What c 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 c changes.
• I can read where a line crosses the y-axis from its rule.
• I can tell from their rules whether two lines are parallel.

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. This lesson follows "What m does to a line": students should already read m as "up for each step right".

Here m stays at 2 and only c changes, so every line is parallel to the dashed y = 2x + 1. The last Modify task is the first time both sliders appear: it is the stretch point of the lesson.

The lesson says "where the line crosses the y-axis". The primer has the formal word, y-intercept, if you want to introduce it at the end.

Running the lesson

1. Predict: take a show of hands. The common wrong answer is 'it gets steeper': students expect any bigger number to steepen the line. Do not correct it yet.
2. Run: ask students to compare the step on the solid line with the dashed line as they move c. It stays 'up 2'. Then read across a column of the table: each y is the same amount more.
3. Investigate: model the worked example live. Put x = 0 into y = 2x + 4 and say 'on the y-axis, x is 0, so 2x is 0 and y is just c'. Then compare both rules at x = 1.
4. Modify: before 'Not on the axis', ask: why can't we just read c off the graph here? (T is not on the y-axis.)
5. Practise: the parallel question plants the same numbers in different places. Ask a student who chose it to work out the steepness of each line.
6. Make: collect two students' answers and check them as a class: work out 2x at P and subtract from y.

When students get stuck

Answers to the checks

1. a bigger c: The line $y = 2x + 1$ has $c = 1$. If $c$ becomes 4, so the rule is $y = 2x + 4$, what happens to the line?
   It moves up by 3 and is just as steep. Every value of y is 3 more than before (2x + 4 instead of 2x + 1), so the whole line moves up 3. m is still 2, so the steepness is the same.
2. change c: Set $c$ to 4. Was your prediction right?
   
3. Where it crosses: Where does the line $y = 3x - 5$ cross the $y$-axis? Give the value of $y$.
   -5
   −5. When x = 0, y = 3 × 0 − 5 = −5, which is c.
4. Find the mistake: Ana decides whether two lines are parallel. Click the line where her reasoning goes wrong.
   line 4
   Line 4: parallel lines need the same m, the number in front of x. Here m is 2 and 5, so one is steeper than the other: they are not parallel.
5. Parallel: Which line is parallel to $y = 3x + 1$?
   $y = 3x - 4$
   y = 3x − 4 has the same m, 3, so it is just as steep: it is y = 3x + 1 moved down 5.
6. Below the axis: Set $c$ so the line passes through $T(0, -2)$.
   
   c = −2: the line crosses the y-axis at −2.
7. Through the origin: Set $c$ so the line passes through $T(2, 4)$.
   
   c = 0: y = 2x passes through (0, 0), the origin.
8. Not on the axis: Set $c$ so the line passes through $T(1, 6)$.
   
   c = 4: at x = 1, y = 2 + 4 = 6.
9. Parallel: Make a line parallel to the dashed line $y = -x + 3$ that passes through $T(0, -1)$.
   
   m = −1 (same as the dashed line, so parallel) and c = −1 (it crosses at T).
10. From the rule: Where does $y = 5x + 7$ cross the $y$-axis? Give the value of $y$.
    7
    7: when x = 0, y = 5 × 0 + 7 = 7.
11. Parallel: Which two lines are parallel?
    $y = 4x + 1$ and $y = 4x - 3$
    Both have m = 4, so they are equally steep. y = 4x + 1 and y = x + 4 use the same numbers but in different places; y = 4x + 1 and y = 2x + 1 cross at (0, 1).
12. From a point: The line $y = 2x + c$ passes through $(3, 10)$. What is $c$?
    4
    At x = 3, 2x = 6. 6 + c = 10, so c = 4.
13. Review: Review: how far does $y = -3x + 2$ go up for each step right? (A negative answer means it goes down.)
    -3
    m = −3: down 3 for each step right.
14. a parallel line: Drag $P$ anywhere, then set $c$ so a line parallel to $y = 2x + 1$ passes through it.
    
    Your line has m = 2, like the dashed one, so they are parallel, and P is on it.

Solutions

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

change c on purpose › Below the axis

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

change c on purpose › Through the origin

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

change c on purpose › Not on the axis

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

change c on purpose › Parallel

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

a parallel line

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

Stretch and support

Stretch: Write the rule of the line parallel to y = 2x + 1 through (5, 0). Then: can two different lines with the same m ever cross? Why not?

Support: Keep the Run board open for the Investigate questions: set c to the number in the rule and read where the line crosses the y-axis. Use squared paper to draw the two parallel lines.

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 c does to a line".

The idea: Effect of intercept and generalising patterns: I can Students investigate the effect of varying the y-intercept b, and generalise patterns including the behaviour of parallel lines.

What understanding looks like:
- Correctly states the effect of varying b on the graph.
- Identifies that lines with equal gradient are parallel.
- States a generalisation supported by evidence from their investigation.

Mistakes students often make here:
- A student says a bigger c makes the line steeper.
- A student reads m as where the line crosses the y-axis.
- A student says lines with the same numbers are parallel (Ana's mistake).

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.
- If I seem to be missing the idea that comes before this one (Effect of gradient parameter on linear graphs), check it quickly, then come back.
- 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 c does to a line".

Exit ticket

Are y = 3x + 2 and y = 3x − 5 parallel? Where does each cross the y-axis?

Yes, both have m = 3. The first crosses at 2, the second at −5.

What each step is for

• a bigger c: Committing to what c will do before moving it shows whether students have separated c from m.
• change c: Moving c while m stays fixed slides the line up and down: the step stays 'up 2', and the line never turns.
• reading c: c is where the line crosses the y-axis (the value of y when x is 0). Lines with the same m are parallel; only c tells them apart.
• change c on purpose: One change at a time: a negative c, c = 0 (through the origin), a point that is not on the y-axis, then both numbers at once for a parallel line.
• read the line: Reading c, recognising parallel lines and finding c from a point are one skill; the review question keeps m from the last lesson fresh.
• a parallel line: Choosing a point and then finding the c that reaches it, with m fixed, reverses the lesson: from the picture to the rule.

Where it leads

Next: solving equations with graphs. 2x + 3 = 11 becomes the point where y = 2x + 3 meets y = 11, which joins this unit to the Equations unit. Then modelling: two plans with different m and c, and where their lines cross.

Where it leads in class: a fixed starting amount plus a rate (a call-out fee and an hourly charge, a deposit and weekly savings). c is the starting amount; m is the rate.