Mathematics • Year 7 • Unit 2 • Lesson 7
Dividing Algebraic Terms in the Real World
Use algebraic division to share things equally, find a missing side of a rectangle when you know the area, work out averages, and check unit prices — all where the numbers are written as letters.
1. Word problems
Each problem uses ideas from Lesson 7: divide the numbers, cancel the variables, subtract the powers. Show your working — answers without working only get half marks.
1.1 — Sharing lollies. A bag holds 12x lollies. The bag is shared equally between 3 friends.
(a) Write an expression for the number of lollies each friend gets.
(b) If x = 5, how many lollies does each friend get? 2 marks
1.2 — Rectangle's missing side. A rectangle has area 12x² and width 3x.
(a) Write and simplify an expression for its length (length = area ÷ width).
(b) Check your answer by multiplying length × width — does it give back 12x²? 3 marks
1.3 — Triangle height. A triangle has area 15x² square units. The area of a triangle = ½ × base × height. The base is 5x.
(a) Write the equation: 15x² = ½ × 5x × height.
(b) Solve for height. (Hint: first multiply both sides by 2 to clear the ½, then divide by 5x.) 3 marks
1.4 — Hot chocolate cost. A school orders 6x cups of hot chocolate for a total cost of 24x dollars.
(a) Write an expression for the cost per cup (total ÷ number of cups).
(b) Why does the x cancel out, and what does that mean in practical terms? 2 marks
1.5 — Stacking blocks. A stack of blocks has total volume 20x³y² cubic centimetres. Each block has volume 4x²y cubic centimetres.
(a) Write an expression for the number of blocks in the stack (total ÷ one block).
(b) Simplify your expression. 3 marks
2. Explain your thinking
This question is about communication, not just symbols. Use full sentences. 4 marks
2.1 A classmate writes: "6x ÷ 2x = 3x — because the x is still there." In your own words, explain (i) why this answer is wrong, (ii) what the correct answer is and why, (iii) when you'd expect the variable to disappear and when you'd expect it to stay. Use a specific number (like x = 4) to demonstrate.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Sharing lollies
(a) Lollies per friend = 12x ÷ 3 = 4x.
(b) Substitute x = 5: 4 × 5 = 20 lollies each.
1.2 — Rectangle's missing side
(a) Length = area ÷ width = 12x² ÷ 3x = (12 ÷ 3)(x² ÷ x) = 4x.
(b) Check: 4x × 3x = (4 × 3)(x × x) = 12x² ✓ — matches the original area.
1.3 — Triangle height
(a) 15x² = ½ × 5x × height.
(b) Multiply both sides by 2: 30x² = 5x × height. Divide both sides by 5x: height = 30x² ÷ 5x = (30 ÷ 5)(x² ÷ x) = 6x.
1.4 — Hot chocolate cost
(a) Cost per cup = 24x ÷ 6x = (24 ÷ 6)(x ÷ x) = 4 × 1 = $4 per cup.
(b) The x cancels because there's an x on top AND on the bottom — anything divided by itself is 1. Practically: the price per cup doesn't depend on how many cups (x) you bought, so x drops out.
1.5 — Stacking blocks
(a) Number of blocks = total volume ÷ one block volume = 20x³y² ÷ 4x²y.
(b) Numbers: 20 ÷ 4 = 5. x³ ÷ x² = x. y² ÷ y = y. Answer: 5xy blocks.
2.1 — Explain your thinking (sample response)
(i) Saying 6x ÷ 2x = 3x is wrong because the student divided the numbers (6 ÷ 2 = 3) but forgot to cancel the variables. There's an x on top AND an x on the bottom, so x ÷ x = 1, and the x disappears.
(ii) The correct answer is 3 (not 3x). Working: (6 ÷ 2)(x ÷ x) = 3 × 1 = 3.
(iii) The variable cancels when it appears on both top and bottom with the same power (like 6x ÷ 2x). It stays when it's only on top (like 6x ÷ 2 = 3x). And if the top has a higher power, you keep what's left over (like 6x² ÷ 2x = 3x).
Check with x = 4: 6x ÷ 2x = 24 ÷ 8 = 3 ✓ (matches our correct answer, not 3x = 12).
Marking: 1 for spotting the missed cancel; 1 for stating correct answer 3 with working; 1 for the cancels-vs-stays distinction; 1 for a sensible numerical check.