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Unit 8 PracticeProblems

Lesson 1Problem 1Find the area of each square. Each grid square represents 1 square unit.

Solution1. 17 square units

2. 20 square units

3. 13 square units

4. 37 square units

Problem 2Find the length of a side of a square if its area is:

1. 81 square inches2. cm3. 0.49 square units4. square units

425

2

m2

Solution1. 9 inches

2. cm

3. 0.7 units

4. units

Problem 3Find the area of a square if its side length is:

1. 3 inches2. 7 units3. 100 cm4. 40 inches5. units

Solution1. 9 square inches

2. 49 square units

3. 10,000 cm

4. 1,600 square inches

5. square units

Problem 4(from Unit 7, Lesson 14)Evaluate . Choose the correct answer:

A.

B.

C.

D.

SolutionC

Problem 5(from Unit 7, Lesson 15)Noah reads the problem, “Evaluate each expression, giving the answer inscientific notation.” The first problem part is: . Noah says, “Ican rewrite as . Now I can add the numbers:

.” Do you agree with Noah’s solution to theproblem? Explain your reasoning.

SolutionAnswers vary. Sample response: I don’t agree with Noah’s solution. Hiscalculations are correct, but his final answer is not in scientific notation. To finishthe problem, he should convert his answer to the form .

Problem 6(from Unit 7, Lesson 6)

25

m

x

2

x2

(3.1 × ) (2 × )104 ⋅ 106

5.1 × 1010

5.1 × 1024

6.2 × 1010

6.2 × 1024

5.4 × + 2.3 ×105 104

5.4 × 105 54 × 104

54 × + 2.3 × = 57.3 ×104 104 104

5.73 × 105

8

Select all the expressions that are equivalent to .

A.

B.

C.

D.

E.

F.

SolutionA, C, D, E

Lesson 2Problem 1A square has an area of 81 square feet. Select all the expressions that equal theside length of this square, in feet.

A. B. C. 9D. E. 3

SolutionB, C

Problem 2Write the exact value of the side length, in units, of a square whose area insquare units is:

1. 362. 373. 4. 5. 0.00016. 0.11

Solution1. 6

2.

3.

4.

5. 0.01

6.

Problem 3Square A is smaller than Square B. Square B is smaller than Square C.

38

(32)4

83

3 3 3 3 3 3 3 3⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅(34)2

36

3-2

36 ⋅ 102

812

81⎯ ⎯⎯⎯√

9⎯⎯√

1009

25

37⎯ ⎯⎯⎯√

103

25

⎯⎯⎯√

0.11⎯ ⎯⎯⎯⎯⎯⎯√

The three squares’ side lengths are , 4.2, and .

What is the side length of Square A? Square B? Square C? Explain how youknow.

SolutionSquare A: units, Square B: 4.2 units, Square C: . I know this because

is between 3 and 4 and is between 5 and 6, so and the side length of A is less than the side length of B is less than the sidelength of C.

Problem 4(from Unit 8, Lesson 1)Find the area of a square if its side length is:

1. cm2. units3. inches4. 0.1 meters5. 3.5 cm

Solution1. cm

2. square units

3. square inches

4. 0.01 square meters

5. 12.25 cm

Problem 5(from Unit 7, Lesson 15)

Solution1. more people

2. The Asian countries ( vs. )

Problem 6

26⎯ ⎯⎯⎯√ 11⎯ ⎯⎯⎯√

11⎯ ⎯⎯⎯√ 26⎯ ⎯⎯⎯√11⎯ ⎯⎯⎯√ 26⎯ ⎯⎯⎯√ < 4.2 <11⎯ ⎯⎯⎯√ 26⎯ ⎯⎯⎯√

1537118

125

2

949

12164

2

Here is a table showing the areas ofthe seven largest countries.

1. How many more people live inRussia than in Canada?

2. The Asian countries on this listare Russia, China, and India.The American countries areCanada, the United States, andBrazil. Which has the greatertotal area: the three Asiancountries, or the three Americancountries?

country area (in km )

Russia

Canada

China

United States

Brazil

Australia

India

2

1.71 × 107

9.98 × 106

9.60 × 106

9.53 × 106

8.52 × 106

6.79 × 106

3.29 × 106

7.12 × 106

2.999 × 107 2.808 × 107

(from Unit 7, Lesson 5)Select all the expressions that are equivalent to .

A. B. C. D. E. F.

SolutionA, C, E, F

Lesson 3Problem 1Decide whether each number in this list is rational or irrational.

SolutionRational: ; Irrational:

Problem 2Which value is an exact solution of the equation ?

