What are the coordinates of the point?

What this sequence teaches

How to read a point’s coordinates off a grid. From the point, go up or down to the x-axis and read the number there; then go across to the y-axis and read that number. Write them in brackets, x-axis number first: (1, 4). The numbers are read, not counted: every value in the teaching and testing sequences sits on a labelled line. Getting the order right is the whole atom; everything else is reading a label.

The teaching sequence

Example 1 — (1, 4). A general point: not on an axis, and with different numbers, so writing them the wrong way round gives a visibly different answer, (4, 1).

Example 2 — the point moves one square across, to (2, 4). Only the first number changes. The 4 doesn’t move in the answer; the 1 becomes a 2.

Example 3 — the point moves three squares down, to (2, 1). Only the second number changes. The bigger move is deliberate: which way the point moves decides which number changes, not how far it goes.

Three examples, no sameness example. Points on an axis, the swapped pair and every reset are left to the testing sequence, where students produce the answer rather than watch it.

Running the sequence

• Keep it fast and uninterrupted — well under a minute.
• Reveal follows the rule in order. The empty brackets and comma appear and the point pulses. A dashed line drops to the x-axis; the number there swells and flies into the first slot. A pause (“what goes in the second slot?”), then a line runs across to the y-axis and that number flies into the second slot. The lines stay, faint, until you move on.
• Moving to Examples 2 and 3 is deliberately slow. First the answer dims and is labelled “Previous answer”. Then the point slides along its grid line to its new place and pulses once. Ask: “How little can you change your answer?” Reveal then plays the same two stages: the number that changed crossfades as the new one flies in, and the one that didn’t just brightens where it is.
• Replay plays everything since the last example: before Reveal, the move; after Reveal, the move, a pause, and then the reveal. On Example 1 it replays the reveal.

The rule and the script

The rule, exactly as Show rule displays it: For the following examples: from the point, go up or down to the x-axis and read the number. Then go across to the y-axis and read the number. Write them in brackets, x-axis number first.

• Example 1 — “Down to the x-axis: 1. Across to the y-axis: 4. The x-axis number comes first: (1, 4).”
• Example 2 — “The point moved across, so only the x-axis number changed: now it’s 2. Across to the y-axis is still 4.”
• Example 3 — “The point moved down, so only the y-axis number changed: now it’s 1. Down to the x-axis is still 2.”

When Why? is open, the page steps through the answer: the line to the x-axis, the number it lands on and the first slot are underlined and pulse; then the same for the y-axis. Colour is never the only cue. Whether students see the rule and the reasons is your call.

Why these conventions?

Why “up or down”, when every point here is above the axis? So the rule is still true when the expansion sequence goes below the x-axis.

Why read from the point, not “along the corridor and up the stairs”? The corridor starts at the origin, which suits plotting. For reading, both numbers sit where a line from the point meets an axis, so that’s where the reveal goes.

Why a single 0 at the origin? It’s the usual convention, and it belongs to both axes: a point on either axis reads its 0 from the same place.

Why no letter on the point? Exam questions usually name the point (“the coordinates of A”). Here the letter would add nothing to the answer, so the point is a plain dot.

Where this atom sits

Before it: reading a number off a labelled number line.

After it: plotting a point from its coordinates (the same rule run backwards); then all four quadrants as an atom of its own; later, reading coordinates as a step inside midpoints, gradients and graphs.

About the testing sequence

Six items in a designed order. The first is a reset: a new point, unlike any teaching example, so students use the rule rather than adapt Example 3’s answer. After that, some items move the point one way from the item before (the previous answer stays up, dimmed), and two more resets start again from nothing. Show the item, students write on mini-whiteboards, then reveal.

What each item is diagnosing

1. (3, 5). Reset. (5, 3) on a board means the order is backwards.
2. (3, 2): the point moves down. Only the second number changes. (2, 5), or a board that changes the 3, means the wrong number was changed.
3. (2, 3): the same two numbers, swapped. The point jumps across the diagonal and, on Reveal, the numbers arc past each other. A student who writes the bigger number first gets exactly one of Items 2 and 3 right. Ask: “Same numbers as last time. Why is the answer different?”
4. (5, 1). Reset. Nothing in it matches the last answer, so a student editing their last board has nothing to edit.
5. (5, 0): the point drops onto the x-axis. “5” or “(5)” means a point on an axis was taken to have one number. The reveal walks along the axis to the 0 at the origin, so the 0 is read like any other number.
6. (0, 4): a point on the y-axis. Reset. (4, 0) means either the order or “the 0 goes second”, a habit Item 5 can plant. The 0 from Item 5’s answer is here in the other slot, so a student editing their last board gets it wrong.

Common confusions

The order is the big one, and on a first-quadrant grid it’s the only one most students show. A student who gets Items 2 and 3 both wrong is reversing consistently; one who gets exactly one of them wrong is using a different wrong rule (bigger number first, say), which is worth asking about. Points on an axis are the second: students expect a point to be “somewhere in the grid”.

On these grids, counting squares from the axes gives the same answers as reading the labels, so nothing here can tell them apart. The expansion sequence can.

Discussion prompts

• “Say the rule, then use it on this one.”
• “Which number did you read first? Which axis was it on?”
• “The point moved. Which number in your answer has to change?”

About the expansion sequence

Eight items that show how far the rule reaches. Each adds one thing; run them in order. The first item is one step from the last test item: the point moves across off the y-axis, and half-square lines appear. It isn’t shown with a previous answer, because nothing carries across cards.

What each step adds

1. (1.5, 4): halves. The point moves off the y-axis, and half-square lines appear. The line down lands between two labels; the reveal shows 1.5 appearing on the axis where it lands. Counting small squares gives 3.
2. (1.5, 3.5): halves on the y-axis too. The point moves down half a square; only the second number changes.
3. (8, 4): the scale goes up in twos. Every value is on a labelled line. Counting squares gives (4, 2), and this is the first item where that fails.
4. (8, 5): a value between two labels. The line across lands halfway between 4 and 6. Reading the nearest label gives 4 or 6; counting squares gives 2.5.
5. (3, 4): a grid that goes below the x-axis. Reset. Two quadrants, but the point is still above the axis, so the only new thing is the grid itself.
6. (3, −4): the point moves down, past the x-axis. Only the second number changes, from 4 to −4. This is where “up or down” in the rule earns its place: the line goes up to the axis. (3, 4) means the minus sign was dropped.
7. (−2, 5): all four quadrants, top left. Reset. The first number is negative. A new grid and a new point at once is the biggest step on the card, and deliberately so: by now it recombines things students have met, apart from the negative first number.
8. (−2, −5): bottom left. The point moves straight down past the x-axis; only the second number changes, from 5 to −5. The climax: both numbers negative, and the order still decides everything.

Where classes stall

• The value between labels. Some write the nearest label. Ask: “What number is halfway between 4 and 6?”
• The scale in twos. Students who have been counting squares all along get (4, 2) and are surprised. That surprise is the point of the item.
• The negatives. Some drop the minus sign, reading distance instead of the label. Point at the label the line lands on.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “Where did the line land? What number is written there?”
• “Why would counting squares go wrong here?”