113r. Review Operations with Decimals

Learning Intentions

• add and subtract decimals
• multiply decimals
• divide decimals
• To understand that multiplying and dividing by powers of $10$ can be done by moving the digits left or right of the decimal point

Pre-requisite Summary

• Know the place value of digits in decimals, including tenths, hundredths and thousandths
• Be able to line up decimal points when writing decimals in columns
• Understand that decimals represent parts of a whole
• Recall multiplication and division facts with whole numbers
• Know that powers of $10$ include $10,100,1000,\dots$
• Understand that multiplying makes groups larger and dividing shares into smaller groups
• Be able to compare decimals and recognise which is greater
• Know that trailing zeros do not change the value of a decimal, for example $2.5=2.50$

Worked Examples

Worked Example 1

Add or subtract the decimals:
a) $3.4+2.7$
b) $8.5-3.2$

Worked Example 2

Add or subtract the decimals:
a) $5.68+1.24$
b) $9.03-4.57$

Worked Example 3

Multiply the decimals:
a) $0.4\times 3$
b) $2.5\times 4$

Worked Example 4

Multiply the decimals:
a) $1.2\times 0.5$
b) $3.6\times 2.4$

Worked Example 5

Divide the decimals:
a) $8.4÷2$
b) $3.6÷4$

Worked Example 6

Use powers of $10$:
a) $4.57\times 10$
b) $83.2÷100$
c) $0.406\times 1000$

Problems

Problem 1

Add or subtract the decimals:
a) $4.3+1.6$
b) $7.9-2.4$

Problem 2

Add or subtract the decimals:
a) $6.75+2.13$
b) $8.04-3.28$

Problem 3

Multiply the decimals:
a) $0.7\times 5$
b) $1.8\times 3$

Problem 4

Multiply the decimals:
a) $1.5\times 0.4$
b) $2.7\times 1.2$

Problem 5

Divide the decimals:
a) $9.6÷3$
b) $4.2÷6$

Problem 6

Use powers of $10$:
a) $5.83\times 10$
b) $64.7÷100$
c) $0.52\times 100$

Exercises

Understanding and Fluency

1. Add the decimals:
   a) $2.4+3.1$
   b) $5.67+1.22$
   c) $0.8+0.35$

2. Subtract the decimals:
   a) $7.5-2.1$
   b) $9.04-3.6$
   c) $6.3-0.78$

3. Add or subtract the decimals:
   a) $4.08+2.7$
   b) $8.9-4.35$
   c) $1.25+0.47$
   d) $10.0-6.28$

4. Multiply the decimals:
   a) $0.6\times 4$
   b) $2.3\times 5$
   c) $1.4\times 0.2$

5. Multiply the decimals:
   a) $3.2\times 1.5$
   b) $0.9\times 0.7$
   c) $4.5\times 2.2$

6. Divide the decimals:
   a) $7.2÷3$
   b) $5.4÷6$
   c) $9.8÷2$

7. Multiply or divide by powers of $10$:
   a) $3.45\times 10$
   b) $6.2\times 100$
   c) $47.8÷10$
   d) $9.36÷100$

8. Multiply or divide by powers of $10$:
   a) $0.58\times 1000$
   b) $72.4÷1000$
   c) $6.007\times 100$
   d) $840÷10$

Reasoning

9. Explain why the decimal points must be lined up when adding or subtracting decimals.

10. A student says that $3.6+0.45=3.105$ because $6+45=105$. Explain the mistake.

11. Noah says that multiplying by $10$ means “add a zero”. Is he correct? Explain your answer using a decimal example.

12. Explain why $4.8÷10=0.48$.

13. A student says that $2.4\times 0.3=7.2$ because $24\times 3=72$. Describe the error.

Problem-solving

14. A runner completed one lap in $1.75$ min and another lap in $1.68$ min. What was the total time?

15. A water tank held $8.5$ L and $2.75$ L was used. How much water was left?

16. A ribbon costs $\mathrm{\$}2.4$ per metre. What is the cost of $3$ m?

17. A machine packs $1.25$ kg of rice into each bag. What is the total mass of $4$ bags?

18. A $6.4$ m rope is cut equally into $4$ pieces. How long is each piece?

19. A number is $0.483$. What number is $100$ times as large, and what number is ${\displaystyle \frac{1}{10}}$ of it?

Potential Misunderstandings

• Students may forget to line up decimal points when adding or subtracting
• Students may treat decimals as whole numbers and ignore place value
• Students may think a longer decimal is always greater or more precise in a calculation
• Students may make regrouping errors when subtracting decimals
• Students may multiply the digits correctly but place the decimal point incorrectly in the answer
• Students may think multiplying by $10$ always means writing an extra zero at the end
• Students may think dividing by $10$ only changes the last digit
• Students may move the decimal point in the wrong direction when multiplying or dividing by powers of $10$
• Students may forget that multiplying by powers of $10$ changes place value, not the digits themselves
• Students may confuse dividing a decimal by a whole number with moving the decimal point automatically
• Students may not check whether their answer is reasonable compared with the original numbers