A. 7B. C. 3.74D.

SolutionB

Problem 3(from Unit 8, Lesson 2)A square has vertices , and . Which of these statementsis true?

A. The square’s side length is 5.

B. The square’s side length is between 5 and 6.

C. The square’s side length is between 6 and 7.

D. The square’s side length is 7.

SolutionB

Problem 4(from Unit 7, Lesson 8)Rewrite each expression in an equivalent form that uses a single exponent.

1.

2.

10-6

11000000

-110000001

106

108 ⋅ 10-2

( )110

6

110 10 10 10 10 10⋅ ⋅ ⋅ ⋅ ⋅

, 0.1234, , -77, - , --133 37⎯ ⎯⎯⎯√ 100⎯ ⎯⎯⎯⎯⎯√ 12⎯ ⎯⎯⎯√

, 0.1234, -77, --133 100⎯ ⎯⎯⎯⎯⎯√ , -37⎯ ⎯⎯⎯√ 12⎯ ⎯⎯⎯√

= 14m2

14⎯ ⎯⎯⎯√

3.74⎯ ⎯⎯⎯⎯⎯⎯√

(0, 0), (5, 2), (3, 7) (-2, 5)

(102)-3

(3-3)2

-5 -5

3.

4.

Solution1. (or equivalent)

2. (or equivalent)

3. (or equivalent)

4. (or equivalent)

Problem 5(from Unit 5, Lesson 5)The graph represents the area of arctic sea ice in square kilometers as a functionof the day of the year in 2016.

1. Give an approximate interval of days when the area of arctic sea ice wasdecreasing.

2. On which days was the area of arctic sea ice 12 million square kilometers?

Solution1. Answers vary. Correct responses should be close to “day 75 to day 250.”

2. Days 135, 350, and 360

Problem 6(from Unit 4, Lesson 14)The high school is hosting an event for seniors but will also allow some juniors toattend. The principal approved the event for 200 students and decided thenumber of juniors should be 25% of the number of seniors. How many juniors willbe allowed to attend? If you get stuck, try writing two equations that eachrepresent the number of juniors and seniors at the event.

Solution40 juniors. Sample reasoning: Solve the system , (orequivalent), where represents the number of seniors and represents thenumber of juniors.

Lesson 4Problem 1

1. Find the exact length of each line segment.

3-5 ⋅ 4-5

25 ⋅ 3-5

10-6

3-6

12-5

( )23

5

s + j = 200 j = 0.25ss j

2. Estimate the length of each line segment to the nearest tenth of a unit.Explain your reasoning.

Solution1. ,

2. is greater than 4 but less than 5 because . is smaller than 6but greater than 5 because .

Problem 2Plot each number on the -axis: . Consider using the grid tohelp.

Solution at 4, a little less than 6, a little more than 8

Problem 3Use the fact that is a solution to the equation to find a decimalapproximation of whose square is between 6.9 and 7.1.

SolutionAnswers vary. Sample responses: 2.63, 2.64, 2.65, 2.66

Problem 4(from Unit 7, Lesson 14)

AB = 17⎯ ⎯⎯⎯√ GH = 32⎯ ⎯⎯⎯√

AB = 1642 GH= 3662

x , , 16⎯ ⎯⎯⎯√ 35⎯ ⎯⎯⎯√ 66⎯ ⎯⎯⎯√

16⎯ ⎯⎯⎯√ 35⎯ ⎯⎯⎯√ 66⎯ ⎯⎯⎯√

7⎯⎯√ = 7x2

7⎯⎯√

Graphite is made up of layers of graphene. Each layer of graphene is about 200picometers, or meters, thick. How many layers of graphene are therein a 1.6-mm-thick piece of graphite? Express your answer in scientific notation.

SolutionAbout . The thickness of the graphite is meters. The number oflayers of graphene is given by . This number, in scientificnotation, is , or about 8 million.

Problem 5(from Unit 6, Lesson 6)Here is a scatter plot that shows the number of assists and points for a group ofhockey players. The model, represented by , is graphed with thescatter plot.

1. What does the slope mean in this situation?

2. Based on the model, how many points will a player have if he has 30assists?

Solution1. For every assist, a player’s points have gone up by 1.5.

2. Approximately 46.2 points

Problem 6(from Unit 3, Lesson 5)The points and lie on a line. What is the slope of the line?

Solution (or 11)

Lesson 5Problem 1

1. Explain how you know that is a little more than 6.

2. Explain how you know that is a little less than 10.

3. Explain how you know that is between 5 and 6.

Solution1. is exactly 6, and is a little more than that.

2. is exactly 10, and is a little less than that.

3. , , and is in between.

