Plotting a rule

Plot each ordered pair from a rule's table as a point, across first and then up. For a rule like y = 2x + 1, the points lie on a straight line.

A table of numbers hides its pattern; a graph shows it at a glance. Plotting the pairs turns a rule into a picture you can read.

In the last lesson you made tables of ordered pairs from rules. Now you plot them. To plot the point $(x, y)$ , start at $(0, 0)$ , go across $x$ , then up $y$ . Then look at the shape the points make.

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.

What shape?

$x$	0	1	2	3
$y = 2x + 1$	1	3	5	7

How much does $y$ go up each time $x$ goes up by 1?

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 upScattered points, no patternA curve that gets steeperA flat line

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

Plot the points

Point	A	B	C	D
$(x, y)$	$(0, 1)$	$(1, 3)$	$(2, 5)$	$(3, 7)$

Drag each point from the bottom of the board to its pair. For each one: across first, then up. What shape do they make?

Try it yourselfYour edits are saved in this browser.

Drag $A$ to $(0, 1)$ , $B$ to $(1, 3)$ , $C$ to $(2, 5)$ and $D$ to $(3, 7)$ .

Use the board. This ticks when you get there.

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

Plotting and describing

Here is how to plot a rule and describe its graph.

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

$y = 2x + 1: \; (0, 1), (1, 3), (2, 5), (3, 7)$ Start with the table of ordered pairs.
$(0, 1): \text{ across } 0, \text{ up } 1$ Plot each pair from $(0, 0)$ : across first, then up.
$(1, 3), (2, 5), (3, 7)$ Plot the rest the same way.
$\text{up } 2 \text{ for each step right}$ Each step of 1 across, $y$ goes up 2: the same step every time.
$\text{a straight line}$ So the points lie on a straight line. Join them with a ruler.
$\text{going up, starting at } 1$ Describe it: it goes up 2 for each step right and crosses the $y$ -axis at 1.

The four points lie on a straight line: up 2 for each step right.

Read Sam's method one line at a time. Which way should he go first?

Which point is

(3, 2)

?The point 3 across and 2 upThe point 2 across and 3 upThe point 3 across and 3 up

Sam plots the point $(2, 5)$ . Click the line where his method goes wrong.

The table for

y = 10 - x

(0, 10)

(1, 9)

(2, 8)

(3, 7)

. Which describes its graph?A curve going downA straight line going up 1 for each step rightA straight line going down 1 for each step right

ModifyChange the problem on purpose and see what happens.

New tables

Each task loads a new table. Drag $A$ to $D$ to the pairs, across then up.

Need a hint?

A negative x means go across to the left first.

Plot it, then look at the steps between the points.

Try it yourselfYour edits are saved in this browser.

$y = 3x - 2$ : drag $A$ to $(0, -2)$ , $B$ to $(1, 1)$ , $C$ to $(2, 4)$ and $D$ to $(3, 7)$ .

Use the board. This ticks when you get there.

$y = x + 2$ : drag $A$ to $(-2, 0)$ , $B$ to $(-1, 1)$ , $C$ to $(0, 2)$ and $D$ to $(1, 3)$ .

Use the board. This ticks when you get there.

$y = 6 - 2x$ : drag $A$ to $(0, 6)$ , $B$ to $(1, 4)$ , $C$ to $(2, 2)$ and $D$ to $(3, 0)$ .

Use the board. This ticks when you get there.

$y = x \times x$ : drag $A$ to $(0, 0)$ , $B$ to $(1, 1)$ , $C$ to $(2, 4)$ and $D$ to $(3, 9)$ . Is it a straight line?

Use the board. This ticks when you get there.

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

Points and graphs

A point is 4 across and 1 up from

(0, 0)

. What are its coordinates?

(1, 4)

(4, 1)

(4, 4)

Which table's points lie on a straight line?(0, 2), (1, 3), (2, 6), (3, 11)(0, 2), (1, 5), (2, 8), (3, 11)(0, 1), (1, 2), (2, 4), (3, 8)

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 $y$ .

Review: for $y = 4x - 1$ , what is the output when the input is 2?

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

Your own straight line

Choose where your line starts. Then place the other points so the line goes up 3 for each step to the right.

Stretch: what rule $y = \ldots$ gives your points? Check it with all three.

Try it yourselfYour edits are saved in this browser.

Need a hint?

Put A anywhere. Then B is 1 across and 3 up from A.

Another hint

C is 1 across and 3 up from B.

Place $A$ , $B$ and $C$ (left to right) on a straight line that goes up 3 for each step right.

Use the board. This ticks when you get there.

For teachers

Students plot ordered pairs from a rule's table, across then up, see that rules adding the same amount each step give straight lines, and describe the line (AC9M7A05, node 3). About 45 minutes.

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

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.

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

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

When students get stuck

A student plots (y, x) by going up first (Sam's mistake). Ask: Which number is x? Which way does the x-axis go? Fix: Say 'along the corridor, then up the stairs': across first, then up.
A student loses count with negative coordinates. Ask: Is x positive or negative? Which way do you go? Fix: Negative x goes left of the y-axis; negative y goes below the x-axis. Count the squares from (0, 0).
A student thinks every set of points makes a straight line. Ask: How much does y go up between each pair of points? Fix: Write the steps under the table. Same step every time: straight. Steps that change: a curve.

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.

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.

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.

Now you can

Plot ordered pairs from a table, across first and then up.
Describe a straight-line graph by which way it goes and how much for each step.
Tell from a table whether its points will lie on a straight line.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson4. Reading a graphA point on a graph of real data pairs two measurements. Find one value on its axis, go to the line, and read across to the other axis.

Source: content/maths/linear-relations/plotting-a-rule/lesson.md