Rules, positions and coordinate evidence

pre1. Explicit-rule check: term 1 is 2, and the rule is add 5 each time. Enter term 6. On paper, label the first six term positions, write the values and show how many additions connect term 1 to term 6.

Answer: 27

Look for:

• Labels 2 as term 1, uses five +5 changes and ties the calculation to the supplied rule rather than treating six positions as six additions.

Teacher response: The values are 2, 7, 12, 17, 22 and 27. Five additions follow the starting term, so 2 + 5 × 5 = 27. The starting value is already term 1.

pre2. Explicit-rule explanation: term 1 is 9, and the rule is add 4 each time. Enter term 8. Explain whether every term under this stated rule is odd or even, using the starting value and increment rather than only the first few values.

Answer: 37

Look for:

• Uses seven changes and justifies oddness with an odd start and repeated even increments. Noticing that a few listed values are odd is incomplete evidence for the claim about every generated term.

Teacher response: Seven additions give 9 + 7 × 4 = 37. Every term is odd: the starting 9 has one unpaired unit, and every added 4 contributes two complete pairs without removing that unpaired unit.

pre3. Coordinate-position check: use axes with origin (0, 0), x increasing right, y increasing up and one-unit intervals. From the origin, move right 4 and up 7. Enter the first coordinate, then write and plot the complete ordered pair and explain its order.

Answer: 4

Look for:

• Writes and plots (4, 7) on labeled one-unit axes and names x before y. Check the plotted point as well as the entered first coordinate.

Teacher response: The point is (4, 7). The first coordinate records the horizontal position 4 units right of the origin; the second records the vertical position 7 units up.

pre4. Coordinate repair: on one-unit axes with x right and y up, start at (1, 8), move right 5 and then down 3. A helper records the endpoint as (5, 6). Enter the correct second coordinate, write the corrected pair and explain which coordinate each movement changed.

Answer: 5

Look for:

• Repairs the pair to (6, 5), changes x only for the horizontal leg and y only for the vertical leg, and explains that a point's coordinates are measured from the origin.

Teacher response: The endpoint is (6, 5). Moving right gives x = 1 + 5 = 6; moving down gives y = 8 - 3 = 5. The helper reversed the endpoint's two coordinates.

pre5. Paired-pattern check: A starts at 0 and adds 2 each time; B starts at 0 and adds 6 each time. Generate the starting pair and the next three pairs in the order (A, B). Plot A horizontally and B vertically with one grid interval for 2 units on both axes. Enter the B term after three additions and explain how B compares with A at corresponding stages.

Answer: 18

Look for:

• Aligns stages, writes the four pairs in (A, B) order, labels the graph and its scale, and explains tripling using both starts and increments.

Teacher response: The pairs are (0, 0), (2, 6), (4, 12) and (6, 18). After three additions B is 3 × 6 = 18. B is three times A because both start at zero and each added 6 is three times each added 2.

pre6. Paired-pattern explanation: A starts at 3 and adds 2 each time; B starts at 3 and adds 4 each time. Write the starting pair and the next two pairs. After two additions, how much greater is B than A? Enter the difference and test the claim that B is twice A at that stage.

Answer: 4

Look for:

• Compares the same-stage terms 7 and 11, gives the difference separately from a multiplicative claim and uses the nonzero starts or the pair itself to reject doubling.

Teacher response: The pairs are (3, 3), (5, 7) and (7, 11). The difference after two additions is 11 - 7 = 4. B is not twice A: twice 7 is 14. Doubling the increments does not double these equal nonzero starting values.

post1. Explicit-rule check: term 1 is 1, and the rule is add 3 each time. Enter term 7. On paper, label the first seven term positions, write the values and show how many additions connect term 1 to term 7.

Answer: 19

Look for:

• Labels 1 as term 1 and records six +3 changes. The written calculation must represent six increments, not seven.

Teacher response: The values are 1, 4, 7, 10, 13, 16 and 19. Six additions follow the starting term, so 1 + 6 × 3 = 19.

post2. Explicit-rule explanation: term 1 is 8, and the rule is add 6 each time. Enter term 9. Explain whether every term under this stated rule is odd or even, using the starting value and increment rather than only the first few values.

Answer: 56

Look for:

• Uses eight changes and explains why an even start plus repeated even increments preserves evenness. A correct distant term without that explanation supplies only the numerical evidence.

Teacher response: Eight additions give 8 + 8 × 6 = 56. Every term is even: eight can be arranged entirely in pairs, and each added six contributes three more complete pairs.

post3. Coordinate-position check: use axes with origin (0, 0), x increasing right, y increasing up and one-unit intervals. From the origin, move right 6 and up 2. Enter the first coordinate, then write and plot the complete ordered pair and explain its order.

Answer: 6

Look for:

• Writes and plots (6, 2), keeps a consistent one-unit scale and explains x before y rather than giving the two numbers in either order.

