# Unit 8: Practice Problem Sets

## Lesson 1

### Problem 1

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

### Problem 2

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

1. 81 square inches
2. $\frac{4}{25}$ cm2
3. 0.49 square units
4. $m^2$ square units

### Problem 3

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

1. 3 inches
2. 7 units
3. 100 cm
4. 40 inches
5. $x$ units

### Problem 4 (from Unit 7, Lesson 14)

Evaluate $(3.1 \times 10^4) \boldcdot (2 \times 10^6)$. Choose the correct answer:

1. $5.1 \times 10^{10}$
2. $5.1 \times 10^{24}$
3. $6.2 \times 10^{10}$
4. $6.2 \times 10^{24}$

### Problem 5 (from Unit 7, Lesson 15)

Noah reads the problem, “Evaluate each expression, giving the answer in scientific notation.” The first problem part is: $5.4 \times 10^5 + 2.3 \times 10^4$. Noah says, “I can rewrite $5.4 \times 10^5$ as $54 \times 10^4$. Now I can add the numbers: $54 \times 10^4 + 2.3 \times 10^4 = 57.3 \times 10^4$.” Do you agree with Noah’s solution to the problem? Explain your reasoning.

### Problem 6 (from Unit 7, Lesson 6)

Select all the expressions that are equivalent to $3^8$.

1. $(3^2)^4$
2. $8^3$
3. $3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3$
4. $(3^4)^2$
5. $\frac{3^6}{3^{\text-2}}$
6. $3^6 \boldcdot 10^2$

## Lesson 2

### Problem 1

A square has an area of 81 square feet. Select all the expressions that equal the side length of this square, in feet.

1. $\frac{81}{2}$

2. $\sqrt{81}$

3. 9

4. $\sqrt{9}$

5. 3

### Problem 2

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

1. 36
2. 37
3. $\frac{100}{9}$
4. $\frac25$
5. 0.0001
6. 0.11

### Problem 3

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

The three squares’ side lengths are $\sqrt{26}$, 4.2, and $\sqrt{11}$.

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

### Problem 4 (from Unit 8, Lesson 1)

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

1. $\frac15$ cm
2. $\frac37$ units
3. $\frac{11}{8}$ inches
4. 0.1 meters
5. 3.5 cm

### Problem 5 (from Unit 7, Lesson 15)

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

1. How many more people live in Russia than in Canada?
2. The Asian countries on this list are Russia, China, and India. The American countries are Canada, the United States, and Brazil. Which has the greater total area: the three Asian countries, or the three American countries?
country area (in km2)
row 1 Russia $1.71 \times 10^7$
row 2 Canada $9.98 \times 10^6$
row 3 China $9.60 \times 10^6$
row 4 United States $9.53 \times 10^6$
row 5 Brazil $8.52 \times 10^6$
row 6 Australia $6.79 \times 10^6$
row 7 India $3.29 \times 10^6$

### Problem 6 (from Unit 7, Lesson 5)

Select all the expressions that are equivalent to $10^{\text-6}$.

1. $\frac{1}{1000000}$
2. $\frac{\text-1}{1000000}$
3. $\frac{1}{10^6}$
4. $10^{8} \boldcdot 10^{\text-2}$
5. $\left(\frac{1}{10}\right)^6$
6. $\frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10}$

## Lesson 3

### Problem 1

Decide whether each number in this list is rational or irrational.

$$\frac {\text{-}13}{3}, \text{ }0.1234, \text{ }\sqrt{37}, \text{ -} 77, \text{ -} \sqrt{100}, \text{ -} \sqrt{12}$$

### Problem 2

Which value is an exact solution of the equation $m^2=14$?

1. 7

2. $\sqrt{14}$

3. 3.74

4. $\sqrt{3.74}$

### Problem 3 (from Unit 8, Lesson 2)

A square has vertices $(0,0), (5,2), (3,7)$, and $(\text-2,5)$. Which of these statements is true?

