Math Skill Builders

Physics A First Course Math Skill Builders

Physics A First CourseTeacher Resource CD-ROM

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Math Skill Builders

Physics A First Course

The Math Skill Builders are listed in the order in which you may use them in your classroom. The chapter or chap-ters for which these skill builders best apply is written in parentheses next to each title. These skill builders were created in order to help you work with students who are concurrently taking algebra and Physics A First Course.

Decimals and Place Value (1)Slope (2)Slope from a Graph (2, 11)Order of Operations, Part 1 (3, 10, 14)Order of Operations, Part 2 (3, 10, 14)Evaluating Algebraic Expressions (3, 10, 13, 14)Inverses (3, 10, 14)Inverse Operations (3, 10, 14)Negative Exponents and Exponents of Zero (4, 8, 17, 18)Scientific Notation: Standard to Scientific (4)Scientific Notation: Scientific to Standard (4)Scientific Notation: Products and Quotients (4)Two-Dimensional Vectors (5)Pythagorean Theorem (5)Pythagorean Triples (5)Special Right Triangles (5)Measuring Angles with a Protractor (5)Constructing Angles with a Protractor (5)Functions: Conversion between Celsius and Fahrenheit (7)Using a Graphing Calculator: Conversion between Celsius and Fahrenheit (7)Ratios (8, 17)Ratio and Proportions in a Recipe (8, 17)Probability (9)Reciprocals and Negative One as an Exponent (19)Problem Solving Boxes (template for solving problems)

Name: Date:

Decimals and Place Value

The place-value chart below is to help you write and read decimals and to understand their values.

The decimal 1364.2895 is shown in the chart at the right. The chart may be extended in either direction. Notice that the columns on either side of the ones columns appear to have similar names. The columns to the right of the decimal point always end with -ths.

Notice that the decimal point separates the ones and tenths places. It is read as and.

The decimal 1,364.2895 is read as one thousand, three hundred sixty-four and two thousand eight hundred ninety-five ten-thousandths.

1. Write thirty and five hundred seventy-four thousandths as a number.

One way to think about this is to see the decimal point as splitting the number in two parts. Now read just the part of the number that comes before the word and. Write that part of the number. Then put a decimal point for the word and. Now read the right side of the number. The number must end in the last column that is named. To help do this correctly, you could draw small segments to fill in the places for the decimal part of the number. If there are not enough digits to end in the last column on the right, place as many zeros as it necessary.30. ___ ___ ___ 574 fits into the lines so no zeros are necessary.30.574

2. Write 345.029 in words.

First you write the words for the part of the number that is to the left of the decimal point. Then write and for the decimal point. Now write the number to the right of the decimal point as if it is a whole number. Finally, write the name of the column that the total number ends in. (Remember the ths at the end of the column name.)Three hundred forty-five and twenty nine thousandths

Page 2 of 2

1. Write each of the following as a decimal:

a. fourteen hundredths

b. four thousand, six hundred and twenty-five thousandths

c. one thousand and one thousandths

d. nine hundred eighty-five and sixty-three hundredths

e. eight thousand thirty-five and four tenths

2. Write each number in words:

a. 105.064

b. 23.0049

c. 36.7

d. 45.003

e. 74.998

3. In the 1996 Olympics, Michael Johnson won both the men’s 200-meter and 400-meter track competitions. These records have held through the 2000 Olympics.

a. His time for the 200-meter competition was 19.32 seconds. Write this decimal in words.

a. His time for the 400-meter competition was forty-three and forty-nine hundredths seconds. Write this as a decimal.

Name: Date:

Slope

Slope is a word used to describe the steepness of a line or the rate of change of a linear relationship. It may have a positive or negative value.

The formula for the slope of the line passing through point 1 with the coordinates (x1, y1) and point 2 with coordinates (x2, y2) is:

A line goes through the points (1, 2) and (5, 4).

To find the slope we do the following:

Although both of these divisions give the same final, positive answer, they can be interpreted in two different ways. One interpretation is that as x changes to the right 2 units then y changes up 1 unit. Another interpretation is that as x changes to the left 2 units y changes down 1 unit.

When the slope is a negative number, such as - 1/2, then the two interpretations could be: (1) that as x changes to the right 2 units then y changes down 1 unit, or (2) that as x changes to the left 2 units y changes up 1 unit.

Find the slope for the following points and then give an interpretation of the slope. The letter m is the usual variable used for slope. It probably comes from the French word monter meaning to climb.

Points Slope (m) Interpretation

Example:

(0, 5), (3, 4)

As x changes ____ unit(s) left or right, y changes _____ unit(s) up or down.

As x changes 3 units to the left, y changes 1 unit up.

1. (-2, 3), (2, 5) m =

2. (0, 0), (-3, 7) m =

3. (-10, -8), (-3, -4) m =

4. (7, 0), (0,7) m =

5. (3.5, 1), (0.5, 0) m =

slope change in ychange in x----------------------------

y2 y1–x2 x1–---------------- rise

run-------- vertical change

horizontal change------------------------------------------= = = =

4 2–5 1–------------ 2

4--- 1

2---= = OR 2 4–

1 5–------------ 2–

4–------ 1–

2–------ 1

2---= = =

m 13–

------=

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Slope from a Graph

To determine the slope of a line choose two points on the line. Then count how many steps up or down you must move to be on the same horizontal line as your second point. Multiply this number by the scale factor.

Put the result along with the positive or negative sign in the numerator of your slope ratio if the scale is one. Then count how many steps you must move right or left to land on your second point. Multiply the number of steps by the scale factor. Place the results in the denominator of your slope ratio.

Find the slope of the line in each of the following graphs:

A The chosen points for the example are (0, 0) and (3, 9). (There are many choices for this graph, but only one slope. If you have the point (0, 0), you should choose it as one of your points.)

It takes 9 vertical steps to move from (0, 0) to (0, 9). Put a 9 in the numerator of your slope ratio (or put 9 – 0). Then count the number of steps to move from (0, 9) to (3, 9). This is your denominator of your slope ratio. Again, you can do this by subtraction (3 - 0).

B The two points that have been chosen for this example are (0, 24) and (6, 15). Be careful of the scales on each of the axes.

It takes 3 vertical steps to go from (0, 24) to (0, 15). But each of these steps has a scale of 3. So you put a -9 into the numerator of the slope ratio. It is negative because you are moving down from one point to the other. Then count the steps over to (6, 15). There are 3 steps but each counts for 2 so you put a 6 into the denominator of the slope ratio.

Graph #1: Graph #2:

9 3 3 1

m = =

9 3 6 2

m − −= =

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Graph #3: Graph #4:

Graph #5: Graph #6:

Graph #7: Graph #8:

Graph #9: Graph #10:

Name: Date:

Order of Operations, Part 1In arithmetic there are four operations: multiplication, division, addition, and subtraction. If you have an expression that has only these symbols, then the rule to evaluate them is to do multiplication or division moving from left to right. When this is completed, go back to the left and do the addition or subtraction moving left to right.

