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# Plotting a rule

Linear relations · Rules, tables and graphs

<!-- Speaker note: Students plot ordered pairs from a rule's table, across then up, see that rules adding the same amount each step give straight lines, and describe the line (AC9M7A05, node 3). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 15 min · Practise and Make 10 min). -->

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## Today

**We are learning to** plot each ordered pair from a rule's table as a point, across first and then up. For a rule like y = 2x + 1, the points lie on a straight line.

- I can plot ordered pairs from a table, across first and then up.
- I can describe a straight-line graph by which way it goes and how much for each step.
- I can tell from a table whether its points will lie on a straight line.

<!-- Speaker note: Before class: Do the Run and Modify boards yourself first. Points start in a row at the bottom of the board and snap to the grid; students drag each to its pair. Students should be able to plot points in all four quadrants (Year 6). The new move is reading a table as points and seeing the line: a rule that adds the same amount for each step right always gives a straight line. The last Modify task (y = x × x) is a non-example: the steps change, so the points curve. Describing the line ("up 2 for each step right, crossing the y-axis at 1") is exactly what m and c will name in the next unit. Teacher: Modify: after the curve, ask: what was different about the steps in the table? -->

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## Why it matters

> A table of numbers hides its pattern; a graph shows it at a glance. Plotting the pairs turns a rule into a picture you can read.

<!-- Speaker note: In the last lesson you made tables of ordered pairs from rules. Now you **plot** them. To plot the point $(x, y)$, start at $(0, 0)$, go **across** $x$, then **up** $y$. Then look at the shape the points make. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**The pairs of $y = 2x + 1$ are $(0, 1)$, $(1, 3)$, $(2, 5)$ and $(3, 7)$. If you plot them, what shape will they make?**

- A straight line going up
- Scattered points, no pattern
- A curve that gets steeper
- A flat line

<!-- Speaker note: What happens: A straight line going up. Each step of 1 across adds the same amount (2) to y, so every point is the same step up from the one before. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Ask a student who chose 'a curve' what made them think so. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Fplotting-a-rule%2Fstep-2&code=widget%3A+graph%0Ax%3A+-1%2C+6%0Ay%3A+-2%2C+9%0Apoint+A%3A+3%2C+-1%2C+drag%0Apoint+B%3A+4%2C+-1%2C+drag%0Apoint+C%3A+5%2C+-1%2C+drag%0Apoint+D%3A+6%2C+-1%2C+drag%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/plotting-a-rule/?full=step-2#step-2)

<!-- Speaker note: | Point | A | B | C | D | |---|---|---|---|---| | $(x, y)$ | $(0, 1)$ | $(1, 3)$ | $(2, 5)$ | $(3, 7)$ | Drag each point from the bottom of the board to its pair. For each one: across first, then up. What shape do they make? -->

<!-- Speaker note: Teacher: as students place each point, ask them to say it: 'across 1, up 3'. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* A straight line going up. Each step of 1 across adds the same amount (2) to y, so every point is the same step up from the one before.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Plotting a rule

Here is how to plot a rule and describe its graph.

$$y = 2x + 1: \; (0, 1), (1, 3), (2, 5), (3, 7)$$

1) $(0, 1): \text{ across } 0, \text{ up } 1$ — Plot each pair from $(0, 0)$: across first, then up.
2) $(1, 3), (2, 5), (3, 7)$ — Plot the rest the same way.
3) $\text{up } 2 \text{ for each step right}$ — Each step of 1 across, $y$ goes up 2: the same step every time.
4) $\text{a straight line}$ — So the points lie on a straight line. Join them with a ruler.
5) $\text{going up, starting at } 1$ — Describe it: it goes up 2 for each step right and crosses the $y$-axis at 1.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write y = 2x + 1: (0, 1), (1, 3), (2, 5), (3, 7) and say "Start with the table of ordered pairs." Line 1: write (0, 1): across 0, up 1 and say "Plot each pair from (0, 0): across first, then up." Line 2: write (1, 3), (2, 5), (3, 7) and say "Plot the rest the same way." Line 3: write up 2 for each step right and say "Each step of 1 across, y goes up 2: the same step every time." Line 4: write a straight line and say "So the points lie on a straight line. Join them with a ruler." Line 5: write going up, starting at 1 and say "Describe it: it goes up 2 for each step right and crosses the y-axis at 1." -->

<!-- Speaker note: Teacher: model the worked example live on a grid: plot each point saying 'across, then up', then join with a ruler. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Plotting a rule

