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

Drawing Angles and Estimating with Benchmarks

E2.4 A1About 40 minutes
Learning goals
  • I can use benchmark angles of 45°, 90°, 135° and 180° to estimate the size of an angle.
  • I can draw an angle of a given size, up to 180°, with a ruler and a protractor.
  • I can check my drawing by measuring it and comparing it with a benchmark.

Words to know

benchmark angle
A well-known angle we use to compare and estimate, like 45°, 90° or 180°.
estimate
A thoughtful guess that is close to the real measure.
construct
To draw something carefully and exactly, using tools like a ruler and protractor.
ray
A straight line that starts at a point and goes on in one direction. The arms of an angle are parts of rays.

Let's start with a hockey shot

A hockey coach says, "Shoot at about a 45° angle to the boards." The player has no protractor on the ice! She has to picture 45° in her head.

Good estimators carry a few benchmark angles in their heads. Then they compare: "Is this angle a bit more, or a bit less, than a benchmark I know?"

Benchmark angles

  • 90°: a right angle, the corner of a sheet of paper.
  • 45°: half a right angle. Fold the corner of a sheet of paper so the edges meet. That fold makes a 45° angle.
  • 180°: a straight angle, a straight line.
  • 135°: a right angle plus half a right angle (90° + 45°).

You can also use a clock. The 12 spaces between the numbers each make 30° (because 360 ÷ 12 = 30). At 1:00 the hands make 30°, and at 2:00 they make 60°.

45°half a right angleright angle90°135°90° + 45°180°straight
The benchmark angles 45°, 90°, 135° and 180°.
90° benchmark
Estimate this angle. The dashed line shows a 90° benchmark from the same vertex.
Worked example

Example 1: Estimate, then measure

Estimate the size of the shaded angle in the picture above, then check it.

  1. Compare with a right angle. The angle opens a little wider than the dashed 90° line.
  2. So it is obtuse, but only just. It is much less than 135°.
  3. Estimate: about 100°.
  4. Measure with a protractor: the angle is 100°. The estimate was very close!

Estimate: about 100°. Measure: 100°.

How to draw an angle with a protractor

Let's draw a 70° angle.

  1. Use a ruler to draw a straight line. This is the first arm. Put a dot at one end. That dot is the vertex.
  2. Put the protractor's centre mark on the vertex, with the baseline along the first arm.
  3. Find the 0 where the first arm is. Count up on that scale to 70. Make a small dot at the edge of the protractor.
  4. Take the protractor away. Use the ruler to join the vertex to your dot. That is the second arm.
  5. Label the angle 70°, then check: is it acute? Is it a bit less than 90°? Yes!
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Step 3: the first arm goes to the right, so we count on the scale that starts at 0 on the right (the inner scale) and put a dot at 70.
70°first arm
Step 4 and 5: join the vertex to the dot and label the angle. The arms can be any length.
Worked example

Example 2: An angle that opens to the left

Draw a 145° angle with the first arm going to the left from the vertex.

  1. Draw the first arm with a ruler. Put the vertex at its right end.
  2. Put the centre mark on the vertex and the baseline on the arm. Now the arm runs through the 0 on the left.
  3. Use the scale that starts at the left 0 (the outer scale). Count up to 145, halfway between 140 and 150. Make a dot.
  4. Join the vertex to the dot. Check with benchmarks: 145° is a bit more than 135°, and wider than a right angle. It looks right.

A 145° obtuse angle, a bit wider than the 135° benchmark.

Worked example

Example 3: Catching a mistake

Lina was asked to draw a 60° angle. Her angle looks wider than a square corner. What happened?

  1. 60° is less than 90°, so it should look narrower than a square corner.
  2. Lina must have counted on the wrong scale. At that spot, the other scale says 120.
  3. She drew a 120° angle by mistake.

Lina's angle is 120°. She should redraw it using the scale that starts at 0 on her first arm.

Estimate first, then measure

Estimating first helps you catch mistakes, like Lina's. If your estimate and your measure are far apart, don't panic. It's a clue! Check which scale you used and whether the vertex is on the centre mark. Mistakes are information.

Just for fun

Riddle: Why should you never argue with a 90° angle?

Because it's always right!

Hands-on activity

The angle guessing game

You need: a protractor, a ruler, pencils and paper, and a partner (a grown-up, brother or sister).

  1. Player 1 secretly draws an angle with a ruler (any size up to 180°).
  2. Player 2 estimates its size in degrees and says which benchmark helped ("It's a bit less than 45°, so about 40°").
  3. Player 1 measures it with the protractor. Player 2 checks the measure too.
  4. Score: within 10° = 2 points, within 20° = 1 point.
  5. Switch roles. Play 5 rounds each.

Bonus round: Player 1 calls out a size (like 125°). Player 2 draws it with the protractor. Player 1 measures to check.

Math talk: "Which benchmark angle did you use most?"

Your turn

1

An angle looks a little smaller than half a right angle. Which estimate is best: 40°, 80° or 140°?

2

Put these steps for drawing an angle in order: (a) join the vertex to the dot, (b) draw the first arm and vertex, (c) count to the number and make a dot, (d) put the centre mark on the vertex and baseline on the arm.

3

You draw a first arm going to the right from the vertex and want a 110° angle. Which scale do you count on?

4

An angle is a little wider than the 135° benchmark. Give a good estimate.

5

Draw a 90° angle. How can you check it without a protractor?

6

Estimate the angle between the hands of a clock at 1:00.

More practice: warm-up

1

How many degrees is half a right angle?

2

Which benchmark is closest to 85°: 45°, 90° or 180°?

3

Which benchmark is closest to 50°: 45°, 90° or 135°?

4

Is 175° closer to 90° or to 180°?

5

What is 90° + 45°?

More practice: on your own

1

Draw a 40° angle. What should it look like?

2

Draw a 155° angle. What should it look like?

3

An angle is wider than a right angle but narrower than 135°. Which is the best estimate: 75°, 110° or 160°?

4

An angle is 20° more than a right angle. How many degrees is it? What type is it?

5

Sam estimated 60° but measured 120°. What should Sam do?

Challenge

1

How can you make a 135° angle with just a square sheet of paper and a fold? Explain.

2

Estimate, then calculate, the angle between the hands of a clock at 5:00.

3

How would you draw a 180° angle? Where is its vertex?

Check your understanding

Quiz

1.

Which benchmark angle is half a right angle?

2.

An angle is a little wider than a right angle. Which is the best estimate?

3.

When drawing an angle, where do you put the protractor's centre mark?

4.

Ava wanted to draw 30°, but her angle is wider than a square corner. What angle did she probably draw?

5.

What is the angle between the hands of a clock at 3:00?

6.

Why is it smart to estimate before you measure?

For parents and teachers

Big idea: The Ontario curriculum (E2.4) asks Grade 5 students to use protractors to construct angles up to 180° and to use benchmark angles to estimate the size of other angles. Benchmarks (45°, 90°, 135°, 180°) give children a mental picture, which helps them catch errors.

Common difficulties:

  • Using the wrong scale when drawing. Remind your child to find the 0 on the first arm and count up from there.
  • Wobbly arms. Always use a ruler for both arms.
  • Not checking. Encourage "Does it look right?" after every angle.

How to help: Play the angle guessing game. When you are out, ask your child to estimate angles: a ramp, a roof, a slide at the park. Social-emotional learning (A1) is woven in: treat mistakes as clues, and praise your child for catching and fixing their own errors.