Example 1:

Example 2:

Evaluate the following expressions:

1. 25 – 5 × 3

2. 15 + 5 × 7

3. 5 + 0.5 × 7

4. 18 – 3 × 2 – 5

5. 18 × 3 – 2 × 5

6. 18 – 5 × 2 – 3

7. 3 × 2 + 5 × 2 + 6

8. 7 × 2 + 3 × 2 + 6 × 2

9. 4 × 2 + 5 × 2 + 6

10. 7 + 3 × 2 – 1

11. 8 – 4 × 2 + 7

12. 25 – 6 + 7 – 2

Step 1 3 + 2 × 4 – 5 = 3 + 8 – 5 Multiply 2 and 4.

Step 2 3 + 8 – 5 = 11 - 5 Do the addition and subtraction moving left to right—add 3 and 8.

Step 3 11 - 5 = 6 Then, subtract 5 from 11.

Step 1 =Multiply and 48. Multiply 6 and 2.

Step 2 = 4 Subtract 12 from 16.

13--- 48 6 2×–× 16 12– 1

3---

16 12–

Page 2 of 2

13. 1/2 × 20 + 4

14. 36 ÷ 3 × 2 – 5

15. 1/4 × 8 + 9 – 1

16. A large computer store has certain software on sale at 4 for $25.00 with a limit of 4 at the sale price. Additional software is available at the regular price of $8.95 each.

a. Write an expression you could use to find the cost of 6 software packages.

b. How much would 6 software packages cost?

17. Valerie is signing up with a new internet provider. The service costs $5.99 a month, which includes 100 hours of access. If she is online for more than 100 hours she must pay an additional $0.95 per hour.

a. Suppose Valerie is online for 120 hours the first month. Write an expression that represents what Valerie must pay for the month.

b. Now evaluate the cost.

18. Most bacteria reproduce by dividing into identical cells. This process is called binary fission. A certain type of bacteria can double its numbers every 20 minutes. Suppose 150 of these cells are in one culture dish and 250 of the cells are in another culture dish.

a. Write an expression that shows the total number of bacteria cells in both dishes after 20 minutes.

b. Now, evaluate the expression to find out how many bacteria are in both dishes after 20 minutes.

19. Jamal and Alexandria are selling tickets for their school talent show. Floor seats cost $7.50 and balcony seats cost $5.00. Alexandria sells 60 floor seats and 70 balcony seats. Jamal sells 50 floor seats and 90 balcony seats.

a. Write an expression to show how much money Alexandria and Jamal have collected for tickets.

b. Evaluate the expression to determine how much they collected.

Name: Date:

Order of Operations, Part 2

In evaluating expressions sometimes grouping symbols such as parentheses, square parentheses, set notation, and fraction bars are used. If these are present then the operations within them are done first. Expressions may also be raised to a power. General steps for evaluating expressions with grouping symbols and values raised to a power are as follows:

Evaluate these expressions using the steps for working with grouping symbols and values raised to a power.

1. (6 – 4) · 3 =

2. (8 + 5) · 2 =

3. 10 + 5 × 6 =

Step 1 Evaluate expressions inside grouping symbols. If grouping symbols enclose grouping symbols start at the innermost part of the expression and work out.

Step 2 Evaluate all power expressions.Step 3 Do all multiplication or division from left to right. Note: Both “·” and “×” can be used to represent

“multiplied by.”Step 4 Do all additions or subtractions from left to right.

A Step 1 2(8) + 5(4 + 3) = 2(8) + 5(7) Add within grouping symbol.

Step 2 2(8) + 5(7) = There are no power expressions to evaluate.

Step 3 2(8) + 5(7) = 16 + 35 This expression means: 2 × 8 + 5 × 7.Multiply moving left to right.

Step 4 16 + 35 = 41 Add 16 and 35.

BStep 1

This means (5 + 42) ÷ (22 × 3).

Step 2= Evaluate the power number in the numerator and denominator.

Step 3= Simplify the numerator and denominator.

Step 4= Reduce the fraction.

2

2

5 + 42 3⋅

2

2

5 + 42 3⋅

5 + 164 3⋅

2112

74

Page 2 of 2

4. 10(5 + 6) =

5. 60 – 12 ÷ 4 =

6. 250 ÷ (5[(3 · 7) + 4]) =

7. 28 ÷ 4 · 2 – 32 =

8. 64 ÷ (2 · 4) + 2 =

9. (6 + 5)(4 + 3) = =

10. =

11. =

12. =

13. =

14. 390 ÷ [5(7 + 6)] =

15. 15 ÷ 3 · 5 – 42 =

16. Use grouping symbols so that the equation is true. If one grouping symbol needs to enclose another, use brackets to enclose parentheses.

a. 10 · 5 + 4 = 90

b. 6 + 5 – 2 · 8 = 30

c. 3 + 4 · 5 – 1 = 28

d. 20 ÷ 2 · 5 + 8 = 10

e. 20 ÷ 2 · 5 + 8 = 58

f. 20 ÷ 2 · 5 + 8 = 130

17. Jared is signing up with a new cable company. The service costs $50.00 a month which includes 100 hours of access. If he is online for more than 100 hours he must pay an additional $3.95 per hour. Suppose Jared is online for 110 hours the first month.

a. What is the expression that represents what Jared must pay for the first month?

b. How much will the bill be for the first month?

15 + 60 30 5−

2

10(3) + 2(3)3 3−

36 9 425 3 6 9

÷ ÷⋅ + +

2 28 2(2 8) + 4

−⋅

Name: Date:

Evaluating Algebraic ExpressionsAlgebraic expressions often contain more than one operation. Algebraic expressions can be evaluated when the values of the variables are known.

Evaluate a2 – (b2 + 4c) when a = 8, b = 4, and c = 5.

1. when m = 0.30 grams and V = 0.040 cm3

2. when v2 = 4 m/sec, v1 = 1 m/sec and t = 1 sec

3. πr2 when π = 3.14 and r = 5 cm

4. 2(L + W) when L = 8 m and W = 12 m

5. when (x1, y1) = (4, 4) and (x2, y2) = (0, 0)

6. The Pyramid of the Sun in Mexico is the third largest pyramid in the world. It stands 60 meters high with a base of 200 meters2. The volume of any pyramid is one-third the product of the area of the base (B) and its height (h).

a. Write an expression that represents the volume (V) of a pyramid.

b. Evaluate your expression using 200 m2 for B and 60 m for h to find the volume of this pyramid.

Step 1 Replace the variables with their values.Step 2 Find the value of the numerical expression using the rules for the order of operations.

Step 1 a2 – (b2 + 4c) = 82 – (42 + 4 · 5) Replace variables.

Step 2 82 – (42 + 4 · 5) = 82 – (16 + 4 · 5) Evaluate 42.

82 – (16 + 4 · 5) = 82 – (16 + 20) Multiply 4 and 5.

82 – (16 + 20) = 82 – (36) Add 16 and 20.

82 – (36) = 64 – (36) Evaluate 82.64 – (36) = 28 Subtract 36 from 64.

mV----

v2 v1–t

----------------

y2 y1–x2 x1–----------------

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Inverses

Every positive rational number can be paired with a negative rational number. These pairs are called opposites. A number and its opposite are additive inverses of each other. When you add two opposites, the sum is always zero.

Two numbers whose product is 1 are called multiplicative inverses or reciprocals.

Zero has no reciprocal because any number times zero equals zero.

