Classifying shapes

What is this skill?

Classifying shapes means sorting them by their properties: the number of equal sides, the kinds of angles, and whether sides are parallel. In fifth grade, the key idea is that shapes belong to families, and that a shape in a smaller family also belongs to the bigger one. A square has four equal sides and four right angles. That makes it a rectangle (four right angles) and also a rhombus (four equal sides), and since both of those have two pairs of parallel sides, it is also a parallelogram.

The reverse does not hold. Every square is a rectangle, but not every rectangle is a square. The practice here asks two kinds of questions. One gives clues and asks which quadrilateral, or which triangle, fits them. The other gives a statement like “Every rhombus is a parallelogram” and asks whether it is true or false. Triangles are sorted by their sides (equilateral, isosceles and scalene) and by having a right angle. In this practice, “isosceles” means at least two equal sides, so an equilateral triangle counts as isosceles in the true or false questions.

A note on trapezoids: some schools define a trapezoid as having exactly one pair of parallel sides, and others as at least one pair. The questions here only ask about “exactly 1 pair of parallel sides,” so both definitions give the same answer.

Why it matters

Classifying shapes is the beginning of mathematical argument. Children have to justify why a shape belongs, using definitions, and notice when a claim is true for every shape of a kind and when it is only true for some. That is the same kind of reasoning that proofs in geometry use later.

It also gives children precise vocabulary. Telling a rhombus from a rectangle, or an isosceles from a scalene triangle, matters in geometry class, in design and in building things.

Common mistakes

• Thinking a square is not a rectangle. It is, because it meets every requirement of a rectangle, and then some.
• Judging by orientation. A square turned on its corner is still a square, even if it looks like a diamond.
• Treating a rhombus as only a “diamond” shape. A rhombus is any four-sided shape with four equal sides, and it does not need right angles.
• Mixing up “opposite sides equal” with “all sides equal.” A parallelogram and a rectangle have the first; a rhombus and a square have the second.
• Choosing a name that is true but not the one the clues point to, when the clues allow only one answer. Read every clue before choosing.

How to help at home

• Cut shapes from paper and sort them by properties: four equal sides, four right angles, parallel sides.
• Draw a family tree of quadrilaterals together, with quadrilateral at the top and square at the bottom.
• Go on a shape hunt around the house. A door is a rectangle, a window might be a square, a slice of pizza is a triangle.
• Ask “always, sometimes or never?” questions. Is a rectangle always a parallelogram? Is a rhombus ever a square?
• Use a ruler and a corner of paper to check equal sides and right angles instead of trusting how a shape looks.

Worked examples

These are real questions from the Tutorsaurus classifying shapes practice, with the answers worked out.

1. Example 1

   Every rhombus is a square.

   Choices: True, False

   Answer: False. A rhombus has 4 equal sides, but its angles do not have to be right angles.

2. Example 2

   Which triangle has one 90° angle?

   Choices: isosceles triangle, scalene triangle, equilateral triangle, right triangle

   Answer: right triangle. A right triangle has one 90° angle.

3. Example 3

   Every equilateral triangle is an isosceles triangle. (Here, isosceles means at least 2 equal sides.)

   Choices: True, False

   Answer: True. Isosceles means at least two equal sides; an equilateral triangle has three.