draw, describe, and analyze solid geometry figures.

Which solid has a net like the one shown?

A. Triangular prism B. Triangular

pyramid C. Rectangular

prism D. Square pyramid

Find the surface area of the cylinder below to the nearest tenth.

E. 1256.6 2cm F. 5026.5 2cm G. 1658.8 2cm H. 2513.3 2cm

Examples



Geometry 1-2 Scope and Sequence

Quarter 1 Quarter 2 Quarter 3 Quarter 4 Points, Lines & Planes • Lines

Distance/Midpoint formula Segment addition Segment congruence Bisectors Collinear

• Angles Angle addition Angle congruence Angle bisectors

• Angle Measures Vertical Linear pair Complementary and

supplementary Obtuse, right or acute

Perpendicular & Parallel Lines • Lines and angles • Proof and perpendicular lines • Parallel lines and transversals • Proving lines are parallel • Using properties of parallel lines • Parallel & Perpendicular Lines in the

coordinate plane Logical Reasoning • Simple formal proof • Counterexample • If...then • Inductive • Deductive • Conjecture • Inverse, converse and contrapositive

Attributes & Properties of Triangles • Triangle Sum = 180 degrees • Types (obtuse, acute, scalene, right,

isosceles, equilateral, equiangular) • Angle relation to side length • Triangle inequality • Distance formula • Pythagorean theorem • Triangle congruence

SSS, SAS, ASA, AAS, HL Label triangles by

corresponding parts Construct a congruent triangle

Similarity • Triangle Similarity

SSS, SAS, AA, Label triangles by

corresponding parts • Scale factor similar Polygons

Proportions Corresponding parts

Quadrilaterals • Hierarchy of Quadrilaterals • Attributes of quad.

Probability & Statistics • Measures of center • Data collection with/without technology • Data analysis • Theoretical probability • Geometric probability • Permutation • Combinations • Network Graphs

Transformations (single and multiple) • Translations, Rotations, Reflections,

Dilations (scale factor)

Right Triangles & Trigonometry • Special Triangles

45-45-90 and 30-60-90 • Right Triangles

Pythagorean theorem Pythagorean inequality

• Trigonometry Set up basic ratios Solve using special triangles

(30/60/45)

2-3 dimensional figures • Changing dimensions • Area of circle and sector • Nets • Volume & Surface Area

Cylinders Cones Sphere Prisms Pyramids

Circles • Segments and angles formed

by circles • Arcs and chords • Secants and tangents • Central, inscribed, and

circumscribed • Degrees of arcs • Identify center and radius


Concept Unit Map Grade Level: Geometry 1 Honors

Using Points, Lines, Planes, and Angles

Key Concepts

Enduring Understanding: Recognizing the relationship of points, lines, and planes and the geometric figures they form is the foundation for applying mathematics to the real world

Identify points, lines, rays, line segments, angles, and planes in a two- and three-dimensional figure and notation.

Solve problems related to complementary, supplementary, congruent angles, bisectors, angles or segment concepts.

Solve problems created by parallel lines cut by a transversal.

Essential Question(s): What is the difference between a postulate and a theorem? How do you use slope to determine the relationship between two lines? What is the distance and midpoint between two points? How can you find the solutions to a system of equations?

Key Vocabulary: collinear/linear coplanar/plane skew point segment ray line angle segment perpendicular bisector linear pair complementary/supplementary vertical angles transversal parallel/perpendicular adjacent angles midpoint postulate theorem angle bisectors congruent segments


Identify lines, segments, points, angles, and arcs of circles.

Determine the distance and midpoint between 2 points in a coordinate system.

Construct and apply concepts of angle bisectors, perpendicular bisectors and congruent segments, in order to solve related problems.

Examples

Given: M is the midpoint of AB . Find AB. 2x+10 4x-6

B M A

Identify a line segment, ray, plane, line, a pair of skewed lines, perpendicular segments, three coplanar points, two collinear points.

F

E

D

C B

A

H

G Cabri Jr. Discovery activities www.timath.com • Angle bisector • Midpoint of a line



Perpendicular and Parallel lines

Enduring Understandings: Parallel and perpendicular lines provide the foundation for most physical structures.

Identify parallel lines, perpendicular lines, skew lines, parallel planes, and a line perpendicular to a plane.

Identify and solve problems using angles formed by perpendicular lines.

Identify congruent angles and supplementary angles formed by parallel lines and a transversal.