Additive inverse examples:

+5 + (-5) = 0 –4 +(+4) = 0

Multiplicative inverse examples:2/3 × 3/2 = 1 8 × 1/8 = 1

2 3/5 × 5/13 = 1 1/4 × 4 = 1

Give the additive inverse for the following:

Give the multiplicative inverse for the following:

1. 29 5. –0.75

2. –101 6. 24.7

3. 5/9 7. 0

4. –4/3 8. – 4.32

1. 5/9 5. -10

2. 2 2/3 6. 0

3. -3/4 7. –1 1/2

4. 0.06 8. 1/0.5

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Inverse OperationsAn inverse operation “undoes” the original operations. Addition and subtraction are the inverse operations of each other. Division and multiplication are the inverse operations of each other.

Example 1:

Example 2:

Example 3:

Example 4:

Find the value of the variable using inverse operations.

1. a – 12 = 12

2. 3a – 12 = 12

3. 3a + 8 = 32

4. b + 7 = 21

5. 2b + 7 = 21

6. 2b – 7 = 17

Step 1 x + 4 = 10 Subtraction is the inverse operation of addition.Step 2 (x + 4) – 4 = 10 – 4 Subtract 4 from both sides of the equation.Step 3 x = 6

Step 1 y – 7 = 12 Addition is the inverse operation of subtraction.Step 2 (y – 7) + 7 = 12 + 7 Add 7 to both sides of the equation.Step 3 y = 19

Step 1 3z = 4 Division is the inverse operation of multiplication.Step 2 Divide both sides of the equation by 3.

Step 3 z = 4

Step 1 Multiplication is the inverse operation of division.

Step 2 Multiply both sides of the equation by 5.

Step 3 t = 5

3z3----- 12

3------=

t5--- 25=

t5--- 5× 25 5×=

Page 2 of 2

7. 3c = 24

8. 3c – 5 = 28

9. 3c + 5 = 23

10.

11.

12.

13.

14.

15.

d4--- 16=

d4--- 8+ 16=

d4--- 4– 16=

59---f 45=

59---f 36– 9=

59---f 36+ 117=

Name: Date:

Negative Exponents and Exponents of Zero

For any nonzero value of b and any value of n,

Negative exponents are not restricted to -1. To find a number or expression to a negative power rewrite the number as a fraction and then invert the numerator and the denominator. Then apply the power.

1. 5-2 =

2. 109-1 =

3. (-1,000,000)0 =

4. (-30)-2 =

5. =

6. =

7. 5–3 =

8. =

b-n =

= bn

b0 = 1 [00 is undefined.]

A.= = =

We invert the two in the numerator with the three in the denominator and then raise each of these to the third power.

B.= =

In this example the exponent only applies to the numerator. The 2–3 is moved into the denominator and becomes 23. A one must be placed into the numerator.

C. 1,0000 = 1 We say that this is true because of the definition of b0.

D.= 1

We say that this is true because of the definition of b0.

1nb

1-nb

32 3

−⎛ ⎞⎜ ⎟⎝ ⎠

33 2

⎛ ⎞⎜ ⎟⎝ ⎠

3

3

32

278

-3 23 3

13 2×

124

0-1 24

⎛ ⎞⎜ ⎟⎝ ⎠

0-2 9

⎛ ⎞⎜ ⎟⎝ ⎠

-22 -3

⎛ ⎞⎜ ⎟⎝ ⎠

5

10(2)

Page 2 of 2

9. =

10. =

11. where x ≠ 0

12. where a and b ≠ 0

13. =

14. where c and d ≠ 0

15. where m and n ≠ 0

16. =

17. =

18. =

19. 36(2–3)(3–2) =

20. =

5

10 (-2)

5

5

2 2

5

5

3 3

xx

2 3

3

a bab

−

−

3

3

77

−

−

2 4

2

82c dc d

−

−

2 3

3 2

0.5m nm n

− −

− −

-2

4

10 10−

4

1000 10−

2

50 5

-1

3

1 52 8−

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

Name: Date:

Scientific Notation: Standard to Scientific

Do you know what 300,000,000 m/sec is the measure of? It’s the speed of light.

Do you recognize what 0.000 000 000 753 kilograms is the measure of? This is the mass of a dust particle.

Scientists have developed a shorthand method for writing very large numbers. This method is called scientific notation. A number is expressed in scientific notation when it is written as a product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10.

The form for scientific notation is written as a × 10n, where 1 ≤ a < 10 and n is an integer.

Example 1:

Express 28,500,000 in scientific notation.

28,500,000 = 2.85 × 107

The decimal point moved 7 places to the left, so the point is between the 2 and the 8. Since 28,500,000 > 1, the exponent is positive. When the decimal point moves left the exponent is positive.

Example 2:

Express 0.0000432 in scientific notation.

0.0000432 = 4.32 × 10-5

The decimal point moved 5 places to the right, so the point is between the 4 and the 3. Since 0.0000432 < 1, the exponent is negative. When the decimal point moves right the exponent is negative.

Example 3:

Express each of the numbers in scientific notation.

0.000781 = 0007.81 × 10n Move the decimal point 4 places to the right.0.000781 = 7.81 × 10-4

21,845,000 = 2.1845 × 10n Move the decimal point 7 places to the left.21,845,000 = 2.1845 × 107

Remember if the number you start with is larger than 1 than you will get a positive exponent in your answer.

Page 2 of 2

1. Convert the number from standard notation to scientific notation:

2. Write the numbers in the following sentences in scientific notation:

a. The national debt in 2000 was about $5,670,000,000,000.

b. In 2000, the U.S. population was 281,000,000.

c. Earth’s crust contains approximately 120 trillion metric tons of gold (120 trillion = 120,000,000,000,000).

d. The mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 91 kilograms.

e. The usual growth rate of human hair is 0.00033 meters per day.

f. The sun burns about 4,400,000 tons of hydrogen per second.

g. In 1995 the population of Iran was about 65,100,000.

h. In the middle layer of the sun's atmosphere, called the chromosphere, the temperature averages 27,800°C.

i. There are approximately 200,000,000,000 stars in the Androeda Galaxy.

j. Alex Rodriguez signed a contract with the Texas Rangers in 2000 that guarantees him $25,200,000 a year for l0 seasons.

k. The Great Pyramid of Giza stands about 137 meters high.

l. A normal, healthy body temperature for a human being is 98.6°F.

Standard Notation Scientific Notation

a. 0.0453

b. 18,700,000.0

c. 0.257000

d. 999.0

e. 264,000

f. 761,000,000

g. 1,030

h. 0.00120

i. 0.03040

j. 0.000 000 000 000 000 052

k. 42,000,000,000,000

l. 7,650,000

m. 0.000999

Name: Date:

Scientific Notation: Scientific to StandardIn this skill sheet, you will practice converting numbers from scientific notation to standard (decimal) numbers.

Example 1:

Express 3.75 × 104 in standard notation.

Example 2:

Express 1.05 × 10-5 in standard notation.

Express the number in each statement in standard notation.

1. An electron has a negative charge of 1.6 × 10-19 coulombs.

2. There are approximately 5.58 × 1021 atoms in a gram of silver.

3. Americans make almost 2 × 1010 phone calls each day.

4. The moon’s average distance from Earth is 2.39 × 105 miles.

5. The mass of a proton is about 1.67 × 10-27 kilograms.

6. 1.5 × 108 km is the approximate distance from Earth to the sun.

