Multiplying and dividing fractions

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

• Slides Download Copy address for Prepare Present from the browser, or export to PowerPoint or PDF.
• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• I can multiply two fractions and simplify the answer.
• I can divide a fraction by a whole number and by a fraction.
• I can say whether a product will be bigger or smaller than the number I started with.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 10 min · Practise and Make 12 min

Before class

Do the Run board yourself first. Students need equivalent fractions and simplifying, and adding fractions (previous lesson) to contrast.

The one idea: a/b × c/d is c/d of a/b; the parts multiply, so the tops and the bottoms multiply. The hinge misconception is that multiplying always makes bigger (Jack's mistake); it matters for percentages next.

Division is introduced as "how many fit", which justifies multiplying by the flipped fraction without a rule to memorise.

Running the lesson

1. Predict: students who chose bigger hold the 'multiplying makes bigger' belief. Keep the vote visible.
2. Run: ask why the overlap is smaller than both shaded parts.
3. Investigate: contrast with adding: no common denominator needed here.
4. Modify: for the last task, ask students to sort multipliers into 'less than 1' and 'more than 1'.
5. Make: collect products equal to 1/4; ask what they all simplify to.

The hard part (on purpose)

Letting go of the belief that multiplying makes bigger. Ask for an estimate (bigger or smaller than the starting number?) before every calculation.

When students get stuck

Answers to the checks

1. half of a third: Is $\tfrac{1}{2} \times \tfrac{1}{3}$ bigger or smaller than $\tfrac{1}{2}$?
   Smaller: 1/2 × 1/3 = 1/6, a third of a half. Multiplying by a number less than 1 takes only part.
2. a third of a half: The board shows $\tfrac{1}{2}$ of the square. Change $c$ and $d$ to show $\tfrac{1}{2} \times \tfrac{1}{3}$. Was your prediction right?
   
   The overlap is 1 of 6 parts: 1/2 × 1/3 = 1/6.
3. Multiply across; divide by flipping: Work out $\tfrac{2}{3} \times \tfrac{3}{5}$ in simplest form.
   2/5
   (2 × 3)/(3 × 5) = 6/15 = 2/5.
4. Find the mistake: Jack compares $\tfrac{3}{4} \times 20$ with 20. Click the line where his reasoning goes wrong.
   line 3
   Line 3: multiplying by a number less than 1 makes smaller. 3/4 of 20 is 15, less than 20.
5. How many fit?: How many quarters are there in 3? Work out $3 \div \tfrac{1}{4}$.
   12
   4 quarters in each whole, so 12 in 3: 3 ÷ 1/4 = 3 × 4 = 12.
6. Three quarters of two fifths: Show $\tfrac{3}{4} \times \tfrac{2}{5}$ on the board.
   
   6 of 20 parts: 6/20 = 3/10.
7. Of a whole number: Work out $\tfrac{2}{5}$ of 30.
   12
   30 ÷ 5 = 6, × 2 = 12.
8. Divide by a whole number: Work out $\tfrac{3}{4} \div 3$ in simplest form.
   1/4
   Share 3 quarters into 3: one quarter each. (3/4 × 1/3 = 3/12 = 1/4.)
9. Divide by a fraction: Work out $\tfrac{2}{3} \div \tfrac{1}{6}$.
   4
   2/3 = 4/6, so four sixths fit: 2/3 × 6 = 4.
10. Bigger or smaller?: Edge case: which of these is bigger than 12?
    $12 \times \tfrac{5}{4}$
    5/4 is more than 1, so 12 × 5/4 = 15 is bigger. Multiplying makes bigger only when the multiplier is more than 1.
11. Multiply: Work out $\tfrac{4}{5} \times \tfrac{5}{8}$ in simplest form.
    1/2
    20/40 = 1/2.
12. Decimals: A bottle holds 0.6 L. How many litres do 4 bottles hold?
    2.4
    0.6 × 4 = 2.4 L.
13. Divide in context: A ribbon is $\tfrac{3}{4}$ m long. How many pieces $\tfrac{1}{8}$ m long can be cut from it?
    6
    3/4 = 6/8: six eighths. (3/4 ÷ 1/8 = 3/4 × 8 = 6.)
14. Review: adding: Review: work out $\tfrac{1}{2} + \tfrac{1}{5}$.
    7/10
    5/10 + 2/10 = 7/10.
15. a quarter, another way: Make the overlap exactly $\tfrac{1}{4}$ of the square, without using $\tfrac{1}{2} \times \tfrac{1}{2}$.
    
    For example 3/4 × 1/3 = 3/12, 2/4 × 1/2 = 2/8, or 2/3 × 3/8 = 6/24.

Solutions

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

one change at a time › Three quarters of two fifths

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

one change at a time › Of a whole number

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

one change at a time › Divide by a whole number

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

one change at a time › Divide by a fraction

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

one change at a time › Bigger or smaller?

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

a quarter, another way

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

Stretch and support

Stretch: Explain with the area square why 2/3 × 3/4 = 3/4 × 2/3.

Support: Use the board for products; strips of paper folded into quarters for division.

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 "Multiplying and dividing fractions".

The idea: Multiplication and division of fractions and decimals: I can Students multiply and divide fractions and decimals using appropriate strategies including area models and algorithm.

What understanding looks like:
- Correctly multiplies two fractions and simplifies the result.
- Correctly divides a fraction by a whole number or simple fraction.
- Does not apply addition rules (common denominator) to multiplication.

Mistakes students often make here:
- A student says multiplying makes bigger (Jack's mistake).
- A student finds a common denominator before multiplying.
- A student multiplies when dividing by a fraction (3 ÷ 1/4 = 3/4).

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.
- If I seem to be missing the idea that comes before this one (Addition and subtraction of fractions and decimals), check it quickly, then come back.
- 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 "Multiplying and dividing fractions".

Exit ticket

Work out 3/5 × 10/12, and 2 ÷ 1/3.

30/60 = 1/2. 2 ÷ 1/3 = 6 (six thirds in 2).

What each step is for

• half of a third: Committing to whether 1/2 × 1/3 is bigger or smaller than 1/2 surfaces the belief that multiplying always makes bigger.
• a third of a half: Shading 1/2 of the columns and 1/3 of the rows shows the overlap is 1/6: smaller than both.
• multiply, and divide: Tops times tops and bottoms times bottoms; Jack's belief that multiplying makes bigger fails for fractions less than 1. Dividing counts how many fit, so dividing by a fraction less than 1 makes bigger.
• one change at a time: One change at a time: a new product on the board, a fraction of a whole number, dividing by a whole number, dividing by a fraction, then which product is bigger than the number.
• multiply and divide: Fractions, a decimal and a division, mixed; 'multiplying makes bigger' is the planted slip; the review keeps adding fractions fresh.
• a quarter, another way: Building a product equal to 1/4 other than 1/2 × 1/2 shows many products share a value, and that simplifying is part of multiplying.

Where it leads

Next: Percentages (node 3), where taking a percentage is multiplying by a fraction of 100.

Where it leads: rates and ratio, and algebraic fractions (Year 10).