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.
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Students will be able to
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.
Timing
Predict and Try 8 min · Worked example 8 min · Vary 10 min · Practise and Make 14 min
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
What you will see
A question to ask
The fix
A student adds members but it still collapses.
Is there a square panel with no diagonal?
One diagonal in every square panel.
A student adds 3 in the rule.
Do supports add ways to move, or take them away?
They take 3 away: members = 2 × joints − 3.
Answers to the checks
the square: Each member of the square can take 5000 N. The wind pushes 500 N sideways at D. The frame… It folds over. The members are strong enough, but a square on pins can lean without any member changing length, so nothing resists.
Test the square: Press Test it.
It collapses: nothing stops it leaning.
Add a diagonal: Add one diagonal (click A then C, or B then D), then test it again.
It holds: two triangles can't change shape.
Which diagonal?: Does it matter which diagonal you added? No: either makes triangles; one pulls, the other would push Either diagonal makes two triangles. With the wind from the left, A-C is stretched (pulls) while B-D would be squashed (pushes).
The rule: A frame has 8 joints, on a pin and a roller. How many members does it need to be rigid with none spare? 13 2 × 8 − 3 = 13.
Find the mistake: Zoe thinks she needs 15 members. Click the line where her working first goes wrong. line 3 Line 3: members + 3 = 2 × joints, so members = 2 × 6 − 3 = 9. She added 3 instead of taking it away.
Two storeys: Brace the two-storey tower with as few members as you can, so it holds when you test it.
Two diagonals, one per panel: 9 members, as the rule says.
Both diagonals: What happens if you add both diagonals to one square? It still holds, with a spare member that shares the load and costs more A spare (redundant) member isn't needed for the shape, but it shares the load and keeps the frame standing if another member fails.
A crane arm: A crane arm has 10 joints, on a pin and a roller. How many members does it need? 17 2 × 10 − 3 = 17.
Rigid?: A frame has 7 joints and 10 members, on a pin and a roller. Is it rigid? No: it needs 11 2 × 7 − 3 = 11, so with 10 it can change shape one way.
A gate: A wooden gate sags at the corner away from its hinges. What should you add? A diagonal brace across the gate A diagonal turns the gate's rectangle into triangles, so it can't lean.
Three storeys: Brace the three-storey tower with as few members as you can, so it holds when you test it.
8 joints: 2 × 8 − 3 = 13 members. One diagonal per panel.
Solutions
One possible solution for each task. Other correct programs are fine.
Vary: a taller tower › Two storeys
Open task: any program that does what the step asks.
Vary: a taller tower › Both diagonals
Open task: any program that does what the step asks.
brace a tower › Three storeys
Open task: any program that does what the step asks.
brace a tower › Evaluate
Open task: any program that does what the step asks.
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.
Exit ticket
A frame has 9 joints on a pin and a roller. How many members does it need to be rigid with none spare?
2 × 9 − 3 = 15.
What each step is for
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.
Where it leads
Next: Design a bridge.
Real examples: gates, scaffolding, electricity pylons, cranes, the Eiffel Tower.