109r. Equivalent Fractions and Simplifying Fractions

Learning Intentions

• To understand what equivalent fractions are
• To understand that fractions have a numerical value, and two distinct fractions, like ${\displaystyle \frac{3}{5}}$ and ${\displaystyle \frac{6}{10}}$, can have the same numerical value
• simplify fractions

Pre-requisite Summary

• Know that a fraction represents part of a whole or part of a collection
• Understand that the numerator tells how many parts are being considered
• Understand that the denominator tells how many equal parts the whole is divided into
• Be able to recognise basic fractions such as ${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{4}}$ and ${\displaystyle \frac{3}{4}}$
• Recall multiplication facts and division facts
• Know that factors are numbers that divide exactly into another number

Worked Examples

Worked Example 1

State whether each pair of fractions is equivalent:
a) ${\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{2}{4}}$
b) ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{6}{8}}$
c) ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{3}{5}}$

Worked Example 2

Find an equivalent fraction for each:
a) ${\displaystyle \frac{2}{5}}$
b) ${\displaystyle \frac{3}{7}}$
c) ${\displaystyle \frac{4}{9}}$

Worked Example 3

Simplify each fraction:
a) ${\displaystyle \frac{6}{8}}$
b) ${\displaystyle \frac{10}{15}}$
c) ${\displaystyle \frac{12}{18}}$

Worked Example 4

Simplify each fraction fully:
a) ${\displaystyle \frac{14}{21}}$
b) ${\displaystyle \frac{16}{24}}$
c) ${\displaystyle \frac{20}{30}}$

Worked Example 5

Use equivalent fractions to compare:
a) ${\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{3}{6}}$
b) ${\displaystyle \frac{2}{4}}$ and ${\displaystyle \frac{5}{10}}$
c) ${\displaystyle \frac{4}{5}}$ and ${\displaystyle \frac{8}{10}}$

Worked Example 6

Write each fraction in simplest form:
a) ${\displaystyle \frac{9}{12}}$
b) ${\displaystyle \frac{15}{25}}$
c) ${\displaystyle \frac{24}{36}}$

Problems

Problem 1

State whether each pair of fractions is equivalent:
a) ${\displaystyle \frac{1}{3}}$ and ${\displaystyle \frac{2}{6}}$
b) ${\displaystyle \frac{2}{5}}$ and ${\displaystyle \frac{4}{10}}$
c) ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{4}{7}}$

Problem 2

Find an equivalent fraction for each:
a) ${\displaystyle \frac{1}{4}}$
b) ${\displaystyle \frac{2}{7}}$
c) ${\displaystyle \frac{5}{8}}$

Problem 3

Simplify each fraction:
a) ${\displaystyle \frac{8}{10}}$
b) ${\displaystyle \frac{9}{12}}$
c) ${\displaystyle \frac{15}{20}}$

Problem 4

Simplify each fraction fully:
a) ${\displaystyle \frac{18}{24}}$
b) ${\displaystyle \frac{21}{28}}$
c) ${\displaystyle \frac{27}{36}}$

Problem 5

Use equivalent fractions to compare:
a) ${\displaystyle \frac{1}{3}}$ and ${\displaystyle \frac{2}{6}}$
b) ${\displaystyle \frac{3}{5}}$ and ${\displaystyle \frac{6}{10}}$
c) ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{4}{6}}$

Problem 6

Write each fraction in simplest form:
a) ${\displaystyle \frac{6}{9}}$
b) ${\displaystyle \frac{18}{30}}$
c) ${\displaystyle \frac{35}{49}}$

Exercises

Understanding and Fluency

1. State whether each pair of fractions is equivalent:
   a) ${\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{3}{6}}$
   b) ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{4}{6}}$
   c) ${\displaystyle \frac{3}{5}}$ and ${\displaystyle \frac{6}{10}}$
   d) ${\displaystyle \frac{2}{7}}$ and ${\displaystyle \frac{3}{7}}$

2. Find an equivalent fraction for each:
   a) ${\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{2}{3}}$
   c) ${\displaystyle \frac{3}{4}}$
   d) ${\displaystyle \frac{4}{5}}$

