Support
Answer and teaching response
16 · 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.
Use a stated rule to predict and prove a pattern
Generate a sequence from an explicit rule, count the changes needed for a distant term and justify an odd-or-even property beyond the displayed terms.
Get student and teacher materialsPrepare the lesson
Learning goal: Produce a term table, a correct distant-term calculation and a rule-based explanation of parity, while distinguishing a supplied rule from a guess based on a short list.
You will need: Paper, pencil and counters or paired dots. Prepare a table with term position and value. All rules are invented for this lesson; term 1 is the stated starting number.
Start with a 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.
Check the reasoning and choose a starting point
13 · Adding three to ten gives thirteen. The stated rule determines the next step; the starting number is already term 1.
Start with support: 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 for core: If the learner gives 13 and cites adding three, begin the main table and ask how many additions connect term 1 to term 8.
Work through an example
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.
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.
- 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.
- 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.
- 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.
- 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.
Run the 40-minute lesson
- 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.
- 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.
- 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.
- 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.
A useful error to discuss
A learner might say: The displayed terms are odd, so any continuation of this finite list must contain only odd numbers.
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.
Practise at your level, then check independently
Use the diagnostic to choose support or core practice, then try the challenge. Complete the independent check without help. Enter a number, fraction or mixed number. The number check does not assess your drawing or explanation.
Core
Answer and teaching response
53 · Six additions give 5 + 6 × 8 = 53. The odd start has one unpaired unit and every added eight contributes only pairs.
Challenge
Answer and teaching response
116 · 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.
Independent check
Answer and teaching response
39 · 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 the idea
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?
Discuss a possible response
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.
Choose the next teaching step
- 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.
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.
Materials for your group
Student pages and teacher answers are separate. Keep the HTML files or print them in your browser.
Revisit a prerequisite
Use Repeated Fraction Groups if the prerequisite gap is connecting repeated addition with multiplication. Its fractional examples develop equal-group meaning; this investigation adds position indexing, integer rules and the separate parity argument.