**Which point is $(3, 2)$?**

- The point 3 across and 2 up
- The point 2 across and 3 up
- The point 3 across and 3 up

* Answer: The point 3 across and 2 up

<!-- Speaker note: The first number is how far across (x); the second is how far up (y). (2, 3) is a different point. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Sam plots the point $(2, 5)$. Click the line where his method goes wrong.**

```
 1  Plot the point (2, 5).
 2  Start at (0, 0).
 3  Go up 2.
 4  Go across 5.
 5  Mark the point there.
```

* Answer: line 3

<!-- Speaker note: Line 3: the first number is x, how far across. Sam should go across 2, then up 5. Going up 2 and across 5 plots (5, 2). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Describing a graph

**The table for $y = 10 - x$ is $(0, 10)$, $(1, 9)$, $(2, 8)$, $(3, 7)$. Which describes its graph?**

- A curve going down
- A straight line going up 1 for each step right
- A straight line going down 1 for each step right

* Answer: A straight line going down 1 for each step right

<!-- Speaker note: y goes down by 1 each time x goes up by 1: the same step each time, so a straight line, going down. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## New tables



- **Subtraction:** Each task loads a new table. Drag $A$ to $D$ to the pairs, across then up.
- **Negative inputs:** 
- **Going down:** 
- **Not a straight line:** Plot it, then look at the steps between the points.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/plotting-a-rule/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a subtraction rule, negative inputs, a line going down, and a rule whose points do not make a straight line. Hints, in order: A negative x means go across to the left first. For students who need more support: Give the table with the pairs already written, and a printed grid. Plot two points together before students plot the rest. For early finishers: Plot y = 2x + 1 and y = x + 3 on the same grid. Where do they meet? What does that point mean for both rules? When students get stuck: A student plots (y, x) by going up first (Sam's mistake). Ask: Which number is x? Which way does the x-axis go? Fix: Say 'along the corridor, then up the stairs': across first, then up. / A student loses count with negative coordinates. Ask: Is x positive or negative? Which way do you go? Fix: Negative x goes left of the y-axis; negative y goes below the x-axis. Count the squares from (0, 0). / A student thinks every set of points makes a straight line. Ask: How much does y go up between each pair of points? Fix: Write the steps under the table. Same step every time: straight. Steps that change: a curve. -->

<!-- Speaker note: Teacher: before the negative inputs, ask: which way is across for a negative x? -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Coordinates

**A point is 4 across and 1 up from $(0, 0)$. What are its coordinates?**

- $(1, 4)$
- $(4, 1)$
- $(4, 4)$

* Answer: $(4, 1)$

<!-- Speaker note: Across is x, up is y: (4, 1). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Straight or not?

**Which table's points lie on a straight line?**

- (0, 2), (1, 3), (2, 6), (3, 11)
- (0, 2), (1, 5), (2, 8), (3, 11)
- (0, 1), (1, 2), (2, 4), (3, 8)

* Answer: (0, 2), (1, 5), (2, 8), (3, 11)

<!-- Speaker note: y goes up 3 every time in the first table. In the others the steps change (1, 3, 5 and 1, 2, 4), so the points curve. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## The next point

**The points of a rule lie on a straight line through $(0, 3)$, $(1, 5)$ and $(2, 7)$. Where is the next point, at $x = 3$? Give its $y$.**

* Answer: 9

<!-- Speaker note: Up 2 for each step right: 7 + 2 = 9. The point is (3, 9). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review

**Review: for $y = 4x - 1$, what is the output when the input is 2?**

* Answer: 7

<!-- Speaker note: 4 × 2 − 1 = 8 − 1 = 7. The pair is (2, 7). -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: your own straight line

Choose where your line starts. Then place the other points so the line goes up 3 for each step to the right.

Stretch: what rule $y = \ldots$ gives your points? Check it with all three.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/plotting-a-rule/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Placing points so they make a chosen straight line reverses the lesson: from the shape back to the steps between pairs. Hints, in order: Put A anywhere. Then B is 1 across and 3 up from A. / C is 1 across and 3 up from B. -->

<!-- Speaker note: Teacher: compare two students' lines: both go up 3 for each step right. Where do they start? -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Plot the pairs of y = 3x − 1 for x = 0 to 3. Describe the graph.**

* (0, −1), (1, 2), (2, 5), (3, 8): a straight line going up 3 for each step right, crossing the y-axis at −1.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: graphs of real data, where the points come from measurements, not a rule. Later in the course, the line's steepness and starting point get names: m and c in y = mx + c. Where it leads in class: science graphs from experiments, and line graphs in any subject. -->