Support
Answer and teaching response
16 · Add 2 to A and 4 to B, giving (8, 16). Applying both rules once preserves the common stage.
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.
Get student and teacher materialsPrepare the lesson
Learning goal: Apply both given rules consistently, compare terms at the same stage and explain when the second term is twice the first using corresponding ordered pairs.
You will need: Squared paper and pencil. For the worked graph, label Pattern A on the horizontal axis from 0 to 8 and Pattern B on the vertical axis from 0 to 16. Use one grid interval for 2 units on both axes. All pattern rules and starting values are supplied; no rule guessing is required.
Start with a 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.
Check the reasoning and choose a starting point
4 · 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.
Start with support: 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 for core: 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.
Work through an example
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?
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.
- 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.
- 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.
- 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.
- 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.
Run the 40-minute lesson
- 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.
- 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.
- 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.
- 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.
A useful error to discuss
A learner might say: Because Pattern B adds 4 and Pattern A adds 2, every B term must be exactly 2 more than its corresponding A term.
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.
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
20 · 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.
Challenge
Answer and teaching response
6 · 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.
Independent check
Answer and teaching response
36 · 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 the idea
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.
Discuss a possible response
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.
Choose the next teaching step
- 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.
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.
Materials for your group
Student pages and teacher answers are separate. Keep the HTML files or print them in your browser.
Revisit a prerequisite
Repeated Fraction Groups develops multiplication as equal repeated groups, using fractions. Use that model if a learner needs help explaining 'twice' or 'three times.' It is a supporting multiplication connection; this investigation teaches corresponding whole-number patterns and coordinate graphs.