Mathematics • Year 7 • Unit 1 • Lesson 9
Multiply & Divide Fractions — Mixed Challenge
Pull every Lesson 9 idea together: multiply straight across, keep-change-flip for division, cross-cancel before multiplying, handle mixed numbers correctly, and spot the classic "flip the wrong fraction" error.
1. Mixed problems — choose the right idea
Each question uses a different part of Lesson 9. Decide which idea applies before you start writing. Show your working. 2 marks each
1.1 2/5 × 15/8. (Cross-cancel before multiplying.)
1.2 7/10 ÷ 14/15. (Keep, change, flip — then cross-cancel.)
1.3 2/3 of 18 (use multiplication).
1.4 2 1/2 × 1 1/4. Convert to improper fractions first.
1.5 3 1/3 ÷ 1 2/3. Convert to improper, then keep-change-flip.
1.6 A jug holds 4 1/2 cups of water. You pour it into glasses that each hold 3/4 of a cup. How many full glasses can you fill?
2. Find the mistake
Another Year 7 student has tried to calculate 1/2 ÷ 1/4. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — calculate 1/2 ÷ 1/4:
Line 1: Rule: "Keep, change, flip" the second fraction.
Line 2: Flip the first fraction: 1/2 becomes 2/1.
Line 3: Change ÷ to ×.
Line 4: Multiply: 2/1 × 1/4 = 2/4 = 1/2.
Line 5: Answer: 1/2.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? "Keep" the first fraction means it stays the same. Only the second fraction flips.3. Open-ended challenge — fraction puzzle
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 Find two different fractions (not equal to each other) that multiply to give 1. Show how you found them, write down the two fractions, and explain in one sentence what property of one fraction makes it the "partner" of the other for multiplication.
Bonus: Find three different fractions that multiply together to give 1 — each fraction must be different.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 2/5 × 15/8
Cross-cancel: 2 and 8 share 2 (→ 1 and 4). 5 and 15 share 5 (→ 1 and 3). Result: 1/1 × 3/4 = 3/4.
1.2 — 7/10 ÷ 14/15
Keep, change, flip: 7/10 × 15/14. Cross-cancel: 7 and 14 share 7 (→ 1 and 2). 10 and 15 share 5 (→ 2 and 3). Result: 1/2 × 3/2 = 3/4.
1.3 — 2/3 of 18
2/3 × 18/1. Cross-cancel: 18 and 3 share 3 (→ 6 and 1). Result: 2/1 × 6/1 = 12.
1.4 — 2 1/2 × 1 1/4
Convert: 2 1/2 = 5/2, 1 1/4 = 5/4. Multiply: 5/2 × 5/4 = 25/8 = 3 1/8.
1.5 — 3 1/3 ÷ 1 2/3
Convert: 3 1/3 = 10/3, 1 2/3 = 5/3. Keep, change, flip: 10/3 × 3/5. Cross-cancel: 3 and 3 share 3 (→ 1 and 1). 10 and 5 share 5 (→ 2 and 1). Result: 2/1 × 1/1 = 2.
1.6 — Jug to glasses
4 1/2 ÷ 3/4. Convert: 4 1/2 = 9/2. Keep, change, flip: 9/2 × 4/3. Cross-cancel: 4 and 2 share 2 (→ 2 and 1). 9 and 3 share 3 (→ 3 and 1). Result: 3/1 × 2/1 = 6.
So she can fill 6 full glasses. (Sense check: each glass is 3/4 cup, so 6 glasses use 6 × 3/4 = 18/4 = 4 1/2 cups exactly. ✓)
2 — Find the mistake
(a) The mistake is on Line 2.
(b) "Keep, change, flip" means keep the first fraction unchanged, change ÷ to ×, and flip only the second fraction. The student flipped the first fraction instead.
(c) Corrected working:
Line 2 (fixed): Keep the first fraction the same: 1/2 stays as 1/2. Flip the second: 1/4 becomes 4/1.
Line 4 (fixed): 1/2 × 4/1 = 4/2 = 2.
Sanity check: how many 1/4's fit in 1/2? Two. ✓
3 — Fraction puzzle (sample solution)
Pick any proper fraction and its reciprocal. Example: 2/3 × 3/2 = 6/6 = 1. Other valid answers: 3/4 × 4/3, 5/7 × 7/5, 1/8 × 8/1, etc.
The property that makes one fraction the "partner" of the other is the reciprocal: flipping the numerator and denominator. Any fraction times its reciprocal = 1.
Bonus: Find three different fractions that multiply to 1. Example: 2/3 × 3/4 × 4/2 = 24/24 = 1. (Or 1/2 × 2/3 × 3/1 = 6/6 = 1.) The chain works because the numerator of each fraction cancels with the denominator of the next, leaving 1 overall.
Marking: 2 marks for any valid pair of reciprocals with multiplication check; 1 mark for naming "reciprocal" as the key property; 1 mark for the bonus triple.