7. 9.29 × 107 miles is the approximate distance from Earth to the Sun.

8. 5 × 10-4 inches is the thickness of a piece of paper.

9. In 1995 the population of the United States was about 2.63 × 108.

10. In 1995 the population of China was about 1.22 × 109.

11. One millimeter equals 1 × 10-3 meters.

12. The speed of sound in air is about 3.4 × 102 m/sec.

3.75 × 104 = 3.75 × 10,000 A positive exponent means the power of ten is greater than one.

= 37,500 Move the decimal 4 places to the right. Add zeros as place holders.

1.05 × 10-5 = A negative exponent means the power of ten is less than one.

= 1.05 × 0.00001= 0.000 010 5 Move decimal 5 places to the left.

Add zeros as place holders.

1.05 1105--------×

Name: Date:

Scientific Notation: Products and QuotientsYou can use properties of powers to compute numbers written in scientific notation.

Example 1:

Evaluate (5.9 × 103)(3 × 10-5). Express the result in scientific and standard notation.

The solution is 1.77 × 10-1 or 0.177.

Example 2:

Evaluate . Express the results in scientific and standard notation.

The solution is 3.45 × 102 or 345.

Evaluate. Express each result in scientific and standard notation.

1.

2.

3. (2.4 × 10-2)(3.0 × 102)

(5.9 × 103)(3 × 10-5) = (5.9 × 3)(103 × 10-5) Associative and communicative properties

= 17.7 × 10-2 Compute the product of factors and the product of powers.

= (1.77 × 101) × 10-2 17.7 = 1.77 × 101

= 1.77 × (101 × 10-2) Associative property

= 1.77 × 10-1 Product of powers

= Associative property

= 0.345 × 103 Quotient of powers (not in scientific notation)

= (3.45 × 10-1) × 103 0.345 = 3.45 × 10-1

= 3.45 × (10-1 × 103) Associative property

= 3.45 × 102

2.45 108×

7.1 105×-------------------------

2.4495 108×

7.1 105×------------------------------- 2.45

7.1----------⎝ ⎠

⎛ ⎞ 108

105--------

⎝ ⎠⎜ ⎟⎛ ⎞

2.5 104×

2 102×----------------------

7 10 12–×

2 10 15–×----------------------

Page 2 of 2

4.

5. (6.60 × 105)(3.10 × 102)

6.

7. (9.1 × 105)(3.1 × 10-4)

8.

9.

10. (4.5 × 10-6)(2.25 × 10-2)

11.

12. Human red blood cells carry oxygen from one place to another in your body. A cubic millimeter of human blood contains about 5 × 106 red blood cells. An adult human body may contain about 5 × 106 cubic millimeters of blood. About how many red blood cells does an adult human body contain?

13. A space probe that is 6.4 × 1012 meters away from Earth sends signals to NASA. If the radio signals travel at the speed of light (3 × 108 m/sec) how long will it take the signals to reach NASA?

14. The minimum distance from Earth to the moon is approximately 2.26 × 105 miles. There are approximately 6.34 × 104 inches in one mile. What is the minimum distance from Earth to the moon in inches?

15. The population of Arizona is about 4.78 × 106. The land area of Arizona is about 1.14 × 105 square miles. What is the population density per square mile?

16. During the year 2000, 1.65 billion credit cards were in use in the United States. During that same year, $1.54 trillion was charged to these cards. ( Hint: 1 trillion = 1 × 1012).

a. Express each of these values in standard and then scientific notation.

b. Find the average amount charged per credit card.

2.35 10 8–×

2.5 10 12–×---------------------------

1.35 108×

2.5 1010×-------------------------

2.2 10 12–×

1.1 10 14–×--------------------------

6 10 4–×

1.5 102×----------------------

5 104×

2.5 10 4–×------------------------

Name: Date:

Two-Dimensional VectorsDefinitions

A vector is a directed line segment that represents a vector quantity.

A scalar is a quantity such as time, speed, or volume that has only magnitude but no direction.

A vector quantity is a quantity such as force, velocity, or displacement that has both magnitude (length) and direction.

The tail of a vector is the point where it begins. The head of a vector is the point where it ends. An arrowhead is drawn at the head of a vector.

(Note: A vector is not a ray. It has two endpoints.)

Two vectors are equal if they have the same magnitude and the same direction. So you may translate a vector without changing it, but you cannot rotate or dilate it.

In the diagram at the left, vectors a and b are equal even though they have different starting and ending points.

The sum of vectors can be determined by representing the vectors as arrows joined in sequence with the initial point of each vector coinciding with the terminal point of the preceding one. The sum is then the vector with the initial point of the first vector and the terminal point at the terminal point of the last vector.

Geometrically, the sum of the vectors a and b is the vector c along the diagonal of the parallelogram determined by a and b.

This is known as the parallelogram law of forces.

Name: Date:

Pythagorean Theorem

The Pythagorean theorem states that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the hypotenuse. The following expression represents the Pythagorean theorem:

a2 + b2 = c2 where c is the hypotenuse of a right triangle and a and b are the measures of the legs.

Geometrically, this theorem is that the area of ABGF in the figure at right is equal to the sum of the areas of ACDE and BCKH.

Use the Pythagorean theorem expression (a2 + b2 = c2) to solve the following problems.

All of the following values apply to right triangles. Find the measure of the missing side of the triangle using the Pythagorean theorem. If the measure has a square root (like ) leave it in the answer.

Example 1: What is the length of c if a = 6 and b = 8?

Example 2: What is the length of a if b = and c = 10?

1. a = 5 b = 12 c =

2. a = b = 15 c = 17

3. a = 7 b = c = 25

4. a = b = 4 c =

5. a = b = 8 c =

6. a = 15 b = 20 c =

62 82+ c2=

36 64+ c2=

100 c2=

100 c2=10 c=

5 3 a2 5 3( )2

+ 102=

a2 75+ 100=

a2 25=

a2 25=a 5=

3

4 2

8 3

Name: Date:

Pythagorean Triples

A Pythagorean triple is three whole numbers that satisfy the equation a2 + b2 = c2 where c is the greatest number. If the measures of the sides of any right triangle are whole numbers, the measures form a Pythagorean triple.

Which of these sets of measures form the sides of a right triangle? Which of these sets form a Pythagorean triple?

• In example 1, the segments form the sides of a right triangle since they satisfy the Pythagorean theorem. The measures are whole numbers and form a Pythagorean triple.

• In example 2, the segments with these measures cannot form a right triangle. Therefore, they do not form a Pythagorean triple.

• In example 3, the segments with these measures form a right triangle. However, the three numbers are not whole numbers. Therefore, they do not form a Pythagorean triple.

Determine if the following segments form a right triangle and determine if they are Pythagorean triples:

1. 5, 12, and 13

2. 8, 15, and 17

3. 1, 2, and 3

4. 12, 35, and 37

5. 4, 16, and

6. 7, 24, and 25

7. When we take multiples of the Pythagorean triples above, they form new Pythagorean triples. These new Pythagorean triples are considered families of the original triple. For example, you found that 3, 4, and 5 formed a triple. If you multiple each of these by a whole number (other than zero) you will find new triples. An example is 18, 24,and 30.

a. How were these formed?

b. Use the triples in #1, to find at two additional triples in the same family.

c. Use the triples in #2, to find two additional triples in the same family.

