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Fraction Classroom · 2.6.0 · Student pages

Connect two generated patterns with ordered pairs

Generate two patterns from supplied rules, pair corresponding terms and use a table and coordinate graph to explain their relationship.

Anonymous group code: __________   Date: __________

A starts at 0 and adds 2; B starts at 0 and adds 4. Each row pairs terms after the same number of additions. Plot the A term horizontally and B vertically; each axis interval represents 2 units.AdditionsPattern APattern BPair (A, B)000(0, 0)124(2, 4)248(4, 8)3612(6, 12)4816(8, 16)
A starts at 0 and adds 2; B starts at 0 and adds 4. Each row pairs terms after the same number of additions. Plot the A term horizontally and B vertically; each axis interval represents 2 units.

Start with a diagnostic

diagnostic. Pattern A starts at 0 and adds 2 each time. Pattern B starts at 0 and adds 4 each time. After one addition to each pattern, write their corresponding pair (A, B), then enter the Pattern B term.

Build the idea

model. Pattern A starts at 0 and repeatedly adds 2. Pattern B starts at 0 and repeatedly adds 4. Generate corresponding pairs from the start through four additions. What Pattern B value is paired with A = 8, and how does the graph help explain the relationship?

  1. Write both starting values in the same table row. Label later rows by the number of additions: 1, 2, 3 and 4. Read both given rules aloud; a term in one row must be compared with the other pattern's term in that same row.
  2. Generate A as 0, 2, 4, 6, 8 and B as 0, 4, 8, 12, 16. Check each adjacent change against its rule. Form the ordered pairs by writing the A term first and its corresponding B term second.
  3. Plot all five pairs on the labeled axes. Use the stated scale: one interval represents 2 units on each axis. Ask partners to identify the pair at A = 6 and read its B value, 12, from the table and graph.
  4. Compare corresponding terms: 4 is twice 2, 8 is twice 4, 12 is twice 6 and 16 is twice 8. Zero also satisfies 0 = 2 × 0. Explain why the rules preserve this: both start at zero, and each added 4 is twice the added 2. Do not infer doubling from different increments alone when the starting values change.

Worked example: The pairs are (0, 0), (2, 4), (4, 8), (6, 12) and (8, 16), so B is 16 when A is 8. At each stage, B is twice A. Plot with A horizontal and B vertical: each move right 2 units is accompanied by a move up 4 units, keeping the same doubling relationship.

Practise and explain

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

p1. Support: keep the worked rules, A starts at 0 and adds 2, B starts at 0 and adds 4. The current pair is (6, 12). Apply each rule once more. Enter the next B term and write the complete next pair.

p2. Core: A starts at 1 and adds 3; B starts at 2 and adds 6. Generate the starting pair and the next three pairs. Enter the B value after three additions, and explain how it compares with its corresponding A value.

p3. Challenge: A starts at 5 and adds 2; B starts at 5 and adds 4. After three additions, how much greater is B than A? Enter the difference and explain whether B is twice A at that stage.

e1. Independent check: A starts at 0 and adds 3; B starts at 0 and adds 9. Generate the first five corresponding pairs, including the starting pair. Plot A horizontally and B vertically using one grid interval for 3 units on both axes. Enter the B term paired with A = 12 and explain the relationship in every pair.

Transfer the idea

transfer. A starts at 0 and adds 2. B starts at 3 and adds 4. A learner says B is always twice A because 4 is twice 2. Use the starting pair and one later pair to test the claim.