E. Spatial Sense · 3D Objects, Directions and Measurement

Congruent Faces, Edges and Objects

E1.3About 30 minutes

Learning goals

I can find faces of a 3D object that are congruent.
I can find edges and angles that are congruent.
I can match parts by tracing, touching or picturing them in my mind.
I can decide if two 3D objects are congruent.

Words to know

congruent
Exactly the same size and the same shape. Congruent things match perfectly, even if one is turned or flipped.

congruent faces
Faces that are the same shape and size. If you traced them, the tracings would match exactly.

congruent edges
Edges that are the same length.

congruent angles
Angles that are the same size, like two square corners.

congruent objects
3D objects that are exactly the same size and shape, like two new cans of the same soup.

Remember from Grade 2

In Grade 2 you learned that two flat shapes are congruent if they are the same shape and the same size. You can check by putting one on top of the other. If they match perfectly, they are congruent, even if you had to turn or flip one first.

This year, we look for congruent parts inside 3D objects: faces, edges and angles.

Let's start with a cereal box

Pick up a cereal box. Look at the big front face. Now turn the box around and look at the back. Are they the same shape and size? Yes! The front and back faces are congruent.

What about the top and bottom? Congruent too. And the two thin ends? Also congruent. So a box has 3 pairs of congruent faces.

A rectangular prism. Face A (front) is congruent to the back. Face B (top) is congruent to the bottom. Face C (right end) is congruent to the left end.

Congruent edges and angles

Edges are congruent when they are the same length. All 12 edges of a cube are the same length.
Angles are congruent when they are the same size. All square corners are congruent to each other. Every corner on every face of a box is a square corner, so they are all congruent.

A cube with edges of 5 cm. All 12 edges are congruent, and all 6 faces are congruent squares.

How to check

Physically: trace one face on paper, then put the other face on the tracing. Or hold two faces against each other. Use a ruler or a strip of paper to compare edges.
Mentally: picture turning the object in your mind and moving one part onto the other. Do they match?

Two objects are congruent if all their matching faces, edges and angles are congruent. Then they are exactly the same size and shape.

Left: two cubes of different sizes. They have the same shape but not the same size, so they are not congruent. Right: the same box, lying down and standing up. They are congruent.

Worked example

Example 1: Pairs in a box

A tissue box is a rectangular prism. Its faces are: front 24 cm by 12 cm, back 24 cm by 12 cm, top 24 cm by 12 cm, bottom 24 cm by 12 cm, and two ends 12 cm by 12 cm. Which faces are congruent?

The front, back, top and bottom are all rectangles 24 cm by 12 cm. They are all congruent to each other.
The two ends are both squares 12 cm by 12 cm. They are congruent to each other.
An end is not congruent to the front. It is a different shape and size.

4 congruent rectangle faces (front, back, top, bottom) and 2 congruent square faces (the ends).

Worked example

Example 2: Are the boxes congruent?

Box P has edges 3 cm, 4 cm and 6 cm. Box Q is lying a different way, and its edges are 6 cm, 3 cm and 4 cm. Box R has edges 3 cm, 4 cm and 7 cm. Which boxes are congruent?

Compare the edge lengths, no matter which way the box is turned.
P: 3, 4, 6. Q: 6, 3, 4, which is the same three lengths. P and Q match.
R has a 7 cm edge instead of 6 cm. It does not match.

P and Q are congruent. Turning a box doesn't change its size or shape. R is not congruent to them.

Worked example

Example 3: A triangular prism

Which faces of a triangular prism are always congruent?

A prism has 2 matching bases, one at each end.
So the 2 triangle faces are congruent.
If all three sides of the triangle are the same length, the 3 rectangle faces are congruent too.

The 2 triangle bases are always congruent.

Same shape is not enough

A small cube and a big cube are the same shape but not the same size, so they are not congruent. Congruent means same shape and same size.

Just for fun

What did one congruent triangle say to the other?

"Same size, same shape... are you my reflection?"

Hands-on activity

Face tracing detectives

You need: a box (cereal, tissue or cracker box), paper, a pencil and scissors.

Grown-up note: help with or supervise cutting.

Stand the box on the paper. Trace around the bottom face. Label it "bottom".
Trace each of the other 5 faces. Label each one.
Cut out your 6 tracings.
Stack the tracings to find the congruent pairs. How many pairs do you find?
Hunt around the house for two congruent objects: two unopened cans of the same food, two dice from a game, two building blocks of the same kind.

Math talk: "How do you know these are congruent and not just the same shape?"

Your turn

What does congruent mean?

How many faces of a cube are congruent to each other?

A cereal box has how many pairs of congruent faces?

Are all the edges of a cube congruent? How do you know?

A big beach ball and a small bouncy ball are both spheres. Are they congruent?

Which faces of a triangular prism are always congruent?

Two juice boxes from the same pack are the same size. Are they congruent?

More practice: warm-up

Congruent edges are the same ___.

Are two square corners congruent?

If you turn a box on its side, does it change size?

How can you check if two faces are congruent?

Name the face of a box that is congruent to the top.

More practice: on your own

A box has edges 2 cm, 5 cm and 8 cm. Another box has edges 8 cm, 2 cm and 5 cm. Are they congruent? Explain.

A rectangular prism has a front face 10 cm by 4 cm. What is the size of the back face? Why?

Two soup cans are the same height, but one is much wider. Are they congruent? Explain.

Kai says a square-based pyramid has 4 congruent faces. Which faces does he mean? Is he always right?

How many edges of a long box with edges 3 cm, 3 cm and 9 cm are 9 cm long? How many are 3 cm long?

Challenge

A box has a square top and square bottom, and 4 rectangle sides. How many faces are congruent to the front face? Explain.

You stack two congruent cubes. Is the new tower congruent to one cube? Is it congruent to a box with edges equal to 1 cube edge, 1 cube edge and 2 cube edges?

Without tracing, how could you check if two closed cereal boxes are congruent?

Check your understanding

Quiz

Congruent means ...

How many pairs of congruent faces does a rectangular prism (like a cereal box) have?

Which two faces of a triangular prism are always congruent?

A small cube and a large cube are ...

Box A has edges 4 cm, 5 cm, 6 cm. Which box is congruent to Box A?

All 12 edges are congruent in a ...

For parents and teachers

Big idea: Children identify congruent lengths (edges), angles and faces in 3D objects, by physically matching (tracing, touching) and by mentally turning objects, and decide if two objects are congruent (Ontario E1.3).

Common difficulties:

Same shape vs congruent. Two cubes of different sizes are not congruent. Stress "same shape and same size".
Orientation. A box lying down and the same box standing up are congruent. Let your child physically turn objects to see this.
Mental turning. Picturing how an object looks from another side takes practice. Encourage your child to predict first, then turn the real object to check.

Safety: supervise scissors when cutting out traced faces.

How to help: In the grocery store, ask "Are these two boxes congruent? How could we check?"