Problem
Different factor pairs, same product
A wire is as long as a 9 cm stick placed end to end 4 times. Each triangle is made of three 2 cm sides. We want the greatest number of such triangles the wire can make.
Operations
Your answer
triangles
Watch out Each triangle needs all three 2 cm sides — that is 6 cm of wire — so dividing the wire by a single 2 cm side would count sides, not whole triangles.
How to solve
Strategy Identify Subproblems — Break the problem into two small steps: first find the total wire length, then find how much wire one triangle needs, and finally see how many triangles fit. Tracking centimeters keeps the multiplication and division lined up correctly.
1STEP 1
Find the total wire length
Four 9 cm lengths joined end to end make 36 cm of wire in all.
9 × 4 = 36
Four equal lengths of 9 cm is just the multiplication fact 9 times 4, which a Grade 3 student knows.
3.OA.B.5Identify Subproblems2STEP 2
Find the wire one triangle needs
One triangle's three 2 cm sides use 6 cm of wire.
2 × 3 = 6
Three sides of the same length is simply 2 added three times, an easy times-three fact.
3.OA.B.5Analyze The Units3STEP 3
Count how many triangles fit
Fitting 6 cm lengths into 36 cm gives 6 triangles, with no wire left over.
36 ÷ 6 = 6
Finding how many equal groups of 6 make 36 is the same as knowing the factor pair 6 and 6 of 36.
4.OA.B.4Identify SubproblemsThe 36 cm wire divides exactly into 6 cm triangle-lengths, with no wire left over.
Why?
Each triangle costs 6 cm of wire and the whole wire is 36 cm, so the number of triangles is just how many 6 cm lengths fit inside 36 cm.
Why?
Asking how many 6 cm lengths fit in 36 cm is the same as asking what number times 6 makes 36, because sharing a total into equal groups is the reverse of building that total by multiplying.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
That number is 6, because six lengths of 6 cm joined end to end make exactly 36 cm with nothing left over.
Why?
Six lengths of 6 cm is 6 counted six times, and adding the same 6 cm over and over gives one fixed total of 36 cm.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
6 triangles
36 ÷ 6 = 6
6 triangles use 6 times 6 = 36 cm, which is exactly the whole wire, so the answer fits with nothing wasted and the magnitude is sensible.
Takeaway
Find the whole length, find one triangle's length, then split into equal groups -- only Grade 4 grouping you already know!
- Find the total wire length
- Find the wire one triangle needs
- Count how many triangles fit
Where next?
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