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Fraction Classroom · 2.3.0 · Teacher answers & responses

Measure an angle by its turn, not its ray length

Use a shared vertex and a circular turn to explain degree measure, then compare angles even when their drawn rays have different lengths.

The labelled turns are 60°, 90° and 120°. Ray endpoints only limit the drawing; the turn at the shared vertex determines angle size.60°90°120°
The labelled turns are 60°, 90° and 120°. Ray endpoints only limit the drawing; the turn at the shared vertex determines angle size.

Build the idea

model. Two angles both turn through 60 one-degree steps. One drawing has much longer rays. Which angle has the greater degree measure?

  1. Find the common endpoint, called the vertex, and follow the direction of each ray from it. The opening between those directions defines the angle.
  2. Imagine rotating one ray around the vertex until it reaches the other ray. A full turn contains 360 equal one-degree steps.
  3. Count the steps in the opening: sixty one-degree turns make a 60-degree angle. Compare it with a 90-degree quarter turn and a 120-degree angle.
  4. Extend either ray without rotating it. The directions are unchanged, so the same sixty steps still connect them. Do not use ray length as the angle measure.

Worked example: Neither. Both measure 60 degrees. Extending a ray changes the drawing's length, not the turn between the directions.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. An angle turns through 75 one-degree steps. What is its measure in degrees?

Answer: 75

Teacher response: Every counted step represents one degree.

p2. A 40-degree angle has both rays extended without rotation. What is its new measure in degrees?

Answer: 40

Teacher response: The directions and the turn between them remain unchanged.

p3. A full turn is 360 degrees. How many degrees are in one quarter of a full turn?

Answer: 90

Teacher response: Four equal quarter turns fill one full turn.

e1. An angle contains 110 one-degree turns. Its rays are shortened in the drawing but keep their directions. State the angle measure and explain why.

Answer: 110

Look for:

  • Locates the vertex and describes a turn between directions.
  • Rejects ray length or page orientation as a measure of the angle.

Teacher response: Shortening the visible segments does not change the opening.

Transfer the idea

transfer. An angle drawing is turned upside down without changing its shape. Does its degree measure change? Explain how you could check.

Answer: No. Rotating the whole drawing preserves the turn between its rays. Align the vertex and one ray with the original to compare the other direction.

Run the 15-minute workshop

teach-1. 3 minutes: Show two equal openings with unequal strip lengths and ask learners to predict whether the angles match.

teach-2. 4 minutes: Align the vertices and one direction; connect the matching opening to the number of one-degree turns.

teach-3. 5 minutes: Complete the practice, explaining what changes in each example and what stays fixed.

teach-4. 3 minutes: Use the final check and have learners describe how to compare angle size without comparing ray lengths.

Respond to the learner

misconception. The angle with longer sides must be the larger angle.

Teacher response: Place the two vertices together and align one ray from each drawing. If the other rays point in the same direction, the turns match even when the visible segments end at different places.

support. Use movable strips and change only one feature at a time: first length, then opening.

extend. Sketch two differently oriented 60-degree angles and explain a way to superimpose them.

Teaching notes