At a glance. One lesson of about 60 minutes. Students bisect a line segment and an angle with compasses, explain why the arcs must have equal and large enough radius, and use halves in simple problems.
Calculator: no calculator. Nothing here needs one. The numbers are halves and doubles, and the construction is physical or on a figure.
What students will learn
Know
- Bisect means cut in half.
- The perpendicular bisector of AB cuts AB at its midpoint at 90° and is the set of points equally far from A and B.
- The steps for bisecting an angle.
Understand
- A compass keeps a distance fixed, so two equal arcs find a point the same distance from two places.
- The radius must be more than half of the segment or the arcs never meet.
- A construction is a proof by drawing: each step can be given a reason.
Do
- Bisect a line segment and an angle with compasses and a ruler.
- Find a half or a whole from a bisector.
- Explain why a construction fails with arcs that are too small or unequal.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the meaning of bisect and the steps (questions 1 to 7).
- Criterion B. Investigating patterns: the radius slider. Students find the smallest radius at which the arcs cross and say why it is half of AB.
- Criterion C. Communicating: writing the steps of a construction in order, with a reason for each, and labelling points correctly (question 6, the worked example).
- Criterion D. Applying mathematics in real-life contexts: finding the middle of an opening or the angle of a fold in craft and building.
The one thing to do with the figure
Ask for the smallest radius before sliding. Most students say “half of AB” and then discover that exactly half only makes the arcs touch.
Then drag past that value and watch P and Q move while M stays put: the line is the same line for every radius that works. In the second figure, ask what the arcs from U and V have in common (the same radius, and the same distance from O).
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with a prediction of the smallest radius. |
| 10 min | Key idea; students bisect a 7 cm segment on paper with compasses, then measure: AM within 1 mm of 3.5 cm, and the angle at M within 1° of 90°. |
| 10 min | Second figure and the angle bisector; worked example and the faded one. |
| 5 min | Question 1, the check. If many choose C 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 | 55°; 60°; 4.5. |
|---|---|
| Worked example | 4.5 cm and 40°. |
| Faded example | 25°. |
| 1. Why the arcs must be big enough | A. B assumes the arcs meet; C ignores the condition on size; D thinks there is only one correct radius. Return to the first figure and drag the radius. |
| 2. Midpoint distance | 5.5. 11 ÷ 2 = 5.5. |
| 3. Equidistant from the ends | B. The perpendicular bisector is the set of points the same distance from A and B. |
| 4. Half an angle | 39. 78 ÷ 2 = 39. |
| 5. Working back from a half | 52. 2 × 26 = 52. |
| 6. Steps of the construction | A. B measures instead of constructing; C has arcs that cannot cross; D uses a protractor. |
| 7. Bisecting twice | 30. 120 ÷ 2 ÷ 2 = 30. |
| 8. Bisecting a segment twice | 9. AM = 6, AN = 3, NB = 12 − 3 = 9. |
| 9. Equal radii | B. A and C miss the equal radius condition; D confuses the centre of the arcs with the point being found. If many choose A, go back to the first figure. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| C or D in question 1 | Thinks any radius works, or that there is a single right radius. | Let them try 3 cm on paper and see the arcs fall short. |
| 11 or 22 in question 2 | Gives the whole segment, or doubles it. | Mark M on a drawn segment and ask which part AM is. |
| A in question 3 | Halves the distance, as for a midpoint. | Ask what P being on the bisector says about its distances from A and B. |
| 156 or 102 in question 4 | Doubles, or subtracts from 180. | Draw the angle and the bisector and write each half's size. |
| 13 or 154 in question 5 | Halves again, or subtracts from 180. | Ask what the two halves add up to. |
| 60 or 15 in question 7 | Stops after one bisection, or bisects three times. | Draw the angle after each bisection and label it. |
| 3 or 6 in question 8 | Gives AN or AM, not NB. | Mark A, N, M, B on a line and ask which distance is wanted. |
Questions worth asking
- Why must the radius be more than half of AB? Each arc reaches only its radius from its end. Two radii that add up to less than AB cannot meet. Listen for “the arcs fall short”.
- What if one arc had a bigger radius than the other? The crossing points are no longer the same distance from A and B, so the line is not the bisector.
- Why do we call it the perpendicular bisector, not only the bisector? It cuts AB in half and meets it at 90°. A line through the midpoint at another angle also bisects AB but is not perpendicular.
- Where have we met “same distance” before? A circle is the points the same distance from a centre; the bisector is where two such circles cross.
Differentiation
A way in for anyone stuck
- Give a pre-drawn segment and arcs on a template, and have students complete the final line.
- Provide a step card: set, arc from A, arc from B, join; with a picture for each.
- Use a rope or a string compass on the floor for the whole class first.
Stretch
- Bisect an angle of 180° to get a perpendicular, then bisect again to make 45°. What angles can be made with this idea alone? (90°, 45°, 22.5°, 135°, and so on.)
- Prove with a construction and a triangle that the bisector of AB is perpendicular to AB, using the idea of two equal triangles.
Students learning in English as an additional language
Say “bisect” with its parts: “bi” means two and “sect” means cut, as in “section”. A compass here is the drawing tool, not the direction finder.
| Word | Plain meaning | Example |
|---|---|---|
| bisect | to cut exactly in half | Bisecting 80° makes two angles of 40° |
| perpendicular bisector | a line that cuts a segment in half at a right angle | The line through M in the first figure |
| compasses | a drawing tool that keeps a fixed distance from a point | Used to draw an arc |
| arc | a part of a circle | The curved marks the compasses leave |
| midpoint | the point exactly halfway along a segment | M in the first figure |
| equidistant | the same distance from two or more things | P is equidistant from A and B |
Sentence starters for explaining aloud or in writing:
- I set the compasses to … because it must be more than half of …
- The arcs meet at …, which is the same distance from … and …
- The bisector makes two angles of … because …
Two things not to say
“Just set the compasses wide and draw.” Wide is not a reason. The condition is more than half of AB, and the compasses must not change between arcs.
“The line is the shortest way between the arcs.” The line joins the two crossing points; its job is to find the midpoint and a right angle.
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.
- HalvesA segment of 14 cm is bisected. How long is each part?
7 cm. - AnglesAn angle of 64° is bisected. What is each part?
32°.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- DesignA craftsperson bisects a 130° angle for a hinge. What is each angle?
65°. - From Unit 1From Unit 1, lesson 7 (rounding and estimating): a segment is 17.6 cm. Estimate its half by rounding 17.6 to the nearest whole number first.
About 9 cm (18 ÷ 2).
3Challenge
Reasoning, generalising and working backwards.
- ReasoningAn angle is bisected and the half is bisected again, giving 17.5°. What was the original angle?
70°. - GeneralizingAfter n bisections of an angle of 360°, what is the size of each part?
360 ÷ 2 to the power n, which for n = 3 is 45°.
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: the figure shows the same marks a pencil leaves.