Session 1 · SOLVE: Link Ideas
Letters for numbers
- 15 minTurn words into a rule: 'three toothpicks for every square, plus one to start' becomes t = 3n + 1. The 3 is the constant increase; the 1 is the extra.
- 15 minWrite rules for your linear patterns and check each against every row of the table.
- 10 minRun it backwards: which step uses exactly 100 toothpicks? Solve 3n + 1 = 100. What does it mean if the answer isn't a whole number?
- 10 minPlot the patterns on a grid or in Desmos. Linear patterns make points in a straight line; how steep is each line, and why?
Session 2 · SOLVE: Link Ideas
Squares, cubes and triangles
- 15 minBuild square numbers (1, 4, 9, 16…) and cube numbers (1, 8, 27, 64…) with tiles and cubes. Write the rules n² and n³.
- 15 minTriangular numbers (1, 3, 6, 10…): place two copies of the same staircase together. What shape do they make? Use it to find the rule n(n + 1) ÷ 2.
- 10 minPlot all three on the same axes. Why do these make curves, not straight lines?
- 10 minTohu asks one question about why one of your rules works.
Getting started
Use the frame 'pieces = ___ × step number + ___' and test it on step 1 and step 2.
Stretch
Show that 2(n + 3) + 1 and 2n + 7 give the same numbers, and draw a tile pattern that matches each way of writing it.
Checkpoint
A correct rule with letters for at least one linear and one non-linear pattern, checked against the table.
Activities for this week
- Squares, cubes and staircases: How do square, cube and triangular numbers grow, and why do their graphs curve?
Useful this week
- Sequences: finding the rule ↗ Step-by-step help for turning a sequence into a rule.
- Desmos graphing calculator ↗ Graph your rule and see whether it is a line or a curve.