Unit 8 – Media Lesson

UNIT 8 – CONNECTING FRACTIONS AND DECIMALS INTRODUCTION As we learned in the last Unit, decimals are indeed fractions written in an alternative form. We want to be able to change fractions in non-decimal form to decimals and vice versa. Our ability to connect these two forms of numbers will aid in our understanding and fluency of working with numbers and operations. The more ways we have to represent, connect, and transform numbers in an equivalent forms, the more tools we have to make sense of mathematics. The table below shows the learning objectives that are the achievement goal for this unit. Read through them carefully now to gain initial exposure to the terms and concept names for the lesson. Refer back to the list at the end of the lesson to see if you can perform each objective.

Learning Objective

Media Examples

You Try

Represent a fraction with a denominator that is not a power of 10 in a decimal grid and convert to a decimal

1

2

Write fractions with denominators that are powers of 10 as decimals

3

4

Write an equivalent fraction with a denominator that is a power of 10 and rewrite as a terminating decimal

5

6

Approximate fractions whose decimal form does not terminate

7

Use a calculator to approximate fractions whose decimal form does not terminate

8

9

Write decimals as simplified fractions

10

11

Write negative decimals as simplified fractions or mixed numbers

12

13

Compare decimals and fraction and represent the result with inequality notation

14

15

1

Unit 8 – Media Lesson

UNIT 8 – MEDIA LESSON SECTION 8.1: VISUALIZING CONVERTING FRACTIONS TO DECIMALS We have already learned that decimals are an alternative way to represent fractions. In this section, we will learn to rewrite a fraction in decimal form. Problem 1

MEDIA EXAMPLE – Writing Fractions as Decimals Using a Decimal Grid

Shade the fractional amount of the grid that is named (The ten by ten decimal grid is the unit). Explain how you knew the region to shade. Write the decimal name for each shaded region. a)

3 4

Decimal Name:___________

Explain how you chose the region to shade:

b)

7 25

Decimal Name:___________

Explain how you chose the region to shade:

c)

3 8

Decimal Name:___________

Explain how you chose the region to shade: 2

Unit 8 – Media Lesson Problem 2

YOU-TRY – Writing Fractions as Decimals Using a Decimal Grid

Shade the fractional amount of the grid that is named (The ten by ten decimal grid is the unit). Explain how you knew the region to shade. Write the decimal name for each shaded region. a)

3 5

Decimal Name:___________

Explain how you chose the region to shade:

b)

11 20

Decimal Name:___________

Explain how you chose the region to shade:

3

Unit 8 – Media Lesson

SECTION 8.2: USING PLACE VALUE TO CONVERT FRACTIONS TO DECIMALS Some fractions are easily written in decimal form because their denominators are powers of ten. In this section, we will convert such fractions to their decimal form. MEDIA EXAMPLE – Writing Fractions with Denominators that are Powers of Ten as Decimals For each of the following fractions use the Place Value chart to write each fraction in decimal form. Problem 3

a)

27 100

Decimal: _________ c)

32 10

Decimal: _________

e)

423 100

Decimal: _________

b)

145 10

Decimal: _________ d)

3 100

Decimal: _________

f)

37 1000

Decimal: _________

Problem 4 YOU-TRY – Writing Fractions with Denominators that are Powers of Ten as Decimals For each of the following fractions use the Place Value chart to write each fraction in decimal form. a)

83 100

Decimal: _________ c)

67 10

Decimal: _________

e)

214 100

Decimal: _________

b)

324 10

Decimal: _________ d)

9 1000

Decimal: _________

f)

76 1000

Decimal: _________ 4

Unit 8 – Media Lesson

SECTION 8.3: USING FACTORING TO CONVERT FRACTIONS TO DECIMALS Some fractions may not be written with a denominator that is a power of 10, but can be rewritten as an equivalent fraction with a denominator that is a power of 10. In this section, we will look at fractions that can and cannot be transformed in this way. Problem 5

MEDIA EXAMPLE – Rewriting Fractions Whose Decimals Terminate

If a simplified fraction can be written as an integer over a power of ten, then its decimal expansion terminates. A decimal is a terminating decimal, if its decimal expansion does not go on to infinity. For each of the following fractions, rewrite the fraction with a denominator that is a power of ten. Then use the Place Value chart to write each fraction in decimal form. a)

3 4

Decimal: _________ b)

7 25

Decimal: _________ c)

