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**Lesson 1: Rational and Irrational Numbers**

**Lesson 2: Sets of real Numbers**

**Lesson 3: Ordering Real Numbers**

**Model Quiz**

**Mixed Review**

**Write each fraction or mixed number as a decimal.**

Question 1.

\(\frac{2}{5}\) =

Answer:

0.4

Explanation:

\(\frac{2}{5}\) = \(\frac{2 × 2}{5 × 2}\) = \(\frac{4}{10}\) = 0.4

Question 2.

\(\frac{8}{9}\) =

Answer:

0.88

Explanation:

\(\frac{8}{9}\) = \(\frac{8 × 10}{9 × 10}\) = \(\frac{80}{9 × 10}\) = \(\frac{8.88}{10}\) = 0.88

Question 3.

3 \(\frac{3}{4}\) =

Answer:

3.75

Explanation:

3 \(\frac{3}{4}\) =\(\frac{15}{4}\) = 3.75

Question 4.

\(\frac{7}{10}\) =

Answer:

0.7

Explanation:

\(\frac{7}{10}\) = 0.7

Question 5.

2 \(\frac{3}{8}\) =

Answer:

2.375

Explanation:

2 \(\frac{3}{8}\) = \(\frac{19}{8}\) = 2.375

Question 6.

\(\frac{5}{6}\) =

Answer:

0.833

Explanation:

\(\frac{5}{6}\) = \(\frac{5 × 10}{6 × 10}\) = \(\frac{50}{6 × 10}\) = \(\frac{8.33}{10}\) = 0.833

**Write each decimal as a fraction or mixed number in simplest form**

Question 7.

0.675

\(\frac{□}{□}\)

Answer:

\(\frac{27}{40}\)

Explanation:

\(\frac{0.675 × 1000}{1 × 1000}\) = \(\frac{675}{1000}\) = \(\frac{675/25}{1000/25}\) = \(\frac{27}{40}\)

Question 8.

5.6

______ \(\frac{□}{□}\)

Answer:

5 \(\frac{3}{5}\)

Explanation:

\(\frac{5.6 × 10}{10}\) = \(\frac{56}{10}\) = 5 \(\frac{6}{10}\) = 5 \(\frac{6/2}{10/2}\) = 5 \(\frac{3}{5}\)

Question 9.

0.44

\(\frac{□}{□}\)

Answer:

\(\frac{11}{25}\)

Explanation:

\(\frac{0.44 × 100}{1 × 100}\) = \(\frac{44}{100}\) = \(\frac{44/4}{100/4}\) = \(\frac{11}{25}\)

Question 10.

0.\(\bar{4}\)

\(\frac{□}{□}\)

Answer:

\(\frac{4}{9}\)

Explanation:

Let x = 0.\(\bar{4}\)

Now, 10x = 4.\(\bar{4}\)

10x – x = 4.\(\bar{4}\) – 0.\(\bar{4}\)

9x = 4

x = \(\frac{4}{9}\)

Question 11.

0.\(\overline { 26 } \)

\(\frac{□}{□}\)

Answer:

\(\frac{26}{99}\)

Explanation:

Let x = 0.\(\overline {26}\)

Now, 100x = 26.\(\overline{26}\)

100x – x = 26.\(\overline{26}\) – 0.\(\overline {26}\)

99x = 26

x = \(\frac{26}{99}\)

Question 12.

0.\(\overline { 325 } \)

\(\frac{□}{□}\)

Answer:

\(\frac{325}{999}\)

Explanation:

Let x = 0.\(\overline {325}\)

Now, 1000x = 325.\(\overline{325}\)

1000x – x = 325.\(\overline{325}\) – 0.\(\overline {325}\)

999x = 325

x = \(\frac{325}{999}\)

**Solve each equation for x**

Question 13.

x^{2} = 144

± ______

Answer:

x=±12

Explanation:

x^{2} = 144

Taking square roots on both the sides

√x2=±√144

x = ±12

Question 14.

x^{2} = \(\frac{25}{289}\)

± \(\frac{□}{□}\)

Answer:

x = ±\(\frac{5}{17}\)

Explanation:

x^{2} = \(\frac{25}{289}\)

Taking square roots on both the sides

√x2=±√\(\frac{25}{289}\)

x = ±\(\frac{5}{17}\)

Question 15.

x^{3} = 216

______

Answer:

x = 6

Explanation:

x^{3} = 216

Taking cube roots on both the sides

^{3}√x^{3}= ^{3}√216

x = 6

**Approximate each irrational number to two decimal places without a calculator.**

Question 16.

\(\sqrt { 5 } \) ≈ ______

Answer:

2.236

Explanation:

x = \(\sqrt { 5 } \)

The 5 is in between 4 and 6

Take square root of each year

√4 < √5 < √6

2 < √5 < 3

√5 = 2.2

(2.2)² = 4.84

(2.25)² = 5.06

(2.5)³ = 5.29

A good estimate for √5 is 2.25

Question 17.

\(\sqrt { 3 } \) ≈ ______

Answer:

1.75

Explanation:

\(\sqrt { 3 } \)

1 < 3 < 4

√1 < √3 < √4

1 < √3 < 2

√3 = 1.6

(1.65)² = 2.72

(1.7)² = 2.89

(1.75)² = 3.06

A good estimate for √3 is 1.75

Question 18.

\(\sqrt { 10 } \) ≈ ______

Answer:

3.15

Explanation:

\(\sqrt { 10 } \)

9 < 10 < 16

√9 < √10 < √16

3 < √10 < 4

√10 = 3.1

(3.1)² = 9.61

(3.15)² = 9.92

(3.2)² = 10.24

A good estimate for √10 is 3.15

Question 19.

