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 ${\displaystyle \frac{1}{10}},{\displaystyle \frac{1}{100}},{\displaystyle \frac{1}{1000}}$
• 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 $4.7$.
b) State the value of the digit $4$.
c) State the value of the digit $7$.

Worked Example 2

For the decimal $3.482$:
a) Name the place value of each digit after the decimal point.
b) State the value of the digit $4$.
c) State the value of the digit $8$.

Worked Example 3

Compare the decimals and decide which is larger:
a) $0.6$ and $0.56$
b) $2.34$ and $2.43$
c) $5.08$ and $5.8$

Worked Example 4

Compare the decimals and write $<,>,$ or $=$:
a) $1.2\mathrm{\_}\mathrm{\_}1.20$
b) $0.09\mathrm{\_}\mathrm{\_}0.9$
c) $4.307\mathrm{\_}\mathrm{\_}4.37$

Worked Example 5

Convert each fraction to a decimal:
a) ${\displaystyle \frac{3}{10}}$
b) ${\displaystyle \frac{47}{100}}$
c) ${\displaystyle \frac{205}{1000}}$

Worked Example 6

Convert each mixed numeral to a decimal:
a) $2{\displaystyle \frac{4}{10}}$
b) $5{\displaystyle \frac{32}{100}}$
c) $7{\displaystyle \frac{9}{1000}}$

Problems

Problem 1

a) Explain the meaning of the decimal point in $6.3$.
b) State the value of the digit $6$.
c) State the value of the digit $3$.

Problem 2

For the decimal $8.571$:
a) Name the place value of each digit after the decimal point.
b) State the value of the digit $5$.
c) State the value of the digit $7$.

Problem 3

Compare the decimals and decide which is larger:
a) $0.4$ and $0.39$
b) $3.27$ and $3.72$
c) $6.05$ and $6.5$

Problem 4

Compare the decimals and write $<,>,$ or $=$:
a) $2.5\mathrm{\_}\mathrm{\_}2.50$
b) $0.07\mathrm{\_}\mathrm{\_}0.7$
c) $9.406\mathrm{\_}\mathrm{\_}9.46$

Problem 5

Convert each fraction to a decimal:
a) ${\displaystyle \frac{6}{10}}$
b) ${\displaystyle \frac{83}{100}}$
c) ${\displaystyle \frac{419}{1000}}$

Problem 6

Convert each mixed numeral to a decimal:
a) $3{\displaystyle \frac{7}{10}}$
b) $4{\displaystyle \frac{58}{100}}$
c) $9{\displaystyle \frac{6}{1000}}$

Exercises

Understanding and Fluency

1. State the value of the underlined digit:
   a) $2.\underset{―}{5}$
   b) $7.\underset{―}{3}4$
   c) $9.4\underset{―}{2}6$

2. Name the place value of each underlined digit:
   a) $1.\underset{―}{8}$
   b) $3.\underset{―}{4}7$
   c) $6.2\underset{―}{9}5$

3. Write each decimal in words:
   a) $4.2$
   b) $0.35$
   c) $7.408$

4. Compare the decimals using $<,>,$ or $=$:
   a) $0.5\mathrm{\_}\mathrm{\_}0.45$
   b) $1.08\mathrm{\_}\mathrm{\_}1.8$
   c) $3.20\mathrm{\_}\mathrm{\_}3.2$

5. Compare the decimals using $<,>,$ or $=$:
   a) $4.17\mathrm{\_}\mathrm{\_}4.107$
   b) $0.09\mathrm{\_}\mathrm{\_}0.12$
   c) $8.005\mathrm{\_}\mathrm{\_}8.05$

6. Convert each fraction to a decimal:
   a) ${\displaystyle \frac{7}{10}}$
   b) ${\displaystyle \frac{18}{100}}$
   c) ${\displaystyle \frac{302}{1000}}$

7. Convert each fraction to a decimal:
   a) ${\displaystyle \frac{9}{10}}$
   b) ${\displaystyle \frac{64}{100}}$
   c) ${\displaystyle \frac{5}{1000}}$

8. Convert each mixed numeral to a decimal:
   a) $1{\displaystyle \frac{3}{10}}$
   b) $6{\displaystyle \frac{25}{100}}$
   c) $2{\displaystyle \frac{407}{1000}}$

9. Convert each mixed numeral to a decimal:
   a) $8{\displaystyle \frac{1}{10}}$
   b) $5{\displaystyle \frac{9}{100}}$
   c) $3{\displaystyle \frac{56}{1000}}$

10. Mixed practice:
    a) Which is larger: $2.6$ or $2.06$?
    b) Convert ${\displaystyle \frac{45}{100}}$ to a decimal.
    c) Convert $4{\displaystyle \frac{8}{10}}$ to a decimal.

Reasoning

11. Explain why $0.7>0.65$ even though $65>7$.

12. A student says $3.4$ is larger than $3.27$ because $27>4$. Explain the mistake.

13. Explain why $1.20=1.2$.

14. A student writes ${\displaystyle \frac{5}{100}}=0.5$. Explain why this is incorrect.

Problem-solving

15. A measuring jug contains $2{\displaystyle \frac{3}{10}}$ litres of water. Write this amount as a decimal.

16. A ribbon is ${\displaystyle \frac{75}{100}}$ m long. Write this length as a decimal.

17. Three runners recorded times of $12.4$ s, $12.38$ s and $12.09$ s. Order these times from shortest to longest.

18. A container holds $5{\displaystyle \frac{42}{100}}$ kg of rice. Write this as a decimal and state the value of the digit $4$.

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 $2.5=2.50$
• Students may think ${\displaystyle \frac{5}{100}}=0.5$ instead of $0.05$
• 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