A square frame on pins folds flat when pushed sideways; a triangle can't change shape without a member stretching. Students brace frames with diagonals, use the counting rule (members = 2 × joints − 3) to find how many they need, and brace a tower with the fewest members.
Why are bridges, cranes and towers full of triangles, and not squares?
A frame on pins can only stay in shape if its members stop the joints moving. A triangle can't change shape unless a side changes length. A square can: it leans over into a diamond, with every side the same length.
Engineers brace frames with diagonals that turn them into triangles. Here a square frame stands on a pin (A) and a roller (B); a 500 N sideways push (the wind) acts at D.
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.
The square
TryTry it on the board and compare the result with your prediction.
Try: brace it
Try it yourselfYour edits are saved in this browser.
Press Test it.
Use the board. This ticks when you get there.
Add one diagonal (click A then C, or B then D), then test it again.
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Worked example: count the members
Worked exampleBefore each step, say what you think comes next.
m+3=2jEach joint gives 2 ways to move (sideways and up); the pin and roller hold 3 of them. Each member holds one.
4+3=7=8One short: the square can move one way (it leans).
5+3=8=2×4✓Add one diagonal: rigid, with none spare.
A frame has 8 joints, on a pin and a roller. How many members does it need to be rigid with none spare?
Zoe thinks she needs 15 members. Click the line where her working first goes wrong.
ModifyChange the problem on purpose and see what happens.
Vary: a taller tower
Need a hint?
Count: 6 joints, so 2 × 6 − 3 = 9 members. You have 7.
Another hint
One diagonal in each square.
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Brace the two-storey tower with as few members as you can, so it holds when you test it.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Rigid or not?
A crane arm has 10 joints, on a pin and a roller. How many members does it need?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Brace a tower
Try it yourselfYour edits are saved in this browser.
Brace the three-storey tower with as few members as you can, so it holds when you test it.
Use the board. This ticks when you get there.
For teachers
Students watch a square frame on pins collapse under a sideways push, brace it with one diagonal, learn the counting rule (members + 3 = 2 × joints for a pin and a roller), and brace two- and three-storey towers with the fewest members. About 40 minutes.
I can explain why triangles keep their shape and squares don't.
I can use the counting rule to say how many members a frame needs.
I can brace a frame with the fewest members.
Before class
Straws and pipe cleaners, or card strips and split pins: a square and a triangle side by side settle this in a minute.
The counting rule m + 3 = 2j is for flat frames on a pin and a roller (three support reactions). It is necessary but not sufficient: members can be in the wrong place (two diagonals in one panel and none in another), so always test. The board finds collapse directly.
Curriculum hooks: Design and Technologies, engineering principles and systems; Maths, linear relations (the rule).
Running the lesson
Predict: hold up a square of straws and push its corner.
Try: let students try both diagonals.
Worked example: count ways to move (2 per joint) against things that stop them (members and supports).
Vary: ask a student who added two diagonals to one panel what went wrong.
The hard part (on purpose)
Students think stronger members fix a wobbly frame. Ask whether any member changes length when the square leans.
When students get stuck
A student adds members but it still collapses. Ask: Is there a square panel with no diagonal? Fix: One diagonal in every square panel.
A student adds 3 in the rule. Ask: Do supports add ways to move, or take them away? Fix: They take 3 away: members = 2 × joints − 3.
Stretch and support
Stretch: Take the pin and roller away and use two pins instead (four support forces). How does the counting rule change?
Support: Give the rule as a fill-in: joints ___, × 2 = ___, − 3 = ___ members.
Where it leads
Next: Design a bridge.
Real examples: gates, scaffolding, electricity pylons, cranes, the Eiffel Tower.
the square: Students trust a square of strong members. Committing first makes the collapse memorable: strength isn't the issue, shape is.
Try: brace it: Watching the square collapse, then adding one diagonal and seeing it hold, makes triangulation concrete.
Worked example: count the members: The counting rule (for a flat frame on a pin and a roller) gives the number of members a frame needs to be rigid with none spare. Zoe adds 3 instead of taking it away, planted after the method.
Vary: a taller tower: The rule predicts the number before students build: two panels, two diagonals.
rigid or not?: The rule both ways, and real braced structures.
brace a tower: Three storeys: students use the rule to plan, then build with the fewest members.
Now you can
Explain why triangles keep their shape and squares don't.
Use the counting rule to say how many members a frame needs.