Defining and non-defining attributes of shapes
A huge red triangle and a tiny blue one turned the other way are both triangles. What makes a shape a triangle, and what about it does not matter?
Where you are 15 in Two-dimensional shapes
Why this exists
Ask a young child to point at a triangle and they will often pick the one that sits flat on its bottom with its point up, like a roof. Turn that same cutout upside down and some children will say it is not a triangle any more. Make it very large, or very thin, and the doubt grows. The child has learned the name from a picture, and a picture carries a lot of things that have nothing to do with the name: its color, its size, which way it happens to be turned.
A name in mathematics comes with a rule. “Triangle” means a closed flat shape with 3 straight sides, and nothing else. Anything that obeys the rule is a triangle, however big, however colored, however turned. Anything that breaks the rule is not, however much it looks like one. The things a shape must have to earn its name are its defining attributes. Everything else about it is a non-defining attribute: true of this one shape, but free to change.
Once a child sorts what matters from what does not, they can name a shape they have never seen before, in any position, and they can make one from a description. That is the whole point of a shape name, and it is how every later lesson on shapes talks.
The lesson runs over four parts, one each class day.
| Part | What it teaches | Examples | Exercises |
|---|---|---|---|
| 1 | What a defining attribute is, with the triangle | 1, 2 | 1 to 7 |
| 2 | Flat shapes, and building and drawing them from their attributes | 3, 4 | 8 to 12 |
| 3 | The same shape in other positions and sizes | 5, 6 | 13 to 16 |
| 4 | Solid shapes: flat faces and curved surfaces | 7, 8 | 17 to 22 |
Exercises 23 to 25 ask for reasons and fit the end of any part from the second on. Have paper cutouts and a felt board for the flat shapes, and a feely bag of solid shapes, a party hat and a soup can for the last part.
Definition
An attribute is anything true about a shape that you could say out loud: “it is red”, “it is bigger than my hand”, “it has 3 sides”, “it is made of felt”, “it points down”.
A defining attribute of a shape is one that every shape with that name must have. Take it away and the name no longer fits.
A non-defining attribute is one that can change while the name still fits. Color, size, which way the shape is turned, where it sits, and what it is made of are non-defining for every shape in this lesson.
Words for the parts of a flat shape
- A side is a straight edge of a flat shape. A curved edge is not counted as a side.
- A corner is a point where two edges meet. (Later grades call it a vertex.)
- A shape is closed when its edges join up all the way round with no gap, so it has an inside and an outside. A shape with a gap is open.
- A square corner is a corner like the corner of a sheet of paper. That one phrase is all this lesson needs about the size of a corner.
The flat shapes
| Shape | Its defining attributes |
|---|---|
| triangle | closed; 3 straight sides; so 3 corners |
| rectangle | closed; 4 straight sides; 4 square corners |
| square | closed; 4 straight sides, all the same length; 4 square corners |
| circle | closed; one curved edge all the way round, every point of it the same distance from the middle; no straight sides and no corners |
| hexagon | closed; 6 straight sides; so 6 corners |
| trapezoid | closed; 4 straight sides, met here as the red pattern block |
A trapezoid needs one more defining attribute, about how two of its sides run, and that waits for later grades. In this lesson the trapezoid and the hexagon are learned mostly by name and by their side counts.
The solid shapes
A solid shape takes up room: you can hold it, and it has an inside you cannot see. Its outside is made of flat faces, like the sides of a box, and curved surfaces, like the side of a ball or a can. A flat face that meets another flat face makes a straight edge.
| Solid | Its defining attributes | Something shaped like it |
|---|---|---|
| cube | 6 flat faces, every one a square | a number cube (a die) |
| rectangular box | 6 flat faces, every one a rectangle | a cereal box |
| cone | 1 flat face that is a circle, 1 curved surface, and a point | a party hat |
| cylinder | 2 flat faces that are circles, 1 curved surface | a soup can |
| sphere | 1 curved surface and no flat faces | a ball |
The same non-defining attributes come along as for flat shapes: a cube is a cube in wood, in red plastic or in sugar, the size of a die or of a moving box, standing on any face.
Intuition
The test for any attribute is a change. Imagine changing it, and ask whether the name still fits.
Paint the triangle blue: still a triangle, so color is non-defining. Photocopy it at twice the size: still a triangle, so size is non-defining. Turn it on the felt board so its point goes down: still a triangle, so the way it is turned is non-defining. Now snip one corner off with scissors. The cut makes a new short side, the shape has 4 straight sides, and it is no longer a triangle. So the number of sides is defining.
Children can run this test with their hands. Turning a cutout, sliding it across the felt board, or swapping it for a bigger one in another color never changes how many straight sides it has. Cutting, bending or tearing can.
