Mathematics • Year 7 • Unit 1 • Lesson 6
Fractions, Mixed Challenge
Pull every Lesson 6 idea together: name the parts, classify proper/improper/mixed, convert in both directions, find fractions of quantities, and spot a classic conversion mistake. End with an open puzzle.
1. Mixed problems, choose the right idea
Each question uses a different part of Lesson 6. Decide which idea applies before you start writing. Show your working. 2 marks each
1.1 Classify each fraction as proper, improper, or mixed: (a) 4/9 (b) 7/7 (c) 11/6 (d) 5 2/3.
1.2 Convert 23/6 to a mixed number.
1.3 Convert 3 5/8 to an improper fraction.
1.4 Find 5/6 of 54.
1.5 Sam ate 3/8 of a chocolate bar; Jordan ate 1/4 of the same chocolate bar. Who ate more? (Hint: rewrite 1/4 as eighths so you can compare.)
1.6 A water bottle holds 600 mL. After PE class, you've drunk 2/3 of it. How many millilitres are left in the bottle?
2. Find the mistake
Another Year 7 student has tried to convert 3 2/5 to an improper fraction. 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, convert 3 2/5 to an improper fraction:
Line 1: Rule: whole × denominator + numerator, keep denominator.
Line 2: Whole = 3, denominator = 5, numerator = 2.
Line 3: 3 × 5 = 15
Line 4: 15 + 2 = 17
Line 5: Answer: 17/2
(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.
3. Open-ended challenge, fraction targets
This question has more than one correct answer. Show one that works and explain. 4 marks
3.1 Use each digit 2, 3, 5 and 8 exactly once across two fractions:
(i) one proper fraction using two digits (one numerator and one denominator);
(ii) one improper fraction using the other two digits (one numerator and one denominator).
Then convert your improper fraction into a mixed number. This conversion uses the same fraction; it does not require extra digits.
Bonus: Of all the proper fractions you could make with two of the digits 2, 3, 5, 8, which is the smallest? Justify your answer using the rule about denominators.
How did this worksheet feel?
What I'll revisit before next class:
1.1, Classify
(a) 4/9 proper; (b) 7/7 improper (equals 1); (c) 11/6 improper; (d) 5 2/3 mixed.
1.2-23/6 as mixed
23 ÷ 6 = 3 remainder 5. So 23/6 = 3 5/6.
1.3-3 5/8 as improper
3 × 8 + 5 = 29. Keep denominator. So 3 5/8 = 29/8.
1.4-5/6 of 54
54 ÷ 6 = 9, then 9 × 5 = 45.
1.5-3/8 vs 1/4
Rewrite 1/4 as eighths: 1/4 = 2/8. So Sam ate 3/8 and Jordan ate 2/8. Sam ate more (3 > 2 with the same denominator).
1.6, Water bottle
Method 1 (find 2/3 then subtract): 600 ÷ 3 = 200, then 200 × 2 = 400 mL drunk. Left = 600 − 400 = 200 mL.
Method 2 (find 1/3 directly): 600 ÷ 3 = 200 mL. ✓ Both methods match.
2, Find the mistake
(a) The mistake is on Line 5.
(b) The rule says keep the denominator. The original was 3 2/5, so the denominator must stay as 5, not become 2. The student wrote 17/2 instead of 17/5.
(c) Corrected working:
Line 5 (fixed): 17/5.
Quick sanity check: convert back: 17 ÷ 5 = 3 remainder 2 = 3 2/5. ✓ Matches the original.
3, Digit puzzle (sample solution)
One complete set uses all four digits exactly once:
(i) Proper fraction: 2/8, since 2 < 8.
(ii) Improper fraction: 5/3, since 5 > 3.
Convert the improper fraction: 5 ÷ 3 = 1 remainder 2, so 5/3 = 1 2/3.
Bonus: The smallest proper fraction is 2/8: it uses the smallest available numerator and the largest available denominator.
Marking: 1 mark for a valid proper fraction; 1 for a valid improper fraction using the remaining digits; 1 for its correct mixed-number conversion; 1 for the bonus justification.