What m does to a line

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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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.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min

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

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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

Answers to the checks

1. a bigger m: 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 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.
2. change m: Set $m$ to 3. Was your prediction right?
   
3. Up for each step right: In $y = 5x + 1$, how far does the line go up for each step of 1 to the right?
   5
   5. Each step right adds one more lot of 5 to 5x, so y goes up by 5 each time.
4. Find the mistake: Lee decides which line is steeper. Click the line where his reasoning goes wrong.
   line 4
   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.
5. Further along: On the line $y = 3x + 1$, how far does $y$ go up from $x = 5$ to $x = 6$?
   3
   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.
6. Steeper: Set $m$ so the line passes through $T(2, 9)$.
   
   m = 4: from (0, 1), up 4 and up 4 again reaches (2, 9).
7. Flat: Set $m$ so the line passes through $T(3, 1)$.
   
   m = 0: y = 0x + 1 = 1 for every x, so the line is flat.
8. Down: Set $m$ so the line passes through $T(2, -3)$.
   
   m = −2: y goes down 2 for each step right. A negative m makes the line go down.
9. Less than 1: Set $m$ so the line passes through $T(4, 3)$.
   
   m = 0.5: up half a square for each step right, so up 2 over 4 steps.
10. From the rule: What is $m$ in $y = 4x - 2$?
    4
    m = 4: the line goes up 4 for each step right.
11. Which way?: Which line goes down from left to right?
    $y = -2x + 5$
    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.
12. From a point: The line $y = mx + 1$ passes through $(3, 7)$. What is $m$?
    2
    From (0, 1) to (3, 7) is up 6 over 3 steps right, so up 2 for each step: m = 2.
13. Review: Review: work out $y$ when $x = 4$ in $y = 3x - 5$.
    7
    3 × 4 − 5 = 12 − 5 = 7.
14. your own line: Choose any $m$, then drag $P$ onto your line, to the right of the $y$-axis.
    
    P is on your line: its coordinates make y = mx + 1 true.

Solutions

One possible solution for each task. Other correct programs are fine.

change m on purpose › Steeper

Open task: any program that does what the step asks.

change m on purpose › Flat

Open task: any program that does what the step asks.

change m on purpose › Down

Open task: any program that does what the step asks.

change m on purpose › Less than 1

Open task: any program that does what the step asks.

your own line

Open task: any program that does what the step asks.

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.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "What m does to a line".

The idea: Effect of gradient parameter on linear graphs: I can Students investigate the effect of varying the gradient m in y = mx + b on the graph and generalise the relationship between m and steepness/direction.

What understanding looks like:
- Correctly describes the effect of varying m on the graph.
- States a general rule for positive, negative and zero gradient.
- Tests the conjecture with at least one new value.

Mistakes students often make here:
- A student says a bigger m moves the line up.
- A student reads c as the steepness (Lee's mistake).
- A student cannot find m for the down task.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "What m does to a line".

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).

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.

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.