Support
Answer and teaching response
2 · The point is (2, 4). The horizontal coordinate is 0 + 2 = 2; the vertical coordinate is 4. Writing (4, 2) reverses the two directions.
Follow a grid route and repair a reversed coordinate
Build a first-quadrant route record with ordered pairs, repair a coordinate reversal and account for each horizontal or vertical leg.
Get student and teacher materialsPrepare the lesson
Learning goal: Locate points by moving along the horizontal axis first and the vertical direction second, then use labeled coordinates to explain a route and its distance.
You will need: One sheet of squared paper, a pencil and a ruler per learner. Prepare perpendicular axes from 0 to 10 with the origin labeled (0, 0), x increasing right and y increasing up. One grid interval is one distance unit on each axis. All journeys are invented; movement follows grid lines.
Start with a diagnostic
From the origin, move 3 units right and 1 unit up. Write the ordered pair, then enter its first coordinate.
Check the reasoning and choose a starting point
3 · 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.
Start with support: 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 for core: 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.
Work through an example
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.
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).
- 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.
- 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.
- 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.
- 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.
Run the 40-minute lesson
- 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.
- 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.
- 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.
- 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.
A useful error to discuss
A learner might say: The coordinates (2, 5) and (5, 2) describe the same point because they contain the same two numbers.
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.
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
8 · 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.
Challenge
Answer and teaching response
2 · 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.
Independent check
Answer and teaching response
13 · 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 the idea
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.
Discuss a possible response
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.
Choose the next teaching step
- 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.
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.
Materials for your group
Student pages and teacher answers are separate. Keep the HTML files or print them in your browser.
Revisit a prerequisite
Number Line Launch practises locating positions by counting equal intervals on one axis. Use it first if intervals and positions are still confused. This investigation introduces two perpendicular axes and ordered pairs; the linked fraction lesson does not teach coordinates.