200 × 10-12

8 × 106 1.6 × 10-3

= 0.008 ×1.6×10-3

200⋅10-12 109

8 × 106

y = 1.5x + 1.2

(12, 23) (14, 45)

222

37⎯ ⎯⎯⎯√

95⎯ ⎯⎯⎯√

30⎯ ⎯⎯⎯√

36⎯ ⎯⎯⎯√ 37⎯ ⎯⎯⎯√

100⎯ ⎯⎯⎯⎯⎯√ 95⎯ ⎯⎯⎯√

= 525⎯ ⎯⎯⎯√ = 636⎯ ⎯⎯⎯√ 30⎯ ⎯⎯⎯√

Problem 2Plot each number on the number line:

Solution

Problem 3Mark and label the positions of two square root values between 7 and 8 on thenumber line.

SolutionAnswers vary. Sample response: and . Since the values are between

and , any square root of a number between 49 and 64 is asolution.

Problem 4(from Unit 8, Lesson 3)Select all the irrational numbers in the list.

Solution

Problem 5(from Unit 8, Lesson 2)Each grid square represents 1 square unit. What is the exact side length of eachsquare?

Solution units

Problem 6(from Unit 7, Lesson 10)For each pair of numbers, which of the two numbers is larger? Estimate howmany times larger.

6, , , , 7.583⎯ ⎯⎯⎯√ 40⎯ ⎯⎯⎯√ 64⎯ ⎯⎯⎯√

55⎯ ⎯⎯⎯√ 60⎯ ⎯⎯⎯√7 = 49⎯ ⎯⎯⎯√ 8 = 64⎯ ⎯⎯⎯√

, , , , , - , -23

-12345 14⎯ ⎯⎯⎯√ 64⎯ ⎯⎯⎯√ 9

1

⎯ ⎯⎯⎯

√ 99⎯ ⎯⎯⎯√ 100⎯ ⎯⎯⎯⎯⎯√

, -14⎯ ⎯⎯⎯√ 99⎯ ⎯⎯⎯√

13⎯ ⎯⎯⎯√

0.37 6 700 4

1. and 2. and 3. and

Solution1. , about 20 times larger

2. , about 30 times larger

3. , about 500 times larger

Problem 7(from Unit 6, Lesson 4)The scatter plot shows the heights (in inches) and three-point percentages fordifferent basketball players last season.

1. Circle any data points that appear to be outliers.

2. Compare any outliers to the values predicted by the model.

Solution1. The point at (85, 14) is an outlier.

2. This point represents a player who had a significantly worse (by about 15%of the attempts) three-point percentage than the model predicts for hisheight.

Lesson 6Problem 1Here is a diagram of an acute triangle and three squares.

SolutionNo, I disagree. Priya's pattern only works for right triangles, and this is an acutetriangle.

Problem 2, , and represent the lengths of the three sides of this right triangle.

0.37 ⋅ 106 700 ⋅ 104

4.87 ⋅ 104 15 ⋅ 105

500, 000 2.3 ⋅ 108

700 ⋅ 104

15 ⋅ 105

2.3 ⋅ 108

Priya says the area of thelarge unmarked square is 26 squareunits because . Do youagree? Explain your reasoning.

9 + 17 = 26

m p z

Select all the equations that represent the relationship between , , and .

A.

B.

C.

D.

E.

F.

SolutionB, C, E, F

Problem 3The lengths of the three sides are given for several right triangles. For each, writean equation that expresses the relationship between the lengths of the threesides.

1. 10, 6, 8

2.

3. 5,

4. 1, , 6

5. 3,

Solution1.

2.

3.

4.

5.

Problem 4(from Grade 8, Unit 4, Lesson 1)Order the following expressions from least to greatest.

Solution

m p z

+ =m2 p2 z2

= +m2 p2 z2

= +m2 z2 p2

+ =p2 m2 z2

+ =z2 p2 m2

+ =p2 z2 m2

, ,5⎯⎯√ 3⎯⎯√ 8⎯⎯√

,5⎯⎯√ 30⎯ ⎯⎯⎯√

37⎯ ⎯⎯⎯√

,2⎯⎯√ 7⎯⎯√

+ =62 82 102

+ =5⎯⎯√ 2 3⎯⎯√ 2 8⎯⎯√ 2

+ =52 5⎯⎯√ 2 30⎯ ⎯⎯⎯√ 2

+ =12 62 37⎯ ⎯⎯⎯√ 2

+ =2⎯⎯√ 2 7⎯⎯√ 2 32

25 ÷ 10 250,000 ÷ 1,000 2.5 ÷ 1,000 0.025 ÷ 1

2.5 ÷ 1,0000.025 ÷ 125 ÷ 10250,000 ÷ 1,000

Problem 5(from Unit 8, Lesson 3)Which is the best explanation for why is irrational?

