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P6 Overlap and cancel

Today’s problem — free all day, every step. Every day has one.

Bunting flags — three overlapping triangles on a string

Mrs Tan pins three identical equilateral triangular bunting flags, Flag 1, Flag 2 and Flag 3, along a straight horizontal string. Each flag points upwards and its bottom edge lies flat on the string. Flag 1 is on the left, Flag 2 is in the middle and Flag 3 is on the right. Flag 2 overlaps Flag 1 and also overlaps Flag 3, but Flag 1 and Flag 3 do not touch each other. On the string:

The shaded parts are the part of Flag 1 not covered by Flag 2 and the part of Flag 3 not covered by Flag 2.

  1. (a) Find the length of one side of a flag.
  2. (b) Find the total perimeter of the two shaded parts.
One step of the worked solution to Bunting flags — three overlapping triangles on a string, drawn in pencil.
6 steps, drawn in pencil

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The pattern behind it: Overlap and cancel

Two shapes share a region, and the shared part cancels out.

In this one. Put every base corner on one number line, and read an overlap as “side length minus shift”.

Counted twiceA + Bgroup Athe two in bothgroup Bthe same two againCounted oncetake the shared part offin all5 + 4 − 2 = 7

Also uses: Gap and difference

How to spot it

Shapes are joined, stacked, or have a piece cut off, and the question compares a perimeter or length before and after, rather than asking for an area from scratch.

Common mistake

Children assume perimeter changes the same way area does. Joining shapes or cutting off a corner can leave the perimeter unchanged, or change it by a different amount from the area.

Try one like it

Challenging PSLE overlapping shapes 4 identical semicircles overlap in a row. A, B and C are the 3 identical shaded overlaps. The unshaded area is 2125 cm² more than …
All 5 with this pattern →

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