The list of defining attributes is short on purpose. It says exactly what the name needs and not one thing more, so that every shape with those attributes gets the name and no shape without them does.
Results
A shape’s name depends only on its defining attributes. Two shapes with the same defining attributes have the same name, whatever their colors, sizes, positions or materials.
Turning, sliding, and making a shape bigger or smaller all keep its name. None of them changes the number of sides, whether a side is straight, or whether the shape is closed.
A closed shape with only straight sides has as many corners as sides. A triangle has 3 of each, a rectangle 4, a hexagon 6.
Why it is true
Turning and sliding. When a paper triangle is turned on the felt board, every part of it moves together. Each straight side is carried along to a new place and is still straight; no side splits in two and no two sides join. The corners stay where the sides meet, and the gap-free edge stays gap-free. So after the turn the shape still has 3 straight sides and is still closed: still a triangle. The only thing that changed is which way it points, and that was never on the list.
Size. A bigger copy of a shape has every side longer by the same amount, but a straight side stays straight and the count stays the same. Color and material are not about the edges at all.
Corners and sides. Walk once around a closed shape with straight sides, starting at a corner. Each side you walk along ends at a corner, where the next side begins. You arrive back where you started, so you passed exactly one corner at the end of each side. The number of corners is the number of sides. This needs the shape to be closed: in an open shape, the two loose ends are not corners.
Worked examples
The huge red triangle
Lauri cuts a huge triangle out of red paper and puts it on the felt board with its point at the bottom. Next to it Bryok puts a small blue triangle with its point at the top. Is the huge red shape a triangle?
Check the defining attributes and nothing else. Run a finger along the edge of the red shape: it has 3 straight sides, and the finger comes back to its start without crossing a gap, so it is closed. Where each pair of sides meets there is a corner, 3 in all. A closed shape with 3 straight sides is a triangle, so it is a triangle.
Now look at what is different between the two shapes. The red one is red, the other is blue: color is non-defining. The red one is far bigger: size is non-defining. The red one points down and the blue one points up: the way it is turned is non-defining. None of the differences touch the rule, so both shapes get the same name. If a child is unsure, turn the red triangle on the board until it points up. Nothing about it was cut or added, so it cannot have changed what it is.
Two shapes that are not triangles
Bryok cuts shape A, which has two straight sides and one curved side, like a slice of pizza. Jaana draws shape B with three straight lines, but the last line stops short of where the first began. Is either one a triangle?
Shape A has three edges and three corners, so it looks close. But a triangle needs 3 straight sides, and one of A’s edges is curved. A curved edge is not a side, so A has only 2 straight sides. It breaks the rule, so it is not a triangle. It has no name in this lesson; “pizza-slice shape” will do.
Shape B has 3 straight lines, so the side count is right. But a triangle must be closed, and B has a gap where the third line stops. A finger running round its edge falls off at the gap, and B has no inside that is shut in. So B is not a triangle either. Draw the missing bit, so that the third line reaches the first, and B becomes a triangle.
Each shape keeps all but one of the defining attributes, and missing one is enough. That is what “defining” means: every item on the list is needed.
Which facts make it a rectangle?
Jaana holds up a green felt rectangle that is longer than her hand. She lists six facts about it:
it is green; it has 4 straight sides; it is longer than my hand; it has 4 square corners; it is made of felt; it is closed.
Which facts must be true of every rectangle, and which are true only of this one?
SolutionRun the change test on each fact. A rectangle cut from blue paper is still a rectangle, so “green” is not needed. A rectangle the size of a stamp or of a door is still a rectangle, so “longer than my hand” is not needed. A rectangle drawn in chalk or cut from cardboard is still a rectangle, so “made of felt” is not needed. Those three are non-defining.
The other three are defining. Snip one corner and the shape has 5 sides, not a rectangle. Push one corner so that it is no longer square and the shape leans over, not a rectangle. Open a gap in one side and the shape is open, not a rectangle. So “4 straight sides”, “4 square corners” and “closed” are the defining attributes, which is exactly the rectangle’s row in the table above.
Drawing a shape from its attributes
Draw a closed shape with 6 straight sides. What is it called, and how many corners does it have? Can two children draw it differently and both be right?
A closed shape with 6 straight sides is a hexagon, because that is the whole of the hexagon’s rule. Its corners follow from the sides: walking round it, each of the 6 sides ends at a corner, so it has 6 corners. The dots in the picture show them, 6 on each shape.
Both hexagons in the picture are right. The left one has all its sides the same length, like the yellow pattern block. The right one is lopsided. Equal sides are not on the hexagon’s list, so the lopsided one fits the name just as well. A child who counts 6 straight sides, checks there is no gap, and draws nothing curved has drawn a hexagon, whatever it looks like.