A. is irrational because it is not rational.

B. is irrational because it is less than zero.

C. is irrational because it is not a whole number.

D. is irrational because if I put into a calculator, I get -3.16227766,which does not make a repeating pattern.

SolutionA

Problem 6(from Unit 7, Lesson 15)A teacher tells her students she is just over 1 and billion seconds old.

1. Write her age in seconds using scientific notation.

2. What is a more reasonable unit of measurement for this situation?

3. How old is she when you use a more reasonable unit of measurement?

Solution1.

2. Years

3. She is about 48 years old. There are 31,536,000 seconds in a year. is about 47.6.

Lesson 7Problem 1

1. Find the lengths of the unlabeled sides.

2. One segment is units long and the other is units long. Find the value of and . (Each small grid square is 1 square unit.)

Solution1.

a. , approximately

b. , exactly 10

2. a. because

- 10⎯ ⎯⎯⎯√

- 10⎯ ⎯⎯⎯√

- 10⎯ ⎯⎯⎯√

- 10⎯ ⎯⎯⎯√

- 10⎯ ⎯⎯⎯√ - 10⎯ ⎯⎯⎯√

12

1.5 × 109

1.5 × ÷ 31,536,000109

n pn p

40⎯ ⎯⎯⎯√ 6.3

100⎯ ⎯⎯⎯⎯⎯√

10⎯ ⎯⎯⎯√ + = 1012 32

⎯ ⎯⎯⎯√ + = 252 2

b. (or 5) because

Problem 2Use the areas of the two identical squares to explain why withoutdoing any calculations.

SolutionAnswers vary. Sample explanation: The areas of the two large squares are thesame since they are both 17 by 17 units. The area of the two rectangles on theleft square are the same as the area of the 4 triangles in the right square (eachpair of triangles makes a rectangle). So the area of the two smaller squares onthe left must be the same as the area of the smaller square on the right. Thismeans .

Problem 3(from Unit 8, Lesson 5)Each number is between which two consecutive integers?

1.

2.

3.

4.

5.

Solution1. 3 and 4

2. 7 and 8

3. 4 and 5

4. 9 and 10

5. 6 and 7

Problem 4(from Unit 8, Lesson 3)

1. Give an example of a rational number, and explain how you know it isrational.

2. Give three examples of irrational numbers.

Solution1. Answers vary. Sample response: is a rational number because rational

numbers are fractions and their opposites and is a fraction.

2. Answers vary. Sample response: , ,

25⎯ ⎯⎯⎯√ + = 2532 42

+ =52 122 132

+ =52 122 132

10⎯ ⎯⎯⎯√

54⎯ ⎯⎯⎯√

18⎯ ⎯⎯⎯√

99⎯ ⎯⎯⎯√

41⎯ ⎯⎯⎯√

23

23

2⎯⎯√ - 12⎯ ⎯⎯⎯√ 1.5⎯ ⎯⎯⎯⎯√

Problem 5(from Unit 7, Lesson 4)Write each expression as a single power of 10.

1.

2.

Solution1.

2.

Problem 6(from Unit 4, Lesson 15)Andre is ordering ribbon to make decorations for a school event. For his design,he needs exactly 50.25 meters of blue and green ribbon. There has to be 50%more blue ribbon than green ribbon, plus an extra 6.5 meters of blue ribbon foraccents. How much of each color of ribbon does Andre need to order?

SolutionAndre needs to order 17.5 meters of green ribbon and 32.5 meters of blueribbon. Strategies vary. Sample strategy: Let represent the length of blueribbon and represent the length of green ribbon. Then and

. Substituting in for in the second equation gives . Solving for gives . Since the two kinds of

ribbon must combine to make 50.25 meters, then , so meters.

Lesson 8Problem 1

Solution (because and )

(because and )

because

because

because

Problem 2(from Unit 8, Lesson 7)

105 ⋅ 100

109

100

105

109

bg b = 1.5g + 6.25

b + g = 50.25 1.5g + 6.25 b(1.5g + 6.25) + g = 50.25 g g = 17.5

b = 50.25−17.5 b = 32.5

Find the exact value of each variablethat represents a side length in a righttriangle.

h = 6 100−64 = 36 = 636⎯ ⎯⎯⎯√

k = 2.5 42.25−36 = 6.25 = 2.56.25⎯ ⎯⎯⎯⎯⎯⎯√

m = 21⎯ ⎯⎯⎯√ 25−4 = 21

n = 90⎯ ⎯⎯⎯√ 100−10 = 90

p = 17⎯ ⎯⎯⎯√ 85−68 = 17

A right triangle has side lengths of , , and units. The longest side has alength of units. Complete each equation to show three relations among , ,and .

