Reasoning · Grade 4-2 Triangles and Angles
Problem
Each cut is equilateral, so its three sides match.
The word 'equilateral' is doing all the work: once you know one edge of a little triangle, you know all three, so a single letter can be written in three places at once.
4.G.A.1Draw A DiagramEach 20 cm side holds two cuts and a leftover.
Each side of the big triangle is a straight 20 cm ruler chopped into three pieces, so the three pieces must add back up to 20 — that is just taking a length apart and putting it together again.
3.MD.D.8Identify SubproblemsAdding the three gives cuts totalling 22 cm.
This is the neat turn: instead of chasing three unknown lengths one at a time, ask only for their total — and the total is exactly what falls out when you add all three side sentences together.
4.OA.A.3Change Focus Count The ComplementAdding all three side sentences at once gives one equation about the cuts, without ever finding them separately.
Each 20 cm side is the pieces along it laid end to end, so every side gives a true statement about the cuts on it.
Adding true equations side by side keeps them true, so the three separate facts can be combined into one.
Adding the leftovers' 16 gives a perimeter of 38 cm.
Perimeter just means 'walk all the way round once and add what you walked', and the three unlabelled sides are exactly the three cut lengths whose total you already know.
3.MD.D.8Identify SubproblemsPinning each cut down also gives 38 cm.
Once you know a + b + c = 22 you can peel off one letter at a time using the sentence it is missing from, and then every number can be checked back against the picture.
4.OA.A.3Guess And CheckYou never have to know how big each snipped corner was — adding the three sides of the big triangle together hands you their total, and that total is exactly the three sides of the hexagon you could not measure.