E. Spatial Sense · Spatial Sense: Angles, Triangles, Transformations and Area

Comparing Angles: Matching and Angle Units

E2.3 A1About 35 minutes

Learning goals

I can explain that the size of an angle is the amount of turn, not the length of its arms.
I can compare two angles by tracing one and matching it on top of the other.
I can measure an angle by counting how many equal angle units (wedges) fit inside it.
I can explain why everyone needs to use the same size of unit.

Words to know

angle
The amount of turn between two arms that meet at a point.

vertex
The point where the two arms of an angle meet. (Plural: vertices.)

arm
One of the two straight lines that make an angle.

non-standard unit
A unit we make ourselves, like a paper wedge, instead of an official unit like the degree.

benchmark
A well-known size we compare with, like a right angle.

Remember from Grade 4

In Grade 4 you sorted angles by comparing them with a right angle (a square corner):

An acute angle is smaller than a right angle.
An obtuse angle is bigger than a right angle but smaller than a straight angle.
A straight angle looks like a straight line.

This year you will go further: you will find out how much bigger or smaller one angle is than another.

Let's start with pizza

Your family orders a pizza cut into 8 equal slices. You take 1 slice and your cousin takes 3 slices that are still joined together. Look at the pointy tips in the middle of the pizza. Your cousin's piece makes a wider angle. In fact, it is exactly 3 times as wide as yours!

That's the big idea of this lesson. We can measure an angle by counting how many equal small angles fit inside it, just like we measure length by counting centimetres.

The size of an angle

An angle is the amount of turn between its two arms. It is not about how long the arms are. You can make the arms longer or shorter, and the angle stays the same size.

There are two ways to compare angles without any special tools:

Matching. Trace one angle on thin paper. Put the traced vertex on the other angle's vertex, and line up one traced arm with one arm of the other angle. Look at where the second arm lands.
Counting angle units. Make a small angle unit, like a paper wedge, and count how many fit inside each angle.

Angle A is shaded. Angle B was traced and placed on top so the vertices and the bottom arms match. B's second arm (dashed) lands outside angle A, so angle B is bigger.

Worked example

Example 1: Compare by matching

Angle A and angle B are on two different pages. Which one is bigger?

Trace angle B on tracing paper (or thin paper held against a window).
Put the traced vertex exactly on the vertex of angle A.
Turn the tracing so that one traced arm lies on one arm of angle A.
Look at the other traced arm. It lands outside angle A. That means B opens wider.

Angle B is bigger. If the second arm had landed inside angle A, then A would be bigger. If it landed right on top, they would be the same size.

Making an angle unit

Matching tells us which angle is bigger. Counting units tells us how much bigger.

Trace around a plate to make a paper circle and cut it out. Fold it in half, in half again, and in half one more time. Open it up. The folds split the circle into 8 equal wedges. Cut out one wedge. Its pointy corner is your angle unit.

A right angle is exactly 2 wedges.
A straight angle is exactly 4 wedges.
A full turn (all the way around) is 8 wedges.

Left: a paper circle folded into 8 equal wedges, with one wedge shaded. Right: an angle that is exactly 3 wedges wide.

Worked example

Example 2: Count the wedges

How big is the angle on the right of the picture above?

Put the wedge's point on the vertex and one straight edge along one arm.
Mark where the other edge of the wedge is. Move the wedge over so it starts at your mark.
Keep going until you reach the other arm. Count: 1, 2, 3 wedges.
Compare with benchmarks: a right angle is 2 wedges, a straight angle is 4. Three wedges is between them.

The angle measures 3 wedges. It is obtuse, because it is bigger than a right angle (2 wedges) and smaller than a straight angle (4 wedges).

Same unit, fair comparison

When you measured length in Grade 2 and 3 with paper clips, everyone had to use the same size of paper clip, or the numbers didn't match. Angles are the same! If Ava uses big wedges and Ben uses wedges half as big, Ben will count twice as many for the same angle. Both are right, but they can only compare numbers if they used the same unit.

Worked example

Example 3: Smaller units

Ben folds his circle one more time, so it has 16 equal wedges. Each of his wedges is half the size of an 8-wedge unit. He measures the same angle from Example 2. What will he count?

Each big wedge holds 2 of Ben's small wedges.
The angle is 3 big wedges, so it holds 3 × 2 = 6 small wedges.

Ben counts 6 small wedges. Smaller units give a bigger count, and they let you measure more exactly.

