MYP 4 mathematics. One lesson on investigating patterns, written for a student working alone.
This is a sample: one lesson, built to the standard a whole unit would be written to. The point of it is that Criterion B is not marked on finding the rule. It is marked on describing it, then on justifying why it has to be true, and most students stop one step too early.
Squares in a row, made from matchsticks. Drag the slider and count what happens to the number of matchsticks each time you add one square.
3 squares, 10 matchsticks
| Squares, n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Matchsticks, m | 4 | 7 | 10 | 13 | 16 |
| First difference | 3 | 3 | 3 | 3 |
The difference is a constant 3, so the relationship is linear and the rule contains 3n. Finding the constant is then one step:
The five data points and the rule, drawn together. If a point sits off the line, the rule is wrong. This is the check, not the proof.
Five points from the table, with m = 3n + 1 through them.
A student who gets to m = 3n + 1 and stops has described a general rule consistent with the findings. That is the middle of the mark band, and it is where most of the class lands.
The top band needs the rule justified, which means explaining why it must be true from the structure of the pattern rather than from the numbers. The explanation is below, and it is three sentences long. Those three sentences are the difference between a 5 and an 8.
The other common loss is at the bottom of the band: a rule written as "add 3 each time". That is a description of the pattern, not a general rule, because it cannot answer "how many matchsticks for 50 squares" without building all fifty.
The first square needs 4 matchsticks. Every square after the first shares its left-hand side with the square before it, so it only needs 3 new matchsticks rather than 4. With n squares there is 1 first square and (n − 1) later ones:
That is a justification, because it explains where the 3 and the 1 come from. The 3 is the three new matchsticks each added square needs. The 1 is the single extra side the very first square needs that none of the others do. Notice that it arrives at the same rule the table gave, which is the point: the numbers suggested it, the structure proves it.
Question. The same matchsticks, but now the squares are built in a 2 by n block, two rows of n squares.
(a) Draw the cases n = 1, 2 and 3 and count the matchsticks. (b) Find a general rule. (c) Justify it from the structure, not from the numbers.
| n | 1 | 2 | 3 |
|---|---|---|---|
| Matchsticks | 7 | 12 | 17 |
| First difference | 5 | 5 |
Count the matchsticks by direction instead of by square. The horizontal ones form 3 rows of n, which is 3n. The vertical ones form (n + 1) columns of 2, which is 2n + 2. Adding them:
Which agrees with part (b), and now explains it: the 5 is 3 horizontal plus 2 vertical matchsticks per column of the block, and the 2 is the extra pair of verticals needed to close the right-hand end.
Part (c) is the whole question. Counting by direction is the move worth learning, because it generalises: it works for 3 by n blocks, for triangles, for hexagons. A student who only ever substitutes into an + c can find rules but can never justify them, and will sit in the middle band every single time.
Every lesson is built like this one: the thing students get wrong named first, something they can push until it breaks, working set out one step per line with the equals signs aligned and division written as a fraction rather than a slash, and an explicit statement of where the achievement levels separate. Task sheets and criterion-referenced rubrics are written alongside.
Keith Spencer. Serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, Bangkok. Thirty years teaching US, UK and IB curricula.