Name a triangle twice: by its sides (scalene, isosceles, equilateral) and by its biggest angle (acute, right, obtuse). Equal sides sit opposite equal angles, so some pairs are impossible: no right-angled equilateral triangle.
Roof trusses, bridges and set squares are all triangles, but not the same kind. Builders name them by their sides and their angles, because that tells you how they fit together.
Every triangle gets two names.
By its sides:
scalene: no sides equal;
isosceles: two sides equal (marked with the same tick);
equilateral: all three sides equal.
By its angles:
acute: every angle is less than 90°;
right-angled: one angle is 90° (marked ∟);
obtuse: one angle is more than 90°.
Equal sides sit opposite equal angles: an isosceles triangle has two equal angles, and an equilateral triangle has three, each 60°.
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.
Right and isosceles?
Think of a square cut in half along a diagonal.
TryTry it on the board and compare the result with your prediction.
Make one
Equal sides get matching ticks; a right angle gets a small square.
Try it yourselfYour edits are saved in this browser.
Drag the corners to make a triangle that is isosceles and right-angled. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Two names
Here is the routine for naming a triangle.
Worked exampleBefore each step, say what you think comes next.
90∘,45∘,45∘A triangle with these angles: two are equal, so the sides opposite them are equal.
isoscelesSides name: two equal sides.
right-angledAngles name: the biggest angle is 90°. Give both: isosceles, right-angled.
60∘,60∘,60∘Equilateral means equal angles too, so an equilateral triangle is always acute.
Now try one: sides 5, 5 and 8.
Read Zara's working. Which angle decides the name?
Zara names a triangle by its angles. Click the line where her reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Move one corner so that no two sides match: the ticks disappear.
Need a hint?
Pull the top far to one side, past the end of the bottom side.
Need a hint?
Corners land on half squares here.
Another hint
Try a bottom side of 4, with the top 3.5 up, half way along.
Try it yourselfYour edits are saved in this browser.
This triangle is isosceles. Move one corner to make it scalene and still acute.
Use the board. This ticks when you get there.
Make an obtuse scalene triangle.
Use the board. This ticks when you get there.
Make a triangle as close to equilateral as the grid allows.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Naming triangles
An isosceles triangle has one angle of 40° between its two equal sides. What is each of the other two angles, in degrees?
Review: a triangle has angles 35° and 85°. What is the third, in degrees?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Obtuse and isosceles
Then explain why the obtuse angle has to be the one between the equal sides.
Try it yourselfYour edits are saved in this browser.
Need a hint?
The obtuse angle has to be between the two equal sides.
Another hint
Try a long flat bottom with the top just above its middle.
Make a triangle that is isosceles and obtuse.
Use the board. This ticks when you get there.
For teachers
Students drag triangles on a dot grid and see them named by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse), combine the two names, and reason about impossible pairs (AC9M7SP02, node 1). About 40 minutes.
I can name a triangle by its sides and by its angles.
I can explain why some triangles are impossible, like a right-angled equilateral triangle.
I can use equal sides and equal angles to find missing angles.
Before class
Do the Run board yourself first. Students need the triangle angle sum (Angles in a triangle) and the words acute, right and obtuse for angles.
The one idea: every triangle has a sides name and an angles name, and equal sides sit opposite equal angles. The common mistake is naming by any acute angle (Zara's mistake): every triangle has at least two.
A square grid cannot make an exact equilateral triangle; the board counts sides within 2% as equal and says so.
Running the lesson
Predict: take a show of hands; ask a 'yes' student to sketch one.
Run: ask what the ticks and the small square mean.
Investigate: for the impossible pair, ask what angles an equilateral triangle must have.
Modify: for nearly equilateral, ask why the grid cannot do it exactly (the height is not a whole number of squares).
Make: ask which angle is obtuse and why.
The hard part (on purpose)
Making a triangle with two names at once on the board. Ask students to say which name they will fix first and what has to stay true while they fix the second, rather than moving corners for them.
When students get stuck
A student calls a triangle acute because it has an acute angle (Zara's mistake). Ask: Which is the biggest angle? Fix: The biggest angle decides: under 90° acute, 90° right, over 90° obtuse.
A student gives only one name. Ask: What do the sides say, and what do the angles say? Fix: Name the sides first, then the biggest angle.
A student thinks a right-angled triangle can't be isosceles. Ask: What happens if you cut a square along its diagonal? Fix: Two equal sides at the right angle: 45°, 45°, 90°.
Stretch and support
Stretch: List all nine pairs (3 side names × 3 angle names). Which are possible? Sketch each possible one.
Support: Keep a two-column table on the board: sides names and angles names with a picture of each.
Where it leads
Next: The quadrilateral family (node 2).
Where it leads: Pythagoras' theorem (Year 8) for right-angled triangles, and congruent and similar triangles (Years 8 and 9).
What each step is for
right and isosceles?: Committing to whether a triangle can be right-angled and isosceles at once shows whether they see the two names as independent.
make one: Dragging corners until the board names the triangle 'isosceles, right-angled' shows the two names are separate choices that can combine.
two names: The sides name and the angles name are separate; Zara names the triangle by an acute angle it has, but acute means every angle is under 90°. Some pairs are impossible.
one change at a time: One change at a time on the board: from isosceles to scalene, then an obtuse scalene triangle, then nearly equilateral on half squares, then the edge case of two right angles.
naming triangles: Naming from sides, from angles and from both, mixed; one acute angle is the planted slip; the review keeps the angle sum fresh.
obtuse and isosceles: Building an obtuse isosceles triangle combines both names on purpose: the equal sides meet at the obtuse angle.
Now you can
Name a triangle by its sides and by its angles.
Explain why some triangles are impossible, like a right-angled equilateral triangle.
Use equal sides and equal angles to find missing angles.