What is the difference between rational and irrational numbers?

Type below:

_____________

Answer:

Rational number can be expressed as a ration of two integers such as 5/2

Irrational number cannot be expressed as a ratio of two integers such as √13

Explanation:

A rational number is a number that can be express as the ratio of two integers. A number that cannot be expressed that way is irrational.

Question 20.

A \(\frac{7}{16}\)-inch-long bolt is used in a machine. What is the length of the bolt written as a decimal?

______ -inch-long

Answer:

0.4375 inch

Explanation:

The length of the bolt is \(\frac{7}{16}\)-inch

Let, x = \(\frac{7}{16}\)

Multiplying by 125 on both nominator and denominator

x = \(\frac{7×125}{16×125}\) = \(\frac{875}{2000}\) =\(\frac{437.5}{1000}\) = 0.4375

Question 21.

The weight of an object on the moon is \(\frac{1}{6}\) its weight on Earth. Write \(\frac{1}{6}\) as a decimal.

______

Answer:

0.1666

Explanation:

The weight of the object on the moon is \(\frac{1}{6}\)

Let, x = \(\frac{1}{6}\)

Multiplying by 100 on both nominator and denominator

x = \(\frac{1×100}{6×100}\) = \(\frac{16.6}{100}\) =0.166

Question 22.

The distance to the nearest gas station is 2 \(\frac{4}{5}\) kilometers. What is this distance written as a decimal?

______

Answer:

2.8

Explanation:

The distance of the nearest gas station is 2 \(\frac{4}{5}\)

Let, x = 2 \(\frac{4}{5}\)

Multiplying by 100 on both nominator and denominator

x = 2 \(\frac{4×100}{5×100}\) = \(\frac{80}{100}\) =0.8

Question 23.

A baseball pitcher has pitched 98 \(\frac{2}{3}\) innings. What is the number of innings written as a decimal?

______

Answer:

98.6

Explanation:

A baseball pitcher has pitched 98 \(\frac{2}{3}\) innings.

98 \(\frac{2}{3}\) = 98 + 2/3

= (294/3) + (2/3)

296/3

98.6

Question 24.

A heartbeat takes 0.8 second. How many seconds is this written as a fraction?

\(\frac{□}{□}\)

Answer:

\(\frac{4}{5}\)

Explanation:

A heartbeat takes 0.8 seconds.

0.8

There are 8 tenths.

8/10 = 4/5

Question 25.

There are 26.2 miles in a marathon. Write the number of miles using a fraction.

\(\frac{□}{□}\)

Answer:

26\(\frac{1}{5}\)

Explanation:

There are 26.2 miles in a marathon.

26.2 miles

262/10

131/5

26 1/5 miles

Question 26.

The average score on a biology test was 72.\(\bar{1}\). Write the average score using a fraction.

\(\frac{□}{□}\)

Answer:

80 \(\frac{1}{9}\)

Explanation:

The average score on a biology test was 72.\(\bar{1}\).

72.\(\bar{1}\)

Let x = 72.\(\bar{1}\)

10x = 10(72.\(\bar{1}\))

10x = 721.1

-x = -0.1

9x = 721

x = 721/9

x = 80 1/9

Question 27.

The metal in a penny is worth about 0.505 cent. How many cents is this written as a fraction?

\(\frac{□}{□}\)

Answer:

\(\frac{101}{200}\)

Explanation:

The metal in a penny is worth about 0.505 cent.

0.505 cent

505 thousandths

505/1000

101/200 cents

Question 28.

Multistep An artist wants to frame a square painting with an area of 400 square inches. She wants to know the length of the wood trim that is needed to go around the painting.

a. If x is the length of one side of the painting, what equation can you set up to find the length of a side?

x^{2} = ______

Answer:

x² = 400

Explanation:

The area of a square is the square of its equal side, x

x² = 400

Question 28.

b. Solve the equation you wrote in part a. How many solutions does the equation have?

x = ± ______

Answer:

x = ± 20

Explanation:

Take the square root on both sides. Solve

x = ± 20

Question 28.

c. Do all of the solutions that you found in part b make sense in the context of the problem? Explain.

Type below:

_____________

Answer:

No. Both values of x do not make sense.

Explanation:

The length cannot be negative, hence negative value does not make sense.

No. Both values of x do not make sense.

Question 28.

d. What is the length of the wood trim needed to go around the painting?

P = ______ inches

Answer:

Length P = 20 + 2y

Question 29.

Analyze Relationships To find \(\sqrt { 15 } \), Beau found 3^{2} = 9 and 4^{2} = 16. He said that since 15 is between 9 and 16, \(\sqrt { 15 } \) must be between 3 and 4. He thinks a good estimate for \(\sqrt { 15 } \) is \(\frac { 3+4 }{ 2 } \) = 3.5. Is Beau’s estimate high, low, or correct? Explain.

_____________

Answer:

3.85

Explanation:

15 is closer to 16

√15 is closer to √16

Beau’s estimate is low.

(3.8)² = 14.44

(3.85)² = 14.82

(3.9)² = 15.21

√15 is 3.85

Question 30.

Justify Reasoning What is a good estimate for the solution to the equation x^{3} = 95? How did you come up with your estimate?

x ≈ ______

Answer:

x ≈ 4.55

Explanation:

^{3}√x = 95

x = ^{3}√95

64 < 95 < 125

Take the cube root of each number

^{3}√64 < ^{3}√95 < ^{3}√125

4 < ^{3}√95 < 5

^{3}√95 = 4.6

(4.5)³ = 91.125

(4.55)³ = 94.20

(4.6)³ = 97.336

^{3}√95 = 4.55

Question 31.