1. 3, 4, and 5

a2 + b2 = c2

32 + 42 = 52

9 + 16 = 2525 = 25

2. 8, 15, and 16

a2 + b2 = c2

82 + 152 = 162

64 + 225 = 256289 ≠ 256

3. , , and 35

------- 65

------- 35---

35

-------⎝ ⎠⎛ ⎞

2 65

-------⎝ ⎠⎛ ⎞

2+ 3

5---⎝ ⎠⎛ ⎞ 2

=

325------ 6

25------+ 9

25------=

3 48

Name: Date:

Special Right Triangles

Two special kinds of right triangles are the 45° – 45° – 90° right triangle and the 30° – 60° – 90° right triangle. They are used so frequently that you should learn the pattern for finding the measures of the sides.

Find the value of x.

In a 45° – 45° – 90° triangle the hypotenuse is times as long as a leg. The legs of these triangles are equal in length. This is indicated by the small tick marks made on each leg.

In a 30° – 60° – 90° triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is times as long as the shorter leg.

Example 1: Since the triangle is a 45° – 45° – 90° triangle, the hypotenuse is times as long as the leg. So, .

Example 2: Since the triangle is a 30° – 60° – 90° triangle, the hypotenuse is times as long as the leg. So, x = 2(8)=16.

2

3

2x 5 2=

3

Page 2 of 2

Find the value of x for each triangle below.

1.

2.

3.

4.

5.

6.

Name: Date:

Measuring Angles with a ProtractorMeasure each of these angles (A - Q) with a protractor. Record the angle measurements in the table below.

Letter Angle Letter AngleA J

B K

C L

D M

E N

F O

G P

H Q

I

Name: Date:

Constructing Angles with a ProtractorUse a protractor to construct the angles listed below. Use the arrow beneath each letter to make the angle.

A. 50° B. 90° C. 130°

D. 35° E. 100° F. 146°

G. 78° H. 15° I. 64°

J. 45° K. 112° L. 160°

Name: Date:

Functions: Conversion between Celsius and Fahrenheit

A function is a relationship between input and output values. Each input has exactly one output. Sometimes people like to describe a function as a “machine.” The idea is that you put a thing in, something happens to it, and some (related) thing comes out.

The point of this picture is not that a function is magic, but that it is an automatic routine. Once we know the routine, we can substitute any number into the place of the input symbol, and we get the corresponding output without any hard work.

If your machine has a rule to square the input number and then add 2 to the results, what will the values for the following output be?

A first degree equation of the form y = ax + b can be used as a function. This means that some value will be input for x. The machine will multiply by the determined a; add a value that has been determined for b; and finally give the output value y.

y = 3x + 10 is the rule for your machine. You may chose any real number for x, and your machine will take your number as the input; multiply it by 3; add 10; and give the final value as output.

Rule: x2 + 2Input Output

10-3.54-2

Step 1 y = ax + b y = 3(x) + 10 Replace variables.Step 2 y = 3(6) + 10 Multiply 3 and 6.

y = 18 + 10 Add 18 and 10.y = 28

Page 2 of 3

Fill in the following table using the rule for the machine (the function).

In science we often want to convert between Fahrenheit and Celsius. There are two formulas, or rules, that we use for these conversions.

We could use these rules for our function machines. One function machine converts Celsius to Fahrenheit. The second machine converts Fahrenheit to Celsius.

If you have trouble remembering which fraction you should be multiplying by think about this:

• When you multiply by a fraction larger than 1 like ( 9/5), the product will be greater than the number you started with. If you are converting from Celsius to Fahrenheit, then you are trying to get a much larger number than the Celsius measure. Therefore, you multiply by 9/5 and then add 32.

• When you multiply by a fraction less than 1 like ( 5/9), then the product will be less than what you started with. The Fahrenheit scale uses larger numbers. Therefore, if you are converting from Fahrenheit to Celsius, you are looking for a smaller number, so you must be multiplying by 5/9 after you reduce the number by 32.

Rule for the machine (the function)y = 3(x) + 10

Input Output

6 28

0

7

– 9/5

– 4

– 5/9

TFahrenheit95---TCelsius 32+= TCelsius

59--- TFahrenheit 32–( )=

Page 3 of 3

Use the temperature conversion formula on the previous page to convert these Fahrenheit values to Celsius. Compare your answers to the thermometer below.

°Fahrenheit °Celsius

45

81

0

–4

–10

100

– 2.5

0

– 18.3

37.8

Name: Date:

Using a Graphing Calculator: Conversion between Celsius and Fahrenheit

Imagine that you are a weather reporter for a television network in your town. During the nightly broadcast you must speak about the temperature in various places around the world. In order to have up-to-date information, you must have your report ready 15 minutes before air time. Since most countries report temperature in degrees Celsius, you need to have a way to do quick conversions to Fahrenheit for your American audience.

The conversion formula for Celsius to Fahrenheit is:

The calculation can be done using a simple calculator. However, a graphing calculator provides other options. Two are being presented here: (1) using the LIST option of your calculator, and (2) using the calculator create a graph and table of the values.

Using the LIST option with a graphing calculator

To use the LIST option (this is found under the STAT button) you should input the values that you have for each location in L1. Then bring your cursor to the top of L2. This will highlight L2. You should type in your function using L1 for your Celsius measure. At the bottom of the second list you would see:

Press enter, and you will have your output values for each of the places in L2. (If you did not remember to put a location’s temperature in your L1, you will need to go through the process again.) Add the missing temperature(s). Then go to the top of L2, and reenter the formula. All the answers will be generated.)

Find the Fahrenheit degree measure for the following places:City (Country) °Celsius (L1) °Fahrenheit (L2)

Athens (Greece) 22Berlin (Germany) 4Buenos Aires (Argentina) 29Calgary (Canada) 7Cairo (Egypt) 18Istanbul (Turkey) 10Melbourne (Australia) 9Mexico City (Mexico) 20Nairobi (Kenya) 33Rio de Janeiro (Brazil) 24Tokyo (Japan) 15

TFahrenheit95---TCelsius 32+=

L2 95---⎝ ⎠

⎛ ⎞L1 32+=

Page 2 of 2

Creating a graph with a graphing calculator

A graph comparing Celsius and Fahrenheit temperatures is a quick way to make conversions. Use x for the Celsius degree measure and y for the Fahrenheit degree measure. Put the conversion equation into y = and set the Window.

Try the following settings for your graph: Xmin = -1, Xmax = 105, Xscale = 5, Ymin = -1, Y max = 215, Yscale = 10. With these dimensions you can see the axis, and you are able to get some values. You could then look at the TABLE of the values for the graph. You should think about the settings for this. You could trace for some values but this is not always as accurate as you would like. You could adjust the increments (∆Tbl) in your table set to see if you get better answers.

Finally, you have the CALC button. You may now select this. Under this menu, you would select the “value” choice. It would give you a prompt to input a Celsius value. Then, it would give you the output value for Fahrenheit.