1. The square’s side length is 5.

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

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

4. The square’s side length is 7.

### Problem 4 (from Unit 7, Lesson 8)

Rewrite each expression in an equivalent form that uses a single exponent.

1. $(10^2)^{\text-3}$
2. $(3^{\text-3})^2$
3. $3^{\text-5} \boldcdot 4^{\text-5}$
4. $2^5 \boldcdot 3^{\text-5}$

### Problem 5 (from Unit 5, Lesson 5)

The graph represents the area of arctic sea ice in square kilometers as a function of the day of the year in 2016.

1. Give an approximate interval of days when the area of arctic sea ice was decreasing.
2. On which days was the area of arctic sea ice 12 million square kilometers?

### Problem 6 (from Unit 4, Lesson 14)

The high school is hosting an event for seniors but will also allow some juniors to attend. The principal approved the event for 200 students and decided the number of juniors should be 25% of the number of seniors. How many juniors will be allowed to attend? If you get stuck, try writing two equations that each represent the number of juniors and seniors at the event.

## Lesson 4

### Problem 1

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

### Problem 2

Plot each number on the $x$-axis: $\sqrt{16},\text{ } \sqrt{35},\text{ } \sqrt{66}$. Consider using the grid to help.

### Problem 3

Use the fact that $\sqrt{7}$ is a solution to the equation $x^2 = 7$ to find a decimal approximation of $\sqrt{7}$ whose square is between 6.9 and 7.1.

### Problem 4 (from Unit 7, Lesson 14)

Graphite is made up of layers of graphene. Each layer of graphene is about 200 picometers, or $200 \times 10^{\text-12}$ meters, thick. How many layers of graphene are there in a 1.6-mm-thick piece of graphite? Express your answer in scientific notation.

### Problem 5 (from Unit 6, Lesson 6)

Here is a scatter plot that shows the number of assists and points for a group of hockey players. The model, represented by $y = 1.5 x + 1.2$, is graphed with the scatter 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 30 assists?

### Problem 6 (from Unit 3, Lesson 5)

The points $(12, 23)$ and $(14, 45)$ lie on a line. What is the slope of the line?

## Lesson 5

### Problem 1

1. Explain how you know that $\sqrt{37}$ is a little more than 6.

2. Explain how you know that $\sqrt{95}$ is a little less than 10.

3. Explain how you know that $\sqrt{30}$ is between 5 and 6.

### Problem 2

Plot each number on the number line: $$6, \sqrt{83}, \sqrt{40}, \sqrt{64}, 7.5$$

### Problem 3

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

### Problem 4 (from Unit 8, Lesson 3)

Select all the irrational numbers in the list. $$\frac23, \frac {\text{-}123}{45}, \sqrt{14}, \sqrt{64}, \sqrt\frac91, \text-\sqrt{99}, \text-\sqrt{100}$$

### Problem 5 (from Unit 8, Lesson 2)

Each grid square represents 1 square unit. What is the exact side length of each square?

### Problem 6 (from Unit 7, Lesson 10)

For each pair of numbers, which of the two numbers is larger? Estimate how many times larger.

1. $0.37 \boldcdot 10^6$ and $700 \boldcdot 10^4$
2. $4.87 \boldcdot 10^4$ and $15 \boldcdot 10^5$
3. $500,000$ and $2.3 \boldcdot 10^8$

### Problem 7 (from Unit 6, Lesson 4)

The scatter plot shows the heights (in inches) and three-point percentages for different 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.

## Lesson 6

### Problem 1

Here is a diagram of an acute triangle and three squares.

Priya says the area of the large unmarked square is 26 square units because $9+17=26$. Do you agree? Explain your reasoning.

### Problem 2

$m$, $p$, and $z$ represent the lengths of the three sides of this right triangle.

Select all the equations that represent the relationship between $m$, $p$, and $z$.

