However you change a triangle, its three angles add to 180°. Tear off the corners and they make a straight line.
A roof truss, a bike frame, a folded paper plane: triangles of every shape. One thing about their angles never changes.
An angle measures a turn, in degrees (°). A full turn is 360°, and the angles on a straight line add to 180°.
A triangle has three corners, so three angles inside it. In this lesson you find out what happens to their total when you change the triangle's shape.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
A stretched triangle
Picture a triangle you could pull out into a long, thin shape. Which angles change?
TryTry it on the board and compare the result with your prediction.
Change the triangle
Drag any corner to a new dot. The board measures each angle and adds them up under the triangle. Make tall, wide and flat triangles. Was your prediction right?
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Drag the corners until one angle is more than 120°. What is the total now?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Why 180°?
Here is the argument, the way you would do it with a paper triangle.
Worked exampleBefore each step, say what you think comes next.
Draw any triangleCut it out of paper, and mark its three angles A, B and C.
Tear off the three cornersEach piece keeps one angle.
Put the points togetherPlace the three corners side by side, all touching at one point.
They fill a straight lineThe outside edges of A and C line up into one straight line.
A+B+C=180°Angles on a straight line add to 180°, so the triangle's angles do too.
On the board, press Tear off the corners, then drag a corner. The pieces change, but they always fill the line.
Read Kai's working one line at a time. Which line ties the angles to the length of the sides?
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Press **Tear off the corners**. Then drag a corner and watch the torn pieces.
Use the board. This ticks when you get there.
Kai compares a small triangle with one twice as big. Click the line where his reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing. First: make a square corner.
Need a hint?
A right angle is a square corner. Use the dots: one side across, one side straight up.
Another hint
Put C straight above A.
Now make one angle very big.
Need a hint?
Pull one corner until the triangle is nearly a line. Tear off the corners to see the pieces.
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Make a triangle with a right angle (90°). What do the other two angles add to?
Use the board. This ticks when you get there.
Make a triangle with an angle of more than 150°. How big are the other two?
Use the board. This ticks when you get there.
A triangle has three equal angles. How big is each one?
PractiseA few on your own, to check you have it.
Use the rule
Review: two angles sit side by side on a straight line. One is 130°. How big is the other?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
One angle equals the other two
Think first: if one angle is the other two added together, what fraction of 180° is it? Then build it.
Stretch: can a triangle have one angle that is exactly twice the sum of the other two? How big is it?
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Need a hint?
If one angle equals the other two added together, it is half of the total.
Another hint
Half of 180° is 90°: make a square corner.
Make a triangle where one angle is exactly the other two added together.
Use the board. This ticks when you get there.
For teachers
Students drag triangles of every shape on a dot grid, see that the three angles always add to 180°, and explain why by tearing off the corners to make a straight line (AC9M7M05, node 1). About 40 minutes.
I can show that the angles of a triangle add to 180°, by measuring and by tearing off the corners.
I can explain why this is true for every triangle, whatever its shape or size.
I can tell whether three angles could be the angles of a triangle.
Before class
Do the Run board and the tear-off task yourself first. Students should know that angles on a straight line add to 180° (Year 6).
The board measures to the nearest degree and shares out the rounding so the shown angles add to the shown total. With a real protractor, totals of 178° to 182° are normal: say so when students measure by hand (Practise has a question on it).
If you have time, do the paper version too: every student draws a different triangle, tears off the corners and lines them up. Seeing 25 different triangles all make a straight line is the strongest evidence. The formal proof with parallel lines comes with AC9M7M04.
Finding a missing angle from the other two is the next step in class (node 3), not this lesson.
Running the lesson
Predict: take a show of hands. Students who chose 'it gets bigger' are thinking of the big angle only. Do not correct it yet.
Run: ask students to make the tallest, the widest and the flattest triangle they can, and to read the total each time.
Investigate: model the worked example live with a paper triangle: mark the three angles, tear them off at writing pace, and put the points together on a ruler's edge, saying each step. Then show the board's torn pieces changing as a corner moves.
Modify: before the right-angle task ask: if one angle is 90°, what must the other two add to? Then check on the board.
Practise: Lia's measurement question is a chance to talk about measuring error, which science lessons need too.
Make: ask a student to explain why the angle must be 90° before anyone builds it.
When students get stuck
A student thinks a bigger triangle has bigger angles (Kai's mistake). Ask: If you make a photocopy twice as big, do the corners get pointier or blunter? Fix: Enlarge a triangle on the board by doubling each corner's coordinates: the angles stay the same.
A student adds the angles and gets 179° or 181° with a protractor. Ask: How accurately can you read a protractor? Fix: Each reading can be a degree out, so three readings can be two or three degrees out. The true total is 180°.
A student says the rule only works for the triangles they tried. Ask: Which triangle could you try that would break it? Fix: Let them try their 'breaking' triangle on the board or on paper. Then point to the torn corners: they always fill a straight line, whatever the shape.
Stretch and support
Stretch: Can a triangle have one angle that is twice the other two added together? (Yes: 120°, since 120 = 2 × 60.) What about three times? (135°.)
Support: Use the Run board throughout, and the preset buttons are on the explain clip's board. Give students paper triangles to tear.
Where it leads
Next: angles in any polygon, by splitting it into triangles.
Where it leads: finding unknown angles with reasons (node 3), the proof with parallel lines (AC9M7M04), and why regular hexagons tessellate.
What each step is for
a stretched triangle: Committing to what happens to the total before dragging makes the board a test of their idea, not a demonstration.
change the triangle: Dragging the corners and watching the total shows that the angles change but their sum never does.
why 180°?: Tearing off the corners and putting them side by side shows why the total is fixed: the three angles fill a straight line, which is 180°. Kai's mistake ties the angles to the side lengths.
one change at a time: One change at a time: a right angle (the other two must make 90°), a very flat triangle (the other two are tiny), then two edge cases that the 180° rule decides.
use the rule: Recognising possible triangles, a measurement that is nearly 180°, and the size question, mixed; the review keeps angles on a straight line from Year 6 fresh.
one angle equals the other two: Building a triangle where one angle equals the other two together reverses the lesson: from the 180° rule to a shape it forces, a right angle.
Now you can
Show that the angles of a triangle add to 180°, by measuring and by tearing off the corners.
Explain why this is true for every triangle, whatever its shape or size.
Tell whether three angles could be the angles of a triangle.