Different factor pairs, same product

3.OA.B.54.OA.B.4 · take · grade 4

Archetype: Decompose a Number into Parts and Factors · step in a 4-type progression

A piece of wire is as long as a $9\ \text{cm}$ stick laid end to end $4$ times. Using this wire, what is the greatest number of triangles you can make if each triangle has three sides that are all $2\ \text{cm}$ long?

Show solution

Understand

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.

Givens

The wire length equals a 9 cm stick laid end to end 4 times.
Each triangle has three sides, each 2 cm long.

Unknowns

The greatest number of 2 cm equilateral triangles the wire can make.

Constraints

Each triangle uses exactly 3 sides of 2 cm.
Whole triangles only (you cannot use part of a triangle).

Plan

#7 Identify Subproblems · also uses: #8 Analyze the Units

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.

Execute

#7 Identify Subproblems 3.OA.B.5

The wire is a 9 cm length repeated 4 times, so multiply 9 by 4.

9 \times 4 = 36

Four equal lengths of 9 cm is just the multiplication fact 9 times 4, which a Grade 3 student knows.

#8 Analyze the Units 3.OA.B.5

Each triangle has three 2 cm sides, so one triangle uses 2 cm three times.

2 \times 3 = 6

Three sides of the same length is simply 2 added three times, an easy times-three fact.

#7 Identify Subproblems 4.OA.B.4

See how many 6 cm pieces fit in 36 cm. Because 6 times 6 equals 36, exactly 6 triangles can be made with no wire left over.

36 \div 6 = 6

Finding how many equal groups of 6 make 36 is the same as knowing the factor pair 6 and 6 of 36.

Answer: 6 triangles

Review

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.

You could repeatedly subtract 6 cm from 36 cm (36, 30, 24, 18, 12, 6, 0) and count the 6 subtractions, which is the Guess-and-Check / repeated-subtraction view of the same division.

Standards · min grade 4

3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Multiplying 9 by 4 to get the total wire length and 2 by 3 for one triangle's wire.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Recognizing that 36 splits into equal groups of 6, giving 6 triangles.

💡 Find the whole length, find one triangle's length, then split into equal groups -- only Grade 4 grouping you already know!