Problem
Smaller unit, more measured counts
A wooden stick is 6 ft long, and a wire is 8 ft long. Find how many times you would lay the wire end to end to cover the same length as 4 lengths of the wooden stick.
Measurement & data
Your answer
times
Watch out It's tempting to think the wire also needs 4 lengths, same as the stick — but a different unit length means a different count.
How to solve
Strategy Analyze the Units — First find the total length to cover (4 sticks), then see how many wires fit inside it. Because the wire is longer than the stick, fewer wires are needed — the smaller unit needs more counts.
1STEP 1
Find the total length of 4 sticks
Laying the 6 ft stick down 4 times covers 24 ft in all.
6 × 4 = 24
Laying the same 6 ft stick down four times is the same as adding 6 four times, so one multiplication, 6 × 4, gives the exact total to cover.
Identify Subproblems2.MD.A.12STEP 2
Count how many wires fit
Dividing by the wire's 8 ft length gives 3 wire-lengths.
24 ÷ 8 = 3
Laying several wires end to end is the same as adding equal 8 ft groups, and since they fill the length with no gaps or overlaps, those pieces add back to the whole 24 ft. So splitting 24 back into 8s — 24 ÷ 8 — undoes that multiplication and tells us exactly how many wires are needed.
Analyze The Units2.MD.A.1Smaller Unit More Measured CountsThe number of 8-ft wires laid end to end that covers 24 ft is found by dividing 24 by 8.
Why?
Covering 24 ft with equal 8-ft wires means finding how many 8-ft lengths fill 24 ft, so we need the number n of wires with n groups of 8 ft making 24 ft.
Why?
Laying n wires each 8 ft long end to end is n equal groups of 8 ft, which is exactly what n times 8 means.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The wires fill the 24 ft with no gaps or overlaps, so their 8-ft pieces add back to the whole 24 ft.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Knowing n groups of 8 make 24, we recover the number of wires n by splitting 24 back into 8s with 24 divided by 8, because division undoes multiplication.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Answer
3 times
24 ÷ 8 = 3
Check — 3 wires × 8 ft = 24 ft, matching 4 sticks × 6 ft = 24 ft.
Takeaway
For the same total length, a shorter unit means more counts, and a longer unit means fewer.
- Total of 4 sticks → 6 × 4 = 24 ft
- Divide 24 ft by the 8 ft wire → 3 times
Where next?
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