At a glance. One lesson of about 60 minutes. The four-quadrant grid, plotting and reading points, quadrants, and distances along grid lines using integer subtraction.
Calculator: no calculator. Every number is a small integer on a grid.
What students will learn
Know
- (x, y): x across, y up or down.
- The quadrants are numbered I to IV anticlockwise from the top right.
- A point on an axis is in no quadrant.
Understand
- Coordinates are two number lines working together.
- Distance along a grid line is the distance between two integers.
Do
- Plot and read points in all four quadrants.
- Find horizontal and vertical distances between points.
- Reflect a point in an axis; find areas of rectangles on the grid.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: questions 2 to 8.
- Criterion B. Investigating patterns: what happens to the coordinates when a point is reflected in each axis? Let students generalize from the figure: (x, y) becomes (x, −y) or (−x, y).
- Criterion C. Communicating: precise use of “origin”, “quadrant” and ordered pairs; the exit ticket asks for the error in a plotting instruction.
- Criterion D. Applying mathematics in real-life contexts: latitude and longitude as signed coordinates; a voyage leg as a change in each coordinate.
The one thing to do with the figure
Ask for predictions before every move. “Where will (−3, −5) be? Which quadrant?” Students say it, then set the sliders.
Then fix attention on Q. Ask how far across and how far down P is from Q, and how they could work it out without counting squares. The answer, a subtraction of integers, connects this lesson to Lessons 2 and 3.
A possible order
| 5 min | Warm-up: distance, subtraction, ordering. |
|---|---|
| 10 min | Two number lines make a map; quadrants; the figure with predictions. |
| 10 min | Worked examples: plotting, distance across. |
| 5 min | Faded example: the rectangle, in pairs. |
| 5 min | Question 1, the check. |
| 20 min | Your turn and exit ticket (question 9). A game of grid battleships if time allows. |
The answers
| Warm-up | 8; 6; −1. |
|---|---|
| Worked examples | A is in Quadrant IV; the distance is 8. |
| Faded example | Width 7, height 3, area 21 square units. |
| 1. Naming a quadrant | C. D tests whether students know a point on an axis is in no quadrant. |
| 2. Reading x | −4; P = (−4, −2). |
| 3. Quadrant | D, Quadrant IV. |
| 4. Reflection | 2; the image is (3, 2). |
| 5. Distance across | 10 = 4 − (−6). |
| 6. Moving a point | 4; A moves to (3, 4). |
| 7. Area on the grid | 30 square units: 6 by 5. |
| 8. Completing a square | 0; the fourth corner is (−1, 0). |
| 9. Order of coordinates | B. A or D: unsure which number is across. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| −2 in question 2; A in question 9 | Swaps x and y. | “Along the corridor, then up the stairs.” Say x first, every time. |
| 4 in question 2 | Ignores the direction. | Ask: is the point left or right of the origin? |
| −2 or 3 in question 4 | Unsure what reflection in the x-axis does. | Fold a printed grid along the x-axis. |
| 2 in question 5 | Subtracted digits across the axis. | Lesson 2: is zero between them? |
| 22 or 6 in question 7 | Perimeter, or one side only. | Ask for the width and the height separately, then decide. |
Questions worth asking
- Why are the quadrants numbered anticlockwise? A convention, the same direction angles turn in later years. Conventions are agreements, like sea level as zero in Lesson 1.
- Which points are exactly 3 units from the origin along the axes? (3, 0), (0, 3), (−3, 0), (0, −3). If you may also turn once, there are others, such as (1, 2) and (−2, −1), because 1 + 2 = 3.
- Is (0, 0) in a quadrant? No: it is on both axes.
Differentiation
A way in for anyone stuck
- Start in Quadrant I only, then extend the axes left and down one at a time.
- A printed grid with the axes labelled every 1, and the phrase “across, then up or down” at the top.
- Questions 2 and 3 before anything with distances.
Stretch
- Question 8, then: three corners of a parallelogram are (0, 0), (4, 1) and (1, 3). Find all three possible fourth corners. ((5, 4), (3, −2) and (−3, 2).)
- Describe all the points with x × y < 0. (Quadrants II and IV.)
Students learning in English as an additional language
Many students know “axis” and “origin” from other meanings. Show the plural “axes” explicitly.
| Word | Plain meaning | Example |
|---|---|---|
| axis (plural axes) | one of the two number lines on the grid | the x-axis goes across |
| origin | where the axes cross, (0, 0) | start every point here |
| ordered pair | two numbers in a fixed order, (x, y) | (−4, 3) |
| coordinate | one of the two numbers in an ordered pair | the x-coordinate of (5, −2) is 5 |
| quadrant | one of the four quarters of the grid | (−3, −5) is in Quadrant III |
| reflect | flip over a mirror line | (3, −2) reflects to (3, 2) in the x-axis |
Sentence starters for explaining aloud or in writing:
- To plot …, I start at the origin, go … then …
- The point is in Quadrant … because x is … and y is …
- The distance is … because … − … = …
Two things not to say
“x is the bottom number.” It confuses the axis with a fraction and breaks down for negative values. Say “across first”.
“Count the squares.” Fine for checking, but the lesson's aim is to subtract integers, which works off the grid too.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- PlottingName the quadrant or axis of (−1, −7), (6, 0) and (−5, 2).
Quadrant III; on the x-axis; Quadrant II. - ReadingA point is 2 right of the origin and 6 down. Write its coordinates.
(2, −6).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- LatitudeA ship sails from latitude 14° south (−14) to latitude 23° north (+23). Through how many degrees of latitude does it sail?
37°: 23 − (−14). - Game boardA counter at (−3, 4) moves 6 right and 9 down. Where does it finish?
(3, −5).
3Challenge
Reasoning, generalising and working backwards.
- Reasoning(a, b) is in Quadrant II. Which quadrant is (−a, b) in? And (b, a)?
Quadrant I; Quadrant IV. - From Lesson 5Plot the points (n, (−1)ⁿ) for n = 1 to 4. What do you notice?
(1, −1), (2, 1), (3, −1), (4, 1): they zigzag between y = −1 and y = 1.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard; “Show me the steps” gives four positions of P and the path from P to Q in words.
Squared paper and a pair of dice (one for x, one for y, a coin for each sign) make a quick game for practice.