At a glance. One lesson of about 60 minutes. Students use angles on a line, at a point and vertically opposite angles to find missing angles, with equations when the angles are expressions, and give a reason for each step.
Calculator: no calculator. Nothing here needs one. The totals are 90, 180 and 360, and every step is a small subtraction a student can do mentally or on paper.
What students will learn
Know
- Angles on a straight line add up to 180°.
- Angles at a point add up to 360°.
- Vertically opposite angles are equal; supplementary pairs add to 180° and complementary pairs to 90°.
Understand
- Each fact is a total, so a missing angle is the total minus the known angles.
- Naming which total applies is the first step; the arithmetic follows.
- A reason, not only an answer, is part of the maths.
Do
- Find unknown angles on a line, at a point and at a crossing.
- Write an equation from an angle fact and solve it.
- State the angle fact used at each step.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding missing angles and naming supplementary and complementary pairs (questions 1 to 6).
- Criterion B. Investigating patterns: the crossing-lines figure. Students move a, record the other three, and state a general rule for all positions.
- Criterion C. Communicating: writing a reason beside every step and setting out an equation clearly (questions 5, 7 and the worked examples).
- Criterion D. Applying mathematics in real-life contexts: the star of eight tiles in the hook (360 ÷ 8 = 45°) and turns in design, where the total must be chosen from the situation.
The one thing to do with the figure
Make students predict the other three angles before they move a. Then press 30° and 150°. The pair swaps roles: the same four numbers, the large pair now at a.
In the second figure, hold p and q and flip between a line and a point. Ask: what changed and what stayed? The two angles are the same; only the total is different. Say “which total?” before any subtraction.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea; the second figure. Students write the reason for each answer. |
| 10 min | Worked example with an equation; students finish the faded one in pairs. |
| 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 | Obtuse; 180; yes, 9 × 20. |
|---|---|
| Worked example | x = 20; the angles are 40°, 70° and 70°. |
| Faded example | x = 35; the angles are 105°, 55°, 110° and 90°, so the largest is 110°. |
| 1. Crossing lines | B. A uses 90, C forgets the opposite angle is equal, D uses 360. Fix any of these by returning to the first figure. |
| 2. Angles on a line | 62. 180 − 118 = 62. |
| 3. Angles at a point | 125. 360 − (140 + 95) = 125. |
| 4. Complementary angles | 52. 90 − 38 = 52. |
| 5. Vertically opposite with an unknown | 20. 4x − 5 = 75, 4x = 80, x = 20. |
| 6. Supplementary | B. 47 + 133 = 180. A is the total 90; C needs equal angles; D is a type of single angle. |
| 7. Equation on a line | 25. 6x + 30 = 180, 6x = 150, x = 25. |
| 8. One angle three times the other | 135. One part is 45°, and 3 × 45 = 135. |
| 9. Opposite or adjacent | B. A swaps neighbours and opposites; C uses the total for a right angle; D mistakes a special case for a rule. If many choose A, go back to the first figure and press 30°. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A or D in question 1 | Uses 90 or 360 as the total for a line. | Ask what a straight line is as a turn, then what total it gives. |
| 242 or 28 in question 2 | Uses 360, or compares with 90, instead of 180. | Draw the line and write 180 above it before any subtraction. |
| 235 or 85 in question 3 | Stops at the sum, or uses a line total for a point. | Mark the three angles round a point and ask what the total of a full turn is. |
| 142 in question 4 | Gives the supplement for the complement. | Write the two totals side by side: complementary 90, supplementary 180. |
| 17.5 or 70 in question 5 | Solves 4x − 5 = 75 with the steps in the wrong order or leaves out the division. | Ask them to put 20 back into 4x − 5. |
| 30 or 35 in question 7 | Drops the constant terms, or adds the 30 instead of subtracting. | Make them collect x terms and numbers in two separate columns. |
Questions worth asking
- Why must the angles round a point add up to 360°? They go once all the way round. Listen for “a full turn” as well as the number.
- Can two vertically opposite angles be different? No. Each is 180° minus the same neighbour. This is the line fact used twice.
- Can three angles on a line all be obtuse? No. Three obtuse angles would total more than 270, but a line is 180.
- What is the smallest angle that can be made by two crossing lines? As close to 0 as you like; there is no smallest, which is why the slider stops at 10.
Differentiation
A way in for anyone stuck
- Give a printed picture of a line and a point with the totals 180 and 360 written on them.
- Allow a number line or a part-whole diagram to subtract: total on top, known angles below, the unknown as the gap.
- Start with numerical questions before the ones with x; use a bar model for the equation.
Stretch
- Question 8, then: two angles on a line are 5 parts and 4 parts of the 180°. Find both. (100° and 80°.)
- Three lines cross at one point, giving six angles. Show the opposite angles are equal in pairs and the six add up to 360°.
Students learning in English as an additional language
Several terms are built from Latin parts. Show each with its picture: “vertically opposite” does not mean up and down, it means across the vertex.
| Word | Plain meaning | Example |
|---|---|---|
| straight line | a line with no bend; as an angle it is exactly 180° | A ruler's edge |
| vertically opposite | the pair of angles across the crossing point from each other | a and c in the first figure |
| supplementary | two angles that add up to 180° | 47° and 133° |
| complementary | two angles that add up to 90° | 38° and 52° |
| adjacent | next to each other, sharing an arm and a vertex | a and b in the first figure |
| reason | the fact that makes a step true | angles on a straight line add up to 180° |
Sentence starters for explaining aloud or in writing:
- The angles on a straight line add up to …, so …
- … is equal to … because they are vertically opposite.
- The missing angle is … − … = …, because …
Two things not to say
“Angles in a line add up to 180, just do 180 minus.” The rule needs the reason: say which angles are on the line, so that students know when it does not apply.
“Opposite angles are Z angles.” Z shapes belong to parallel lines (lesson 3). Vertically opposite angles need no parallel lines at all.
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.
- On a lineAngles on a straight line are 73° and x. Find x.
107°. - ComplementFind the complement and the supplement of 41°.
49° and 139°.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- DesignFive equal tiles meet at a point with no gaps. What is the angle of each tile at the point?
72°: 360 ÷ 5. - From Unit 1From Unit 1, lesson 3 (factors, multiples and primes): angles at a point are 30°, 45° and x. Find x, then say whether x is a multiple of 5.
x = 285, and 285 = 5 × 57, so yes.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningAngles on a line are (x + 10)°, (2x − 10)° and 3x°. Find the smallest angle.
x = 30, so the angles are 40°, 50° and 90°, and the smallest is 40°. - GeneralizingTwo lines cross and one angle is a°. Write the other three in terms of a.
180 − a, a and 180 − a.
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? Two crossed strips of card and a split pin show every case.