The volume of a sphere is 36π ft^{3}. What is the radius of the sphere? Use the formula V = \(\frac { 4 }{ 3 } \)πr^{3} to find your answer.

r = ______

Answer:

r = 3

Explanation:

V = 4/3 πr³

36π = 4/3 πr³

r³ = 36π/π . 3/4

r³ = 27

r = ^{3}√27

r = 3

**FOCUS ON HIGHER ORDER THINKING**

Question 32.

Draw Conclusions Can you find the cube root of a negative number? If so, is it positive or negative? Explain your reasoning.

_____________

Answer:

Yes

Explanation:

Yes. The cube root of a negative number would be negative. Because the product of three negative signs is always negative.

Question 33.

Make a Conjecture Evaluate and compare the following expressions.

\(\sqrt { \frac { 4 }{ 25 } } \) and \(\frac { \sqrt { 4 } }{ \sqrt { 25 } } \) \(\sqrt { \frac { 16 }{ 81 } } \) and \(\frac { \sqrt { 16 } }{ \sqrt { 81 } } \) \(\sqrt { \frac { 36 }{ 49 } } \) and\(\frac { \sqrt { 36 } }{ \sqrt { 49 } } \)

Use your results to make a conjecture about a division rule for square roots. Since division is multiplication by the reciprocal, make a conjecture about a multiplication rule for square roots.

Expressions are: _____________

Answer:

Evaluating and comparing

√4/25 = 2/5

√16/81 = 4/9

√36/49 = 6/7

Conjecture about a division rule for square roots

√a/√b = √(a/b)

Conjecture about a multiplication rule for square roots

√a × √b

Question 34.

Persevere in Problem Solving

The difference between the solutions to the equation x^{2} = a is 30. What is a? Show that your answer is correct.

_____

Answer:

30

Explanation:

x^{2} = a

x = ±√a

√a – (-√a) = 30

√a + √a = 30

2√a = 30

√a = 15

a = 225

x^{2} = 225

x = ±225

x = ±15

15 – (-15) = 15 + 15 = 30

**Write all names that apply to each number.**

Question 1.

\(\frac{7}{8}\)

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

Question 2.

\(\sqrt { 36 } \)

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

Explanation:

\(\sqrt { 36 } \) = 6

Question 3.

\(\sqrt { 24 } \)

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

e. Irrational Numbers

Question 4.

0.75

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

Question 5.

0

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

Question 6.

−\(\sqrt { 100 } \)

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

c. Integers

Explanation:

−\(\sqrt { 100 } \) = – 10

Question 7.

5.\(\overline { 45 } \)

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

Question 8.

−\(\frac{18}{6}\)

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

c. Integers

Explanation:

−\(\frac{18}{6}\) = -3

**Tell whether the given statement is true or false. Explain your choice.**

Question 9.

All whole numbers are rational numbers.

i. True

ii. False

Answer:

i. True

Explanation:

All whole numbers are rational numbers.

Whole numbers are a subset of the set of rational numbers and can be written as ratio of the whole number to 1.

Question 10.

No irrational numbers are whole numbers.

i. True

ii. False

Answer:

i. True

Explanation:

True. Whole numbers are ration numbers.

**Identify the set of numbers that best describes each situation. Explain your choice.**

Question 11.

the change in the value of an account when given to the nearest dollar

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

c. Integer Numbers

Explanation:

The change can be a whole dollar amount and can be positive, negative or zero.

Question 12.

the markings on a standard ruler

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

b. Rational Numbers

Explanation:

The ruler is marked every 1/16t inch.

**ESSENTIAL QUESTION CHECK-IN**

Question 13.

What are some ways to describe the relationships between sets of numbers?

Answer:

There are two ways that we have been using until now to describe the relationships between sets of numbers

- Using a scheme or a diagram as the one on page 15.
- Verbal description, for example, “All irrational numbers are real numbers.”

**Write all names that apply to each number. Then place the numbers in the correct location on the Venn diagram.**

Question 14.

\(\sqrt { 9 } \)

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

Explanation:

\(\sqrt { 9 } \) = 3

Question 15.

257

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

Question 16.

\(\sqrt { 50 } \)

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

e. Irrational Numbers

Question 17.

8 \(\frac{1}{2}\)

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

Question 18.

16.6

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

Question 19.

\(\sqrt { 16 } \)

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

Explanation:

\(\sqrt { 16 } \) = 4

**Identify the set of numbers that best describes each situation. Explain your choice.**

Question 20.

the height of an airplane as it descends to an airport runway

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

d. Whole Numbers

Explanation:

Whole. The height of an airplane as it descents to an airport runway is a whole number greater than 0

Question 21.

the score with respect to par of several golfers: 2, – 3, 5, 0, – 1

Options:

a. Real Numbers

b. Rational Numbers

c. Integer Numbers

d. Whole Numbers

e. Irrational Numbers

Answer:

c. Integer Numbers

Explanation:

Integers. The scores are counting numbers, their opposites, and zero.

Question 22.

Critique Reasoning Ronald states that the number \(\frac{1}{11}\) is not rational because, when converted into a decimal, it does not terminate. Nathaniel says it is rational because it is a fraction. Which boy is correct? Explain.

i. Ronald

ii. Nathaniel

Answer:

ii. Nathaniel

Explanation:

Nathaniel is correct.

