Problem Solving: Working Systematically and Finding Rules

1. 1 Display the toolkit: (1) Draw a picture. (2) Make a table. (3) Look for a pattern. (4) Work backwards. (5) Try a simpler case. 'Mathematicians don't just guess — they are systematic.'
2. 2 'There are 5 people in a room. Each person shakes hands with everyone else exactly once. How many handshakes?' Draw a diagram, make a table for 1,2,3,4,5 people. Pattern: 0,1,3,6,10... differences increase by 1 each time. Extend to 10 people.
3. 3 'I think of a number. I add 7, then double it, then subtract 3. The answer is 25. What was my number?' Work backwards: 25+3=28, 28÷2=14, 14−7=7. Check forward: 7+7=14, 14×2=28, 28−3=25 ✓.
4. 4 Display a pattern of shapes (e.g. squares in an L-shape growing each step). Count squares for each term and record in a table. Find the nth term rule. Predict the 10th term. Verify if possible.
5. 5 'How many squares are there on a standard 8×8 chessboard?' (Not 64 — count 1×1, 2×2, 3×3... squares.) Start with a 3×3 board to find the pattern. Use a table. Generalise for an n×n board.
6. 6 Different groups share their approach. 'Did everyone reach the same answer? Which strategy made it easiest? Could you have used a different approach?' Compare efficiency of different methods.
7. 7 Students write: 'The strategy I found most useful today was ___ because ___. I would use it again when ___.'

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