Determine if lines are parallel or perpendicular based on slope or angle relationships.

Essential Questions: What angle relationships are formed by parallel lines cut by a transversal? What angle relationships are formed by perpendicular lines? How does slope determine the relationship between two lines? How can angle relationships be used to determine measures of other angles?

Key Vocabulary parallel lines perpendicular lines skew lines parallel planes line perpendicular to a plane complementary angles supplementary angles transversal corresponding angles alternate interior angles alternate exterior angles same-side interior angles congruent converse theorem postulate perpendicular bisectors

Key Concepts

Apply properties, theorems, and constructions about parallel lines, perpendicular lines, and angles to prove theorems.

Resources Algebra Help http://www.algebrahelp.com/index.jsp Textbook resources http://classzone.com Geometry lesson on parallel and perp. http://www.homeschoolmath.net/teaching/g/parallel_and_perpendicular.php

Examples


110°

( )2 10x − °( )5x + °

( )2 25x + °


Cabri Jr. Discovery activities www.timath.com • Vertical and

adjacent angles • Transversals



Logical Reasoning

Enduring Understandings: Logical reasoning is essential to synthesize information, put the world in perspective, and apply knowledge.

Use inductive reasoning to find patterns and use them to make conjectures.

Prove a conjecture false by finding a counterexample.

Draw valid conclusions from conditional statements using Law of Detachment and Syllogism.

Determine triangle congruency using SSS, SAS, ASA, and AAS congruence postulates.

Essential Questions: What conjecture can be made based on an observation? How can you use a counter-example to prove a conjecture false? How can a logical conclusion be made using Law of Detachment and Syllogism? How can you use triangle congruence theorems to prove two triangles are congruent? How do you write a paragraph proof?

Key Vocabulary conjecture inductive reasoning counter-example patterns deductive reasoning conditional statement hypothesis conclusion Law of Detachment syllogism inverse converse contrapositive if-then statement reflexive property symmetric property transitive property proofs writing proofs paragraph proofs two-column proofs

Examples

Key Concepts

State the inverse, converse, and contrapositive of a given if-then statement and determine if it is true or false.

Critique inductive and deductive arguments concerning geometric ideas and relationships.

Using the Law of Syllogism, which statement follows from the pair of true statements? If I do not get paid this week, then I cannot pay my bills. If I cannot pay my bills, then the bill collectors will call.

What is the next number you expect? 45, 90, 135, 180……..

Construct a formal deductive proof.

Cabri Jr. Discovery activities www.timath.com • Conditional statements • ASA,SAS,SSS Triangle

congruence




Attributes and Properties of Triangles

Enduring Understandings: Triangles are the support system for most objects that we use on a daily basis. Are you sitting on a triangle now?

Classify triangles according to lengths of sides and angle measures.

Solve problems using triangle sum and exterior angles theorems.

Solve problems related to base angles, equilateral, and equiangular triangle theorems.

Use the Pythagorean Theorem and Distance formula to find unknown lengths.

Essential Questions: How are triangles classified? How do triangle attributes help solve problems related to angle

measures and side lengths? How are the Pythagorean Theorem and Distance formula related? What is the relationship between segments formed by the intersection of

the medians of a triangle? Can I show two triangles are congruent using AAS, ASA, SSS, SAS, or

HL?

Key Vocabulary triangle equilateral triangle, isosceles triangle, scalene triangle, right triangle, obtuse triangle, acute triangle, equiangular triangle vertex length measure corollary interior angles exterior angles base legs base angles hypotenuse Pythagorean Theorem square square root distance classify median of a triangle centroid triangle inequality coordinate proof corresponding midsegment Angle-Angle-Side Theorem

(AAS) Angle-Side-Angle Postulate

(ASA) Side-Side-Side Postulate (SSS) Side-Angle-Side Postulate (SAS) Hypotenuse-Leg Theorem (HL)

Key Concepts

Determine and solve problems using angle and side length relationships.

Solve problems using median and centroids of a triangle.

Solve problems using the triangle inequality theorem.

Identify midsegments and the points of concurrency of a triangle.

Show that two triangles are congruent using AAS and HL Theorems, and ASA, SSS, and SAS postulates.

Decide whether a triangle could have the given angle measures. Explain your reasoning. 18°, 34°, 128°

Resources Algebra Help http://www.algebrahelp.com/index.jsp Textbook resources http://classzone.com Geometry Interactive activities http://www.shodor.org/interactivate/activities/

Examples

Classify the triangle with the most specific name.