A fraction is a rational real number, even if it is not a terminating decimal.

Question 23.

Critique Reasoning The circumference of a circular region is shown. What type of number best describes the diameter of the circle? Explain your answer.

Options:

a. Real Numbers

b. Rational Numbers

c. Irrational Numbers

d. Integers

e. Whole Numbers

Answer:

e. Whole Numbers

Explanation:

Circumference of the circle

A = 2πr

π = 2πr

Diameter is twice the radius

2r = 1

Whole

Question 24.

Critical Thinking A number is not an integer. What type of number can it be?

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

b. Rational Numbers

e. Irrational Numbers

Question 25.

A grocery store has a shelf with half-gallon containers of milk. What type of number best represents the total number of gallons?

Options:

a. Real Numbers

b. Rational Numbers

c. Integers

d. Whole Numbers

e. Irrational Numbers

Answer:

b. Rational Numbers

**FOCUS ON HIGHER ORDER THINKING**

Question 26.

Explain the Error Katie said, “Negative numbers are integers.” What was her error?

Type below:

_______________

Answer:

Her error is that she stated that all negative numbers are integers. Some negative numbers are integers such as -4 but some are not such an -0.8

Question 27.

Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational? Explain.

Type below:

_______________

Answer:

Not always.

Explanation:

Not always.

If the calculator shows a terminating decimal, the number is rational but otherwise, it is not possible as you can only see a few digits.

Question 28.

Draw Conclusions The decimal 0.\(\bar{3}\) represents \(\frac{1}{3}\). What type of number best describes 0.\(\bar{9}\) , which is 3 × 0.\(\bar{3}\)? Explain.

Type below:

_______________

Answer:

1

Explanation:

let x = 0.9999999

10x = 9.99999999

10x = 9 + 0.999999999

10x = 9 + x

9x = 9

x=1.

Question 29.

Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form. Why is this?

Answer:

Because irrational numbers are nonrepeating, otherwise they could be represented as a fraction. Although a potential counter-example to this claim is that some irrational numbers can only be represented in decimal form, for example, 0.1234567891011121314151617…, 0.24681012141618202224…, 0.101101110111101111101111110… are all irrational numbers.

**Compare. Write <, >, or =.**

Question 1.

\(\sqrt { 3 } \) + 2 ________ \(\sqrt { 3 } \) + 3

Answer:

\(\sqrt { 3 } \) + 2 < \(\sqrt { 3 } \) + 3

Explanation:

\(\sqrt { 3 } \) is between 1 and 2

\(\sqrt { 3 } \) + 2 is between 3 and 4

\(\sqrt { 3 } \) + 3 is between 4 and 5

\(\sqrt { 3 } \) + 2 < \(\sqrt { 3 } \) + 3

Question 2.

\(\sqrt { 11 } \) + 15 _______ \(\sqrt { 8 } \) + 15

Answer:

\(\sqrt { 11 } \) + 15 > \(\sqrt { 8 } \) + 15

Explanation:

\(\sqrt { 11 } \) is between 3 and 4

\(\sqrt { 8 } \) is between 2 and 3

\(\sqrt { 11 } \) + 15 is between 18 and 19

\(\sqrt { 8 } \) + 15 is between 17 and 18

\(\sqrt { 11 } \) + 15 > \(\sqrt { 8 } \) + 15

Question 3.

\(\sqrt { 6 } \) + 5 _______ 6 + \(\sqrt { 5 } \)

Answer:

\(\sqrt { 6 } \) + 5 < 6 + \(\sqrt { 5 } \)

Explanation:

\(\sqrt { 6 } \) is between 2 and 3

\(\sqrt { 5 } \) is between 2 and 3

\(\sqrt { 6 } \) is between 7 and 8

\(\sqrt { 5 } \) is between 8 and 9

\(\sqrt { 6 } \) + 5 < 6 + \(\sqrt { 5 } \)

Question 4.

\(\sqrt { 9 } \) + 3 _______ 9 + \(\sqrt { 3 } \)

Answer:

\(\sqrt { 9 } \) + 3 < 9 + \(\sqrt { 3 } \)

Explanation:

\(\sqrt { 9 } \) + 3

9 + \(\sqrt { 3 } \)

\(\sqrt { 3 } \) is between 1 and 2

\(\sqrt { 9 } \) + 3 = 3 + 3 = 6

9 + \(\sqrt { 3 } \) is between 10 and 11

\(\sqrt { 9 } \) + 3 < 9 + \(\sqrt { 3 } \)

Question 5.

\(\sqrt { 17 } \) – 3 _______ -2 + \(\sqrt { 5 } \)

Answer:

\(\sqrt { 17 } \) – 3 > -2 + \(\sqrt { 5 } \)

Explanation:

\(\sqrt { 17 } \) is between 4 and 5

\(\sqrt { 5 } \) is between 2 and 3

\(\sqrt { 17 } \) – 3 is between 1 and 2

-2 + \(\sqrt { 5 } \) is between 0 and 1

\(\sqrt { 17 } \) – 3 > -2 + \(\sqrt { 5 } \)

Question 6.

10 – \(\sqrt { 8 } \) _______ 12 – \(\sqrt { 2 } \)

Answer:

10 – \(\sqrt { 8 } \) < 12 – \(\sqrt { 2 } \)

Explanation:

\(\sqrt { 8 } \) is between 2 and 3

\(\sqrt { 2 } \) is between 1 and 2

10 – \(\sqrt { 8 } \) is between 8 and 7

12 – \(\sqrt { 2 } \) is between 11 and 10

10 – \(\sqrt { 8 } \) < 12 – \(\sqrt { 2 } \)

Question 7.

