Use a stated rule to predict and prove a pattern

diagnostic. The rule is start at 4 and add 3 each time. The first three terms are 4, 7, 10. Enter term 4 and explain which instruction you used.

Answer: 13

Teacher response: Adding three to ten gives thirteen. The stated rule determines the next step; the starting number is already term 1.

model. Use this exact rule: start at 3 and add 6 each time. What is term 8? Will every term generated by this rule be odd? Justify the prediction without checking every term.

1. Write the rule before extending the list. Label the starting 3 as term 1, then generate 9, 15, 21 and 27 as terms 2 to 5. Write +6 over each arrow so the instruction remains visible.
2. Count changes rather than positions. Term 1 uses no additions; term 2 uses one. Term 8 therefore uses seven additions, giving 3 + 7 × 6 = 45. Compare the calculation with an extended list: 33, 39, 45 are terms 6, 7 and 8.
3. Represent 3 as one pair and one unpaired counter. Each added 6 brings three complete pairs, so the one unpaired counter remains. The same argument works at each step, which justifies odd terms under the supplied rule rather than merely noticing five odd entries.
4. Separate the rule from the evidence in a short list. If only 3, 9, 15 were given, one possible rule is always add 6; another is add 6 twice, then add 7 thereafter. Their fourth terms are 21 and 22. Both fit the displayed odd terms, yet only the first keeps every term odd. Ask for the rule when the intended continuation is not specified.

Worked example: Term 8 is 45: seven additions of six follow the starting 3, so 3 + 7 × 6 = 45. Every term is odd because the start is odd and adding an even number preserves one unpaired unit.

p1. Support: start at 4 and add 3 each time. Enter term 5, draw the four changes, and state whether the first five values stay even or alternate parity.

Answer: 16

Teacher response: Four additions give 4 + 4 × 3 = 16. The values 4, 7, 10, 13, 16 alternate even and odd because adding an odd number changes parity.

p2. Core: start at 5 and add 8 each time. Enter term 7 and explain why the stated rule produces only odd terms.

Answer: 53

Teacher response: Six additions give 5 + 6 × 8 = 53. The odd start has one unpaired unit and every added eight contributes only pairs.

p3. Challenge: start at 2 and add 6 each time. Enter term 20. A partner claims every term is even and therefore every term is a multiple of six. Accept or reject each part with an example or an explanation.

Answer: 116

Teacher response: Nineteen additions give 2 + 19 × 6 = 116. All terms are even, but the multiple-of-six claim fails already at the starting term 2. Adding six keeps two extra units beyond complete groups of six.

e1. Independent check: start at 7 and add 4 each time. Enter term 9, show how many changes are used, and justify whether every term under this rule is odd or even.

Answer: 39

Look for:

• Keeps term positions separate from the number of additions.
• Explains parity from the stated starting value and step instead of relying only on displayed examples.
• Independent rubric: one criterion each for 39, eight additions, and odd start plus even increments preserves oddness; all three demonstrate the prediction and justification.
• Recognises that a finite list can satisfy more than one rule.

Teacher response: There are eight changes, so 7 + 8 × 4 = 39. Every term is odd: start with one unpaired unit in seven and add only pairs through each +4 step.

transfer. Two rules both begin 3, 9, 15. Rule A starts at 3 and always adds 6. Rule B starts at 3, adds 6 twice, then adds 7 at every later step. Give each fourth term and its parity. What can the three displayed values alone establish about the intended fourth term?

Answer: Rule A gives 21, which is odd; Rule B gives 22, which is even. Both match all three displayed values, so those values alone establish neither a unique intended fourth term nor its parity. Once a rule is supplied, follow it exactly; a prediction from a guessed rule should be labelled as conditional.

teach-1. 5 minutes: Collect the diagnostic and listen for use of the stated +3 rule. Use position cards with learners who need support; ask ready learners to distinguish a starting term from the first change.

teach-2. 10 minutes: Build the main term table together. Partners count arrows to term 8, compare 3 + 7 × 6 with an extended list, then model an odd starting value and an even increment using pairs of counters.

teach-3. 15 minutes: Spend five minutes on the support or core rule, five on explaining a parity claim to a partner, and five on the challenge. Require each claim to name both the start and the repeated change. Ask partners to test whether an observed property follows from the rule.

teach-4. 10 minutes: Allow six minutes for the independent term and parity explanation, then four minutes to compare the transfer's two explicit rules. Assess three exit criteria: correct term, one fewer change than position, and a parity justification tied to the start and increment. Revisit position cards or pairing according to the missing criterion.

misconception. The displayed terms are odd, so any continuation of this finite list must contain only odd numbers.

Teacher response: Ask which rule is being used. Under start at 3 and always add 6, pairing proves every term is odd. A finite list alone also fits a rule that changes its step later; observed terms do not uniquely determine what must come next.

support. Use five position cards and four arrow cards. Put a value under each position, then use paired dots to show what adding three does to even and odd values before discussing a more distant term.

extend. Compare start at 3 and add 6 with start at 3 and add 5. Their term-8 values are 45 and 38. Explain why the first rule preserves oddness while the second alternates parity.

support-route. If the learner repeats a number, changes the step or counts 4 as a change, lay out position cards 1 to 4 and make one +3 arrow between neighbouring positions. Recheck the four values 4, 7, 10, 13 before the support task.

ready-route. If the learner gives 13 and cites adding three, begin the main table and ask how many additions connect term 1 to term 8.