8 9

10


51°x°

102°

Cabri Jr. Discovery activities www.timath.com • Triangle Inequalities • Ratio of Right Triangles • Distance in the coordinate

plane


Concept Unit Map

Grade Level: Geometry 1 Honors

Quadrilaterals

Enduring Understandings: The building blocks of society are based on polygons, especially four sided objects called quadrilaterals.

Identify and classify polygons.

Identify properties of parallelograms.

Prove that a given quadrilateral is a parallelogram.

Solve problems using properties of trapezoids.

Essential Questions: What information on quadrilaterals are used to identify and classify polygons? Given some measures of quadrilaterals, how do I find missing measures of quadrilaterals? Can I apply properties of parallelograms, trapezoids and special quadrilaterals to problem-solve? Can I show that a quadrilateral is a Parallelogram? How do I justify the classification of a quadrilateral?

Key Vocabulary polygon side of a polygon vertex of a polygon diagonal of a polygon parallelogram rhombus rectangle square trapezoid kite bases, legs, and base angles of a trapezoid isosceles trapezoid midsegment of a trapezoid convex concave equilateral equiangular

Solve problems using properties of special quadrilaterals including rhombuses, rectangles, and squares.

Justify the classification of a quadrilateral using coordinate geometry.

Find the angle measures of quadrilaterals.

Key Concepts


Examples Classify the figure in as many way as possible Find the value of x and y.

• y x 14

9

Cabri Jr. Discovery activities www.timath.com • Properties of

parallelograms • Angles in a quadrilateral

Trapezoid JKLM

(11x)° 89°

(4y + 3)°

M L

K J 103°

x = y =

JKLM is a rhombus. Find the value of x.

50°

5x°

J

K

L

M


Concept Unit Map


Transformations

Enduring Understandings: A clear understanding of transformations allows students to visualize the movement of objects in three-dimensional space and on a plane.

Essential Questions: What does the word transformations means? What is the relationship between transformations and symmetry? Can you use more than one transformation to describe the relationship?

Key Vocabulary transformation translation image pre-image reflection line of symmetry dilation glide reduction enlargement rotation center of rotation angle of rotation rotational symmetry tessellation isometry plan figure scale factor vector frieze pattern

Resources Cabri Jr. Java Applet Sketchpad Computer TI Calculator Power Point Textbook Resources www.classzone.com

Examples

RRoottaattiioonn 4455°°

RReefflleeccttiioonn

TTrraannssllaattiioonn

DDiillaattiioonn

TRANSFORMATIONS

Determine the effects of a single transformation on linear or area measurements of a planar geometric figure.

Cabri Jr. Discovery activities www.timath.com • Students can use the

transformation tool on Cabri Jr. to explore translation, reflections, rotation or dilations

Sketch and identify the properties of a planar figure that is the result of two or more transformations

Determine the new coordinates of a point when a single transformation is performed on a planar geometric figure.

Determine whether a given pair of figures on a coordinate plane represents a translation, reflection, rotation, or dilation.

Classify transformations based on whether they produce congruent or similar figures

Discover the line of reflection, the center of rotation, and the center of dilation.

Key Concepts



Probability

Enduring Understandings: Understanding theoretical probability makes you better informed when making future decisions.

Determine the theoretical probability of events, estimate probabilities using experiments, and compare the two.

Use concepts and formulas of area to calculate geometric probabilities.

Determine the number of possible outcomes of an event.

Make predictions and solve problems based on theoretical probability models.

Essential Questions: How do you determine the number of possible outcomes of an event? What is the difference between theoretical and experimental probability? How can you determine the number of arrangements that can be made given an event? How can you determine the number of combinations in an event? How can you use theoretical probability to make predictions?

Key Vocabulary outcome theoretical probability experimental probability event counting principle factorial notation arrangements combinations permutations measures of centers biased vs. unbiased sample size

Apply the addition and multiplication principles of counting, and represent these principles algebraically using factorial notation. Teacher A would like to select 3

students from a group of 10 to form a team. Teacher B wants to pick a President, Vice President and a Secretary from a group of 10 students. Teacher A thinks their way of selecting students provides more possibilities than Teacher B. Which teacher is correct and why?

Resources Algebra Help http://www.algebrahelp.com/index.jsp Textbook resources http://classzone.com Combinations and permutations calculator http://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html

Examples Mynda was playing a game at the state fair. She had to throw darts in the shaded area to win a prize. If the length of each side of the outside square is 6 feet, what is the probability that she would win a prize?

