Terminating and Repeating Decimals

[Pages:4]Terminating and Repeating Decimals

Student Probe

Write 5 as a decimal. 8

Answer: 0.625

Lesson Description

In this lesson students use long division to convert fractions into repeating or terminating decimals. Calculator use in encouraged.

Rationale

All rational numbers, including mixed numbers, can be

converted into repeating or terminating decimals by

dividing the numerator by the denominator. This division

process can be done with or without the use of a

calculator. It is important for students to realize that

there situations where a fractional representation is

more efficient and there are times when a decimal

representation is more efficient. Students need to be

flexible in their thinking to decide which is more

appropriate in a given situation.

Additionally, some fractions and their decimal

equivalents are used so often that it is efficient for

students to know them without having to convert from

one form to another. Some examples of these include:

1 0.5, 1 0.25, 3 0.75, 1 0.125 1 0.33, 2 0.66.

2

4

4

8

3

3

With repeated use students will become familiar with

these conversions, so it is not necessary to require

memorization.

At a Glance

What: Converting fractions into repeating or terminating decimals Common Core Standard: CC.7.NS.2d. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. (d) Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Mathematical Practices: Use appropriate tools strategically. Attend to precision. Look for and express regularity in repeated reasoning. Who: Students who cannot convert a fraction into a repeating or terminating decimal Grade Level: 7 Prerequisite Vocabulary: numerator, denominator, rational number, equivalent fractions Prerequisite Skills: division of whole numbers Delivery Format: Individual, small group Lesson Length: 30 minutes Materials, Resources, Technology: calculator Student Worksheets: none

Preparation

Allow students to have access to calculators. Simple, four-function calculators are appropriate for this lesson.

Lesson

The teacher says or does...

1. What are some fractions that are equivalent to 1 ? 2

2. How can we write 5 as a 10

decimal? 3. So we are saying 1 5 0.5 ,

2 10 or 1 0.5 .

2 4. What does the "fraction bar"

mean? That means 1 is the same as

2 1 2. Put 1 2 in your calculator and tell me what answer it returns. That is another way to figure out 1 0.5 .

2 5. What does 3 mean?

8 Use your calculator to find the decimal equivalent of 3 .

8

6. Find the decimal equivalent of 1 . 3

Expect students to say or do...

Answers will vary, but listen for 5 .

10

0.5

If students do not, then the teacher says or does... What about 5 ? Is it

10 equivalent to 1 ?

2 Prompt students, if necessary.

Division. 0.5

What operation does the fraction bar indicate?

Monitor students.

38

3 8 0.375 . 1 3 0.333333333 ...

What operation is the fraction bar telling us to do?

Monitor students. Monitor students. Prompt if necessary.

The teacher says or does...

7. Notice that the 3's keep going! This is called a repeating decimal. Can we write all of the 3's? Mathematicians have a way to show that the 3's repeat forever.

It is written like this: 0.33 . The bar is placed over the part of the decimal that repeats to show that it repeats those digits forever. 8. There are two kinds of rational numbers: the ones that repeat such as 1 , and

3 the ones that end, or terminate, like 1 and 3 .

2 8 9. Find the decimal equivalent

of 5 . 6

Is it a repeating decimal or a terminating decimal? How do you know? 10. Find the decimal equivalent of 5 .

16 Is it a repeating decimal or a terminating decimal? How do you know? 11. Let's find the decimal equivalent of 2 3 . We know

5 we have 2 wholes, so we only need to consider the 3 .

5 Is it terminating or repeating?

Expect students to say or do... No.

5 0.83 6 Repeating, because the 3 repeats forever. 5 0.3125 16 Terminating, because it ends. 2 3 2 3 5 2 0.6 2.6

5 Terminating

If students do not, then the teacher says or does...

5 means 5 6 . 6 5 means 5 16 . 16 3 3 5. 5

The teacher says or does...

12. Repeat the steps above with a variety of fractions and mixed numbers. Include both terminating and repeating decimals.

Expect students to say or do...

If students do not, then the teacher says or does...

Teacher Notes

1. This lesson relies on the interpretation of fractions as indicted division, k k n . This can n

seem unusual to students who have been thinking of fractions as parts of wholes. 2. Make sure that students understand the notation for repeating decimals and use it

correctly. 3. When converting mixed numbers to decimals, students should understand that only the

fractional part of the number is converted. The whole number remains unchanged. 4. Notice that any fraction with a denominator of 7 will have a repeating block of 6 digits.

Students will sometimes fail to notice all of the digits.

Variations

Students may be interested in the types of rational numbers that have repeating or terminating decimal forms. Let students create a table of the first 20 unit fractions, 1 , 1 , 1 ,..., 1 , and their

2 3 4 20 decimal equivalents (using a calculator). Students should then investigate the patterns that emerge. (Teacher Note: denominators of fractions which terminate have prime factors of only 2 and/or 5.)

Formative Assessment

Write 5 as a decimal. 12

Answer: 0.416

References

Mathematics Preparation for Algebra. (n.d.). Retrieved August 10, 2010, from Doing What Works: Van de Walle, J. A., & Lovin, L. H. (2006). Teaching Student-Centered Mathematics Grades 5-8 Volume 3. Boston, MA: Pearson Education, Inc.

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