112r. Review Decimals, Place Value and Conversions

Learning Intentions

• To understand place value in a decimal
• compare two or more decimals to decide which is largest
• convert decimals to fractions
• convert some simple fractions to decimals

Pre-requisite Summary

• Know that whole numbers use place value columns such as ones, tens and hundreds
• Understand that the decimal point separates whole-number parts from fractional parts
• Know that tenths, hundredths and thousandths are parts of a whole
• Be able to compare whole numbers by place value
• Understand that fractions represent parts of a whole
• Recall simple equivalent fractions such as ${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{5}}$ and ${\displaystyle \frac{1}{10}}$
• Know that a fraction can be written in different forms with the same value
• Be able to simplify fractions

Worked Examples

Worked Example 1

State the value of the underlined digit:
a) In $4.7$, the digit $7$
b) In $12.35$, the digit $3$
c) In $8.406$, the digit $6$

Worked Example 2

Write each decimal in expanded form:
a) $3.4$
b) $5.27$
c) $9.083$

Worked Example 3

Compare the decimals and state which is largest:
a) $0.5$ and $0.45$
b) $2.17,2.071,2.7$
c) $4.09$ and $4.9$

Worked Example 4

Convert each decimal to a fraction:
a) $0.3$
b) $0.75$
c) $1.2$

Worked Example 5

Convert each simple fraction to a decimal:
a) ${\displaystyle \frac{1}{2}}$
b) ${\displaystyle \frac{3}{10}}$
c) ${\displaystyle \frac{1}{4}}$

Worked Example 6

Convert each simple fraction to a decimal:
a) ${\displaystyle \frac{3}{5}}$
b) ${\displaystyle \frac{7}{10}}$
c) ${\displaystyle \frac{3}{4}}$

Problems

Problem 1

State the value of the underlined digit:
a) In $6.8$, the digit $8$
b) In $14.26$, the digit $2$
c) In $7.309$, the digit $9$

Problem 2

Write each decimal in expanded form:
a) $4.6$
b) $8.34$
c) $2.057$

Problem 3

Compare the decimals and state which is largest:
a) $0.6$ and $0.56$
b) $3.14,3.041,3.4$
c) $5.08$ and $5.8$

Problem 4

Convert each decimal to a fraction:
a) $0.4$
b) $0.62$
c) $2.5$

Problem 5

Convert each simple fraction to a decimal:
a) ${\displaystyle \frac{1}{10}}$
b) ${\displaystyle \frac{1}{5}}$
c) ${\displaystyle \frac{3}{4}}$

Problem 6

Convert each simple fraction to a decimal:
a) ${\displaystyle \frac{2}{5}}$
b) ${\displaystyle \frac{9}{10}}$
c) ${\displaystyle \frac{1}{2}}$

Exercises

Understanding and Fluency

1. State the place value of the underlined digit:
   a) In $3.8$, the digit $8$
   b) In $7.54$, the digit $5$
   c) In $6.209$, the digit $9$
   d) In $12.406$, the digit $4$

2. Write each decimal in expanded form:
   a) $2.5$
   b) $4.18$
   c) $9.307$
   d) $15.04$

3. Compare each pair of decimals using $<,>$ or $=$:
   a) $0.7◻0.67$
   b) $1.25◻1.205$
   c) $3.09◻3.9$
   d) $5.40◻5.4$

4. Put these decimals in order from smallest to largest:
   a) $0.8,0.08,0.18$
   b) $2.4,2.04,2.14$
   c) $5.09,5.9,5.19$

5. Convert each decimal to a fraction:
   a) $0.2$
   b) $0.9$
   c) $0.45$
   d) $1.6$

6. Convert each decimal to a fraction in simplest form where possible:
   a) $0.5$
   b) $0.25$
   c) $0.8$
   d) $0.75$

7. Convert each simple fraction to a decimal:
   a) ${\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{1}{4}}$
   c) ${\displaystyle \frac{1}{5}}$
   d) ${\displaystyle \frac{9}{10}}$

8. Convert each simple fraction to a decimal:
   a) ${\displaystyle \frac{3}{4}}$
   b) ${\displaystyle \frac{2}{5}}$
   c) ${\displaystyle \frac{7}{10}}$
   d) ${\displaystyle \frac{4}{5}}$

Reasoning

9. Explain why $0.6$ is greater than $0.56$ even though $56$ is greater than $6$.

10. A student says that $3.07$ is larger than $3.7$ because $07$ is larger than $7$. Explain the mistake.

11. Noah says that $0.5$ and $0.50$ are different numbers because one has more digits. Is he correct? Explain.

12. Explain why $0.75={\displaystyle \frac{75}{100}}={\displaystyle \frac{3}{4}}$.

13. A student says that ${\displaystyle \frac{1}{4}}=0.4$ because the denominator is $4$. Describe the error.

Problem-solving

14. A bottle contains $1.25$ L of water and another contains $1.205$ L. Which bottle contains more water?

15. A runner completed races in $12.4$ s, $12.37$ s and $12.09$ s. Which time was the greatest?

16. A ribbon is $0.75$ m long. Write this length as a fraction of a metre in simplest form.

17. A container is filled to ${\displaystyle \frac{3}{5}}$ of its capacity. Write this as a decimal.

18. A student scored $0.8$ on one task and ${\displaystyle \frac{3}{4}}$ on another. Which score is greater?

19. A number has $4$ ones, $3$ tenths, $0$ hundredths and $6$ thousandths. Write the decimal and then write it as a fraction.

Potential Misunderstandings

• Students may think digits after the decimal point work like whole numbers rather than place values
• Students may confuse tenths and hundredths
• Students may think a longer decimal is always greater
• Students may compare decimals by ignoring the place value of each digit
• Students may not recognise that trailing zeros do not change the value of a decimal, for example $0.5=0.50$
• Students may write a decimal as a fraction but forget to use the correct denominator such as $10$, $100$ or $1000$
• Students may not simplify decimal fractions when required
• Students may think converting a fraction to a decimal changes the value
• Students may confuse ${\displaystyle \frac{1}{4}}$ with $0.4$ instead of $0.25$
• Students may not see that some simple fractions can be converted by using known equivalent fractions with denominator $10$ or $100$