3 8

Decimal: _________ Problem 6

YOU-TRY –– Rewriting Fractions Whose Decimals Terminate

For each of the following fractions, rewrite the fraction with a denominator that is a power of ten. Then use the Place Value chart to write each fraction in decimal form. a)

3 5

Decimal: _________ b)

11 20

Decimal: _________ c)

1 8

Decimal: _________

5

Unit 8 – Media Lesson Problem 7 MEDIA EXAMPLE –Approximating Fractions with Decimals that do not Terminate If a simplified fraction cannot be written as an integer over a power of ten, then its decimal expansion repeats. A decimal is a repeating decimal, if its decimal expansion eventually repeats the same pattern of digits to infinity. 1

a) Use the decimal grid to approximate to four decimal places. 3

Decimal Approximation: _________

b) Use the decimal grid to approximate

8 11

to four decimal places.

Decimal Approximation: _________

6

Unit 8 – Media Lesson Problem 8 MEDIA EXAMPLE –Approximating Fractions as Decimals with a Calculator When the corresponding decimal for a fraction doesn’t terminate, we will frequently use a calculator to approximate the decimal by dividing and then rounding.

Approximate the following fractions with decimals by dividing on your calculator. Give approximations to one, two, and three decimal places.

a)

8

b) −

21

13

c) −4

17

5 7

one decimal place: _________

one decimal place: _________

one decimal place: _________

two decimal places: ________

two decimal places: ________

two decimal places: ________

three decimal places: _______

three decimal places: _______

three decimal places: _______

Problem 9

YOU-TRY –– Approximating Fractions with Decimals that do not Terminate

Approximate the following fraction with a decimal by dividing on your calculator. Give approximations to one, two, and three decimal places. −3

4 13

one decimal place: _________ two decimal places: __________ three decimal places: ___________

7

Unit 8 – Media Lesson

SECTION 8.4: CONVERTING DECIMALS TO FRACTIONS We have already written decimals in a fraction form with denominators that are powers of ten. In this section, we will also write these fractions in simplest form. Problem 10

MEDIA EXAMPLE –Writing Decimals as Simplified Fractions

When we rewrite a decimal as a simplified fraction, we will start by writing it as a fraction based on its place value, a power of ten. Observe that 10’s prime factorization is 2 ∙ 5. So any power of 10 is just a product of 2’s and 5’s. This will make the process of simplification easier because we will only have to check the numerator for factors of 2’s and 5’s. Complete the table below. Show all of your work for simplifying the fraction. Decimal Fraction Simplified Fraction a) 0.8 b)

0.65

c)

0.44

d)

0.002

Problem 11

YOU-TRY –– Writing Decimals as Simplified Fractions

Complete the table below. Show all of your work for simplifying the fraction. a)

Decimal 0.6

b)

0.85

c)

0.042

Fraction

Simplified Fraction

8

Unit 8 – Media Lesson Problem 12

MEDIA EXAMPLE – Writing Decimals as Fractions or Mixed Numbers

Complete the table below. Show all of your work for simplifying the fraction. Decimal a)

−0.4

b)

3.25

c)

6.008

d)

−7.024

Problem 13

Fraction or Mixed Number

Simplify Fraction

Final Answer

YOU-TRY –– Writing Decimals as Fractions or Mixed Numbers

Complete the table below. Show all of your work for simplifying the fraction. Decimal a)

−1.2

b)

6.45

c)

−7.016

Fraction or Mixed Number

Simplify Fraction

Final Answer

9

Unit 8 – Media Lesson

SECTION 8.5: COMPARING DECIMALS AND FRACTIONS In this section, we will discuss methods to compare decimals and fractions and use inequality notation to express their order. We will use a few methods to accomplish this. Given a fraction and a decimal, we can determine their order in the following ways 1. Rewrite the decimal as a fraction and use methods for ordering fractions. 2. Rewrite the fraction as a decimal and use methods for ordering decimals. 3. Use benchmarks (such as one half) when possible to order the numbers. Problem 14

MEDIA EXAMPLE – Comparing Decimals and Fractions

Determine which number is greater. Use the symbols, to express this relationship. 𝑎)

𝑑)

5 ___0.5 8

4 ___0.44 9

Problem 15

3 ____0.64 8

𝑏)

𝑒)

7 ____0.64 11

𝑐)

0.28______

5 7

0.62______

8 13

𝑐) 0.35______

6 17

𝑓)

YOU-TRY –– Comparing Decimals and Fractions

Determine which number is greater. Use the symbols, to express this relationship.

𝑎) 0.5 ___

5 12

𝑏)

5 ____0.56 9

10