Connect two generated patterns with ordered pairs

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.

Answer: 4

Teacher response: The corresponding pair is (2, 4). Both terms come after exactly one addition. Pairing the A term 2 with the B starting term 0 would mix stages.

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.

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.

Answer: 16

Teacher response: Add 2 to A and 4 to B, giving (8, 16). Applying both rules once preserves the common stage.

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.

Answer: 20

Teacher response: The pairs are (1, 2), (4, 8), (7, 14) and (10, 20). After three additions B is 2 + 3 × 6 = 20, twice the A value 10. Both the starting value and each increment in B are twice those in A.

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.

Answer: 6

Teacher response: After three additions A is 11 and B is 17, so the difference is 6. B is not twice A: twice 11 would be 22. The equal nonzero starting values prevent the doubling claim even though B's increment is twice A's.

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.

Answer: 36

Look for:

• Independent rubric, 1 point: generates exactly (0, 0), (3, 9), (6, 18), (9, 27) and (12, 36), matching stages and keeping the order (A, B).
• Independent rubric, 1 point: plots the five pairs on axes labeled Pattern A and Pattern B, with origin and a consistent three-unit interval on both axes.
• Independent rubric, 1 point: enters 36 and explains tripling using the zero starts and the relationship 9 = 3 × 3 between increments. A correct entered number alone does not establish the table, graph or explanation.

Teacher response: The pairs are (0, 0), (3, 9), (6, 18), (9, 27) and (12, 36). A reaches 12 after four additions, so B is 4 × 9 = 36. B is three times A because both start at zero and each added 9 is three times the added 3.

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.

Answer: The starting pair is (0, 3), and after one addition the pair is (2, 7). Three is not twice zero, and seven is not twice two. In this example B is always 3 more than twice A: the extra starting 3 remains while the increments have the doubling relationship.

teach-1. 5 minutes: Give the diagnostic and ask how the learner matched stages. Follow the support or ready route. Keep the two supplied rules visible throughout the lesson.

teach-2. 10 minutes: Generate the worked table together, checking one row at a time. Learners plot all five pairs while the teacher checks axis labels and the two-unit scale. Compare table and graph values at A = 6 and A = 8.

teach-3. 15 minutes: Support learners complete p1 and then build p2 as an aligned table. Ready pairs complete p2 and p3, testing whether doubling still holds when starting values change. Ask pairs to support one relationship claim with at least two corresponding pairs and the supplied rules.

teach-4. 10 minutes: Give e1 independently for six minutes, then inspect the five pairs, graph and explanation with the three-point rubric. Discuss the transfer counterexample. Revisit alignment or starting values with learners earning 0–2 points; learners earning 3 explain why the independent rules preserve tripling.

misconception. Because Pattern B adds 4 and Pattern A adds 2, every B term must be exactly 2 more than its corresponding A term.

Teacher response: Compare two rows: at (2, 4) the difference is 2, but at (4, 8) it is 4. Ask which comparison stays true: adding 2 or multiplying by 2. On the graph, pair each horizontal position with its vertical value rather than comparing two neighboring points as if they were corresponding terms.

support. Provide a table with one column for each pattern and a row labeled 'start.' Keep a finger on one row while comparing terms. Add arrows labeled +3 and +6 for p2, then remove the arrows when the learner can generate and pair the next terms independently.

extend. Write two new supplied rules whose corresponding second terms are always four times the first. State both starts and increments, generate four pairs and explain why the relationship persists. Then change just one starting value and find a pair that breaks the claim.

support-route. If the learner pairs different stages or treats 'add 4' as 'multiply by 4,' make a two-row table with columns 'start' and 'one addition.' Add each stated amount once and align the results. Recheck the next pair, (4, 8), before plotting.

ready-route. If the learner gives (2, 4) and identifies a common stage, ask for the next two pairs, (4, 8) and (6, 12). Ask what 'twice' means in those pairs, then continue to the worked table and graph.