A. 13

B. 14

C. 12

D. 712

3 ft

4 ft

• Use simulators to generate experimental probability data

• Compute the number of possible arrangement with permutations and combinations


Key Concepts



Data Analysis

Enduring Understandings: Data analysis is used to make important decisions in all facets of our lives, including education and corporate America.

Draw inferences about data sets, from lists, tables, matrices, and plots.

Organize collected data into an appropriate graphical representation.

Display data, make predictions and observations.

Make inferences by comparing data sets.

Essential Questions: Can I draw inferences about data sets from lists, tables, matrices and

plots? Can I organize collected data into an appropriate graphical

representation? Can I make inferences by comparing data sets? Can I identify misrepresentations and distortions in displays of data?

Key Vocabulary measures of center mean, median, mode, range collect data data analysis frequency tables matrices lists make predictions and observations make inferences interval sample population misrepresentation and distortions of data display simple experiments network graphs adjacency matrices Vertex, edges

Key Concepts

Determine the most appropriate measure of center for a given situation.

Identify misrepresentations and distortions in displays of data.

Understand and design simple experiments.

Resources Textbook resources http://classzone.com Comparing data activity http://www.northcanton.sparcc.org/~technology/excel/files/comparing_data.html

ExamplesYou want to find the average ages of each student in your PE class. 1). What measure of center would you use? 2). Which data representation would be the best visual represenation?

• Enter data sets into lists to create graphical representations, then use 1-VAR Stats under LISTS to compute the measures of central tendency


U of A hosts football camps for HS athletes. Teams play a round-robin tournament in the morning to determine who qualifies for the final elimination round. The matrix below represents the results of a morning round-robin tournament. A “1” indicates a win for the team in that row. A “0” indicates a loss for the team in that row. Team

W Team X Team Y Team Z

Team W - 0 1 0 Team X 1 - 0 0 Team Y 0 1 - 1 Team Z 1 1 0 - If the final round consists of two teams with the most round-robin wins, then which two teams qualify for the final round? A) W and X B) W and Z C) X and Y D) Y and Z

Solve network problems using graphs and matrices


Concept Unit Map


Similarity

Enduring Understandings: Properties of similar shapes can be used to solve problems and create and critique inductive and deductive arguments.

Use ratios and proportions to problem sove.

Use the Triangle Proportionality Theorem to problem solve. Essential Questions:

How can I use ratios and proportions to solve similiariy problems? What makes polygons similar? Can I apply the Triangle Proportionality Theorem ? Can I show two triangles are similar using AA, SSS, or SAS postulates?

Key Vocabulary ratio proportion cross product property similar polygons scale factor midsegment of a triangle similarity triangle proportionality theorem the midsegment theorem enlargement reduction means extremes geometric mean dilation image pre-image transformation Angle-Angle (AA) Similarity Theorem Side-Side-Side (SSS) Similarity

Theorem Side-Angle-Side (SAS)

Similarity Theorem Proportionality Theorems Geometric mean

Key Concepts

Identify similar polygons in order to problem solve.

Show that two triangles are similar using AA, SSS, and SAS postulates.

Compare perimeter and area of similar figures.

Use proportionality theorems to solve problems involving segment length calculations and parallel lines.

Resources Textbook resources http://classzone.com similarity lesson http://www.gogeometry.com/geometry/similarity_ratio_proportion_index.html

Examples

Cabri Jr. Discovery activities www.timath.com • Constructing

similar triangles

A). 14 B). 7 C). 127

D). 1412

Solve the proportion 5 72x x=

−

Write the ratio of the values of 4 nickels to the values of 10 dimes?

A). 15

B). 25

C). 410

D). 104

28 34

40

What value of x will make the two triangles similar? x

51

42


Concept Unit Map


Right Triangles and Trigonometry

Key Concepts

Enduring Understandings: Demonstrate understanding of Pythagorean Theorem, sine, cosine, and tangents to solve problems involving right triangles and use them in real life situations.

Solve problems using right triangle trigonometry

Investigate the fundamental concepts behind trigonometry; three basic trig functions, and how to determine which trig function to use.

Essential Questions: Could we use Pythagorean Theorem in other than right triangles? How do you determine if the right triangle has a hypotenuse? How do you determine the magnitude and direction of a vector? How do you add vectors together?