1. $m^2+p^2=z^2$
2. $m^2=p^2+z^2$
3. $m^2=z^2+p^2$
4. $p^2+m^2=z^2$
5. $z^2+p^2=m^2$
6. $p^2+z^2=m^2$

### Problem 3

The lengths of the three sides are given for several right triangles. For each, write an equation that expresses the relationship between the lengths of the three sides.

1. 10, 6, 8
2. $\sqrt5, \sqrt3, \sqrt8$
3. 5, $\sqrt5, \sqrt{30}$
4. 1, $\sqrt{37}$, 6
5. 3, $\sqrt{2}, \sqrt7$

### Problem 4 (from Unit 4, Lesson 1)

Order the following expressions from least to greatest.

$25\div 10$

$250,\!000 \div 1,\!000$

$2.5 \div 1,\!000$

$0.025\div 1$

### Problem 5 (from Unit 8, Lesson 3)

Which is the best explanation for why $\text-\sqrt{10}$ is irrational?

1. $\text- \sqrt{10}$ is irrational because it is not rational.

2. $\text- \sqrt{10}$ is irrational because it is less than zero.

3. $\text- \sqrt{10}$ is irrational because it is not a whole number.

4. $\text- \sqrt{10}$ is irrational because if I put $\text- \sqrt{10}$ into a calculator, I get -3.16227766, which does not make a repeating pattern.

### Problem 6 (from Unit 7, Lesson 15)

A teacher tells her students she is just over 1 and $\frac{1}{2}$ 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?

## Lesson 7

### Problem 1

1. Find the lengths of the unlabeled sides.

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

### Problem 2

Use the areas of the two identical squares to explain why $5^2+12^2=13^2$ without doing any calculations.

### Problem 3 (from Unit 8, Lesson 5)

Each number is between which two consecutive integers?

1. $\sqrt{10}$

2. $\sqrt{54}$

3. $\sqrt{18}$

4. $\sqrt{99}$

5. $\sqrt{41}$

### Problem 4 (from Unit 8, Lesson 3)

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

2. Give three examples of irrational numbers.

### Problem 5 (from Unit 7, Lesson 4)

Write each expression as a single power of 10.

1. $10^5 \boldcdot 10^0$
2. $\frac{10^9}{10^0}$

### 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 for accents. How much of each color of ribbon does Andre need to order?

## Lesson 8

### Problem 1

Find the exact value of each variable that represents a side length in a right triangle.

### Problem 2 (from Unit 8, Lesson 7)

A right triangle has side lengths of $a$, $b$, and $c$ units. The longest side has a length of $c$ units. Complete each equation to show three relations among $a$, $b$, and $c$.

1. $c^2=$

2. $a^2=$

3. $b^2=$

### 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.

### Problem 4 (from Unit 7, Lesson 15)

In 2015, there were roughly $1 \times 10^6$ high school football players and $2 \times 10^3$ professional football players in the United States. About how many times more high school football players are there? Explain how you know.

### Problem 5 (from Unit 7, Lesson 6)

Evaluate:

1. $\left(\frac{1}{2}\right)^3$
2. $\left(\frac{1}{2}\right)^{\text-3}$

### Problem 6 (from Unit 6, Lesson 6)

Here is a scatter plot of weight vs. age for different Dobermans. The model, represented by $y = 2.45x + 1.22$, is graphed with the scatter plot. Here, $x$ represents age in weeks, and $y$ 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 to be?

## Lesson 9

### Problem 1

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

### Problem 2

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

### Problem 3 (from Unit 8, Lesson 8)

In each part, $a$ and $b$ represent the length of a leg of a right triangle, and $c$ represents the length of its hypotenuse. Find the missing length, given the other two lengths.

1. $a=12, b=5, c={?}$
2. $a={?}, b=21, c=29$

### Problem 4 (from Unit 8, Lesson 6)

For which triangle does the Pythagorean Theorem express the relationship between the lengths of its three sides?

