Fraction Classroom · 2.6.0 · Student pages

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.

Anonymous group code: __________ Date: __________

The stated rule is start at 3 and always add 6. There is one fewer change than the term position. Pairing, together with the rule, explains the parity.

Start with a diagnostic

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.

Build the idea

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.

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.

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.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

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.

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

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.

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.

Transfer the idea

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?