040. Decimals and Place Value

Learning Intentions

• To understand the meaning of the decimal point
• To know the place value of digits after a decimal point
• decide if one decimal is larger or smaller than another decimal
• convert proper fractions and mixed numerals to decimals, when their denominators are powers of ten

Pre-requisite Summary

• Understand whole-number place value, including ones, tens and hundreds
• Know that a fraction can represent part of a whole
• Recognise fractions with denominators that are powers of ten, such as
• Understand that the decimal point separates whole-number parts from fractional parts
• Be able to compare whole numbers by place value
• Be able to read and write simple fractions and mixed numerals

Worked Examples

Worked Example 1

a) Explain the meaning of the decimal point in .

b) State the value of the digit .

c) State the value of the digit .

Worked Example 2

For the decimal :

a) Name the place value of each digit after the decimal point.

b) State the value of the digit .

c) State the value of the digit .

Worked Example 3

Compare the decimals and decide which is larger:

a) and

b) and

c) and

Worked Example 4

Compare the decimals and write or :

a)

b)

c)

Worked Example 5

Convert each fraction to a decimal:

a)

b)

c)

Worked Example 6

Convert each mixed numeral to a decimal:

a)

b)

c)

Problems

Problem 1

a) Explain the meaning of the decimal point in .

b) State the value of the digit .

c) State the value of the digit .

Problem 2

For the decimal :

a) Name the place value of each digit after the decimal point.

b) State the value of the digit .

c) State the value of the digit .

Problem 3

Compare the decimals and decide which is larger:

a) and

b) and

c) and

Problem 4

Compare the decimals and write or :

a)

b)

c)

Problem 5

Convert each fraction to a decimal:

a)

b)

c)

Problem 6

Convert each mixed numeral to a decimal:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

State the value of the underlined digit:

a)

b)

c)

Exercise 2.

Name the place value of each underlined digit:

a)

b)

c)

Exercise 3.

Write each decimal in words:

a)

b)

c)

Exercise 4.

Compare the decimals Use or :

a)

b)

c)

Exercise 5.

Compare the decimals using or :

a)

b)

c)

Exercise 6.

Convert each fraction to a decimal:

a)

b)

c)

Exercise 7.

Convert each fraction to a decimal:

a)

b)

c)

Exercise 8.

Convert each mixed numeral to a decimal:

a)

b)

c)

Exercise 9.

Convert each mixed numeral to a decimal:

a)

b)

c)

Exercise 10.

Mixed practice:

a) Which is larger: or ?

b) Convert to a decimal.

c) Convert to a decimal.

Reasoning

Exercise 11.

Explain why even though .

Exercise 12.

A student says is larger than because . Explain the mistake.

Exercise 13.

Explain why .

Exercise 14.

A student writes . Explain why this is incorrect.

Problem-solving

Exercise 15.

A measuring jug contains litres of water. Write this amount as a decimal.

Exercise 16.

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

Exercise 17.

Three runners recorded times of s, s and s. Order these times from shortest to longest.

Exercise 18.

A container holds kg of rice. Write this as a decimal and state the value of the digit .

Potential Misunderstandings

• Students may think the decimal point is just punctuation rather than a separator between whole numbers and fractional parts
• Students may confuse tenths, hundredths and thousandths
• Students may think the number with more digits after the decimal point is always larger
• Students may compare digits without considering place value from left to right
• Students may not understand that trailing zeros do not change the value, for example
• Students may think instead of
• Students may place the decimal point incorrectly when converting fractions with denominators that are powers of ten
• Students may forget that a mixed numeral has both a whole-number part and a fractional part when converting to a decimal

Next: 041. Rounding Decimal Numbers