### Problem 5 (from Unit 4, Lesson 5)

Andre makes a trip to Mexico. He exchanges some dollars for pesos at a rate of 20 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 $\frac{1}{10}$ the amount he started with. Find how many dollars Andre exchanged for pesos and explain your reasoning. If you get stuck, try writing an equation representing Andre’s trip using a variable for the number of dollars he exchanged.

## Lesson 10

### Problem 1

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

### Problem 2

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

### Problem 3 (from Unit 8, Lesson 9)

Select all the sets that are the three side lengths of right triangles.

1. 8, 7, 15

2. 4, 10, $\sqrt{84}$

3. $\sqrt{8}$, 11, $\sqrt{129}$

4. $\sqrt{1}$, 2, $\sqrt{3}$

### Problem 4 (from Unit 7, Lesson 10)

For each pair of numbers, which of the two numbers is larger? How many times larger?

1. $12 \boldcdot 10^9$ and $4 \boldcdot 10^9$
2. $1.5 \boldcdot 10^{12}$ and $3 \boldcdot 10^{12}$
3. $20 \boldcdot 10^4$ and $6 \boldcdot 10^5$

### Problem 5 (from Unit 3, Lesson 10)

A line contains the point $(3,5)$. If the line has negative slope, which of these points could also be on the line?

1. $(2,0)$
2. $(4,7)$
3. $(5,4)$
4. $(6,5)$

### Problem 6 (from Unit 3, Lesson 4)

Noah and Han are preparing for a jump rope contest. Noah can jump 40 times in 0.5 minutes. Han can jump $y$ times in $x$ minutes, where $y = 78x$. If they both jump for 2 minutes, who jumps more times? How many more?

## Lesson 11

### Problem 1

The right triangles are drawn in the coordinate plane, and the coordinates of their vertices are labeled. For each right triangle, label each leg with its length.

### Problem 2

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

1. $M=(0,\text-11)$ and $P=(0,2)$
2. $A=(0,0)$ and $B=(\text-3, \text-4)$
3. $C=(8,0)$ and $D=(0, \text-6)$

### Problem 3 (from Unit 2, Lesson 10)

Which line has a slope of 0.625, and which line has a slope of 1.6? Explain why the slopes of these lines are 0.625 and 1.6.

### Problem 4 (from Unit 3, Lesson 7)

Write an equation for the graph.

## Lesson 12

### Problem 1

1. What is the volume of a cube with a side length of
1. 4 centimeters?
2. $\sqrt[3]{11}$ feet?
3. $s$ units?
2. What is the side length of a cube with a volume of
1. 1,000 cubic centimeters?
2. 23 cubic inches?
3. $v$ cubic units?

### Problem 2

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

1. $\sqrt[3]{1}$
2. $\sqrt[3]{216}$
3. $\sqrt[3]{8000}$
4. $\sqrt[3]{\frac{1}{64}}$
5. $\sqrt[3]{\frac{27}{125}}$
6. $\sqrt[3]{0.027}$
7. $\sqrt[3]{0.000125}$

### Problem 3 (from Unit 8, Lesson 11)

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

1. $X=(5,0)$ and $Y=(\text-4,0)$
2. $K=(\text-21,\text-29)$ and $L=(0,0)$

### Problem 4 (from Unit 8, Lesson 9)

Here is a 15-by-8 rectangle divided into triangles. Is the shaded triangle a right triangle? Explain or show your reasoning.

### Problem 5 (from Unit 8, Lesson 10)

Here is an equilateral triangle. The length of each side is 2 units. A height is drawn. In an equilateral triangle, the height divides the opposite side into two pieces of equal length.

1. Find the exact height.
2. Find the area of the equilateral triangle.
3. (Challenge) Using $x$ for the length of each side in an equilateral triangle, express its area in terms of $x$.

## Lesson 13

### Problem 1

Find the positive solution to each equation. If the solution is irrational, write the solution using square root or cube root notation.