1. 2. 3.

Solution1. or

2.

3.

Problem 3(from Unit 8, Lesson 7)What is the exact length of each line segment? Explain or show your reasoning.(Each grid square represents 1 square unit.)

1.

2.

3.

Solution1. 4 units. The segment is along the gridlines, so count the squares.

2. because

3. because

Problem 4(from Unit 7, Lesson 15)In 2015, there were roughly high school football players and professional football players in the United States. About how many times morehigh school football players are there? Explain how you know.

SolutionThere are approximately 500 times more high school football players.

Problem 5(from Unit 7, Lesson 6)Evaluate:

1.

a b cc a b

c

=c2=a2

=b2

= +c2 a2 b2 = +c2 b2 a2

= −a2 c2 b2

= −b2 c2 a2

20⎯ ⎯⎯⎯√ + = 2042 22

41⎯ ⎯⎯⎯√ + = 4142 52

1 × 106 2 × 103

= 0.5 × = 5 ×1×106

2×103 103 102

( )12

3

-3

2.

Solution1.

2. 8

Problem 6(from Unit 6, Lesson 6)Here is a scatter plot of weight vs. age for different Dobermans. The model,represented by , is graphed with the scatter plot. Here, represents age in weeks, and represents weight in pounds.

1. What does the slope mean in this situation?

2. Based on this model, how heavy would you expect a newborn Doberman tobe?

Solution1. The slope means that a doberman can be expected to gain 2.45 pounds

per week.

2. 1.22 pounds (the -intercept of the function).

Lesson 9Problem 1Which of these triangles are definitely right triangles? Explain how you know.(Note that not all triangles are drawn to scale.)

SolutionB, D, and E are right triangles. A and C are not.

A: is false

( )12

-3

18

y = 2.45x + 1.22 xy

y

+ 2 =92 121 142

+ =2 2 2

B: is true

C: is false

D: is true

E: is true

Problem 2A right triangle has a hypotenuse of 15 cm. What are possible lengths for the twolegs of the triangle? Explain your reasoning.

SolutionAnswers vary. Sample responses: and 5; and 10. If the legs of thetriangle are and , then we can set up the equation . This means

and must sum to 225. If , then . If , then .

Problem 3(from Unit 8, Lesson 8)In each part, and represent the length of a leg of a right triangle, and represents the length of its hypotenuse. Find the missing length, given the othertwo lengths.

1.

2.

Solution1. . If and then .

2. . If and then .

Problem 4(from Unit 8, Lesson 6)For which triangle does the Pythagorean Theorem express the relationshipbetween the lengths of its three sides?

SolutionB

+ =50⎯ ⎯⎯⎯√ 2 50⎯ ⎯⎯⎯√ 2 102

+ =162 302 352

+ =102 10.52 14.52

+ =3⎯⎯√ 2 13⎯ ⎯⎯⎯√ 2 42

200⎯ ⎯⎯⎯⎯⎯√ 125⎯ ⎯⎯⎯⎯⎯√a b + =a2 b2 152

a2 b2 = 25a2 b = 200 = 100a2

= 125b2

a b c

a = 12, b = 5, c = ?

a = ?, b = 21, c = 29

c = 13 a = 12 b = 5 + =122 52 c2

a = 20 b = 21 c = 29 + =a2 212 292

Problem 5(from Unit 4, Lesson 5)Andre makes a trip to Mexico. He exchanges some dollars for pesos at a rate of20 pesos per dollar. While in Mexico, he spends 9000 pesos. When he returns,he exchanges his pesos for dollars (still at 20 pesos per dollar). He gets back the amount he started with. Find how many dollars Andre exchanged for pesosand explain your reasoning. If you get stuck, try writing an equation representingAndre’s trip using a variable for the number of dollars he exchanged.

Solution500 dollars. Sample reasoning: , where represents the numberof dollars he exchanged. Rewrite the equation as , and thensolve to find .

Lesson 10Problem 1A man is trying to zombie-proof his house. He wants to cut a length of wood thatwill brace a door against a wall. The wall is 4 feet away from the door, and hewants the brace to rest 2 feet up the door. About how long should he cut thebrace?

SolutionAround 4.5 feet. Solving , we get , which is approximately4.5.

Problem 2At a restaurant, a trash can's opening is rectangular and measures 7 inches by 9inches. The restaurant serves food on trays that measure 12 inches by 16inches. Jada says it is impossible for the tray to accidentally fall through the trashcan opening because the shortest side of the tray is longer than either edge ofthe opening. Do you agree or disagree with Jada’s explanation? Explain yourreasoning.

