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Sorting and matching shapes

Geometry · level 1, lesson 4 of 5

A question to start

Here is a messy pile of shapes. How would you tidy them into groups?

Some people sort them by colour: all the orange ones, all the blue ones, all the green ones. Some sort them by size: big ones and small ones. Some sort them by shape: circles, triangles, and four-sided shapes.

Which way is right? Here is the surprise: all of them. There is no one right way to sort. What matters is that you choose a rule, and then follow it for every shape. Mathematicians sort things all the time, and choosing a good rule is one of the most useful skills in all of math.

The idea

Sorting means choosing a rule

To sort means to put things into groups so that everything in a group is alike in some way. The way they are alike is the rule for sorting.

A good sorting rule has two jobs:

  1. Every shape must fit in some group. None left over.
  2. Every shape must fit in only one group. No shape can be in two groups at once.

Here is the pile sorted by the rule "count the corners".

no corners 3 corners 4 corners
  • No corners: the circles. There are 4 of them.
  • 3 corners: the triangles. There are 4 of them.
  • 4 corners: the squares and rectangles. There are 5 of them.

Every shape went in exactly one group. 4 + 4 + 5 = 13 shapes, and there were 13 in the pile. That is a good way to check you did not lose any.

Notice that this rule ignores colour and size. A tiny blue circle and a big green circle go in the same group, because they both have no corners. Choosing a rule means choosing what to pay attention to and what to ignore.

Sorting the same pile another way

Now sort the same pile by colour.

  • Orange: a circle, a turned square, a rectangle, and an upside-down triangle. That is 4.
  • Blue: two squares, a small circle, a tipped triangle and a tiny triangle. That is 5.
  • Green: two circles, a small square and a triangle. That is 4.

Again 4 + 5 + 4 = 13. Same pile, different rule, different groups. Both are correct.

In geometry, we usually care about the shape, not the colour. Colour is not part of a shape. If you paint a triangle red, it is still the same triangle.

Same shape, different size

Look at these triangles.

small, middle, big turned squashed

The first three are the same shape in three sizes: small, middle and big. If you looked at them through a magnifying glass, the small one would look exactly like the big one. They have the same "look": the same pointy top leaning to the same side, the same sharp corners. One is just a bigger copy of the other, like a photo printed small and printed big.

Grown-ups have a special word for "same shape, different size": similar. Mathematicians use it in a very exact way, and you will meet it again when you are older.

The blue triangle is also the same shape, only turned round. Turning does not change the shape. If you cut it out and turned it back, it would sit exactly on top of the orange one of the same size.

The green triangle on the right is different. It is squashed: it is long and flat, and its corners are not the same as the others. You could not make it by turning or making the orange one bigger or smaller. It is still a triangle (3 sides, 3 corners), but it is a different kind of triangle.

So there are two questions you can ask about two shapes:

  1. Do they have the same name? (Both triangles? Both rectangles?) Count sides and corners.
  2. Are they the same shape exactly? Could one be a bigger, smaller or turned copy of the other?

All squares are the same shape exactly, because every square has equal sides and square corners. A small square is just a shrunk big square. But rectangles are not all the same shape: a long thin rectangle and a nearly square rectangle look different, even though both are rectangles.

Matching

To match means to find two things that are the same. Same shape, same size: if you cut one out, you could put it exactly on top of the other. Turning it is allowed. Flipping it over is allowed too.

A jigsaw puzzle is a matching game. Each hole has exactly one piece that fits. You may have to turn the piece to make it fit, but you never have to stretch it.

The odd one out

An "odd one out" puzzle asks: which one does not belong with the others? To solve it, look for a rule that fits all of them but one.

A B C D E

Which is the odd one out? Think about it before you read on.

  • B looks different, because it is standing on a corner. But turning does not change a shape. B is a square.
  • C is small and E is big. But size does not change the shape. They are squares.
  • D is the odd one out. It is a rectangle with two long sides and two short sides. All the others are squares.

The rule "is it a square?" fits A, B, C and E, and only D breaks it.

Worked examples

Example 1. Sort by sides. Sort these into two groups: a door, a slice of sandwich cut corner to corner, a window, a yield road sign, a book.

  1. Choose the rule: "how many sides?"
  2. Door: 4 sides. Window: 4 sides. Book: 4 sides.
  3. Sandwich slice: 3 sides. Yield sign: 3 sides (it is a triangle pointing down).
  4. Groups: 3 sides: sandwich slice, yield sign. 4 sides: door, window, book.

Example 2. Same shape? A sticky note is a small square. A floor tile is a big square. Are they the same shape?

  1. Both are squares: 4 equal sides, 4 square corners.
  2. If you made the sticky note bigger and bigger, keeping its sides equal, it would become exactly like the tile.
  3. So yes, they are the same shape in different sizes. Every square is the same shape as every other square.

Example 3. Same shape? A sheet of paper and a long ruler are both rectangles. Are they the same shape?

  1. Both have 4 sides and 4 square corners, so both are rectangles. Same name.
  2. But a ruler is very long and thin. A sheet of paper is much wider for its length.
  3. If you made the ruler bigger, it would stay long and thin. It would never look like the sheet of paper.
  4. So no, they are not the same shape. Same name, different shape.

Try it

  1. Sort these by number of corners: a clock face, a tile, a slice of sandwich cut corner to corner, a coin, a book.
  2. In a bag there are 3 circles, 5 triangles and 2 squares. How many shapes are in the bag?
  3. Jo sorted some shapes into two groups: "red" and "triangles". Why is that a bad sorting rule? (Hint: think about a red triangle.)
  4. Is a small circle the same shape as a big circle?
  5. Find the odd one out: a ball, an orange, a dice, a marble.
  6. Find the odd one out: a triangle, a square, a rectangle, a book cover.
  7. Two triangles are both pointing up, but one is tall and thin and the other is short and wide. Same name? Same shape?
  8. Think of your own sorting rule for your toys. Does every toy fit in exactly one group?
Answers
  1. No corners: the clock face and the coin (circles). 3 corners: the sandwich slice (a triangle). 4 corners: the tile and the book.
  2. 3 + 5 + 2 = 10 shapes.
  3. A red triangle would fit in both groups. A good rule puts every shape in only one group. Better: sort by colour (red, not red) or by shape (triangle, not triangle), not by a mix of the two.
  4. Yes. Every circle is round in exactly the same way, so a big circle is just a bigger copy of a small one. All circles are the same shape.
  5. The dice. It is a cube, with flat faces and corners. The ball, the orange and the marble are all spheres.
  6. The triangle. It has 3 sides. The square, the rectangle and the book cover all have 4 sides. (A book cover is a rectangle.) The rule "has 4 square corners" fits the other three, and only the triangle breaks it. Saying your rule out loud is what makes an odd-one-out answer right.
  7. Same name: both are triangles. Not the same shape: a tall thin triangle cannot be turned or resized into a short wide one. Its corners are different.
  8. Any rule that works! For example: "toys with wheels" and "toys without wheels". Check: every toy either has wheels or does not, so every toy fits in exactly one group.

Common mistake

Thinking a turned shape or a bigger shape is a different shape. A square on its corner, or a giant square, is still a square. Turning and changing size do not change the shape.

Thinking every rectangle is the same shape. All squares are the same shape, and all circles are the same shape. But rectangles (and triangles) come in many different shapes: some long and thin, some nearly square. Having the same name is not the same as being the same shape.

Where it leads