Teacher response: The point is (6, 2). The first coordinate is the horizontal position, six units right, and the second is the vertical position, two units up.

post4. Coordinate repair: on one-unit axes with x right and y up, start at (2, 9), move right 6 and then down 5. A helper records the endpoint as (4, 8). Enter the correct second coordinate, write the corrected pair and explain which coordinate each movement changed.

Answer: 4

Look for:

• Repairs the pair to (8, 4), explains the unchanged y during the first leg and unchanged x during the second, and retains the order (x, y).

Teacher response: The endpoint is (8, 4). The horizontal movement gives x = 2 + 6 = 8 and the downward movement gives y = 9 - 5 = 4. The helper's pair reverses those coordinate values.

post5. Paired-pattern check: A starts at 0 and adds 4 each time; B starts at 0 and adds 8 each time. Generate the starting pair and the next four pairs in the order (A, B). Plot A horizontally and B vertically with one grid interval for 4 units on both axes. Enter the B term after four additions and explain how B compares with A at corresponding stages.

Answer: 32

Look for:

• Matches stages in all five pairs, plots A horizontally and B vertically on the stated four-unit scale and explains doubling with both starts and increments.

Teacher response: The pairs are (0, 0), (4, 8), (8, 16), (12, 24) and (16, 32). After four additions B is 4 × 8 = 32. B is twice A because both start at zero and every increment of 8 is twice the increment of 4.

post6. Paired-pattern explanation: A starts at 4 and adds 3 each time; B starts at 4 and adds 9 each time. Write the starting pair and the next two pairs. After two additions, how much greater is B than A? Enter the difference and test the claim that B is three times A at that stage.

Answer: 12

Look for:

• Compares 10 and 22 from the same stage, distinguishes the additive difference from a multiplier and uses the start or explicit counterexample to reject tripling.

Teacher response: The pairs are (4, 4), (7, 13) and (10, 22). The difference after two additions is 22 - 10 = 12. B is not three times A: three times 10 is 30. Tripling an increment does not triple these equal nonzero starting values.

transfer. Pattern A has term 1 equal to 2 and adds 3 each time. Pattern B has term 1 equal to 5 and adds 6 each time. On a graph, A is horizontal and B is vertical. A helper records the term-4 pair as (23, 11) and says B is twice A. Enter the correct B coordinate for term 4. Write the repaired pair, explain how many additions were used and test the helper's relationship claim.

Answer: 23

Teacher response: Term 4 uses three additions: A is 2 + 3 × 3 = 11 and B is 5 + 3 × 6 = 23. The correct pair is (11, 23), with A first and B second. B is not twice A because twice 11 is 22. Here B is one more than twice A: the start 5 is one more than twice 2, and every added 6 is twice each added 3. Inspect the change count, repaired pair and relationship explanation separately from the entered 23.

lesson-1. Use a stated rule to predict and prove a pattern · 40 minutes. Before: Review pre1 and pre2. Separate an arithmetic slip from counting the starting term as an addition. Prepare position cards and one fewer arrow than positions for learners who need that distinction. Use the existing opening diagnostic to check the proposed support route. Focus: Run the existing start-at-3, add-6 investigation. Label the start as term 1, generate the table and explain why term 8 uses seven additions. Use paired counters to justify parity from both the start and increment. Discuss why a short list without a supplied rule does not determine a unique continuation. Keep the existing support/core/challenge phase, assigning work according to the observed difficulty.

Teacher response: Evidence: Collect post1 and post2 during the final ten minutes. Inspect the position-to-change records as well as 19 and 56. For post2, look for an even start plus an even increment preserving evenness, rather than a claim based only on displayed terms. Record the numerical result and explanation separately. If secure: Proceed to the coordinate lesson when the learner can use a supplied rule, count one fewer change than term position and explain a property from the start and increment. Ask them to keep the words 'position' and 'value' visible in their table as they encounter another kind of position next. If not secure: Use the explicit-rule routing recheck after a brief position-card or pairing model. If the fresh response still counts one addition too many, retain the cards and schedule further rule work; coordinate work can still begin with small whole-number movements, while the pattern gap stays recorded for the final lesson.

lesson-2. Follow a grid route and repair a reversed coordinate · 40 minutes. Before: Review pre3 and pre4 for a reversed pair, movement from the wrong starting point or a changing fixed coordinate. Prepare first-quadrant axes with origin (0, 0), x increasing right, y increasing up and one-unit intervals. Say explicitly that a term's position in a list and a point's position on a grid are different uses of the word position. Focus: Run the existing A (1, 2) to B (5, 2) to C (5, 6) to D (2, 6) route. Repair the helper's reversed B, explain which coordinate changes on each leg and distinguish location from distance traveled. Keep the original worked example and differentiated practice; use the exact pair data to check a sketch rather than estimating from its appearance.

