At a glance. One lesson of about 60 minutes. Students build triangles from three sides with compasses, find when three lengths fail to make a triangle, and construct 60° and 30° angles from an equilateral triangle.
Calculator: no calculator. Nothing here needs one. The numbers are whole-number sums and halves, and the constructions are on paper or on the figure.
What students will learn
Know
- How to construct a triangle from three sides.
- Three lengths make a triangle only if the two shorter add up to more than the longest.
- An equilateral triangle gives 60°, and bisecting gives 30°.
Understand
- Three sides fix a triangle, including all of its angles, which add up to 180°.
- The compass keeps a distance fixed, so equal arcs make equal sides and equal angles.
- An angle is a turn, so it does not change with the size of the drawing.
Do
- Construct a triangle from three given sides.
- Test whether three lengths can make a triangle and explain why.
- Construct 60° and 30° and combine bisecting and adding to make other angles.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the triangle condition, angle values from constructions (questions 1 to 6).
- Criterion B. Investigating patterns: the two sliders. Students record which pairs of sides fail, and state the rule in their own words, then test it with a new base.
- Criterion C. Communicating: writing a construction as a sequence of steps with a reason for each, and justifying why the angle is 60° (question 6, worked examples).
- Criterion D. Applying mathematics in real-life contexts: copying a tile, cutting a triangular panel, where a measured length is more reliable than a measured angle.
The one thing to do with the figure
Ask for a prediction of the failures. “Which pairs of arcs will not meet?” Let students list them before they slide, then compare the list with the figure.
In the second figure, change the radius, then step through again, and ask what has not changed. The angle. Then ask why: three equal sides. That sentence is the proof, and it uses the angle sum from lesson 4.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure: predictions of which pairs of arcs fail. |
| 10 min | Key idea; students construct a 6, 7, 8 cm triangle on paper with compasses. |
| 10 min | Second figure; worked example and the faded one. If time allows, build the 5 cm, 50°, 7 cm triangle with a protractor and measure BC. |
| 5 min | Question 1, the check. If many choose A or D, go back to the first figure. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 6 cm; 60°; 8.5. |
|---|---|
| Worked example | Yes: 4 + 5 = 9 is more than 8. |
| Faded example | No: 3 + 4 = 7 is not more than 7, so the triangle is flat. |
| 1. Can these sides make a triangle? | B. 4 + 5 = 9 < 10. A is the common belief; C confuses a triangle with an equilateral one; D adds all three sides. |
| 2. Equilateral triangle angle | 60. 180 ÷ 3 = 60. |
| 3. Which lengths make a triangle | C. Only 6 + 7 is more than the longest side. B and D are exactly equal, A is less. |
| 4. Bisect 60 twice | 15. 60 ÷ 2 ÷ 2 = 15. |
| 5. Angle left on a line | 120. 180 − 60 = 120. |
| 6. Why 60 for every radius | A. B confuses the size of the drawing with the angle; C credits the tool; D names a step, not a reason. |
| 7. Smallest angle by bisecting | 7.5. 60 ÷ 8 = 7.5; 90 ÷ 8 = 11.25. |
| 8. Two equilateral triangles | 120. 60 + 60 = 120. |
| 9. Arcs that do not meet | B. A, C and D are all ways of trying to rescue it. If many choose D, return to the first figure and set 4 and 3. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Thinks any three lengths work, or adds all three. | Have them draw it: an 8 cm line and two arcs of 3 and 4. |
| 120 or 30 in question 2 | Uses 360, or halves. | Ask what the three angles add up to. |
| A, B or D in question 3 | Does not compare the two shorter sides with the longest. | Ask them to name the longest side first and add the other two. |
| 30 or 7.5 in question 4 | Stops after one bisection, or does three. | Draw the angle after each bisection and label it. |
| 30 or 300 in question 5 | Halves, or uses the full turn. | Draw the straight line and mark the 180° side. |
| 15 or 11.25 in question 7 | Does only two bisections, or starts at 90. | Write 60, 30, 15, 7.5 and 90, 45, 22.5, 11.25 side by side. |
| 60 or 180 in question 8 | Counts one triangle's angle, or its sum. | Draw the shape and count the angles at one corner. |
Questions worth asking
- What is the smallest value the third side can have? More than the difference of the other two. With 8 and 5 it must be more than 3 and less than 13.
- Why do two sides that add up exactly to the third give a flat triangle? The two arcs meet on the base itself, at height 0.
- Why is a construction better than measuring for a copy? Lengths and equal arcs are exact; an angle read from a protractor is only as exact as the eye.
- Can you build 90° with this idea? Yes, by bisecting a straight angle: the perpendicular bisector from lesson 5.
Differentiation
A way in for anyone stuck
- Provide strips of card of given lengths: students pin them and see which sets close into a triangle.
- Give a step card for the construction with a picture of the compass position each time.
- Use the first figure with the sliders fixed on one pair, and ask for the angles to be read.
Stretch
- Question 7, then: which angles can you make by bisecting 90° once, twice or three times? (45°, 22.5° and 11.25°.)
- How many different whole-number triangles are there with a longest side of 8 cm? (Count the pairs b and a, with b ≤ a ≤ 8 and b + a > 8: 2 + 4 + 6 + 8 = 20.)
Students learning in English as an additional language
Say “construct” as “build with a ruler and compasses”. “Equilateral” contains “equal” and “lateral” (side). “Arc” is part of a circle.
| Word | Plain meaning | Example |
|---|---|---|
| construct | to draw exactly with only a straight edge and compasses | Constructing a 60° angle |
| equilateral | with all sides equal; all angles are 60° | A triangle with sides 6, 6 and 6 cm |
| base | the side a triangle is drawn on | AB in the first figure |
| arc | a part of a circle | The marks left by the compasses |
| flat triangle | three points on one line, so the triangle has no height | Sides 3, 4 and 7 cm |
| inequality | a statement that one amount is more or less than another | 4 + 5 is more than 8 |
Sentence starters for explaining aloud or in writing:
- The two shorter sides add up to …, which is more than / equal to / less than …
- The triangle is equilateral because …, so each angle is …
- I bisected the angle of …, which gives …
Two things not to say
“Make the compass bigger until the arcs meet.” The radii are the side lengths. If they are fixed, the sides decide whether the arcs meet.
“The angle is 60° because I measured it.” The construction gives 60° exactly; a measurement would only give about 60°.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- TestCan a triangle have sides 7 cm, 8 cm and 9 cm?
Yes: 7 + 8 = 15 is more than 9. - AnglesWhat is the angle at O if you bisect the 60° angle made in the figure?
30°.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- FenceThree lengths of fencing, 6 m, 7 m and 14 m, are to make a triangular pen. Is it possible?
No: 6 + 7 = 13 is less than 14. - From Unit 1From Unit 1, lesson 5 (common factors and multiples): sides of 12 cm and 18 cm are needed. What is the longest piece of string that divides both exactly?
6 cm, the highest common factor.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningTwo sides are 5 cm and 9 cm. Between what whole numbers of centimetres can the third side be?
From 5 to 13, since it must be more than 4 and less than 14. - GeneralizingState in words the rule for three lengths to make a triangle, and say why equal to is not enough.
The two shorter must add up to more than the longest; equal makes a flat triangle with no height.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use a real pair of compasses, a ruler and plain paper; or strips of card pinned with split pins.