114. Recurring Decimals and Rounding

Learning Intentions

• To know the meaning of the terms terminating decimal and recurring decimal
• To understand the different notations for recurring decimals (involving dots and dashes)
• convert a fraction to a recurring decimal Use division
• round decimals to a given number of decimal places by first finding the critical digit

Pre-requisite Summary

• Know that decimals Use place value columns such as tenths, hundredths and thousandths
• Be able to divide whole numbers and decimals using written division
• Understand that a fraction can be written as a division
• Know that some fractions convert to exact decimals
• Be able to Identify the digit in a given decimal place
• Understand that rounding depends on the value of the digit to the right of the required place
• Know that place value determines the size of each digit in a decimal
• Be familiar with writing equivalent numerical forms of the same value

Worked Examples

Worked Example 1

State whether each decimal is terminating or recurring:

a)

b)

c)

Worked Example 2

Write each recurring decimal using recurring decimal notation:

a)

b)

c)

Worked Example 3

Convert each fraction to a decimal using division:

a)

b)

c)

Worked Example 4

Convert each fraction to a decimal using division:

a)

b)

c)

Worked Example 5

Round each decimal to the stated number of decimal places by first identifying the critical digit:

a) to decimal places

b) to decimal places

c) to decimal place

Worked Example 6

Round each decimal to the stated number of decimal places by first identifying the critical digit:

a) to decimal places

b) to decimal places

c) to decimal places

Problems

Problem 1

State whether each decimal is terminating or recurring:

a)

b)

c)

Problem 2

Write each recurring decimal using recurring decimal notation:

a)

b)

c)

Problem 3

Convert each fraction to a decimal using division:

a)

b)

c)

Problem 4

Convert each fraction to a decimal using division:

a)

b)

c)

Problem 5

Round each decimal to the stated number of decimal places by first identifying the critical digit:

a) to decimal places

b) to decimal places

c) to decimal place

Problem 6

Round each decimal to the stated number of decimal places by first identifying the critical digit:

a) to decimal places

b) to decimal places

c) to decimal places

Exercises

Understanding and Fluency

Exercise 1.

State whether each decimal is terminating or recurring:

a)

b)

c)

d)

Exercise 2.

State whether each decimal is terminating or recurring:

a)

b)

c)

d)

Exercise 3.

Write each recurring decimal using appropriate recurring decimal notation:

a)

b)

c)

Exercise 4.

Write each recurring decimal using appropriate recurring decimal notation:

a)

b)

c)

Exercise 5.

Convert each fraction to a decimal using division:

a)

b)

c)

d)

Exercise 6.

Convert each fraction to a decimal using division:

a)

b)

c)

d)

Exercise 7.

Round each decimal to the stated number of decimal places:

a) to decimal places

b) to decimal places

c) to decimal place

d) to decimal places

Exercise 8.

Round each decimal to the stated number of decimal places:

a) to decimal places

b) to decimal places

c) to decimal place

d) to decimal places

Reasoning

Exercise 9.

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

Exercise 10.

A student says that is terminating because the digits repeat in a pattern. Explain the mistake.

Exercise 11.

Noah says that means both digits repeat. Is he correct? Explain.

Exercise 12.

Explain why the critical digit is important when rounding a decimal.

Exercise 13.

A student rounds to decimal places and writes . Describe the error.

Problem-solving

Exercise 14.

A calculator display shows after dividing. State whether the decimal is terminating or recurring, and write it in recurring decimal notation.

Exercise 15.

A fraction is divided and gives . Write this decimal using recurring notation and identify the repeating block.

Exercise 16.

A measurement is m. Round this measurement to decimal places.

Exercise 17.

A price is $ . Round the price to decimal places.

Exercise 18.

A student converts to a decimal using division. What decimal should they obtain, and what type of decimal is it?

Exercise 19.

A decimal rounded to decimal place is . Give two possible original decimals.

Potential Misunderstandings

• Students may think a recurring decimal stops because the repeating digits are predictable
• Students may confuse terminating decimals with decimals that only show a few digits on a calculator
• Students may think any decimal with repeated digits is terminating
• Students may misunderstand recurring notation and not know which digit or block is repeating
• Students may think dots or dashes mean the decimal is approximate rather than recurring
• Students may stop the division too early when converting a fraction to a decimal
• Students may not recognise that a repeated remainder in division leads to a recurring decimal
• Students may confuse the last kept digit with the critical digit when rounding
• Students may look at the wrong digit when rounding to a given number of decimal places
• Students may think rounding always makes a number larger
• Students may forget that digits after the rounded place become zero in value or are removed from the written decimal
• Students may make errors when the rounded digit becomes and needs regrouping, such as in

Next: 115r. Review Percentages and Percentage Calculations