Here is a growing tile pattern: pattern 1, then pattern 2, then pattern 3, each one a little bigger than the last. Pattern 1 uses 3 tiles, pattern 2 uses 5 tiles, pattern 3 uses 7 tiles.

This machine takes a term-number in and hands back a value using a hidden rule. We feed in 1, 2 and 3, read off what comes out, and work out the rule each time. Watch the jump between the values.

Today we work through this growing pattern together: term 1 gives 4, term 2 gives 7, term 3 gives 10. I will send the next term-numbers through the machine and build a two-row table on the board as the values land, while you watch and call out each value. Then we will agree on the rule in words together.

Now we crack the rule for three of these patterns by sending term-numbers through the machine and watching the values. Once we have the rule, we will use it to leap straight to a far term, without drawing every step in between. Here are our three patterns:

• Pattern A: term 1 gives 2, term 2 gives 4, term 3 gives 6.
• Pattern B: term 1 gives 5, term 2 gives 8, term 3 gives 11.
• Pattern C: term 1 gives 2, term 2 gives 6, term 3 gives 10 — and once we have the rule, we leap all the way to term 10.

What we learned

• A two-row table of term-number against value lays a pattern out so its rule is easy to spot.
• The step between values tells you the number to multiply the term by; the rest of the rule is the adjustment that sets where the pattern starts.
• Once you have the rule in words, you can leap straight to any term without drawing every step.