---
marp: true
theme: default
html: true
---

# What m does to a line

Linear relations · Lines and their rules

<!-- Speaker note: Students vary m in y = mx + 1 on a graph board and find that m is how far the line goes up for each step right (AC9M8A04, node 1). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

---

## Today

**We are learning to** in y = mx + c, the number in front of x sets how steep the line is and which way it goes.

- I can describe what happens to the line y = mx + c when m gets bigger, is 0, or is negative.
- I can read m from a rule as how far the line goes up for each step right.
- I can find m for a line through (0, 1) and another point.

<!-- Speaker note: Before class: Do the Run and Modify boards yourself first. Students should already be able to plot a line from a rule or a table, and work with negative numbers. Only m changes in this lesson: c stays at 1, so every line passes through (0, 1). What c does, and parallel lines, is the next lesson. If students ask about c, say "hold that thought": separating the two is the point. The lesson says "up for each step right" rather than "rise over run" or "gradient". The primer has the formal words if you want to introduce them at the end. -->

---

## Why it matters

> Ramps, roofs, ski runs and phone bills all have a steepness: how much more for each step along. In a rule like y = 3x + 1, one number decides it.

<!-- Speaker note: A rule such as $y = 3x + 1$ gives a **straight line**: work out $y$ for each value of $x$, plot the points, and join them. Every rule of the form $y = mx + c$ gives a straight line. $m$ is the number in front of $x$, and $c$ is the number on its own. In this lesson $c$ stays at 1, and you find out what $m$ does. -->

---

<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 line $y = x + 1$ has $m = 1$. If $m$ becomes 3, so the rule is $y = 3x + 1$, what happens to the line?**

- It moves up by 2, and is just as steep
- It gets less steep
- It gets steeper, and still crosses the $y$-axis at 1
- It moves 2 to the right

<!-- Speaker note: What happens: It gets steeper. When x = 0, y is still 1 (3 × 0 + 1 = 1), so the line still crosses the y-axis at 1, but now y goes up 3 for each step right instead of 1. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands for each option. The common wrong answer is 'moves up by 2': students expect a bigger number to lift the line. Do not correct it yet. -->

---

<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%2Fgradient%2Fstep-2&code=widget%3A+graph%0Ax%3A+-1%2C+5%0Ay%3A+-4%2C+16%0Aslider+m%3A+0%2C+4%2C+1%0Aplot%3A+y+%3D+mx+%2B+1%0Atable%3A+0%2C+1%2C+2%2C+3%0Astep%3A+1%0A&fresh=1}}

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

<!-- Speaker note: Change $m$ with the slider, or the − and + buttons. Watch three things: the line, the table, and the orange step, which shows how far the line goes up for one step to the right. -->

<!-- Speaker note: Teacher: ask students to watch the point where the line crosses the y-axis as they move m. It never moves. Then point at the orange step and the table: both show the same number as m. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 25px; }</style>

---

<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?

* It gets steeper. When x = 0, y is still 1 (3 × 0 + 1 = 1), so the line still crosses the y-axis at 1, but now y goes up 3 for each step right instead of 1.

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

---

<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>

## Up for each step right

Here is how to read $m$ from a rule, by working out $y$ one step at a time.

$$y = 3x + 1$$

1) $x = 0: \; y = 3(0) + 1 = 1$ — Work out $y$ where $x$ is 0.
2) $x = 1: \; y = 3(1) + 1 = 4$ — One step right: $x$ goes from 0 to 1.
3) $4 - 1 = 3$ — $y$ went up by 3.
4) $x = 2: \; y = 3(2) + 1 = 7, \; 7 - 4 = 3$ — Another step right: up 3 again.
5) $m = 3$ — Each step right adds one more lot of 3, so $y$ goes up by $m$, the number in front of $x$.

<!-- 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 = 3x + 1 and say "How far does this line go up for each step right? Start with the rule." Line 1: write x = 0: y = 3(0) + 1 = 1 and say "Work out y where x is 0." Line 2: write x = 1: y = 3(1) + 1 = 4 and say "One step right: x goes from 0 to 1." Line 3: write 4 - 1 = 3 and say "y went up by 3." Line 4: write x = 2: y = 3(2) + 1 = 7, 7 - 4 = 3 and say "Another step right: up 3 again." Line 5: write m = 3 and say "Each step right adds one more lot of 3, so y goes up by m, the number in front of x." -->

<!-- Speaker note: Teacher: model the worked example live. Write y = 3x + 1, then work out y at x = 0, 1 and 2 at writing pace, saying 'one step right, how far up?' each time. Reveal the prepared lines afterwards to compare. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 21px; }</style>

---

<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>

## Up for each step right

**In $y = 5x + 1$, how far does the line go up for each step of 1 to the right?**

* Answer: 5

<!-- Speaker note: 5. Each step right adds one more lot of 5 to 5x, so y goes up by 5 each time. -->

---

<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

**Lee decides which line is steeper. Click the line where his reasoning goes wrong.**

```
 1  Which line is steeper: y = 2x + 4 or y = 3x + 1?
 2  y = 2x + 4: when x = 0, y = 4
 3  y = 3x + 1: when x = 0, y = 1
 4  4 is bigger than 1, so y = 2x + 4 is steeper.
```

