Week 28 · Protractors and measuring angles · 3 lessons
1
Paper models of angles
Discover Mathematics P4: Term 2, Unit 13, Lessons 5–8
GoalMake paper models of 45°, 90° and 135° angles by folding, and use them to measure and compare angles.
New words
paper model · an angle we make by folding paper to compare with other anglesfold · the crease in folded paper; it divides an angle in two equal partscompare · to see if an angle is smaller than, equal to or larger than another angle
Explanation
You can measure angles without a tool by folding paper. Cut a circle from paper. Fold it in half: the fold is a straight line, a 180° angle at the centre. Fold it in half again: the corner at the centre is a right angle, 90°. Fold once more: the corner is 45°. If you open the paper, the folds cut the circle into 8 equal angles of 45°. To measure an angle with your model, put the centre of the model on the vertex and one fold along one arm. Look at where the other arm falls. If it falls between the 90° fold and the 135° fold, the angle is between 90° and 135°. We can also build bigger angles from smaller ones: 45° + 90° = 135°.
Number of folds
1
2
3
Angle at the centre
180°
90°
45°
Equal angles in the circle
2
4
8
Worked examples
Fold a paper circle three times. The first fold makes 2 equal parts of 180°. The second fold makes 4 equal parts: 180° ÷ 2 = 90°. The third fold makes 8 equal parts: 90° ÷ 2 = 45°. The paper circle now has all the benchmark angles.
Measure angle ABC with the paper model. Put the centre on B and one fold along BA. The arm BC falls between the 90° fold and the 135° fold, and it is closer to the 135° fold. So the angle is between 90° and 135°. It is obtuse, and about 120°.
Put two paper models side by side. A 90° model and a 45° model share one arm. Together they make an angle of 90° + 45° = 135°. This is how we add angles: by placing them next to each other without gaps or overlaps.
Class activity
Make an angle tester. Cut a circle of paper and fold it three times to get eight 45° corners. Open it and write 0° at one fold, then 45°, 90°, 135° and 180° on the next folds. Hold your tester on three angles from your home (a book corner, the hands of a watch, a door opened a little). Say which two folds each angle falls between.
Practice
1
A paper circle is folded in half three times. How many equal angles are made at the centre, and how big is each?
8 equal angles of 45° each (360° ÷ 8 = 45°).
2
Two paper models of 45° are placed side by side. What is the angle they make together?
45° + 45° = 90°, a right angle.
3
An arm falls between the 45° fold and the 90° fold. Is the angle acute, right or obtuse?
Acute: it is between 45° and 90°, so it is smaller than a right angle.
4
How many 45° models make a straight angle?
180° ÷ 45° = 4 models.
5
Hana says an angle that falls between the 90° and 135° folds measures 150°. Is she right?
No. 150° is larger than 135°. The angle must be between 90° and 135°, for example 110°.
