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.