Think like a mathematician

Measuring angles with wedges can be fiddly. The wedge slides, or the last bit doesn't fit exactly. That's normal! Take a breath, hold the paper down with one hand, and try again. If the angle ends partway through a wedge, say "about 2 and a half wedges". Careful, patient measuring is a real math skill.

Just for fun

Riddle: Why was the small angle invited to every party?

Because it was so a-cute!

Hands-on activity

Make an angle measurer

You need: a paper plate or a sheet of paper, a plate to trace, scissors (ask a grown-up to help with cutting), a pencil and some tracing paper or thin paper.

Trace a plate, cut out the circle and fold it in half three times to make 8 wedges. Cut out one wedge. Write "1 unit" on it.
Find 5 angles around your home: an open book, a partly open door, a pair of scissors opened a little (a grown-up opens and holds them; keep your fingers away from the blades), the corner of a picture frame, a laptop screen.
Estimate each angle in wedges first. Then measure by moving your wedge along. Record both numbers in a table.
Which angle was the biggest? Which was exactly 2 wedges (a right angle)?
Trace two of the angles and compare them by matching too. Did matching agree with counting?

Math talk: "Why doesn't it matter how long the arms of the angle are?"

Your turn

Angle P measures 4 wedges. Angle Q measures 2 wedges (same size of wedge). Which angle is bigger? How many times as big?

Using wedges that are 1/8 of a circle, how many wedges make a right angle?

How many of those wedges make a straight angle?

You trace angle A and place it on angle B, matching the vertices and one arm. B's other arm lands inside your tracing of A. Which angle is bigger?

An angle measures 3 wedges (1/8-circle wedges). Is it acute, right or obtuse?

Two angles both measure 1 wedge. One has arms 10 cm long. The other has arms 3 cm long. Which angle is bigger?

Jo measures an angle with big wedges and gets 2. Sam measures the same angle with wedges half as big. What does Sam get?

More practice: warm-up

What do we call the point where the two arms of an angle meet?

An angle measures 1 wedge (1/8 of a circle). Is it smaller or bigger than a right angle?

Which angle is bigger: 5 wedges or 3 wedges (same unit)?

If you make the arms of an angle longer, does the angle get bigger?

How many 1/8-circle wedges fit all the way around a circle?

More practice: on your own

A pizza is cut into 8 equal slices. How big is the angle at the tip of 3 joined slices, in wedges? Compare it with a right angle.

An angle measures 6 small wedges, where each small wedge is 1/16 of a circle. How many 1/8-circle wedges is that?

Angle A is 3 wedges, angle B is 1 wedge and angle C is 2 wedges. Put them in order from smallest to largest. Which one is a right angle?

Lia measured an angle and got 4. Omar measured the same angle and got 8. Neither made a mistake. How is that possible?

An angle is more than 2 wedges but less than 3 wedges. Estimate its size. Is it acute or obtuse?

Challenge

Maya folds a circle into 12 equal wedges (like the 12 spaces between the numbers on a clock). How many of her wedges make a right angle? A straight angle?

At 2:00, the hands of a clock point at the 12 and the 2. Using Maya's 12-wedge unit, how big is the angle between the hands? Is it acute, right or obtuse?

Ali says angle X is bigger than angle Y, because X measured 6 and Y measured 4. But Ali measured X with 1/16-circle wedges and Y with 1/8-circle wedges. Is Ali right?

Check your understanding

Quiz

What does the size of an angle depend on?

Using 1/8-circle wedges, a right angle measures:

Angle M is 5 wedges and angle N is 3 wedges (same unit). Which is true?

You trace angle R and lay it on angle S with vertices and one arm matching. The second arms land exactly on top of each other. What can you say?

Why should everyone use the same wedge size when comparing angles?

An angle measures 3 wedges (1/8-circle wedges). What kind of angle is it?

For parents and teachers

Big idea: The Ontario curriculum (E2.3) asks Grade 5 students to compare angles and determine their relative size by matching them and by measuring them with non-standard units. This builds the idea that angles can be measured with equal units, which gets them ready for degrees and the protractor in the next lesson.

Common difficulties:

Thinking longer arms mean a bigger angle. Draw two equal angles with different arm lengths and match them with tracing paper.
Leaving gaps or overlaps between wedges. Show how to mark the edge of the wedge before moving it, like marking the end of a ruler.
Comparing counts from different units. Link it to measuring a table with spoons versus pencils.

How to help: Help with the scissors when cutting the circle. Point out angles at home (a door, a laptop, clock hands) and ask "About how many wedges is that?" Social-emotional learning (A1) is woven in: praise patience and careful re-trying when the wedge slips.