\(\sqrt { 7 } \) + 2 _______ \(\sqrt { 10 } \) – 1

Answer:

\(\sqrt { 7 } \) + 2 > \(\sqrt { 10 } \) – 1

Explanation:

\(\sqrt { 7 } \) is between 2 and 3

\(\sqrt { 10 } \) is between 3 and 4

\(\sqrt { 7 } \) + 2 is between 4 and 5

\(\sqrt { 10 } \) – 1 is between 2 and 3

\(\sqrt { 7 } \) + 2 > \(\sqrt { 10 } \) – 1

Question 8.

\(\sqrt { 17 } \) + 3 _______ 3 + \(\sqrt { 11 } \)

Answer:

\(\sqrt { 17 } \) + 3 > 3 + \(\sqrt { 11 } \)

Explanation:

\(\sqrt { 17 } \) is between 4 and 5

\(\sqrt { 11 } \) is between 3 and 4

\(\sqrt { 17 } \) + 3 is between 7 and 8

3 + \(\sqrt { 11 } \) is between 6 and 7

\(\sqrt { 17 } \) + 3 > 3 + \(\sqrt { 11 } \)

Question 9.

Order \(\sqrt { 3 } \), 2 π, and 1.5 from least to greatest. Then graph them on the number line.

\(\sqrt { 3 } \) is between _________ and _____________ , so \(\sqrt { 3 } \) ≈ ____________.

π ≈ 3.14, so 2 π ≈ _______________.

From least to greatest, the numbers are ______________, _____________________ ,_________________.

Type below:

___________

Answer:

1.5, \(\sqrt { 3 } \), 2 π

Explanation:

\(\sqrt { 3 } \) is between 1.7 and 1.75

π = 3.14; 2 π = 6.28

1.5, \(\sqrt { 3 } \), 2 π

Question 10.

Four people have found the perimeter of a forest using different methods. Their results are given in the table. Order their calculations from greatest to least.

Type below:

___________

Answer:

\(\sqrt { 17 } \) – 2, 1+ π/2, 2.5, 12/5

Explanation:

\(\sqrt { 17 } \) – 2

\(\sqrt { 17 } \) is between 4 and 5

Since, 17 is closer to 16, the estimated value is 4.1

1+ π/2

1 + (3.14/2) = 2.57

12/5 = 2.4

2.5

\(\sqrt { 17 } \) – 2, 1+ π/2, 2.5, 12/5

**ESSENTIAL QUESTION CHECK-IN**

Question 11.

Explain how to order a set of real numbers.

Type below:

___________

Answer:

Evaluate the given numbers and write in decimal form. Plot on number line and arrange the numbers accordingly.

**Order the numbers from least to greatest.**

Question 12.

\(\sqrt { 7 } \), 2, \(\frac { \sqrt { 8 } }{ 2 } \)

Type below:

____________

Answer:

\(\frac { \sqrt { 8 } }{ 2 } \), 2, \(\sqrt { 7 } \)

Explanation:

\(\sqrt { 7 } \), 2, \(\frac { \sqrt { 8 } }{ 2 } \)

\(\sqrt { 7 } \) is between 2 and 3

Since 7 is closer to 9, (2.65)² = 7.02, hence the estimated value is 2.65

\(\frac { \sqrt { 8 } }{ 2 } \)

\(\sqrt { 8 } \) is between 2 and 3

Since 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85

2.85/2 = 1.43

\(\frac { \sqrt { 8 } }{ 2 } \), 2, \(\sqrt { 7 } \)

Question 13.

\(\sqrt { 10 } \), π, 3.5

Type below:

____________

Answer:

π, \(\sqrt { 10 } \), 3.5

Explanation:

\(\sqrt { 10 } \), π, 3.5

\(\sqrt { 10 } \) is between 3 and 4

Since, 10 is closer to 9, (3.15)² = 9.92, hence the estimated value is 3.15

π = 3.14

3.5

π, \(\sqrt { 10 } \), 3.5

Question 14.

\(\sqrt { 220 } \), −10, \(\sqrt { 100 } \), 11.5

Type below:

____________

Answer:

-10, √100, 11.5, √220

Explanation:

\(\sqrt { 220 } \), −10, \(\sqrt { 100 } \), 11.5

196 < 220 < 225

√196 < √220 < √225

14 < √220 < 15

√220 = 14.5

√100 = 10

-10, √100, 11.5, √220

Question 15.

\(\sqrt { 8 } \), −3.75, 3, \(\frac{9}{4}\)

Type below:

____________

Answer:

−3.75, \(\frac{9}{4}\), \(\sqrt { 8 } \)

Explanation:

\(\sqrt { 8 } \), −3.75, 3, \(\frac{9}{4}\)

\(\sqrt { 8 } \) is between 2 and 3

Since, 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85

-3.75 = 3

9/4 = 2.25

−3.75, \(\frac{9}{4}\), \(\sqrt { 8 } \)

Question 16.

Your sister is considering two different shapes for her garden. One is a square with side lengths of 3.5 meters, and the other is a circle with a diameter of 4 meters.

a. Find the area of the square.

_______ m^{2}

Answer:

(3.5)² = 12.25

Explanation:

Area of the square = x²

Area = (3.5)² = 12.25

Question 16.

b. Find the area of the circle.

_______ m^{2}

Answer:

π(2)² = 12.56

Explanation:

Area of the circle = πr² where r = d/2 = 4/2 = 2

Area = π(2)² = 12.56

Question 16.

c. Compare your answers from parts a and b. Which garden would give your sister the most space to plant?