3. Complete each statement with an equivalent fraction:
   a) ${\displaystyle \frac{1}{3}}={\displaystyle \frac{◻}{6}}$
   b) ${\displaystyle \frac{2}{5}}={\displaystyle \frac{◻}{10}}$
   c) ${\displaystyle \frac{3}{4}}={\displaystyle \frac{◻}{8}}$
   d) ${\displaystyle \frac{4}{7}}={\displaystyle \frac{◻}{14}}$

4. Simplify each fraction:
   a) ${\displaystyle \frac{4}{6}}$
   b) ${\displaystyle \frac{6}{9}}$
   c) ${\displaystyle \frac{8}{12}}$
   d) ${\displaystyle \frac{10}{15}}$

5. Simplify each fraction fully:
   a) ${\displaystyle \frac{12}{16}}$
   b) ${\displaystyle \frac{14}{20}}$
   c) ${\displaystyle \frac{18}{27}}$
   d) ${\displaystyle \frac{30}{45}}$

6. Write each fraction in simplest form:
   a) ${\displaystyle \frac{16}{20}}$
   b) ${\displaystyle \frac{24}{32}}$
   c) ${\displaystyle \frac{28}{42}}$
   d) ${\displaystyle \frac{40}{60}}$

7. Use equivalent fractions to decide whether each statement is true or false:
   a) ${\displaystyle \frac{2}{4}}={\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{3}{6}}={\displaystyle \frac{2}{3}}$
   c) ${\displaystyle \frac{4}{8}}={\displaystyle \frac{1}{2}}$
   d) ${\displaystyle \frac{5}{10}}={\displaystyle \frac{1}{2}}$

8. Fill in the missing number:
   a) ${\displaystyle \frac{2}{3}}={\displaystyle \frac{6}{◻}}$
   b) ${\displaystyle \frac{3}{5}}={\displaystyle \frac{◻}{15}}$
   c) ${\displaystyle \frac{4}{7}}={\displaystyle \frac{12}{◻}}$
   d) ${\displaystyle \frac{5}{8}}={\displaystyle \frac{◻}{24}}$

Reasoning

9. Explain why ${\displaystyle \frac{2}{4}}$ and ${\displaystyle \frac{1}{2}}$ have the same numerical value.

10. A student says that ${\displaystyle \frac{3}{5}}$ and ${\displaystyle \frac{6}{10}}$ are different because the numbers in the numerator and denominator are different. Explain the mistake.

11. Noah says that to simplify a fraction, you can subtract the same number from the numerator and denominator. Is he correct? Explain.

12. Explain why ${\displaystyle \frac{4}{6}}$ and ${\displaystyle \frac{2}{3}}$ are equivalent.

13. A student simplifies ${\displaystyle \frac{12}{18}}$ to ${\displaystyle \frac{6}{12}}$ and says the fraction is fully simplified. Describe the error.

Problem-solving

14. A recipe uses ${\displaystyle \frac{2}{4}}$ of a cup of milk. Another recipe uses ${\displaystyle \frac{1}{2}}$ of a cup of milk. Do they use the same amount? Explain.

15. A class shaded ${\displaystyle \frac{6}{8}}$ of a shape. Write this fraction in simplest form.

16. A student writes two fractions, ${\displaystyle \frac{4}{10}}$ and ${\displaystyle \frac{2}{5}}$. Show whether they represent the same amount.

17. A packet is ${\displaystyle \frac{9}{12}}$ full. Write this in simplest form and explain what it means.

18. Liam says that ${\displaystyle \frac{5}{15}}$ is equivalent to ${\displaystyle \frac{1}{3}}$. Show how to check whether he is correct.

19. Write three different fractions that are all equivalent to ${\displaystyle \frac{1}{2}}$.

Potential Misunderstandings

• Students may think fractions are only equivalent if the numerator and denominator are exactly the same
• Students may focus on the appearance of the numbers rather than the numerical value of the fraction
• Students may think equivalent fractions must have the same denominator
• Students may forget that to make an equivalent fraction, the numerator and denominator must both be multiplied or divided by the same non-zero number
• Students may try to add or subtract the same number to the numerator and denominator to make equivalent fractions
• Students may simplify only one part of the fraction instead of both parts
• Students may stop simplifying before the fraction is in simplest form
• Students may confuse simplifying a fraction with changing its value
• Students may not recognise that a simplified fraction and the original fraction represent the same amount