Plotting a rule

Classroom resources

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Students will be able to

• 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.

Timing

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

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.

Running the lesson

1. Predict: take a show of hands. Ask a student who chose 'a curve' what made them think so.
2. Run: as students place each point, ask them to say it: 'across 1, up 3'.
3. Investigate: model the worked example live on a grid: plot each point saying 'across, then up', then join with a ruler.
4. Modify: before the negative inputs, ask: which way is across for a negative x?
5. Modify: after the curve, ask: what was different about the steps in the table?
6. Make: compare two students' lines: both go up 3 for each step right. Where do they start?

When students get stuck

Answers to the checks

1. what shape?: 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. Each step of 1 across adds the same amount (2) to y, so every point is the same step up from the one before.
2. plot the points: Drag $A$ to $(0, 1)$, $B$ to $(1, 3)$, $C$ to $(2, 5)$ and $D$ to $(3, 7)$.
   
3. Plotting a rule: Which point is $(3, 2)$?
   The point 3 across and 2 up
   The first number is how far across (x); the second is how far up (y). (2, 3) is a different point.
4. Find the mistake: Sam plots the point $(2, 5)$. Click the line where his method goes wrong.
   line 3
   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).
5. Describing a graph: The table for $y = 10 - x$ is $(0, 10)$, $(1, 9)$, $(2, 8)$, $(3, 7)$. Which describes its graph?
   A straight line going down 1 for each step right
   y goes down by 1 each time x goes up by 1: the same step each time, so a straight line, going down.
6. Subtraction: $y = 3x - 2$: drag $A$ to $(0, -2)$, $B$ to $(1, 1)$, $C$ to $(2, 4)$ and $D$ to $(3, 7)$.
   
   A straight line going up 3 for each step right, crossing the y-axis at −2.
7. Negative inputs: $y = x + 2$: drag $A$ to $(-2, 0)$, $B$ to $(-1, 1)$, $C$ to $(0, 2)$ and $D$ to $(1, 3)$.
   
   Negative inputs go to the left. Still a straight line: up 1 for each step right.
8. Going down: $y = 6 - 2x$: drag $A$ to $(0, 6)$, $B$ to $(1, 4)$, $C$ to $(2, 2)$ and $D$ to $(3, 0)$.
   
   A straight line going down 2 for each step right.
9. Not a straight line: $y = x \times x$: drag $A$ to $(0, 0)$, $B$ to $(1, 1)$, $C$ to $(2, 4)$ and $D$ to $(3, 9)$. Is it a straight line?
   
   Not a straight line: y goes up 1, then 3, then 5. The steps are not the same, so the points curve upwards.
10. Coordinates: A point is 4 across and 1 up from $(0, 0)$. What are its coordinates?
    $(4, 1)$
    Across is x, up is y: (4, 1).
11. Straight or not?: Which table's points lie on a straight line?
    (0, 2), (1, 5), (2, 8), (3, 11)
    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.
12. 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 $y$.
    9
    Up 2 for each step right: 7 + 2 = 9. The point is (3, 9).
13. Review: Review: for $y = 4x - 1$, what is the output when the input is 2?
    7
    4 × 2 − 1 = 8 − 1 = 7. The pair is (2, 7).
14. your own straight line: Place $A$, $B$ and $C$ (left to right) on a straight line that goes up 3 for each step right.
    
    Each point is 3 up for each step across from the one before, so they lie on one straight line going up 3.

Solutions

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

new tables › Subtraction

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

new tables › Negative inputs

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

new tables › Going down

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

new tables › Not a straight line

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

your own straight line

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

Stretch and support

Stretch: Plot y = 2x + 1 and y = x + 3 on the same grid. Where do they meet? What does that point mean for both rules?

Support: Give the table with the pairs already written, and a printed grid. Plot two points together before students plot the rest.

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 7 student in Australia. Help me understand ONE idea only, from the lesson "Plotting a rule".

The idea: Plotting and describing relationships on the Cartesian plane: I can Students plot ordered pairs from a table of values on the Cartesian plane and describe the resulting relationship.

What understanding looks like:
- Correctly plots ordered pairs from a table on the Cartesian plane.
- Draws a line through plotted points for a linear relationship.
- Describes the relationship shown in the graph using appropriate language.

Mistakes students often make here:
- A student plots (y, x) by going up first (Sam's mistake).
- A student loses count with negative coordinates.
- A student thinks every set of points makes a straight line.

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 (Generating tables from function rules), 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 "Plotting a rule".

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.

What each step is for

• what shape?: Committing to a shape before plotting makes the straight line a discovery, and sets up the question of why.
• plot the points: Dragging each point to its pair, across then up, and seeing them line up settles the prediction and shows why: the same step each time.
• plotting and describing: Across first, then up; and a rule that adds the same amount for each step makes a straight line, described by which way it goes and how fast.
• new tables: 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.
• points and graphs: Reading and plotting points in order, and telling a straight-line rule from its table, mixed; the review is making a table from a rule.
• your own straight line: Placing points so they make a chosen straight line reverses the lesson: from the shape back to the steps between pairs.

Where it 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.