___________

Answer:

12.25 < 12.56

The circle will give more space

Question 17.

Winnie measured the length of her father’s ranch four times and got four different distances. Her measurements are shown in the table.

a. To estimate the actual length, Winnie first approximated each distance to the nearest hundredth. Then she averaged the four numbers. Using a calculator, find Winnie’s estimate.

______

Answer:

7.4815

Explanation:

\(\sqrt { 60 } \) = 7.75

58/8 = 7.25

7.3333

7 3/5 = 7.60

Average = (7.75 + 7.25 + 7.33 + 7.60)/4 = 7.4815

Question 17.

b. Winnie’s father estimated the distance across his ranch to be \(\sqrt { 56 } \) km. How does this distance compare to Winnie’s estimate?

____________

Answer:

They are nearly identical

Explanation:

\(\sqrt { 56 } \) = 7.4833

They are nearly identical

**Give an example of each type of number.**

Question 18.

a real number between \(\sqrt { 13 } \) and \(\sqrt { 14 } \)

Type below:

____________

Answer:

A real number between \(\sqrt { 13 } \) and \(\sqrt { 14 } \)

Example: 3.7

Explanation:

\(\sqrt { 13 } \) = 3.61

\(\sqrt { 13 } \) = 3.74

A real number between \(\sqrt { 13 } \) and \(\sqrt { 14 } \)

Example: 3.7

Question 19.

an irrational number between 5 and 7

Type below:

____________

Answer:

An irrational number between 5 and 7

Example: \(\sqrt { 29 } \)

Explanation:

5² = 25 and 7² = 49

An irrational number between 5 and 7

Example: \(\sqrt { 29 } \)

Question 20.

A teacher asks his students to write the numbers shown in order from least to greatest. Paul thinks the numbers are already in order. Sandra thinks the order should be reversed. Who is right?

_____________

Answer:

Neither are correct

Explanation:

\(\sqrt { 115 } \), 115/11, 10.5624

\(\sqrt { 115 } \) is between 10 and 11

Since, 115 is closer to 121, (10.7)² = 114.5, hence the estimated value is 10.7

115/11 = 10.4545

10.5624

Neither are correct

Question 21.

Math History

There is a famous irrational number called Euler’s number, symbolized with an e. Like π, its decimal form never ends or repeats. The first few digits of e are 2.7182818284.

a. Between which two square roots of integers could you find this number?

Type below:

_____________

Answer:

The square of e lies between 7 and 8

2.718281828

(2.72)² = 7.3984

Hence, it lies between \(\sqrt { 7 } \) = 2.65 and \(\sqrt { 8 } \) = 2.82

Question 21.

b. Between which two square roots of integers can you find π?

Type below:

_____________

Answer:

3.142

(3.14)² = 9.8596

Hence. it lies between \(\sqrt { 9 } \) = 3 and \(\sqrt { 10 } \) = 3.16

**H.O.T.**

**FOCUS ON HIGHER ORDER THINKING**

Question 22.

Analyze Relationships

There are several approximations used for π, including 3.14 and \(\frac{22}{7}\). π is approximately 3.14159265358979 . . .

a. Label π and the two approximations on the number line.

Type below:

_____________

Answer:

Question 22.

b. Which of the two approximations is a better estimate for π? Explain.

Type below:

_____________

Answer:

As we can see from the number line, 22/7 is closer to π, so we can conclude that 22/7 is a better estimation for π.

Question 22.

c. Find a whole number x so that the ratio \(\frac{x}{113}\) is a better estimate for π than the two given approximations.

Type below:

_____________

Answer:

355/113 is a better estimation for π, because 355/113 = 3.14159292035 = 3.14159265358979 = π

Question 23.

Communicate Mathematical Ideas

What is the fewest number of distinct points that must be graphed on a number line, in order to represent natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers? Explain.

_______ points

Answer:

2 points

Explanation:

There need to be plotting of at least 2 points because a rational number can never be equal to an irrational number. So let’s say 5 points are the same among six but the 6th will be different as there both rational numbers and irrational numbers included.

Question 24.

Critique Reasoning

Jill says that 12.\(\bar{6}\) is less than 12.63. Explain her error.

Type below:

_____________

Answer:

12.\(\bar{6}\) = 12.666

12.\(\bar{6}\) > 12.63

**Write each fraction as a decimal or each decimal as a fraction.**

Question 1.

\(\frac{7}{20}\)

_______

Answer:

0.35

Explanation:

\(\frac{7}{20}\) = 0.35

Question 2.

1.\(\overline { 27} \)

______ \(\frac{□}{□}\)

Answer:

1\(\frac{28}{99}\)

Explanation:

1.\(\overline { 27} \)

x = 1.\(\overline { 27} \)

100x = 100(1.\(\overline { 27} \))

100x = 127(\(\overline { 27} \))

x = .\(\overline { 27} \)

99x = 127

x = 127/99

x = 1 28/99

Question 3.

1 \(\frac{7}{8}\)

______

Answer:

1.875

Explanation:

1 \(\frac{7}{8}\)

1 + 7/8

8/8 + 7/8

15/8 = 1.875

**Solve each equation for x.**

Question 4.

x^{2} = 81

± ______

Answer:

± 9

Explanation:

x^{2} = 81

x = ± 81

x = ± 9

Question 5.

x^{3} = 343

______

Answer:

x = 7

Explanation:

x^{3} = 343

x = 7

Question 6.

x^{2} = \(\frac{1}{100}\)

± \(\frac{□}{□}\)

Answer:

± \(\frac{1}{10}\)

Explanation:

x^{2} = \(\frac{1}{100}\)

x = ± \(\frac{1}{10}\)

Question 7.