It is important for you to know how the output value is calculated. Once you understand this, it is not necessary for you to do the calculations by hand every time.

Use the graphing method to fill in the values for temperature in Fahrenheit in the table below.

°Fahrenheit °Celsius °Fahrenheit °Celsius

32 38

-4 100

0 -28

-273 31.8

25.8 50

99 -3

-24 15.5

Name: Date:

Ratios

A ratio is a comparison of two quantities that tells the scale between them. Ratios may be expressed as quotients, fractions, decimals, percents, or given in the form a:b. The word per in a ratio means “for every.”

1. The ratio of females to males on a swim team is 3 to 4, or 3/4.

2. The airplane is flying at a constant speed of 150 meters per second or 150 meters for every second flown or .

A ratio can be a comparison of a part to a whole or a part to a part. In the example above about the swim team members, the team is made up of two parts, females and males. The whole is the total number of people on the team. There are three ratios that can be constructed given what we need. They are: , , or .

Because ratios are not always fractions, we can have numbers in either position without regard to size. This also allows us to use two different notations when comparing males and females. We need to be concerned with the order in which the ratio is stated.

• In the above example, we have . We know from the information that this is 3/4.

• However, we also know what would be from the same statement. It would be 4/3.

The first quantity mentioned in the comparison is the numerator—the number on the upper position of the fraction.

Reduce all fractions as you solve the following problems.

1. A research study shows that 3 out of every twenty pet owners got their pets from a breeder, 14 out of the twenty got their pets from an animal rescue agency, 1 got their pet from the street, and 2 got their pets from a friend or relative.

a. What is the ratio of pets received from a breeder compared to the total number?

b. What is the ratio of pets received from a friend compared to an animal rescue agency?

c. What is the ratio of pets from an animal rescue agency compared to the total number of pets?

d. Do you know how many pets were in this research study? Explain your answer.

2. The ratio of two measurements having different units of measure is called a rate. Examples of this are a price of $1.99 per dozen, a speed of 55 miles per hour, and a salary of $30,000 per year. A ratio or rate called a scale is used when making a model or drawing of something that is too large or too small to be drawn conveniently at actual size. Maps and blueprints are two commonly used scale drawings.

Give a ratio (or rate, or scale) for the following situations:

a. The scale of a map for Blue Hills Regional Park is 2 inches = 9 miles.

150 meters1 second

--------------------------

MalesTeam-------------- Females

Males-------------------- Females

Team--------------------

FemalesMales

--------------------

MalesFemales--------------------

Page 2 of 2

b. Alicia goes on a 30-mile bike ride every Saturday. She rides the distance in 4 hours.

c. The Tremblays’ minivan requires 5 gallons of gasoline to travel 112 miles.

d. A blueprint for a house states that 2.5 inches equals 10 feet.

e. A collector's model racecar is scaled so that 1 inch on the model equals 63/8 feet on the actual car.

f. On average, the basketball player scores 19 points per game.

g. These cookies have 35 grams of fat in one serving of 5 cookies.

h. Wynonna’s prepaid calling card cost $5 and allows her to make 80 minutes worth of calls.

i. Bill's 85-pound Husky eats 40 pounds of dog food every two weeks.

3. Use the following chart to answer the questions:

a. Write the ratio of right-handed boys to right-handed girls.

b. Write the ratio of left-handed boys to the total number of students in the class.

c. Write the ratio of ambidextrous girls to the total number of girls in the class.

d. Write the ratio of ambidextrous girls to ambidextrous boys.

e. Write the ratio of right-handed boys to ambidextrous boys.

f. Write the ratio of boys in the class to girls in the class.

g. Write the ratio of right-handed girls to the entire class.

Mr. Little’s Class

Right-handed Left-handed Ambidextrous

Boys 11 5 1

Girls 8 4 2

Name: Date:

Ratio and Proportions in a RecipeDouble Fudge Brownies

Ingredients:

Makes 16 brownies.

1. What is the ratio of milk to chocolate chips?

2. When we know the ratios, we can make proportions by setting two ratios equal to one another. This will help us to find missing answers.

Suppose Patricia only needs 8 brownies. She does not want to ruin her diet by having leftovers. Find out the ingredients she needs. The original recipe will make 16 brownies. You will use the ratio of 8/16 = 1/2 to find the amount for each of the ingredients. Use cross-multiplication to solve the proportions.

For flour:

1. What is the ratio of unsalted butter to eggs?

For every __________ tablespoons of butter, you will need __________ eggs.

2. What is the ratio of flour to baking soda?

For every __________ cups of flour, you will need __________ teaspoons of baking soda.

3/4 c. sugar 2 eggs

6 tablespoons unsalted butter 1 teaspoon vanilla extract

2 tablespoons milk 3/4 cup all-purpose flour

2 cups semi-sweet chocolate chips 1/3 teaspoon baking soda1/4 teaspoon salt 2 tablespoons confectioner’s sugar

Step 1

Step 2

Step 3

Step 4

Step 5 Patricia needs 3/8 cup of flour to make 8 brownies.

2 tablespoons2 cups

--------------------------------

816------ x

3 4⁄----------=

8 34---× 16x=

6 16x=

616------ 16x

16---------=

38--- x=

Page 2 of 2

3. What is the ratio of salt to flour?For every __________ teaspoon(s) of salt, you will need __________ cups of flour

4. Find the correct amount of each ingredient to make 8 brownies (1/2 of the recipe).

5. Why are the ingredients, eggs and confectioner’s sugar, easy to work with to make 8 brownies?

6. Patricia has a surplus of all the ingredients. How many brownies can be made using 3 cups of chocolate chips?

7. How much vanilla will she need when she makes the batch of brownies using 3 cups of chocolate chips?

Ingredient Amount

Flour 3/8 cup

Sugar

Butter

Milk

Chocolate chips

Eggs

Vanilla extract

Baking soda

Salt

Confectioner’s sugar

Name: Date:

ProbabilityThe probability of an event is the chance of getting a possible outcome of a system. Probability is defined as the ratio of the number of ways an event can occur to the number of possible outcomes.

What is the probability of rolling a number less than 5 when you roll a six-sided die?

Answer: There are four numbers less than 5 on the die out of 6 possible outcomes. Therefore, the probability of the number rolled being less than 5 is 4/6 or 2/3.

1. A box of colored pencils contains 2 red ones, 5 blues, 3 greens, and 2 yellows. The total number of pencils in the box is ______. If you choose at random, what is the probability that you will pick each of the following?

a. a yellow pencil

b. a blue pencil

c. a red or green pencil

d. a pencil that is not blue

e. a pencil that is not red

2. A card is chosen randomly from a deck of 52 cards. A standard deck of cards has four suits: spades, clubs, diamonds, and hearts. The spades and clubs are black; the hearts and diamonds are red. Each suit has 13 cards. Each type of card has one from each suit, so there are 4 cards of each type. For example, there are 4 nines. Find the probability of choosing each of the following.

a. a red card

b. the queen of spades

c. an ace

d. a black 9

e. not a diamond

f. a heart

3. A spinner is divided into 5 equal pie shapes. Each one is of the same size. The shapes are colored red, purple, red, yellow, and red. Find the probability of spinner landing on each of the following.

a. a red section

b. a yellow section

c. a blue section

d. a purple or red section

e. not a red section

probability of an event number of ways the event can occurnumber of possible outcomes

--------------------------------------------------------------------------------------=

Name: Date:

Reciprocals and Negative One as an Exponent

All real numbers, except zero, have a reciprocal. To find the reciprocal, you write the number as a fraction and then invert the numerator and the denominator. The reciprocal is another name for the multiplicative inverse.

