044. Multiplying Decimals

Learning Intentions

• multiply decimals
• To understand that it is helpful to estimate to check the position of the decimal point in the final answer

Pre-requisite Summary

• Understand place value in decimal numbers, including tenths, hundredths and thousandths
• Be able to multiply whole numbers using written or mental methods
• Know that digits in a decimal still follow place value rules
• Be able to estimate products using rounding
• Understand that an answer should be reasonable in size compared with the numbers being multiplied

Worked Examples

Worked Example 1

a) Calculate $0.4\times 3$
b) Estimate the product first
c) Explain how the estimate helps check the decimal point

Worked Example 2

a) Calculate $2.5\times 4$
b) Estimate the product first
c) Check whether the final answer is reasonable

Worked Example 3

a) Calculate $1.2\times 0.5$
b) Estimate the product first
c) Explain why the answer must be less than $1.2$

Worked Example 4

a) Calculate $3.4\times 2.6$
b) Estimate the product by rounding
c) Use the estimate to check the position of the decimal point

Worked Example 5

a) Calculate $0.75\times 0.8$
b) Estimate the product first
c) Explain why an answer of $6.0$ would be unreasonable

Worked Example 6

a) Calculate $4.26\times 1.3$
b) Estimate the product first
c) Use the estimate to justify the final decimal placement

Problems

Problem 1

a) Calculate $0.6\times 5$
b) Estimate the product first
c) Explain how the estimate helps check the decimal point

Problem 2

a) Calculate $3.5\times 2$
b) Estimate the product first
c) Check whether the final answer is reasonable

Problem 3

a) Calculate $1.4\times 0.3$
b) Estimate the product first
c) Explain why the answer must be less than $1.4$

Problem 4

a) Calculate $2.7\times 3.2$
b) Estimate the product by rounding
c) Use the estimate to check the position of the decimal point

Problem 5

a) Calculate $0.45\times 0.6$
b) Estimate the product first
c) Explain why an answer of $2.7$ would be unreasonable

Problem 6

a) Calculate $5.18\times 1.4$
b) Estimate the product first
c) Use the estimate to justify the final decimal placement

Exercises

Understanding and Fluency

1. Multiply each decimal:
   a) $0.3\times 4$
   b) $0.7\times 6$
   c) $1.5\times 2$

2. Multiply each decimal:
   a) $2.4\times 3$
   b) $4.6\times 2$
   c) $3.5\times 5$

3. Multiply each decimal:
   a) $0.2\times 0.4$
   b) $0.6\times 0.5$
   c) $0.8\times 0.7$

4. Multiply each decimal:
   a) $1.2\times 0.3$
   b) $2.5\times 0.4$
   c) $3.6\times 0.2$

5. Multiply each decimal:
   a) $2.3\times 1.4$
   b) $4.1\times 1.2$
   c) $5.4\times 2.3$

6. Multiply each decimal:
   a) $0.75\times 0.4$
   b) $1.25\times 0.6$
   c) $2.08\times 0.5$

7. Estimate first, then multiply:
   a) $3.2\times 2.9$
   b) $4.7\times 1.8$
   c) $0.48\times 0.9$

8. Estimate first, then multiply:
   a) $6.15\times 1.1$
   b) $2.34\times 0.7$
   c) $0.96\times 0.8$

Reasoning

9. Explain why estimating $3.8\times 2.1$ as about $4\times 2=8$ helps check the final answer.

10. A student says $0.4\times 0.5=2.0$. Explain the mistake.

11. Explain why multiplying by a decimal less than $1$ makes the product smaller than the starting number.

12. A student calculates $2.6\times 1.4=364$. Explain why the estimate shows this cannot be correct.

Problem-solving

13. A ribbon is $2.5$ m long. Four equal pieces are cut from the same type of ribbon. Find the total length of ribbon needed.

14. A bottle holds $0.75$ L of juice. How much juice is in $6$ bottles?

15. A runner completes $1.2$ km per lap. How far does the runner travel in $5$ laps?

16. A shop sells fabric for $3.4 per metre. What is the cost of $2.5$ m of fabric?

17. A tank is filled at a rate of $0.8$ L per minute for $15$ minutes. How much water is added?

18. A packet weighs $0.45$ kg. What is the total mass of $8$ packets?

Potential Misunderstandings

• Students may ignore place value and multiply decimals as though they were whole numbers without adjusting the final answer
• Students may place the decimal point incorrectly in the product
• Students may think the total number of decimal places rule works without checking whether the answer is reasonable
• Students may forget to estimate before calculating, so they miss an unreasonable answer
• Students may think multiplying by a decimal less than $1$ should make the answer larger
• Students may confuse multiplication by $10$ or $100$ with general decimal multiplication
• Students may line up decimal points in a multiplication algorithm as if they were adding
• Students may not recognise that an estimate can be used to reject an answer with the decimal point in the wrong place

Next: 045. Dividing Decimals