A square patio has an area of 200 square feet. How long is each side of the patio to the nearest 0.05?

______ feet

Answer:

14.15 feet

Explanation:

The area of a square is found by multiplying the side of the square by itself. Therefore, to find the side of the square, we have to take the square root of the area.

Let’s denote with A the area of the patio and with s each side of the square.

We have:

A = 200

A = s.s

s = \(\sqrt { A } \) = \(\sqrt { 200 } \)

Following the steps as in “Explore Activity” on page 9, we can make an estimation for the irrational number:

196 < 200 < 225

\(\sqrt { 196 } \) < \(\sqrt { 200 } \) < \(\sqrt { 225 } \)

14 < \(\sqrt { 200 } \) < 15

We see that 200 is much closer to 196 than to 225, therefore the square root of it should be between 14 and 14.5. To make a better estimation, we pick some numbers between 14 and 14.5 and calculate their squares:

(14.1)² = 198.81

(14.2)² = 201.64

14.1 < \(\sqrt { 200 } \) < 14.2

\(\sqrt { 200 } \) = 14.15

We see that 200 is much closer to 14.1 than to 14.2, therefore the square root of it should be between 14.1 and 14.15. If we round to the nearest 0.05, we have:

s = 14.15

**1.2 Sets of Real Numbers**

**Write all names that apply to each number.**

Question 8.

\(\frac { 121 }{ \sqrt { 121 } }\)

Type below:

___________

Answer:

Rational, whole, integer, real numbers

Explanation:

\(\frac { 121 }{ \sqrt { 121 } }\)

121/11 = 11

Question 9.

\(\frac{π}{2}\)

Type below:

___________

Answer:

Irrational, real numbers

Question 10.

Tell whether the statement “All integers are rational numbers” is true or false. Explain your choice.

___________

Answer:

True

Explanation:

“All integers are rational numbers” is true, because every integer can be expressed as a fraction with a denominator equal to 1. The set of integer A a subset of rational numbers.

**1.3 Ordering Real Numbers**

**Compare. Write <, >, or =.**

Question 11.

\(\sqrt { 8 }\) + 3 _______ 8 + \(\sqrt { 3 }\)

Answer:

\(\sqrt { 8 }\) + 3 < 8 + \(\sqrt { 3 }\)

Explanation:

4 < 8 < 9

\(\sqrt { 4 }\) < \(\sqrt { 8 }\) < \(\sqrt { 9 }\)

2 < \(\sqrt { 8 }\) < 3

1 < 3 < 4

\(\sqrt { 1 }\) < \(\sqrt { 3 }\) < \(\sqrt { 4 }\)

1 < \(\sqrt { 3 }\) < 2

\(\sqrt { 8 }\) + 3 is between 5 and 6

8 + \(\sqrt { 3 }\) is between 9 and 10

\(\sqrt { 8 }\) + 3 < 8 + \(\sqrt { 3 }\)

Question 12.

\(\sqrt { 5 }\) + 11 _______ 5 + \(\sqrt { 11 }\)

Answer:

\(\sqrt { 5 }\) + 11 > 5 + \(\sqrt { 11 }\)

Explanation:

\(\sqrt { 5 }\) lies in between 2 and 3

\(\sqrt { 11 }\) lies in between 3 and 4

\(\sqrt { 5 }\) + 11 lies in between 13 and 14

5 + \(\sqrt { 11 }\) lies in between 8 and 9

\(\sqrt { 5 }\) + 11 > 5 + \(\sqrt { 11 }\)

**Order the numbers from least to greatest.**

Question 13.

\(\sqrt { 99 }\), π^{2}, 9.\(\bar { 8 }\)

Type below:

_______________

Answer:

π^{2}, 9.\(\bar { 8 }\), \(\sqrt { 99 }\)

Explanation:

\(\sqrt { 99 }\), π^{2}, 9.\(\bar { 8 }\)

99 lies between 9² and 10²

99 is closer to 100, hence \(\sqrt { 99 }\) is closer to 10

(9.9)² = 98.01

(9.95)² = 99.0025

(10)² = 100

\(\sqrt { 99 }\) = 9.95

π² = 9.86

9.88888 = 9.89

π^{2}, 9.\(\bar { 8 }\), \(\sqrt { 99 }\)

Question 14.

\(\sqrt { \frac { 1 }{ 25 } } \), \(\frac{1}{4}\), 0.\(\bar { 2 }\)

Type below:

____________

Answer:

\(\sqrt { \frac { 1 }{ 25 } } \), 0.\(\bar { 2 }\), \(\frac{1}{4}\)

Explanation:

\(\sqrt { \frac { 1 }{ 25 } } \), \(\frac{1}{4}\), 0.\(\bar { 2 }\)

\(\sqrt { \frac { 1 }{ 25 } } \) = 1/5 = 0.2

1/4 = 0.25

0.\(\bar { 2 }\) = 0.222 = 0.22

\(\sqrt { \frac { 1 }{ 25 } } \), 0.\(\bar { 2 }\), \(\frac{1}{4}\)

**Essential Question**

Question 15.

How are real numbers used to describe real-world situations?

Type below:

_______________

Answer:

In real-world situations, we use real numbers to count or make measurements. They can be seen as a convention for us to quantify things around, for example, the distance, the temperature, the height, etc.

Question 1.

The square root of a number is 9. What is the other square root?

