--- title: "Triangles hold their shape" subtitle: "Structures · Frames and trusses · Lesson 2" --- # Triangles hold their shape **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Why are bridges, cranes and towers full of triangles, and not squares? :::teacher-note **In one sentence:** 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. **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict :::keep-together **Each member of the square can take 5000 N. The wind pushes 500 N sideways at D. The frame…** ☐ Slides off the roller ☐ Holds easily: 500 N is small ☐ Breaks one member ☐ Folds over: it changes shape without any member stretching My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** It folds over. The members are strong enough, but a square on pins can lean without any member changing length, so nothing resists. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Worked example: count the members :::keep-together ### The rule 1. $\text{joints } j = 4,\; \text{members } m = 4$ — The square: 4 joints, 4 members. 2. $m + 3 = 2j$ — Each joint gives 2 ways to move (sideways and up); the pin and roller hold 3 of them. Each member holds one. 3. $4 + 3 = 7 \ne 8$ — One short: the square can move one way (it leans). 4. $5 + 3 = 8 = 2 \times 4 \;\checkmark$ — 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?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 13 2 × 8 − 3 = 13. ::: :::keep-together ### Find the mistake **Zoe thinks she needs 15 members. Click the line where her working first goes wrong.** ``` 1 Zoe's frame has 6 joints, on a pin and a roller. 2 The rule: members + 3 = 2 × joints. 3 So members = 2 × 6 + 3 = 15. 4 She needs 15 members to make it rigid. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 Line 3: members + 3 = 2 × joints, so members = 2 × 6 − 3 = 9. She added 3 instead of taking it away. ::: :::typical ## Vary: a taller tower **Two storeys.** **Both diagonals.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Rigid or not? :::keep-together ### A crane arm **A crane arm has 10 joints, on a pin and a roller. How many members does it need?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 17 2 × 10 − 3 = 17. ::: :::keep-together ### Rigid? **A frame has 7 joints and 10 members, on a pin and a roller. Is it rigid?** ☐ No: it needs 11 ☐ Yes ☐ It has a spare member ::: :::teacher-note **Answer:** No: it needs 11 2 × 7 − 3 = 11, so with 10 it can change shape one way. ::: :::keep-together ### A gate **A wooden gate sags at the corner away from its hinges. What should you add?** ☐ A diagonal brace across the gate ☐ A third hinge on the same side ☐ Paint ☐ Thicker rails ::: :::teacher-note **Answer:** A diagonal brace across the gate A diagonal turns the gate's rectangle into triangles, so it can't lean. ::: ## Make: brace a tower **Three storeys.** **Evaluate.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A frame has 9 joints on a pin and a roller. How many members does it need to be rigid with none spare?** {{lines-3}} ::: :::teacher-note **Answer:** 2 × 9 − 3 = 15. :::