046. Decimals, Fractions and Recurring Decimals

Learning Intentions

• convert decimals to fractions
• convert fractions to decimals
• To understand the symbols used to indicate recurring decimals

Pre-requisite Summary

• Understand place value in decimals, including tenths, hundredths and thousandths
• Know that a decimal can be written as a fraction with denominator , , , \dots
• Be able to Simplify fractions Use common factors
• Understand that division can be written as a fraction
• Know that some fractions give terminating decimals and some give repeating decimal patterns
• Be able to compare equivalent forms of the same number

Worked Examples

Worked Example 1

Convert each decimal to a fraction in simplest form:

a)

b)

c)

Worked Example 2

Convert each decimal to a fraction in simplest form:

a)

b)

c)

Worked Example 3

Convert each fraction to a decimal:

a)

b)

c)

Worked Example 4

Convert each fraction to a decimal:

a)

b)

c)

Worked Example 5

Write each recurring decimal using the correct notation and Describe the repeating part:

a)

b)

c)

Worked Example 6

For each number, State whether the decimal is terminating or recurring:

a)

b)

c)

Problems

Problem 1

Convert each decimal to a fraction in simplest form:

a)

b)

c)

Problem 2

Convert each decimal to a fraction in simplest form:

a)

b)

c)

Problem 3

Convert each fraction to a decimal:

a)

b)

c)

Problem 4

Convert each fraction to a decimal:

a)

b)

c)

Problem 5

Write each recurring decimal using the correct notation and describe the repeating part:

a)

b)

c)

Problem 6

For each number, state whether the decimal is terminating or recurring:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Convert each decimal to a fraction in simplest form:

a)

b)

c)

Exercise 2.

Convert each decimal to a fraction in simplest form:

a)

b)

c)

Exercise 3.

Convert each decimal to a fraction in simplest form:

a)

b)

c)

Exercise 4.

Convert each fraction to a decimal:

a)

b)

c)

Exercise 5.

Convert each fraction to a decimal:

a)

b)

c)

Exercise 6.

Convert each fraction to a decimal:

a)

b)

c)

Exercise 7.

Write each recurring decimal using the correct notation:

a)

b)

c)

Exercise 8.

State whether each decimal is terminating or recurring:

a)

b)

c)

Reasoning

Exercise 9.

Explain why before simplification.

Exercise 10.

A student says . Explain the mistake.

Exercise 11.

Explain why gives a terminating decimal but gives a recurring decimal.

Exercise 12.

A student writes as . Explain whether this notation is correct and what it means.

Problem-solving

Exercise 13.

A measuring jug contains L of water. Write this amount as a fraction in simplest form.

Exercise 14.

A ribbon is m long. Write this length as a decimal.

Exercise 15.

A battery charge is shown as of full capacity. Write this as a fraction in simplest form.

Exercise 16.

A runner completes of a lap. Write this as a decimal.

Exercise 17.

A calculator displays . Write this using recurring decimal notation and state the repeating digit.

Exercise 18.

A number is . Write it as a decimal and Identify whether it is terminating or recurring.

Potential Misunderstandings

• Students may confuse the place value of the last decimal digit when converting a decimal to a fraction
• Students may forget to simplify the fraction after writing the decimal over a power of ten
• Students may think every fraction converts to a terminating decimal
• Students may stop a decimal expansion too early and assume it terminates
• Students may not Recognise that a mixed or improper fraction can still be converted to a decimal
• Students may misunderstand recurring decimal notation and think the dots Apply to only one digit when a block of digits repeats
• Students may write the denominator incorrectly when converting decimals with two or three decimal places
• Students may confuse with even though one terminates and the other recurs

Next: 047. Percentages and Decimals