Options:

a. -9

b. -3

c. 3

d. 81

Answer:

a. -9

Explanation:

We know that every positive number has two square roots, one positive and one negative. We are given the principal square root (9), so the other square root would be its negative (-9). To prove that, we square both numbers and we compare the results:

9 • 9 = 81

(-9). (-9)= 81

Question 2.

A square acre of land is 4,840 square yards. Between which two integers is the length of one side?

Options:

a. between 24 and 25 yards

b. between 69 and 70 yards

c. between 242 and 243 yards

d. between 695 and 696 yards

Answer:

b. between 69 and 70 yards

Explanation:

The area of a square is found by multiplying the side of the square by itself. Therefore, to Bud the side of the square, we have to take the square root of the area.

Let’s denote with A the area of the land and with each side of the square. We have:

A = 4840

A = s . s

A = s²

s = √A = √4840

Following the steps as in °Explore Activity on page 9, we can make an estimation for the irrational number:

4761 < 4840 < 4900

\(\sqrt { 4761 }\) < \(\sqrt { 4840 }\) < \(\sqrt { 4900 }\)

69 < \(\sqrt { 4840 }\) < 70

Each side of the land is between 69 and 70 yards.

Question 3.

Which of the following is an integer but not a whole number?

Options:

a. -9.6

b. -4

c. 0

d. 3.7

Answer:

b. -4

Explanation:

Whole numbers are not negative

-4 is an integer but not a whole number

Question 4.

Which statement is false?

Options:

a. No integers are irrational numbers.

b. All whole numbers are integers.

c. No real numbers are irrational numbers.

d. All integers greater than 0 are whole numbers.

Answer:

c. No real numbers are irrational numbers.

Explanation:

Rational and irrational numbers are real numbers.

Question 5.

Which set of numbers best describes the displayed weights on a digital scale that shows each weight to the nearest half pound?

Options:

a. whole numbers

b. rational numbers

c. real numbers

d. integers

Answer:

b. rational numbers

Explanation:

The scale weighs nearest to 1/2 pound.

Question 6.

Which of the following is not true?

Options:

a. π^{2} < 2π + 4

b. 3π > 9

c. \(\sqrt { 27 }\) + 3 > 172

d. 5 – \(\sqrt { 24 }\) < 1

Answer:

c. \(\sqrt { 27 }\) + 3 > 172

Explanation:

a. π^{2} < 2π + 4

(3.14)² < 2(3.14) + 4

9.86 < 10.28

True

b. 3π > 9

9.42 > 9

True

c. \(\sqrt { 27 }\) + 3 > 172

5.2 + 3 > 8.5

8.2 > 8.5

False

d. 5 – \(\sqrt { 24 }\) < 1

5 – 4.90 < 1

0.1 < 1

True

Question 7.

Which number is between \(\sqrt { 21 }\) and \(\frac{3π}{2}\) ?

Options:

a. \(\frac{14}{3}\)

b. 2 \(\sqrt { 6 }\)

c. 5

d. π + 1

Answer:

Explanation:

a. \(\sqrt { 21 }\) and \(\frac{3π}{2}\)

\(\sqrt { 21 }\) = 4.58

\(\frac{3π}{2}\) = 4.71

14/3 = 4.67

b. 2\(\sqrt { 6 }\) = 4.90

c. 5

d. π + 1 = 3.14 + 1 = 4.14

Question 8.

What number is shown on the graph?

Options:

a. π+3

b. \(\sqrt { 4 }\) + 2.5

c. \(\sqrt { 20 }\) + 2

d. 6.\(\overline { 14 } \)

Answer:

c. \(\sqrt { 20 }\) + 2

Explanation:

6.48

a. π+3 = 3.14 + 3 = 6.14

b. \(\sqrt { 4 }\) + 2.5 = 2 + 2.5 = 4.5

c. \(\sqrt { 20 }\) + 2 = 4.47 + 2 = 6.47

d. 6.\(\overline { 14 } \) = 6.1414

Question 9.

Which is in order from least to greatest?

Options:

a. 3.3, \(\frac{10}{3}\), π, \(\frac{11}{4}\)

b. \(\frac{10}{3}\), 3.3, \(\frac{11}{4}\), π

c. π, \(\frac{10}{3}\), \(\frac{11}{4}\), 3.3

d. \(\frac{11}{4}\), π, 3.3, \(\frac{10}{3}\)

Answer:

d. \(\frac{11}{4}\), π, 3.3, \(\frac{10}{3}\)

Explanation:

10/3 = 3.3333333

11/4 = 2.75

**Mini-Task**

Question 10.

The volume of a cube is given by V = x^{3}, where x is the length of an edge of the cube. The area of a square is given by A = x^{2}, where x is the length of a side of the square. A given cube has a volume of 1728 cubic inches.

a. Find the length of an edge.

______ inches

Answer:

12 inches

Explanation:

V = x^{3}

A = x^{2}

1728 = x^{3}

x = 12

The length of an edge = 12 in

Question 10.

b. Find the area of one side of the cube.

______ in^{2}

Answer:

144 in^{2}

Explanation:

A = (12)² = 144

Area of the side of the cube = 144 in^{2}

Question 10.

c. Find the surface area of the cube.

______ in^{2}

Answer:

864 in^{2}

Explanation:

SA = 6 (12)² = 864

The surface area of the cube = 864 in^{2}

Question 10.

d. What is the surface area in square feet?

______ ft^{2}

Answer:

6 ft^{2}

Explanation:

SA = 864/144 = 6

The surface area of the cube = 6 ft^{2}