This is the other direction of the lesson. Naming reads the defining attributes off a shape; drawing starts from them and makes a shape that has them.
A square turned onto its corner
Bryok takes a paper square off the felt board and puts it back standing on one corner. Is it still a square?
Yes, it is still a square. Turning moved the whole cutout at once. Its 4 sides are the same 4 straight edges they were before, still all the same length, and its 4 corners are the same square corners. The paper was not cut or folded, so every defining attribute of a square is still there. What changed is which way it is turned, and that is non-defining.
Standing on a corner, a square is sometimes called a “diamond”. That word names the way the square is turned, not a different shape. To test it, lay a corner of a fresh sheet of paper into any corner of the turned square: it fits exactly, as it did before the turn.
Triangles on the felt board
Count the triangles on the felt board. How many corners do the triangles have altogether?
Go through the shapes one at a time and check the rule: closed, 3 straight sides.
- A, the big one pointing down: closed, 3 straight sides. A triangle.
- B, small and tipped: closed, 3 straight sides. A triangle.
- C, long and thin, lying on its side: closed, 3 straight sides. A triangle.
- D has 4 sides and E has 4 sides. Not triangles.
- F, the circle, has no straight sides at all. Not a triangle.
- G has 3 straight lines but a gap, so it is open. Not a triangle.
- H has a curved edge, so only 2 straight sides. Not a triangle.
So there are 3 triangles: A, B and C. They differ in size, in shape (one is long and thin), and in which way they point, and none of that matters.
Each triangle has 3 corners, so altogether
The triangles have 9 corners.
The feely bag
Lauri reaches into the feely bag and, without looking, feels a solid with two flat faces and one curved surface. Each flat face is round. Which solid is it? Then Jaana pulls out a solid with one round flat face, a curved surface and a point. Which is that?
SolutionA hand in the bag cannot see color, so it goes by what it can feel: how many flat faces, what shape they are, and whether there is a curved surface. Use the table of solids.
Two flat faces that are circles and one curved surface is the rule for a cylinder. A sphere has no flat faces, a cone has only one, and a cube or rectangular box has six, so it can only be a cylinder. It is shaped like the soup can.
One flat face that is a circle, a curved surface and a point is the rule for a cone. It is shaped like the party hat.
Both solids roll when laid on their curved surface, because a curved surface touches the table along a line and can turn. A cube cannot roll that way: every face it could land on is flat.
A tall can and a short can
Bryok holds a tall thin can of tennis balls. Jaana holds a short wide can of tuna. Are both cylinders?
Check each against the cylinder’s rule: 2 flat faces that are circles, and 1 curved surface. The tall can has a round top, a round bottom and a curved side. So does the short can. Both are cylinders.
How tall a cylinder is, and how wide its circles are, is not on the list, so a cylinder can be tall and thin or short and wide. Compare the square, where “all 4 sides the same length” is on the list: stretch a square in one direction only and it stops being a square. Each name decides for itself which measurements matter, and the cylinder’s name does not care about any.
Boundaries
Only the stated attributes count. A shape that looks like a triangle but has a curved edge, or a gap, is not one. A shape that looks odd but meets the rule is one. The eye is a guide, the rule decides.
Size means the same shape made bigger. Making a square bigger keeps all its sides equal, so it stays a square. Stretching it in one direction only is a different change: the sides stop being equal, and the result is a rectangle that is not a square.
Some names need more than a count. Counting sides tells a triangle from a hexagon, but squares, rectangles and trapezoids all have 4 sides, and telling them apart takes square corners and side lengths. By the table, a square also has everything a rectangle needs; whether one shape can carry two names is taken up later in shapes in more than one category.
Real objects are only close to the solids. A party hat is open at the bottom, so it has no flat face at all; the solid it is shaped like has one. A soup can has rims and a pull tab. A shape name belongs to the ideal solid, and an object is “shaped like” the solid it is closest to.
A drawing of a solid is flat. The picture of a cube on paper is a flat drawing with lines on it. Faces are counted on the solid itself, including the ones the drawing hides behind dashed lines.
Where this goes next
This lesson builds on sorting shapes by sides and corners, where children first count sides and corners and group shapes by them.
Next in the unit, equal shares cuts circles and rectangles into halves and fourths, and the names of the pieces depend on attributes in just the same way. Composing and decomposing shapes puts shapes together to make new ones, and the new shape is named by its own defining attributes, not by the pieces it came from.