Teacher response: Evidence: Collect post3 and post4 in place of the original exit and transfer items. Look for (6, 2) and the repaired endpoint (8, 4), with an explanation that x is horizontal and y is vertical. The requested numerical entries are 6 and 4; neither entry alone establishes correct ordered-pair notation or plotting. If secure: Proceed to paired patterns when the learner plots and reads ordered pairs with a stable axis order and can explain why a horizontal move changes only x and a vertical move changes only y. Introduce Pattern A as the future horizontal coordinate and Pattern B as the future vertical coordinate. If not secure: Use the coordinate-position routing recheck after tracing one horizontal and one vertical movement. Keep x and y labels visible and reduce the number of stops. Before paired-pattern graphing, confirm the learner can repair the new endpoint and explain the coordinate order; meanwhile, they can generate paired terms in an aligned table.

lesson-3. Connect two generated patterns with ordered pairs · 40 minutes. Before: Review pre5 and pre6 alongside the earlier rule and coordinate evidence. Prepare a row for each common stage and clearly separate 'term 1' from 'zero additions': both name the starting stage in this sequence. Use the existing lesson's stated zero starts and +2/+4 rules before changing any starts or increments. Focus: Run the existing two-pattern investigation, generating both supplied rules at the same stage and writing each pair in the order (A, B). Plot A horizontally and B vertically using the stated scale. Explain doubling from the zero starts and the relationship between increments, then use the original changed-start challenge to test a claim that different increments alone determine the relationship.

Teacher response: Evidence: Collect post5 and post6 during the final ten minutes. Inspect the ordered-pair tables, labeled graphs and starting-value arguments in addition to 32 and 12. A graph made from mismatched stages or an unexplained 'three times' claim needs follow-up even when its entered number is correct. If secure: Use the unit transfer after the sequence to combine term-position counting, ordered-pair orientation and a relationship with an offset. Treat the result as evidence for selecting the next task, alongside the written work from each skill; do not convert it into a general mastery claim. If not secure: Choose the paired-patterns routing recheck for stage alignment or an unsupported multiplier claim. Return to the earlier skill route when the difficulty is an off-by-one change count or reversed coordinates. Reassess with a fresh independent response after the targeted model, keeping the original response in the record.

route-1. explicit-rule: If the numerical response uses one extra increment, place the starting value under term 1 and count arrows rather than cards. If the value is correct but the parity claim cites only examples, pair the starting counters and each increment. Give the fresh recheck without copying the earlier table; ask separately how many additions were used. Recheck: Term 1 is 6, and the stated rule is add 5 each time. Enter term 6. Label the changes and explain why the terms alternate even and odd rather than staying even.

Answer: 31

Teacher response: Five additions give 6 + 5 × 5 = 31. Adding the odd increment 5 changes parity at every step; the first six values are 6, 11, 16, 21, 26 and 31.

route-2. coordinate-position: If coordinates are reversed, trace horizontal movement from the origin before vertical movement and keep axis labels visible. If a relative move is treated as a new coordinate, mark the actual starting point and change only the relevant coordinate. After modeling, use the new endpoint below and ask for the complete pair as well as the numerical entry. Recheck: On one-unit axes with x right and y up, start at (4, 7), move left 2 and up 1. A helper writes (8, 2). Enter the correct second coordinate, write the complete endpoint and explain the repair.

Answer: 8

Teacher response: The endpoint is (2, 8): x changes from 4 to 2 and y from 7 to 8. The helper has put the vertical coordinate first.

route-3. paired-patterns: If terms from different stages are paired, align both starts in one table row and apply both rules once per row. If a relationship is inferred from increments alone, compare the starting values as well. Repair the table before plotting; then request a fresh relationship explanation using the recheck's supplied rules. Recheck: A starts at 0 and adds 5 each time; B starts at 0 and adds 15 each time. Enter B after two additions. Write and plot the starting pair and the next two pairs, using A horizontally, B vertically and one grid interval for 5 units on both axes. Explain the relationship between corresponding terms.

Answer: 30

Teacher response: The pairs are (0, 0), (5, 15) and (10, 30). B is 2 × 15 = 30 after two additions. B is three times A because both start at zero and each added 15 is three times the added 5.

ready. Choose the next application when the collected work shows correct use of an explicit rule and change count, stable coordinate order with a labeled plot, and a corresponding-term relationship justified by starts and increments. Use the actual written evidence and, where needed, an independent recheck. This is a classroom routing decision; no overall numerical score or unvalidated cutoff is used to claim mastery.

support. Route the specific missing evidence: position/change confusion to explicit-rule, reversed axes or relative-movement errors to coordinate-position, and mismatched stages or unsupported multiplier claims to paired-patterns. When a number is correct but an explanation or graph is missing, ask for that evidence before selecting a route. Keep numerical responses and explanation/model observations separate and preserve original and recheck work.

extension. Ask learners to supply two explicit starting values and increments that preserve a chosen doubling or tripling relationship, then change one starting value and explain a counterexample. Have them generate labeled pairs, plot them and compare a distant pair using a counted number of changes. Request the rule whenever only a short list is presented.