1. $t^3=216$

2. $a^2=15$

3. $m^3=8$

4. $c^3=343$

5. $f^3=181$

### Problem 2

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

1. $\sqrt[3]{11}$
2. $\sqrt[3]{80}$
3. $\sqrt[3]{120}$
4. $\sqrt[3]{250}$

### Problem 3

Arrange the following values from least to greatest: $$\sqrt[3]{530},\;\sqrt{48},\;\pi,\;\sqrt{121},\;\sqrt[3]{27},\;\frac{19}{2}$$

### Problem 4 (from Unit 8, Lesson 8)

Find the value of each variable, to the nearest tenth.

1.
2.
3.

### Problem 5 (from Unit 8, Lesson 10)

A standard city block in Manhattan is a rectangle measuring 80 m by 270 m. A resident wants to get from one corner of a block to the opposite corner of a block that contains a park. She wonders about the difference between cutting across the diagonal through the park compared to going around the park, along the streets. How much shorter would her walk be going through the park? Round your answer to the nearest meter.

## Lesson 14

### Problem 1

Andre and Jada are discussing how to write $\frac{17}{20}$ as a decimal.

Andre says he can use long division to divide $17$ by $20$ to get the decimal.

Jada says she can write an equivalent fraction with a denominator of $100$ by multiplying by $\frac{5}{5}$, 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 $\frac{17}{20}$ as a decimal. Explain or show your reasoning.

### Problem 2

Write each fraction as a decimal.

1. $\sqrt{\frac{9}{100}}$

2. $\frac{99}{100}$

3. $\sqrt{\frac{9}{16}}$

4. $\frac{23}{10}$

### Problem 3

Write each decimal as a fraction.

1. $\sqrt{0.81}$

2. 0.0276

3. $\sqrt{0.04}$

4. 10.01

### Problem 4 (from Unit 8, Lesson 13)

Find the positive solution to each equation. If the solution is irrational, write the solution using square root or cube root notation.

1. $x^2=90$

2. $p^3=90$

3. $z^2=1$

4. $y^3=1$

5. $w^2=36$

6. $h^3=64$

### Problem 5 (from Unit 8, Lesson 10)

Here is a right square pyramid.

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

2. What is the surface area of the pyramid?

## Lesson 15

### Problem 1

Elena and Han are discussing how to write the repeating decimal $x = 0.13\overline{7}$ as a fraction. Han says that $0.13\overline{7}$ equals $\frac{13764}{99900}$. “I calculated $1000x = 137.77\overline{7}$ because the decimal begins repeating after 3 digits. Then I subtracted to get $999x = 137.64$. Then I multiplied by $100$ to get rid of the decimal: $99900x = 13764$. And finally I divided to get $x = \frac{13764}{99900}$.” Elena says that $0.13\overline{7}$ equals $\frac{124}{900}$. “I calculated $10x = 1.37\overline{7}$ because one digit repeats. Then I subtracted to get $9x = 1.24$. Then I did what Han did to get $900x = 124$ and $x = \frac{124}{900}$.”

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

### Problem 2

How are the numbers $0.444$ and $0.\overline{4}$ the same? How are they different?

### Problem 3

1. Write each fraction as a decimal.
1. $\frac{2}{3}$

2. $\frac{126}{37}$

2. Write each decimal as a fraction.

1. $0.\overline{75}$

2. $0.\overline{3}$

### Problem 4

Write each fraction as a decimal.

1. $\frac{5}{9}$

2. $\frac{5}{4}$

3. $\frac{48}{99}$

4. $\frac{5}{99}$

5. $\frac{7}{100}$

6. $\frac{53}{90}$

### Problem 5

Write each decimal as a fraction.

1. $0.\overline{7}$

2. $0.\overline{2}$

3. $0.1\overline{3}$

4. $0.\overline{14}$

5. $0.\overline{03}$

6. $0.6\overline{38}$

7. $0.52\overline{4}$

8. $0.1\overline{5}$

### Problem 6

$2.2^2 = 4.84$ and $2.3^2 = 5.29$. This gives some information about $\sqrt 5$.

Without directly calculating the square root, plot $\sqrt{5}$ on all three number lines using successive approximation.