Drawing a shape from what it must have grows into drawing shapes with given attributes. Later, the lists get longer and more exact: rhombuses, rectangles and squares and Shapes in more than one category, and quadrilaterals in none sort four-sided shapes by their corners and sides, and the hierarchy of flat shapes fits them into one family tree. The solids return as Three-dimensional shapes, cylinders and cones and Spheres. Every one of those lessons rests on the habit begun here: say what a name requires, and check only that.
Exercises
Part 1: triangle or not?
For Exercises 1 to 6, use the shapes below. Is the shape a triangle? Give the reason.
- Shape A.
- Shape B.
- Shape C.
- Shape D.
- Shape E.
- Shape F.
- Which of these could change without a triangle stopping being a triangle: its color, the number of its sides, its size, whether its sides are straight, which way it points?
Part 2: flat shapes
- A closed shape has 4 straight sides, all the same length, and 4 square corners. What is it called?
- How many straight sides and how many corners does a trapezoid have?
- Draw a closed shape with 4 straight sides that is not a square and not a rectangle.
- How many straight sides does a circle have? How many corners?
- Bryok puts 2 triangles and 1 square on the felt board. How many corners do the three shapes have altogether?
Part 3: other positions and sizes
For Exercises 13 and 14, use the felt board below.
- How many triangles are on the felt board?
- How many shapes on the felt board have no corners at all?
- Jaana turns a paper rectangle so that its long sides go up and down instead of across. Is it still a rectangle?
- Lauri cuts one square as big as the felt board and another as small as her thumbnail. Do they have the same name?
Part 4: solid shapes
- Which solid has no flat faces?
- Which solid has 6 flat faces that are all squares?
- In the feely bag, a solid has exactly one flat face, which is a circle, and a point. Which solid is it?
- A cereal box and a number cube are put together on the table. Name the solid each is shaped like. How many flat faces do the two have altogether?
- Which of the cube, cone, cylinder, sphere and rectangular box can roll? What do those have that the others do not?
- How many flat faces do a soup can and a cube have altogether?
Reasons
- Is every shape with 3 corners a triangle? Explain.
- Explain why “points up” can never be one of the defining attributes of a triangle.
- Can a closed shape with only straight sides have 5 sides and 4 corners? Explain.
Answers
- Yes. Shape A is closed and has 3 straight sides. Being tall and thin does not matter.
- No. Shape B has 4 straight sides, and a triangle has 3.
- Yes. Shape C is closed and has 3 straight sides; being tipped over is only which way it is turned.
- No. Shape D has a bend in its left side, which makes 4 straight sides and 4 corners.
- No. Shape E has 3 straight sides but a gap, so it is not closed.
- Yes. Shape F is closed and has 3 straight sides. Being tiny does not matter.
- Its color, its size and which way it points can change. The number of sides and whether they are straight cannot: change either and the shape is no longer a triangle.
- A square. Those are exactly the square’s defining attributes.
- 4 straight sides and 4 corners. It is closed with only straight sides, so it has as many corners as sides.
- Any closed shape with 4 straight sides where at least one corner is not a square corner, for example a trapezoid like the red pattern block, or a lopsided four-sided shape like Shape B.
- 0 straight sides and 0 corners. A circle’s edge is one curve with nowhere for two edges to meet.
- 10 corners. Each triangle has 3 and the square has 4, so 3 + 3 + 4 = 10.
- 4 triangles: the small one pointing up, the one pointing left, the tipped one and the one pointing down. Each is closed with 3 straight sides.
- 2 shapes, the two circles. The half circle has 2 corners where its curved edge meets its straight side, and every other shape has corners too.
- Yes. Turning it does not change its 4 straight sides or its 4 square corners.
- Yes, both are squares. Size is non-defining; each is closed with 4 equal straight sides and 4 square corners.
- The sphere. It has 1 curved surface and nothing else.
- The cube.
- The cone, the shape of the party hat.
- The cereal box is a rectangular box and the number cube is a cube. Each has 6 flat faces, so together they have 6 + 6 = 12.
- The cone, the cylinder and the sphere. Each has a curved surface, and the cube and the rectangular box have only flat faces.
- 8 flat faces. The soup can, a cylinder, has 2 and the cube has 6: 2 + 6 = 8.
- No. The pizza-slice shape of Example 2 has 3 corners but one curved edge, so it is not a triangle. A shape needs 3 corners and straight sides, closed up, to be a triangle.
- Turning a triangle makes it point a different way without changing its sides or corners. If “points up” were on the list, the same paper triangle would stop being a triangle every time it was turned, though nothing about its edges had changed.
- No. Walking once around the shape, each side ends at a corner where the next side begins, so a closed shape with straight sides has exactly as many corners as sides. 5 sides means 5 corners.