In algebra, we use a negative one as an exponent for a number or expression to indicate the reciprocal. If the numerator or denominator has more than a single number then parentheses must be used to indicate what the base of the exponent is.

Example 1:

Example 2:

Example 3:

=

Convert the fraction to a number without a negative exponent. Notice that because of the parentheses, the negative exponent only applies to the numerator of this fraction.

=This fraction can easily be expressed as a reciprocal with a negative exponent.

=

=The reciprocal is found by inverting the numerator and denominator.

37---⎝ ⎠

⎛ ⎞ 1– 7

3---⎝ ⎠

⎛ ⎞ Notice that the number did not become negative.=

5a( ) 1– 15a------=

12---⎝ ⎠

⎛ ⎞

3----------

1–1

3 12---⎝ ⎠⎛ ⎞

------------

1 32---

---------

32---⎝ ⎠⎛ ⎞ 1–

23---

Page 2 of 2

Find the reciprocal for the following numbers:

1. 3/4

2. 1/5

3. 7

4. 81/4

5. –1/4

6. –41/8

7. –6/13

8. –100

Convert the following to numbers that do not have a negative exponent.

9. 20-1

10.

11. (-3b)-1

12.

13.

14. (6d)-1

15. 6d(-1)

16.

17. (-8)-1

18. 8(3a)-1

19.

23---⎝ ⎠

⎛ ⎞ 1–

34---⎝ ⎠

⎛ ⎞

5----------

1–

523---⎝ ⎠

⎛ ⎞ 1–

2 8×( )3 6×( )

-----------------⎝ ⎠⎛ ⎞ 1–

35---⎝ ⎠

⎛ ⎞ 1– 615------⎝ ⎠⎛ ⎞ 1–

×

Name: Date:

Problem Solving Boxes

Looking for Solution

Given

Relationships

Looking for Solution

Given

Relationships

Looking for Solution

Given

Relationships

1

Math Skill Builder Answer KeysDecimals and Place Value

1. Answers are:a. 0.14b. 4,600.025c. 1,000.001d. 985.63e. 8,035.4

2. Answers are:a. one hundred five and sixty-four thousandthsb. twenty-three and forty nine ten-thousandthsc. thirty-six and seven tenthsd. forty-five and three thousandthse. seventy-four and nine hundred ninety-eight thousandths

3. Answers are:a. nineteen and thirty-two hundredthsb. 43.49

Slope

1. m = 2/4 = 1/2As x changes 4 units to the left, y changes 2 units down. ORAs x changes 4 units to the right, y changes 2 units up.

2. m = 7/-3As x changes 3 units to the left, y changes 7 units up. ORAs x changes 3 units to the right, y changes 7 units down.

3. m = 4/7As x changes 7 units to the left, y changes 4 units down. ORAs x changes 7 units to the right, y changes 4 units up.

4. m = 7/-7 = m = -1As x changes 1 units to the left, y changes 1 units up. ORAs x changes 1 units to the right, y changes 1 units down.

5. m = 1/3As x changes 3 units to the left, y changes 1 units down. ORAs x changes 3 units to the right, y changes 1 units up.

Slope from a Graph

Numbers correlate to graph numbers:

1.

2.

3.

4.

5.

6.

7. m = 0

8.

9. m = 2

10.

Order of Operations, Part 1

1. 102. 503. 8.54. 75. 446. 57. 228. 329. 2410. 1211. 712. 2413. 14

14. 1915. 1016. Answers are:

a. $25.00 + 2 × $8.95b. $42.90

17. Answers are:a. $5.99 + 20 × $0.95b. $24.99

18. Answers are:a. 2 × 150 + 2 × 250b. 800 cells of bacteria

19. Answers are:a. 60 × $7.50 + 70 × $5.00 + 50 × $7.50 + 90 × $5.00b. $1,625.00

m 3– 3×6 2×

--------------- 3–4

------= =

m 2 3×–4 2×

--------------- 3–4

------= =

m 59---=

m 42--- 2= =

m 33--- 1= =

m 2–2

------ 1–= =

m 34---=

m 1–2

------=

2

Order of Operations, Part 2

1. 62. 263. 404. 1105. 576. 27. -188. 109. 7710. 311. 612. 1/90 = 0.111

13. 314. 615. -1716. Answers are:

a. 10 · (5 + 4) = 90b. 6 + [(5 – 2) · 8] = 30c. (3 + 4) · (5 – 1) = 28d. 20 ÷ (2 · 5) + 8 = 10e. (20 ÷ 2) · 5 + 8 = 58f. (20 ÷ 2) · (5 + 8) = 130

17. Answers are:a. $50.00 + (10 × $3.95)b. $89.50

Evaluating Algebraic Expressions

1. 7.5 g/cm3

2. 3 m/sec2

3. 78.5 cm2

4. 40 m

5. 16. Answers are:

a. 1/3(B × h)

b. 4000 m3

Inverses

Additive inverse: Multiplicative inverses:

Inverse Operations

1. a = 242. a = 83. a = 84. b = 145. b = 76. b = 127. c = 88. c = 11

9. c = 610. d = 6411. d = 3212. d = 8013. f = 8114. f = 8115. f = 145.8

Negative Exponents and Exponents of Zero

1. 1/252. 1/1093. 14. 1/9005. 16. 9/47. 1/125

8.

9.10. 111. 112. a13. 1

1. -29 5. 0.752. 101 6. – 24.73. –5/9 7. Zero has no reciprocal.4. 4/3 8. 4.32

1. 9/5 5. -0.12. 3/8 6. Zero has no reciprocal.3. –4/3 7. – 2/34. 100/6 = 50/3 8. 0.5

516------

516–

---------

3

14. 4d3

15.

16. 100

17. 10,000,00018. 219. 1/220. 1/5

Scientific Notation: Standard to Scientific

1. Answers are:a. 4.53 × 10-2

b. 1.87 × 107

c. 2.57 × 10-1

d. 9.99 × 102

e. 2.64 × 105

f. 7.61 × 108

g. 1.03 × 103

h. 1.20 × 10-3

i. 3.04 × 10-2

j. 5.2 × 10-17

k. 4.2 × 1013

l. 7.658 × 106

m. 9.99 × 10-4

2. Answers are:a. $5.67 × 1012

b. 2.81 × 108 peoplec. 1.2 × 1014 metric tonsd. 9.1 × 10-31 kge. 3.3 × 10-4 mf. 4.4 × 106 tonsg. 6.51 × 107 peopleh. 2.78 × 104 °Ci. 2.0 × 1011 starsj. $2.52 × 107

k. 1.37 × 102 ml. 9.86 × 101 °F

Scientific Notation: Scientific to Standard

1. 0.000 000 000 000 000 000 16 coulombs2. 5,580,000,000,000,000,000,000 atoms3. 20,000,000,000 phone calls4. 239,000 miles5. 0.000 000 000 000 000 000 000 000 001 67 kg6. 150,000,000 km