SolutionI disagree. Explanations vary. Sample explanation: It is impossible for the tray tofall through the opening, but not for the reason Jada gives. The longestdimension of the trash can opening is the diagonal. The diagonal is incheslong, because . The diagonal is between 11 and 12 inches long,because . The tray cannot fall through the opening because thediagonal is a little shorter than the shortest dimension of the tray.

Problem 3(from Unit 8, Lesson 9)Select all the sets that are the three side lengths of right triangles.

110

=20x−900020

x10 x

20x−9000 = 2xx = 500

+ =22 42 b2 b = 20⎯ ⎯⎯⎯√

130⎯ ⎯⎯⎯⎯⎯√+ = 13072 92

< 130 <112 122

A. 8, 7, 15

B. 4, 10,

C. , 11,

D. , 2,

SolutionB, C

Problem 4(from Unit 7, Lesson 10)For each pair of numbers, which of the two numbers is larger? How many timeslarger?

1. and

2. and

3. and

Solution1. , 3 times larger

2. , 2 times larger

3. , 3 times larger

Problem 5(from Unit 3, Lesson 10)A line contains the point . If the line has negative slope, which of thesepoints could also be on the line?

A. B. C. D.

SolutionC

Problem 6(from Unit 3, Lesson 4)Noah and Han are preparing for a jump rope contest. Noah can jump 40 times in0.5 minutes. Han can jump times in minutes, where . If they bothjump for 2 minutes, who jumps more times? How many more?

SolutionNoah jumps 160 times and Han jumps 156 times, so Han jumps 4 more times.

Lesson 11Problem 1The right triangles are drawn in the coordinate plane, and the coordinates of theirvertices are labeled. For each right triangle, label each leg with its length.

84⎯ ⎯⎯⎯√

8⎯⎯√ 129⎯ ⎯⎯⎯⎯⎯√

1⎯⎯√ 3⎯⎯√

12 ⋅ 109 4 ⋅ 109

1.5 ⋅ 1012 3 ⋅ 1012

20 ⋅ 104 6 ⋅ 105

12 ⋅ 109

3 ⋅ 1012

6 ⋅ 105

(3, 5)

(2, 0)(4, 7)(5, 4)(6, 5)

y x y = 78x

Solution

Problem 2Find the distance between each pair of points. If you get stuck, try plotting thepoints on graph paper.

1. and

2. and

3. and

Solution1. 13

2. 5

3. 10

Problem 3(from Unit 2, Lesson 10)

M = (0, -11) P = (0, 2)

A = (0, 0) B = (-3, -4)

C = (8, 0) D = (0, -6)

Which line has a slope of 0.625, andwhich line has a slope of 1.6? Explainwhy the slopes of these lines are

SolutionSlope of 0.625:

Slope of 1.6:

Construct triangles perpendicular to the axes whose hypotenuses lie on their lineto find the slopes. The slopes of the lines are then the quotient of the length ofthe vertical edge by the length of the horizontal edge.

Problem 4(from Unit 3, Lesson 7)Write an equation for the graph.

0.625 and 1.6.

Solution

Lesson 12Problem 1

1. What is the volume of a cube with a side length ofa. 4 centimeters?

b. feet?

c. units?

2. What is the side length of a cube with a volume ofa. 1,000 cubic centimeters?

b. 23 cubic inches?

c. cubic units?

Solution1.

a. 64 cubic centimeters

b. 11 cubic feet

c. cubic units

2. a. 10 cm

b. inches

c. units

Problem 2Write an equivalent expression that doesn’t use a cube root symbol.

1. 2.

y = 2x + 1.5

11⎯ ⎯⎯⎯√3

s

v

s3

23⎯ ⎯⎯⎯√3

v⎯⎯√3

1⎯⎯√3216⎯ ⎯⎯⎯⎯⎯√3⎯ ⎯⎯⎯⎯⎯⎯⎯√

3. 4.

5.

6. 7.

Solution1. 1

2. 6

3. 20

4.

5.

6. 0.3

7. 0.05

Problem 3(from Unit 8, Lesson 11)Find the distance between each pair of points. If you get stuck, try plotting thepoints on graph paper.

1. and

2. and

Solution1. 9

2.

Problem 4(from Unit 8, Lesson 9)Here is a 15-by-8 rectangle divided into triangles. Is the shaded triangle a righttriangle? Explain or show your reasoning.