Key Vocabulary radical radicand 45°-45°-90° triangle 30°-60°-90° triangle trigonometric ratio leg opposite an angle leg adjacent to an angle tangent sine cosine magnitude of a vector direction of a vector sum of two vectors

If we know the two legs of a right triangle, we can solve for the hypotenuse using the formula:

where a and b are the lengths of the two legs of the triangle, and h is the hypotenuse

Resources Cabri Jr. Java Applet Sketchpad Computer TI Calculator Power Point Textbook Resources www.classzone.com

Examples Pythagorean Theorem

Relative to angle A, this is how the sides of a right triangle would be labeled

Know how to find the magnitude and direction of a vector.

Cabri Jr. Discovery activities www.timath.com • Investigating special

triangles • Use the calculator to find

sine, cosine, or tangents

Use SOHCAHTOA to memorize the three main trigonometric functions.

SOH → oppositehypotenuse

sin x =

CAH → adjacenthypotenuse

cos x =

TOA → oppositeadjacent

tan x =

Know that recognizing special right triangles in geometry can help you to problem solve. A special right triangle is a right triangle whose sides are in a particular ratio. ( 30 60 90, , ) & ( 45 45 90, , )


Concept Unit Map


Circles

Enduring Understandings: Circular objects are abundant in our universe. They are formed by nature; specifically by the force of gravity.

Use properties of tangents to solve problems.

Use properties of arcs and central angles to solve problems; Identify and name.

Determine how inscribed angles are related to central angles.

Essential Questions: Can I identify segments and lines related to circles? Can I use properties of a tangents, arcs, and chords to problem solve with a circle? Can I use properties of inscribed angles to problem solve? How do you find the measures of angles formed by chords, tangents and secants?

Key Vocabulary chord secant tangent minor arc major arc arc length inscribed angle intercepted arc diameter radius semi circle circumscribed point of tangency circumference area inscribed polygon locus


Key Concepts

Find the circumference and area of circles.

Use properties of inscribed angles and polygons.

Identify and name parts of a circle.

Use properties of arcs & chords of circles to solve problems.

Use the diagram below to find the intercepted arc or inscribed angle

Examples

C

100°

E

A

D

B

70°

37°

146

Find the measures of angles formed by chords, tangents and secants?

Cabri Jr. Discovery activities www.timath.com • Relationship between

circumference and diameter

• Measure angles and related arc

The diagram below show special segments and lines of a circle.

J

G

F

E

DB

A

CI

H



Two-Dimensional Figures

Enduring Understandings: Analyze characteristics and properties of two-geometric shapes and develop mathematical arguments about the relationships of geometric figures in the real world.

Compare and contrast two- dimensional figures.

Identify and classify polygons using manipulatives and create two-dimensional figures.

Identify and describe properties of a circle, kite, trapezoid, parallelogram, rectangle, square, and rhombus.

Essential Questions: What attributes define a polygon? What is the relationship between the angles and the number of sides in a polygon? How do you calculate the measure of the interior and exterior angles of a polygon? How do you measure area and perimeter?

Key Vocabulary equilateral polygon equiangular polygon regular polygon concave/convex apothem area perimeter circumference circle center of the circle radius diameter sector area central angle interior angle exterior angle height of trapezoid midsegment of trapezoid Polygon names triangle quadrilateral parallelogram rhombus square rectangle trapezoid kite pentagon hexagon heptagon octagon decagon n-gon Find the area of the parallelogram.

a). 16 m2 b). 36 m2 c). 81 m2 d). 13 m2

Identify and describe polygons (concave, convex, regular, pentagon, hexagon, and n-gonal, circles and sector areas.

Resources Cabri Jr. Sketchpad Computer TI Calculator Power Point Textbook Resources www.classzone.com

Examples

Key Concepts

Identify how many vertices, sides, and edges of a polygon.

Use congruent relationships of 2 two-dimensional figures to determine unknown values, such as angles, side lengths, perimeter or circumference and areas.

Apply the interior and exterior angle sum of convex polygons to solve problems.

Find the area of a circle with a radius of 10 millimeters. Round to the nearest tenth. a) 31.4 mm2 b) 314.2 mm2

c) 314.1 mm2 d) 3,142 mm2

Find the area of the figure.

a). 42 ft2 b). 30 ft2 c). 60 ft2 d). 36 ft2


Find the area of a circle and the area of a sector of a circle



Three-Dimensional Figures

Enduring Understandings: Analyze characteristics and properties of three-dimensional geometric shapes and develop mathematical arguments about geometric relationships in the real world.