* Answer: line 4

<!-- Speaker note: Line 4: 4 and 1 are where the lines cross the y-axis, not how steep they are. Steepness is the number in front of x: 3 is more than 2, so y = 3x + 1 is steeper. -->

---

<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>

## Further along

**On the line $y = 3x + 1$, how far does $y$ go up from $x = 5$ to $x = 6$?**

* Answer: 3

<!-- Speaker note: At x = 5, y = 16; at x = 6, y = 19. Up 3, the same as from x = 0 to x = 1: m is the same all along the line. -->

---

<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>

## Change m on purpose



- **Steeper:** Each task puts a point $T$ on the board. Find the $m$ that takes the line through $T(2, 9)$. Change only $m$.
- **Flat:** Now take the line through $T(3, 1)$. What does the line do?
- **Down:** $T(2, -3)$ is below the start of the line. Which way must it go?
- **Less than 1:** Take the line through $T(4, 3)$. The slider now moves in halves.

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

<!-- Speaker note: What this step is for: Changing one thing at a time shows the whole rule: a bigger m is steeper, m = 0 is flat, a negative m goes down, and m need not be a whole number. Hints, in order: The line starts at (0, 1). How far up is 9 from 1? / That rise happens over 2 steps right. How far up for each step? / T is at the same height as where the line starts. How far up for each step right? / T is below where the line starts. Which way must the line go? / From 1 down to −3 is down 4, over 2 steps right. / From (0, 1) to T is up 2 over 4 steps right. How far up for one step? For students who need more support: Use the Run board for the Investigate questions: set m to the number in the rule and read the table. Give squared paper to count steps by hand. For early finishers: Find m for a line through (0, 1) and (6, 4). Then: which is steeper, y = −3x + 1 or y = 2x + 1? (Steepness is the size of m; the sign gives the direction.) When students get stuck: A student says a bigger m moves the line up. Ask: Where does the line cross the y-axis when m is 1? When m is 3? Fix: Work out y at x = 0 for both rules: 1 × 0 + 1 and 3 × 0 + 1 are both 1. The crossing point stays; only the steepness changes. / A student reads c as the steepness (Lee's mistake). Ask: Which number changes how much y goes up each time x goes up by 1? Fix: Make a two-row table for x = 0 and x = 1 and find the difference in y. It is the number in front of x. / A student cannot find m for the down task. Ask: Is T above or below where the line starts? Which way must the line go? Fix: Count from (0, 1) to T: how far down, over how many steps right? Down means a negative m. / A student moves the slider until it works without noticing why. Ask: Before you move it, what value of m do you think you need, and why? Fix: Have them say the count aloud first ('2 steps right, up 8, so 4 each step'), then set the slider. -->

<!-- Speaker note: Teacher: each task changes one thing. Before the flat and down tasks, ask: what m do you think it will be? Then let them check. -->

<style scoped>pre, marp-pre { font-size: 0.5em; line-height: 1.25; } section { font-size: 26px; }</style>

---

<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>

## From the rule

**What is $m$ in $y = 4x - 2$?**

* Answer: 4

<!-- Speaker note: m = 4: the line goes up 4 for each step right. -->

<!-- Speaker note: Teacher: the 'which way' question plants a minus sign in c. Ask a student who chose y = 3x − 2 to work out y at x = 0 and x = 1. -->

---

<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>

## Which way?

**Which line goes down from left to right?**

- $y = 0x + 4$
- $y = 3x - 2$
- $y = -2x + 5$

* Answer: $y = -2x + 5$

<!-- Speaker note: Only a negative m makes a line go down: y = −2x + 5 goes down 2 for each step right. The −2 in y = 3x − 2 is c, so that line still goes up 3 for each step; y = 0x + 4 is flat. -->

---

<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>

## From a point

**The line $y = mx + 1$ passes through $(3, 7)$. What is $m$?**

* Answer: 2

<!-- Speaker note: From (0, 1) to (3, 7) is up 6 over 3 steps right, so up 2 for each step: m = 2. -->

---

<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: work out $y$ when $x = 4$ in $y = 3x - 5$.**

* Answer: 7

<!-- Speaker note: 3 × 4 − 5 = 12 − 5 = 7. -->

---

<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 line

Choose your own steepness with the slider. Then use counting (1 right, $m$ up) to find a point on your line and drag $P$ there. Try a negative $m$ too.

Stretch: without the board, give a point on $y = 3x + 1$ where $x = 10$.

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

<!-- Speaker note: What this step is for: Choosing m first and then finding a point on that line reverses the lesson: from the number to the picture. Hints, in order: Pick a value of m first, say m = 2. / Start at (0, 1) and count: 1 right, 2 up. Drag P to that point, or count further along. -->

<!-- Speaker note: Teacher: collect a few students' m and P. Check one as a class: start at (0, 1), count 1 right, m up. -->

---

<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

**Without plotting, describe the line y = −2x + 1: where does it cross the y-axis, which way does it go, and how steep is it?**

* It crosses the y-axis at 1 and goes down 2 for each step to the right (m = −2).

<!-- 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: what c does (the line moves up and down without turning) and parallel lines (same m). Then solving equations with graphs, where 2x + 3 = 11 becomes the point where y = 2x + 3 meets y = 11, and modelling with two lines. Where it leads in class: any rate. Dollars per hour, kilometres per hour, litres per minute: the steepness of the graph is the rate, and m in the rule is the same number. -->