7. 92,900,000 miles8. 0.0005 inches9. 263,000,000 people10. 1,220,000,000 people11. 0.001 meters12. 340 m/sec

Scientific Notation: Products and Quotients

1. 1.25 × 102 = 1252. 3.5 × 103 = 3,5003. 7.2 × 100 = 7.24. 9.4 × 103 = 9,4005. 2.05 × 108 = 205,000,0006. 5.4 × 10-3 = 0.00547. 2.8 × 102 = 2808. 2.0 × 102 = 2009. 4 × 10-6 = 0.00000410. 1.0 × 10-7 = 0.000 000 111. 2 × 108 = 200,000,000

12. (5 × 106 red blood cells/mm3) × (5 × 106 mm3)= 2.5 × 1013 = 25,000,000,000,000 red blood cells

13. (6.4 × 1012 meters) ÷ (3 × 108 m/sec)= 2.1 × 104 sec = 21,000 sec

14. (2.26 × 105 miles) × (6.34 × 104 in/mile) = 1.43 × 1010 inches = 14,300,000,000 inches

15. (4.78 × 106 people) ÷ (1.14 × 105 mi2) = 4.19 × 101 people/mi2 = 41.9 people/mi2

16. Answers are:a. 1.65 billion credit cards =

1,650,000,000 credit cards = 1.65 × 109 credit cards$1.54 trillion = $1,540,000,000,000 = $1.54 × 1012

b. ($1.54 × 1012) ÷ (1.65 × 109 credit cards) = $9.33 × 102 = $933

Two-Dimensional Vectors

There are no questions to answers for this skill builder.

0.5mn

------------

4

Pythagorean Theorem

1. c = 132. a = 83. b = 24

4. a = 45. c = 166. c = 25

Pythagorean Triples

1. 52 + 122 = 132 = 169; yes, this is a Pythagorean triple.2. 82 + 152 = 172 = 289; yes, this is a Pythagorean triple.3. 12 + 22 = 5; 32 = 9; no, this is not a Pythagorean triple.4. 122 + 352 = 372 = 1,369; yes, this is a Pythagorean triple.5. 42 + 162 = 272; = 432; no, this is not a Pythagorean

triple and one of the values is not a whole number.

6. 72 + 242 = 252 = 625; yes, this is a Pythagorean triple.7. Answers are:

a. The triple (18, 24, 30) was formed by multiplying eachnumber in the triple (3, 4, 5) by the whole number 6.

b. Example answer: (10,24,26) by multiplying by 2; and (15,36, 39) by multiplying by 3.

c. Example answer: (16, 30, 34) by multiplying by 2; and(24, 45, 51) by multiplying by 3.

Special Right Triangles

1.2. x = 8 cm3. x = 10 m

4. x = 40 ft5.6. x = 14 yards

Measuring Angles with a Protractor

Answers are:

Constructing Angles with a Protractor

There are no questions to answers for this skill builder; studentsneed only to construct the angles listed.

3 48( )2

x 7 2 mm=

x 16 2 inches=

Letter Angle Letter AngleA 56° J 153°

B 110° K 131°

C 10° L 148°

D 96° M 81°

E 167° N 90°

F 122° O 73°

G 34° P 27°

H 45° Q 139°

I 19°

5

Functions: Conversions between Celsius and Fahrenheit

Example:

Practice set 1:

Practice set 2:

Using a Graphing Calculator: Conversions between Celsius and Fahrenheit

Practice set 1: Practice set 2:

Ratios

1. Answers are:a. 3/20

b. 2/14 = 1/7

c. 14/20 = 7/10

d. No, you only know the ratios and not the actual numbers.The total number of pets could be 20 or 20,000—youcan’t tell from a ratio.

2. Answers are:

Input Output10 102-3 11.5 2.254 18-2 6

Rule for the machine: y = 3(x) + 10

Input Output6 280 107 31

– 9/5 43/5– 4 -2

– 5/9 81/3

°Fahrenheit °Celsius45 7.281 27.20 -17.8–4 -20–10 -23.3212 10027.5 -2.532 0–1 -18.3100 37.8

City (Country) °Celsius (L1) °Fahrenheit (L2)Athens (Greece) 22 71.6Berlin (Germany) 4 39.2Buenos Aires (Argentina)

29 84.2

Calgary (Canada) 7 44.6Cairo (Egypt) 18 64.4Istanbul (Turkey) 10 50Melbourne (Australia) 9 48.2Mexico City (Mexico) 20 68Nairobi (Kenya) 33 91.4Rio de Janeiro (Brazil) 24 75.2Tokyo (Japan) 15 59

°Fahrenheit °Celsius89.6 3224.8 -432 0-459.4 -27378.4 25.8210.2 99-11.2 -24100.4 38212 100-18.4 -2889.24 31.8122 5026.6 -359.9 15.5

6

a. scale: 2 inches/9 miles = 0.22 inches per mile

b. 30 miles/4 hours = 7.5 mph

c. 112 miles/5 gallons = 22.4 mpg

d. scale: 2.5 inches/10 feet = 0.25 inches per foot

e. scale = 1 inch/6 3/8 feet = 0.157 inches per footf. 19 points per gameg. 35 grams per 5 cookies = 7 grams fat/cookie

h. $5/80 minutes = $0.0625 per minute or 6.25 cents per minute

i. 40 pounds/2 weeks = 20 pounds of dog food per week3. Answers are:

a. 11/8b. 5/31c. 2/14 = 1/7d. 2/1e. 11/1f. 17/14

Ratios and Proportions in a Recipe

1. Answers for the blanks in order: 6 tablespoons; 2 eggs2. Answers for the blanks in order: 3/4 cup; 1/3 teaspoon3. Answers for the blanks in order: 1/4 teaspoon; 3/4 cup4. Table answers: 5. To make 16 brownies, you need two eggs and 2 table spoons

of sugar. Therefore, to make 8 brownies, you only need 1 ofeach unit for each ingredient: 1 egg and 1 tablespoon.

6. Since 8 brownies requires 1 cup of chocolate chips, 3 cups ofchocolate chips will make 24 brownies.

7. 1.5 teaspoons vanilla is needed to make 24 brownies.

Probability

1. 12 totala. 2/12 = 1/6b. 5/12c. 5/12d. 7/12e. 10/12 = 5/6

2. Answers are:a. 26/52 = 1/2b. 1/52c. 4/52 = 1/13

d. 2/52 = 1/26e. 39/52 = 3/4f. 13/52 = 1/4

3. Answers are:a. 3/5b. 1/5c. 0d. 4/5e. 2/5

Reciprocals and Negative One as an Exponent

1. 4/32. 53. 1/74. 4/335. -46. -8/337. -13/68. -1/1009. 1/2010. 3/2

11. -1/3b12. 4/1513. 3/1714. 1/6d15. 6/d16. 9/817. -1/818. 8/3a19. 25/6

Butter 3 tablespoonsMilk 1 tablespoonChocolate chips 1 cupEggs 1 eggVanilla extract 1/2 teaspoon

Baking soda 1/6 teaspoon

Salt 1/8 teaspoon

Confectioner’s sugar 1 tablespoon