SolutionNo, it is not. Use the Pythagorean Theorem to find the length of the interior sidesof the triangle: the lengths are and 10. The longest side is 15, the length ofthe rectangle. Now check whether this triangle’s side lengths make .Because , not 225, the converse of the Pythagorean Theoremstates this triangle is not a right triangle.

Problem 5(from Unit 8, Lesson 10)Here is an equilateral triangle. The length of each side is 2 units. A height isdrawn. In an equilateral triangle, the height divides the opposite side into twopieces of equal length.

1. Find the exact height.

√8000⎯ ⎯⎯⎯⎯⎯⎯⎯√3

164

⎯ ⎯⎯⎯√3

27125

⎯ ⎯⎯⎯⎯⎯√3

0.027⎯ ⎯⎯⎯⎯⎯⎯⎯⎯√30.000125⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√3

14

35

X = (5, 0) Y = (-4, 0)

K = (-21, -29) L = (0, 0)

1282⎯ ⎯⎯⎯⎯⎯⎯⎯√

145⎯ ⎯⎯⎯⎯⎯√+ =a2 b2 c2

145 + 100 = 245

2. Find the area of the equilateral triangle.

3. (Challenge) Using for the length of each side in an equilateral triangle,express its area in terms of .

Solution1. units

2. units

3. The area is units

Lesson 13Problem 1Find the positive solution to each equation. If the solution is irrational, write thesolution using square root or cube root notation.

1.

2.

3.

4.

5.

Solution1.

2.

3.

4.

5.

Problem 2For each cube root, find the two whole numbers that it lies between.

1.

2.

3.

4.

Solution1. 2 and 3

xx

3⎯⎯√

3⎯⎯√ 2

x2 3√4

2

= 216t3

= 15a2

= 8m3

= 343c3

= 181f 3

t = 6

a = 15⎯ ⎯⎯⎯√

m = 2

c = 7

f = 181⎯ ⎯⎯⎯⎯⎯√3

11⎯ ⎯⎯⎯√3

80⎯ ⎯⎯⎯√3

120⎯ ⎯⎯⎯⎯⎯√3

250⎯ ⎯⎯⎯⎯⎯√3

2. 4 and 5

3. 4 and 5

4. 6 and 7

Problem 3Arrange the following values from least to greatest:

Solution, , , , ,

Problem 4(from Unit 8, Lesson 8)Find the value of each variable, to the nearest tenth.

1.

2.

3.

Solution1.

2.

3.

Problem 5(from Unit 8, Lesson 10)A standard city block in Manhattan is a rectangle measuring 80 m by 270 m. Aresident wants to get from one corner of a block to the opposite corner of a blockthat contains a park. She wonders about the difference between cutting acrossthe diagonal through the park compared to going around the park, along thestreets. How much shorter would her walk be going through the park? Roundyour answer to the nearest meter.

Solution

, , π, , ,530⎯ ⎯⎯⎯⎯⎯√3 48⎯ ⎯⎯⎯√ 121⎯ ⎯⎯⎯⎯⎯√ 27⎯ ⎯⎯⎯√3 192

27⎯ ⎯⎯⎯√3 π 48⎯ ⎯⎯⎯√ 530⎯ ⎯⎯⎯⎯⎯√3 192 121⎯ ⎯⎯⎯⎯⎯√

x ≈7.1

f ≈5.4

d ≈15.1

The walk through the park is about 68 m shorter than the walk around the park.Along the streets: . Along the diagonal: solve and get approximately 282. .

Lesson 14Problem 1Andre and Jada are discussing how to write as a decimal.

Andre says he can use long division to divide by to get the decimal.

Jada says she can write an equivalent fraction with a denominator of bymultiplying by , then writing the number of hundredths as a decimal.

1. Do both of these strategies work?

2. Which strategy do you prefer? Explain your reasoning.

3. Write as a decimal. Explain or show your reasoning.

Solution1. Yes, both strategies are effective.

2. Answers vary. Sample responses:I prefer Jada's method because I can calculate it mentally.

I prefer Andre's method because it always works, even if thedenominator is not a factor of 100.

3. . Explanations vary. Sample explanation: , so equals0.85.

Problem 2Write each fraction as a decimal.

1.

2.

3.

4.

Solution1. 0.3

2. 0.99

3. 0.75

4. 2.3

Problem 3Write each decimal as a fraction.

1.

2. 0.0276

3.

80 + 270 = 350 + =802 2702 x2

350−282 = 68

1720

17 20

10055

1720

0.85 =1720 ⋅ 5

585

1001720

9100

⎯ ⎯⎯⎯⎯⎯√99

100

916

⎯ ⎯⎯⎯√2310

0.81⎯ ⎯⎯⎯⎯⎯⎯√

0.04⎯ ⎯⎯⎯⎯⎯⎯√

4. 10.01

Solution1.

