Mathematics • Year 7 • Unit 1 • Lesson 6
Understanding Fractions
Build the basics: name the numerator and denominator, tell proper apart from improper and mixed numbers, and find a fraction of a quantity by dividing then multiplying.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Find 2/5 of 30.
Step 1 — Identify the numerator and denominator.
Numerator = 2 (top), Denominator = 5 (bottom).
Reason: the denominator tells us how many equal parts the whole is split into; the numerator tells us how many of those parts we want.
Step 2 — Divide the quantity by the denominator.
30 ÷ 5 = 6
Reason: splitting 30 into 5 equal groups makes each group worth 6.
Step 3 — Multiply that result by the numerator.
6 × 2 = 12
Reason: we want 2 of those groups of 6, so we have 6 + 6 = 12.
Step 4 — State the answer with units in the original context.
2/5 of 30 = 12
Answer: 12. (Sanity check: 12 is less than 30, as expected because 2/5 is less than 1.)
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Find 3/4 of 28.
Step 1 — Identify numerator and denominator:
Numerator = _____ Denominator = _____
Step 2 — Divide 28 by the denominator:
28 ÷ _____ = _____
Step 3 — Multiply by the numerator:
_____ × _____ = _____
Step 4 — State the answer:
3/4 of 28 = _____
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation, the middle two are standard, and the last two are extension.
Foundation — single step
3.1 Classify each as proper, improper, or mixed: (a) 5/8 (b) 9/4 (c) 2 1/3. 1 mark
3.2 A pizza is cut into 8 equal slices. You eat 5 slices. What fraction did you eat? What fraction is left? 1 mark
3.3 Find 1/3 of 18. 1 mark
3.4 Find 1/4 of 20. 1 mark
Standard — combine two ideas
3.5 Find 3/5 of 40. Show every step. 2 marks
3.6 Convert 17/5 to a mixed number. (Hint: how many whole 5's fit into 17, and what is left over?) 2 marks
Extension — push your thinking
3.7 Convert 4 1/3 to an improper fraction and explain in one sentence why the rule (whole × denominator + numerator) works. 3 marks
3.8 A bag holds 48 lollies. You give away 5/8 of the lollies. How many do you have left? Show two methods: (a) find 5/8 of 48 then subtract, (b) find 3/8 of 48 directly. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (3/4 of 28)
Step 1: numerator = 3, denominator = 4.
Step 2: 28 ÷ 4 = 7.
Step 3: 7 × 3 = 21.
Step 4: 3/4 of 28 = 21.
3.1 — Classify
(a) 5/8 is proper (top < bottom).
(b) 9/4 is improper (top > bottom).
(c) 2 1/3 is a mixed number (whole + proper fraction).
3.2 — Pizza fractions
You ate 5/8. Left over: 8 − 5 = 3 slices, so 3/8 is left. The numerators add to 8, the total slices, which is the check.
3.3 — 1/3 of 18
18 ÷ 3 = 6, then 6 × 1 = 6.
3.4 — 1/4 of 20
20 ÷ 4 = 5, then 5 × 1 = 5.
3.5 — 3/5 of 40
Step 1: numerator 3, denominator 5.
Step 2: 40 ÷ 5 = 8.
Step 3: 8 × 3 = 24.
Check: 24 is less than 40, as expected (3/5 < 1).
3.6 — 17/5 as a mixed number
17 ÷ 5 = 3 remainder 2. So 17/5 = 3 2/5. (Three whole fives fit into 17, with 2 fifths left over.)
3.7 — 4 1/3 as improper
Whole × denominator + numerator = 4 × 3 + 1 = 13. Keep the denominator. So 4 1/3 = 13/3.
Why the rule works: the 4 wholes are each made of 3 thirds, so they contribute 4 × 3 = 12 thirds. Add the extra 1 third to get 13 thirds total.
3.8 — Lollies given away
(a) 5/8 of 48: 48 ÷ 8 = 6, then 6 × 5 = 30 given away. Left over: 48 − 30 = 18.
(b) 3/8 of 48 directly: 48 ÷ 8 = 6, then 6 × 3 = 18. ✓ Both methods match.