Part A — Growing Patterns & Rules
Every growing pattern hides two rules. The term-to-term rule tells you how to jump from one term to the next. The more powerful position-to-term rule lets you find any term straight away, even the 100th, without drawing it.
1
How many dots are in Term 4 and Term 5? Sketch Term 4 to check.
Term 4: Term 5:
2
Describe the term-to-term rule (how you get from one term to the next) in words.
3
Find the position-to-term rule that connects the term number n to the number of dots.
Number of dots =
4
Use your rule to predict the number of dots in the 10th term and the 50th term.
10th term: 50th term:
5
Which term has exactly 99 dots? Show how you worked backwards from the rule.
6
Prove it. Will any term in this pattern ever have an even number of dots? Explain your reasoning.
Part B — Function Machines & Sequences
A function machine takes an input number, applies one or more operations in order, and produces an output. Reading a table of inputs and outputs lets us discover the hidden rule — and reversing the operations lets us work backwards.
1
A function machine does ×3 then −1. Complete the output column.
| Input | Output |
|---|---|
| 1 | |
| 2 | |
| 5 | |
| 10 |
2
Look at this input/output table and find the one-step or two-step rule.
| Input | Output |
|---|---|
| 2 | 9 |
| 3 | 13 |
| 5 | 21 |
Rule:
3
Work backwards. A machine does ×2 then +5. Its output is 23. What was the input?
4
Open challenge. Design a two-step function machine that turns 4 into 20. Then find a different two-step machine that also turns 4 into 20.