2. (or equivalent)

3. (or equivalent)

4. (or equivalent)

Problem 4(from Unit 8, Lesson 13)Find the positive solution to each equation. If the solution is irrational, write thesolution using square root or cube root notation.

1. 2. 3. 4. 5. 6.

Solution1.

2.

3.

4.

5.

6.

Problem 5(from Unit 8, Lesson 10)Here is a right square pyramid.

1. What is the measurement of the slant height of the triangular face of thepyramid? If you get stuck, use a cross section of the pyramid.

2. What is the surface area of the pyramid?

Solution1. 17 units ( and )

2. 800 square units (The pyramid is made from a square and four triangles.The square’s area, in square units, is . Each triangle’s area, insquare units, is . The surface area, in square units, is

.)

910

27610000

15

1001100

= 90x2

= 90p3

= 1z2

= 1y3

= 36w2

= 64h 3

x = 90⎯ ⎯⎯⎯√

p = 90⎯ ⎯⎯⎯√3

z = 1

y = 1

w = 6

h = 4

ℓ

+ = 289152 82 = 17289⎯ ⎯⎯⎯⎯⎯√

= 256162

16 17 = 13612 ⋅ ⋅

256 + 4 136 = 800⋅

Lesson 15Problem 1Elena and Han are discussing how to write the repeating decimal as afraction. Han says that equals . “I calculated because the decimal begins repeating after 3 digits. Then I subtracted to get

. Then I multiplied by to get rid of the decimal: . And finally I divided to get .” Elena says that

equals . “I calculated because one digit repeats. Then Isubtracted to get . Then I did what Han did to get and

.”

Do you agree with either of them? Explain your reasoning.

SolutionBoth strategies are valid. Han and Elena both get fractions that are equal to

. These are equivalent fractions, but Elena's fraction has fewer commonfactors in the numerator and denominator. The equivalent fraction with the lowestpossible denominator is .

Problem 2How are the numbers and the same? How are they different?

SolutionAnswers vary. Sample response: They are the same in that they are both rationalnumbers between 0.4 and 0.5, and the first three digits in their decimalexpansions are the same. They are different in that is greater than because it has a greater digit in the ten-thousandths place. is a terminatingdecimal, while is an infinitely repeating decimal.

Problem 31. Write each fraction as a decimal.

a.

b.

2. Write each decimal as a fraction.

a.

b.

Solution1.

a.

b.

2. a. (or equivalent)

b. (or equivalent)

Problem 4Write each fraction as a decimal.

1. 2.

x = 0.137⎯ ⎯⎯

0.137⎯ ⎯⎯ 1376499900 1000x = 137.777⎯ ⎯⎯

999x = 137.64 10099900x = 13764 x = 13764

99900 0.137⎯ ⎯⎯124900 10x = 1.377⎯ ⎯⎯

9x = 1.24 900x = 124x = 124

900

0.137⎯ ⎯⎯

31225

0.444 0.4⎯ ⎯⎯

0.4⎯ ⎯⎯ 0.4440.444

0.4⎯ ⎯⎯

23

12637

0.75⎯ ⎯⎯⎯⎯⎯

0.3⎯ ⎯⎯

0.6⎯ ⎯⎯

3.405⎯ ⎯⎯⎯⎯⎯⎯⎯⎯

7599

13

595448

3. 4. 5. 6.

Solution1.

2.

3.

4.

5.

6.

Problem 5Write each decimal as a fraction.

1. 2. 3. 4. 5. 6. 7. 8.

Solution1. (or equivalent)

2. (or equivalent)

3. (or equivalent)

4. (or equivalent)

5. (or equivalent)

6. (or equivalent)

7. (or equivalent)

8. (or equivalent)

Problem 6 and . This gives some information about .

Without directly calculating the square root, plot on all three number linesusing successive approximation.

48995997

1005390

0.5⎯ ⎯⎯

1.25

0.48⎯ ⎯⎯⎯⎯⎯

0.05⎯ ⎯⎯⎯⎯⎯

0.07

0.58⎯ ⎯⎯

0.7⎯ ⎯⎯

0.2⎯ ⎯⎯

0.13⎯ ⎯⎯

0.14⎯ ⎯⎯⎯⎯⎯

0.03⎯ ⎯⎯⎯⎯⎯

0.638⎯ ⎯⎯⎯⎯⎯

0.524⎯ ⎯⎯

0.15⎯ ⎯⎯

79

29

215

1499

399

632990

472900

1490

= 4.842.22 = 5.292.32 5⎯⎯√

5⎯⎯√

Solution