Equivalent fractions

Classroom resources

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Students will be able to

• I can find equivalent fractions by multiplying or dividing the top and the bottom by the same number.
• I can write a fraction in its simplest form using the highest common factor.
• I can explain with a picture why two fractions are equal.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need fractions as parts of a whole (Year 6) and factors and highest common factors (Year 7 Number).

The one idea: equal fractions shade the same amount; the same multiplication or division on the top and the bottom keeps the amount. The common mistake is adding the same number to both (2/3 = 3/4); it is planted in Lily's working.

Simplest-form answers are marked by form: an equal fraction that can still be simplified is told so, not marked right.

Running the lesson

1. Predict: take a show of hands. Students who chose 3/4 added 1 to both.
2. Run: ask what happened to each third when the orange bar went to 6 parts.
3. Investigate: model the worked example with the bars: split each part (multiply), join parts (divide).
4. Modify: before the fewest-parts task, ask: how will you know you cannot join any more?
5. Make: collect pairs and ask for the simplest form of each.

When students get stuck

Answers to the checks

1. the same as 2/3: Which fraction is the same amount as $\tfrac{2}{3}$?
   4/6. Split each third into 2: 6 parts, and the 2 shaded thirds become 4 shaded sixths. 3/4 (add 1 to both) is more than 2/3.
2. match the bars: The top bar is $\tfrac{2}{3}$. Set the orange bar to 6 parts ($d = 6$) and shade the same amount. Was your prediction right?
   
3. Simplest form: Write $\tfrac{12}{18}$ in its simplest form.
   2/3
   The highest common factor of 12 and 18 is 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 2/3.
4. Find the mistake: Lily looks for a fraction equal to $\tfrac{2}{3}$. Click the line where her working goes wrong.
   line 3
   Line 3: adding changes the amount (3/4 is more than 2/3). Doing the same to the top and the bottom means multiplying or dividing: 2/3 = 4/6.
5. Why they are equal: Why is $\tfrac{2}{3} = \tfrac{8}{12}$?
   Each third is split into 4 twelfths, so 2 thirds are 8 twelfths
   × 4 on the top and the bottom: the same amount, cut into 4 times as many parts.
6. Twelfths: The top bar is $\tfrac{3}{4}$. Make the orange bar an equal fraction with 12 parts.
   
   3/4 = 9/12: × 3 on the top and the bottom.
7. A missing number: Fill the gap: $\tfrac{3}{5} = \tfrac{?}{20}$.
   12
   5 × 4 = 20, so 3 × 4 = 12.
8. Simplify: Write $\tfrac{15}{25}$ in its simplest form.
   3/5
   Divide by 5: 3/5.
9. Fewest parts: The top bar is $\tfrac{8}{12}$. Make the orange bar the same amount with as few parts as possible.
   
   8/12 = 2/3: divide by 4, the highest common factor.
10. Already simplest: Edge case: can $\tfrac{7}{9}$ be simplified?
    No: 7 and 9 have no common factor but 1
    7 and 9 share no factor but 1, so 7/9 is already in simplest form.
11. Simplify: Write $\tfrac{18}{24}$ in its simplest form.
    3/4
    HCF 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4.
12. Missing number: Fill the gap: $\tfrac{5}{8} = \tfrac{15}{?}$.
    24
    × 3: 8 × 3 = 24.
13. Compare: Which is bigger, $\tfrac{2}{3}$ or $\tfrac{3}{5}$? Write both with 15 parts.
    2/3 (10/15 is more than 9/15)
    2/3 = 10/15 and 3/5 = 9/15, so 2/3 is bigger.
14. Review: integers: Review: work out $-3 + 5$.
    2
    Start at −3 and move 5 right: 2.
15. neither in lowest terms: Make the two bars equal, with neither fraction in lowest terms.
    
    For example 2/4 and 3/6, or 4/6 and 8/12: both simplify to the same fraction.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Twelfths

Open task: any program that does what the step asks.

one change at a time › A missing number

Open task: any program that does what the step asks.

one change at a time › Simplify

Open task: any program that does what the step asks.

one change at a time › Fewest parts

Open task: any program that does what the step asks.

one change at a time › Already simplest

Open task: any program that does what the step asks.

neither in lowest terms

Open task: any program that does what the step asks.

Stretch and support

Stretch: Find all the fractions equal to 3/4 with a bottom number less than 50. What do their bottoms have in common?

Support: Use the bars for every question; physical fraction walls help too.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Equivalent fractions".

The idea: Equivalence and simplification of fractions: I can Students find equivalent fractions using common multiples and express fractions in simplest form using common divisors.

What understanding looks like:
- Generates multiple equivalent fractions for a given fraction.
- Correctly simplifies a fraction to lowest terms.
- Can explain why the fractions are equivalent (multiplying/dividing numerator and denominator by the same value).

Mistakes students often make here:
- A student adds the same number to the top and the bottom (Lily's mistake).
- A student simplifies in steps and stops too early (6/9 for 12/18).
- A student cannot find the missing number in 3/5 = ?/20.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Equivalent fractions".

Exit ticket

Write 20/32 in its simplest form, and give one other fraction equal to it.

5/8 (÷ 4). For example 10/16 or 15/24.

What each step is for

• the same as 2/3: Committing to a fraction equal to 2/3 shows whether they multiply both numbers or add the same number to both.
• match the bars: Matching the shaded amount with 6 parts shows that splitting every part multiplies both the number of parts and the number shaded.
• multiply or divide both: Multiplying or dividing the top and the bottom by the same number keeps the amount; dividing by the highest common factor gives lowest terms in one step. Lily's mistake adds the same number to both.
• one change at a time: One change at a time: a different fraction on the board, a missing number, simplifying, simplifying on the board, then the edge case of a fraction already in lowest terms.
• equal fractions: Simplifying, finding a missing number and comparing with a common denominator, mixed; adding to both is the planted slip; the review keeps integers fresh.
• neither in lowest terms: Building two equal fractions where neither is in lowest terms shows that one amount has many names, all multiples of its simplest one.

Where it leads

Next: Fractions, decimals and percentages (node 2), then Fractions and decimals on the line (node 3).

Where it leads: adding and subtracting fractions with different denominators (common denominators), and ratio.