Follow a grid route and repair a reversed coordinate

diagnostic. From the origin, move 3 units right and 1 unit up. Write the ordered pair, then enter its first coordinate.

Answer: 3

Teacher response: The point is (3, 1): the first coordinate gives the horizontal movement from the origin, and the second gives the vertical movement. The requested first coordinate is 3.

model. A grid journey visits A (1, 2), then B, then C (5, 6), then D (2, 6). Its directions are A to B: right 4; B to C: up 4; C to D: left 3. A helper records B as (2, 5). Correct B and find the total distance traveled along these three legs.

1. Label the origin and both axes before plotting. Locate A by going right 1 from the origin and up 2. An ordered pair describes a position relative to the origin; it is not an instruction to move that amount from the previous stop.
2. Follow the first leg from A: move right 4, so x becomes 1 + 4 = 5 and y remains 2. Write B (5, 2). Plot the helper's (2, 5) separately and ask which point actually lies four units to the right of A.
3. Continue up 4 to C (5, 6), keeping x fixed, then left 3 to D (2, 6), keeping y fixed. Have partners read each pair aloud and check which coordinate changes on each leg.
4. Count the intervals traveled, not the points visited: A to B is 5 - 1 = 4, B to C is 6 - 2 = 4, and C to D is 5 - 2 = 3. Add 4 + 4 + 3 = 11 units. Check the final position separately; its coordinates (2, 6) are not the total journey length.

Worked example: B is (5, 2), and the route travels 11 units: 4 + 4 + 3. Moving right from A changes x from 1 to 5 while y stays 2. The helper reversed the coordinates. The complete record is A (1, 2), B (5, 2), C (5, 6), D (2, 6).

p1. Support: from (0, 0), move 2 units right and 4 units up. Enter the first coordinate, then write the complete ordered pair and label x and y on your grid.

Answer: 2

Teacher response: The point is (2, 4). The horizontal coordinate is 0 + 2 = 2; the vertical coordinate is 4. Writing (4, 2) reverses the two directions.

p2. Core: a route goes from (2, 1) to (7, 1), then to (7, 4), using straight horizontal and vertical legs. Enter the total distance in units and label each leg on paper.

Answer: 8

Teacher response: The horizontal leg is 7 - 2 = 5 units and the vertical leg is 4 - 1 = 3 units. The total is 8 units. The shared coordinate identifies the direction of each leg.

p3. Challenge: start at (1, 5), move right 6 and then down 3. A helper writes the endpoint as (2, 7). Enter the correct second coordinate and write the corrected endpoint with an explanation.

Answer: 2

Teacher response: The endpoint is (7, 2): x becomes 1 + 6 = 7 and y becomes 5 - 3 = 2. The helper has swapped the endpoint's coordinates. Moving down reduces y while leaving x unchanged.

e1. Independent check: start at P (2, 3). Travel right 6 to Q, up 4 to R and left 3 to S. Enter the total distance in units. On paper, write Q, R and S as ordered pairs, plot them on labeled one-unit axes and label all three leg lengths.

Answer: 13

Look for:

• Independent rubric, 1 point: writes Q (8, 3), R (8, 7) and S (5, 7), with x before y in every pair.
• Independent rubric, 1 point: plots the three points consistently on axes labeled x and y, with origin (0, 0) and one-unit intervals.
• Independent rubric, 1 point: labels legs 6, 4 and 3 and gives 13 distance units. The number checker assesses the entered total; inspect the written coordinate and grid evidence separately.

Teacher response: Q is (8, 3), R is (8, 7), and S is (5, 7). The three leg lengths are 6, 4 and 3 units, giving 13 units. The coordinates describe positions; their sum is not the journey length.

transfer. Must every horizontal-and-vertical route from the worked A (1, 2) to D (2, 6) travel 11 units? Give another route and its distance using the same one-unit grid.

Answer: No. Travel right 1 from (1, 2) to (2, 2), then up 4 to (2, 6): the distance is 5 units. Both routes finish at D, but the worked journey goes farther right and then returns left. An endpoint alone does not record all movement along a route.

teach-1. 5 minutes: Give the diagnostic individually and collect both the pair and first coordinate. Use the support or ready route. Confirm that every learner has labeled axes and equal one-unit intervals.

teach-2. 10 minutes: Model A to B and plot the helper's reversed pair. Ask learners to repair B before revealing it. Partners complete C and D, then explain which coordinate stayed fixed on each leg. Record the exact route table.

teach-3. 15 minutes: Learners needing support complete p1 with finger tracing and then p2 with labeled legs. Ready pairs complete p2 and p3, exchanging route records to check positions from the origin. Ask each pair to explain one coordinate choice and one interval count; discuss the reversed-pair error together.

teach-4. 10 minutes: Allow six minutes for e1 without help, then compare the answer and all three rubric criteria. Use the transfer question for the final discussion. Revisit coordinate order or interval counting with learners earning 0–2 points; learners earning 3 explain a second route to the same endpoint.

misconception. The coordinates (2, 5) and (5, 2) describe the same point because they contain the same two numbers.

Teacher response: Mark both points from the same origin. Ask the learner to name the horizontal distance before the vertical distance for each. Then test the route instruction: only (5, 2) is four units right of A (1, 2). Keep the labels x and y visible while the learner repairs the record.

support. Provide axes with the origin and one-unit intervals already marked. Use separate horizontal and vertical finger movements; say the pair before plotting. Cover later route instructions so the learner can record one stop at a time. Recheck (2, 4), then reduce help for p2.

extend. Create two routes from (2, 3) to (5, 7), using only horizontal and vertical legs in the first quadrant. Give their ordered pairs and show one distance of 7 units and one longer distance. Explain where the extra travel occurs.

support-route. If the learner writes (1, 3), names 1 as the first coordinate or counts grid lines instead of intervals, trace the x-axis from the origin with a finger. Say 'right 3, then up 1,' mark the point and repeat with right 2, up 4. Recheck for (2, 4) before the main route.

ready-route. If the learner writes (3, 1) and explains the coordinate order, ask them to locate (1, 3) as a second point and explain why the two are different. Continue to repairing the route record.