043. Multiplying and Dividing Decimals by Powers of Ten

Learning Intentions

• To understand that a power of ten is a number like , , etc.
• multiply a decimal by a power of ten
• divide a decimal by a power of ten

Pre-requisite Summary

• Understand the meaning of the decimal point as separating whole-number parts from fractional parts
• Know the place value of digits to the left and right of the decimal point
• Recognise powers of ten such as as
• Understand that multiplying or dividing by changes place value
• Be able to compare the size of decimals before and after a calculation
• Know that placeholder zeros may be needed to show place value correctly

Worked Examples

Worked Example 1

a) Explain what a power of ten is.

b) State three examples of powers of ten.

c) Explain how multiplying by a power of ten affects place value.

Worked Example 2

Multiply each decimal by a power of ten:

a)

b)

c)

Worked Example 3

Multiply each decimal by a power of ten:

a)

b)

c)

Worked Example 4

Divide each decimal by a power of ten:

a)

b)

c)

Worked Example 5

Divide each decimal by a power of ten:

a)

b)

c)

Worked Example 6

a) Compare , , and .

b) Explain why one result is greater and one result is smaller than the starting number.

c) State the role of the decimal point in each calculation.

Problems

Problem 1

a) Explain what a power of ten is.

b) State three examples of powers of ten.

c) Explain how dividing by a power of ten affects place value.

Problem 2

Multiply each decimal by a power of ten:

a)

b)

c)

Problem 3

Multiply each decimal by a power of ten:

a)

b)

c)

Problem 4

Divide each decimal by a power of ten:

a)

b)

c)

Problem 5

Divide each decimal by a power of ten:

a)

b)

c)

Problem 6

a) Compare , , and .

b) Explain why one result is greater and one result is smaller than the starting number.

c) State the role of the decimal point in each calculation.

Exercises

Understanding and Fluency

Exercise 1.

State whether each number is a power of ten:

a)

b)

c)

Exercise 2.

Write the next three powers of ten after:

a)

b)

c)

Exercise 3.

Multiply each decimal by :

a)

b)

c)

Exercise 4.

Multiply each decimal by :

a)

b)

c)

Exercise 5.

Multiply each decimal by :

a)

b)

c)

Exercise 6.

Divide each decimal by :

a)

b)

c)

Exercise 7.

Divide each decimal by :

a)

b)

c)

Exercise 8.

Divide each decimal by :

a)

b)

c)

Exercise 9.

Complete the table:

a)

b)

Exercise 10.

Mixed practice:

a)

b)

c)

d)

Reasoning

Exercise 11.

Explain why .

Exercise 12.

A student says . Explain the mistake.

Exercise 13.

Explain why dividing by gives a smaller number than .

Exercise 14.

A student says . Explain why this is incorrect.

Problem-solving

Exercise 15.

A bottle holds L of juice. Express this amount in decilitres by multiplying by .

Exercise 16.

A ribbon is m long. Express this length in centimetres by multiplying by .

Exercise 17.

A container has mL of water. Express this amount in litres by dividing by .

Exercise 18.

A runner records a distance of km. Express this in metres by multiplying by .

Potential Misunderstandings

• Students may think multiplying by a power of ten means “adding zeros” without considering decimal place value
• Students may move digits rather than understanding that the value of each digit changes place
• Students may confuse multiplying by with dividing by
• Students may think the decimal point moves on its own rather than the digits changing place value around it
• Students may forget to Use placeholder zeros when needed, for example in
• Students may believe that dividing by makes a number larger because more digits appear after the decimal point
• Students may write answers without checking whether the result should be larger or smaller than the original number
• Students may not recognise that powers of ten are specifically and not just any multiple of

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