Essential Questions: How do you determine which formula to use with a given solid? How does changing a dimension affect the volume? How do I find lateral areas of prisms and cylinders? How do I find lateral areas of cones? How do I identify and use three-dimensional figures and use nets to draw them?

Key Vocabulary polyhedron base face edge surface area lateral area slant height height prism pyramid cylinder cone sphere hemisphere net

Key Concepts

Find the volume of the rectangular prism below.

A. 25 cm3 B. 200 cm3 C. 392 cm3 D. 480 cm3

Resources Cabri Jr. Java Applet Sketchpad Compute TI Calculator Power Point Textbook Resources www.classzone.com

Examples Which solid has a net like the one shown?

A. triangular prism B. triangular

pyramid C. rectangular

prism D. square pyramid

Find the surface area of the cylinder below to the nearest tenth.

E. 1256.6 cm2 F. 5026.5 cm2 G. 1658.8 cm2 H. 2513.3 cm2

Examples

Identify how many vertices, sides, edges, and/or faces of a three-dimensional figure.

Calculate the surface area and volume of three-dimensional geometric figures

Use manipulatives to create three-dimensional figures

Draw three-dimensional figures with appropriate labels and make a three-dimensional model from a net.

Compare and contrast three-dimensional figures.

Draw, describe, and analyze solid geometry figures.



Geometry 1-2 Honors Scope and Sequence

Quarter 1 Quarter 2 Quarter 3 Quarter 4 Points, Lines & Planes • Lines

Distance/midpoint formula Segment addition Segment congruence Bisectors Collinear || Lines cut by transversal o AIA/AEA/Corresponding

Angles/Consecutive Interior Angles

• Angles Angle addition, Angle

congruence & Angle bisectors • Angle Measures

Vertical & Linear pair Complementary and

supplementary Obtuse, right, or acute

Perpendicular & Parallel Lines • Lines and angles • Proof and perpendicular lines • Parallel lines and transversals • Proving lines are parallel • Using properties of parallel lines • Parallel and perpendicular Lines in the

coordinate plane Structure & Logic • Logical chain • Converse/Contrapositive/Inverse • Valid and Invalid arguments • Counterexamples • Deductive / Inductive Reasoning • Proofs with parallel lines • Proofs of congruent & similar polygons • Proofs of congruent triangles

Attributes & Properties of Triangles • Triangle Sum = 180 degrees • Types (obtuse, acute, scalene, right,

isosceles, equilateral, equiangular) • Angle relation to side length • Triangle inequality • Special segments/points in triangles

Mid-segments Altitude and median Perpendicular bisector -

circumcenter Angle bisector - incenter

• Triangle congruence SSS, SAS, ASA, AAS, HL CPCTC Label triangles by

corresponding parts Construct a congruent triangle

Quadrilaterals • Quadrilaterals

Hierarchy and attributes Midsegments of Trapezoids

Transformations (single and multiple) • Translations, Rotations, Reflections,

Dilations (scale factor) • Vectors

Probability & Statistics • Statistics

Measures of Center Misrepresentations

• Probability Counting Principle Permutations Combinations Geometric Probability

• Network Graphs Similar Triangles • Triangle Similarity

SSS, SAS, AA, Side splitting Label triangles by corresponding

parts Ratios of medians, altitudes, and

angle bisectors Geometric mean

• Similar Polygons (Proportions & Corresponding parts)

Scale factor

Right Triangles & Trigonometry • Special Triangles

30 60 90− − 45 45 90− −

• Right Triangles Pythagorean Theorem and

Inequality • Trigonometry

Set up basic ratios Solve using special triangles

(30/60/45) Law of Sines and Cosines

(excluding the ambiguous case)

Circles • Segments & Angles formed by

Circles • Secants • Tangents • Chords • Central • Inscribed/Circumscribed

• Degrees of Arcs • Square inscribed in

a circle • Circle Equations

• Identify center and radius

Two-dimensional Figures • Circles

Area and circumference Area of sectors and arc

length • Polygons

Sum of interior/exterior angles

Measure of interior/exterior angles in a regular polygon

• Area and perimeter of polygons

Irregular and regular Quadrilateral, hexagon,

triangle Three-dimensional figures • Nets • Volume and surface area

Cylinders Cones Sphere Prisms Pyramids