What m does to a line

In y = mx + c, the number in front of x sets how steep the line is and which way it goes.

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.

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.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

A bigger m

y = x + 1 \quad\longrightarrow\quad y = 3x + 1

Think about two points on each line: where $x = 0$ , and where $x = 1$ . Which one moves?

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 steepIt gets less steepIt gets steeper, and still crosses the

y

-axis at 1It moves 2 to the right

TryTry it on the board and compare the result with your prediction.

Change m

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.

Try it yourselfYour edits are saved in this browser.

Set $m$ to 3. Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Reading m

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

Worked exampleBefore each step, say what you think comes next.

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

y = 3x + 1: from x = 1 to x = 2, y goes from 4 to 7: up 3. — $y = 3x + 1$ : from $x = 1$ to $x = 2$ , $y$ goes from 4 to 7: up 3.

Read Lee's working one line at a time. Which line compares the wrong thing?

Does the line go up by the same amount everywhere, or only near the start?

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

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

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

ModifyChange the problem on purpose and see what happens.

Change m on purpose

Each task puts a point $T$ on the board. Find the $m$ that takes the line through $T(2, 9)$ . Change only $m$ .

Need a hint?

The line starts at (0, 1). How far up is 9 from 1?

Another hint

That rise happens over 2 steps right. How far up for each step?

Now take the line through $T(3, 1)$ . What does the line do?

Need a hint?

T is at the same height as where the line starts. How far up for each step right?

$T(2, -3)$ is below the start of the line. Which way must it go?

Need a hint?

T is below where the line starts. Which way must the line go?

Another hint

From 1 down to −3 is down 4, over 2 steps right.

Take the line through $T(4, 3)$ . The slider now moves in halves.

Need a hint?

From (0, 1) to T is up 2 over 4 steps right. How far up for one step?

Try it yourselfYour edits are saved in this browser.

Set $m$ so the line passes through $T(2, 9)$ .

Use the board. This ticks when you get there.

Set $m$ so the line passes through $T(3, 1)$ .

Use the board. This ticks when you get there.

Set $m$ so the line passes through $T(2, -3)$ .

Use the board. This ticks when you get there.

Set $m$ so the line passes through $T(4, 3)$ .

Use the board. This ticks when you get there.

PractiseA few on your own, to check you have it.

Read the line

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

Which line goes down from left to right?

y = 0x + 4

y = 3x - 2

y = -2x + 5

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

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

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

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 $x = 10$ .

Try it yourselfYour edits are saved in this browser.

Need a hint?

Pick a value of m first, say m = 2.

Another hint

Start at (0, 1) and count: 1 right, 2 up. Drag P to that point, or count further along.

Choose any $m$ , then drag $P$ onto your line, to the right of the $y$ -axis.

Use the board. This ticks when you get there.

For teachers

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.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

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.

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.

Running the lesson

Predict: 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.
Run: 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.
Investigate: 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.
Modify: each task changes one thing. Before the flat and down tasks, ask: what m do you think it will be? Then let them check.
Practise: 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.
Make: collect a few students' m and P. Check one as a class: start at (0, 1), count 1 right, m up.

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.

Stretch and support

Stretch: 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.)

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.

Where it 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.

What each step is for

a bigger m: Committing to how the line will change before touching the slider makes the board a test of their idea about m, not a demonstration.
change m: Moving m while c stays fixed shows the line turning about the point (0, 1): m changes the steepness, not where the line crosses the y-axis.
reading m: m is how far y goes up for each step of 1 to the right. The table and the step both show it, and it is the same all along the line.
change m on purpose: 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.
read the line: Reading m from a rule, from a description and from a point are the same skill; the review question keeps substitution from the Equations unit fresh.
your own line: Choosing m first and then finding a point on that line reverses the lesson: from the number to the picture.

Now you can

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

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson7. What c does to a lineIn y = mx + c, the number on its own moves the line up or down without changing its steepness, so lines with the same m are parallel.

Source: content/maths/linear-relations/gradient/lesson.md