KS3 · Maths

Plotting coordinates generated from a rule

A doorkeeper has one rule: y must be x add 2. (3, 5) gets in. (3, 6) doesn't. The points that get in make a straight line.

Drag along the rule y = x + 2

-4-2024-20246xy(0, 2)y minus x: 2

x: 0. y: 2. y minus x: 2

Drag the dot. Try stopping halfway between two whole numbers.

Watch out: The picture has to stop at the edge of the screen, but the line does not. Unless a question restricts it, a line goes on infinitely.

Maths · Algebra

Your turn: make a coordinate from a rule

A rule says: 'the y coordinate is 4 lots of the x coordinate, subtract 5'. Turn it into an equation, then use it to make a coordinate.

  1. The rule in words: the y coordinate is 4 lots of the x coordinate, subtract 5.
  2. missing step
Which line is step 2?

Maths · Algebra

Spot the rule

Three coordinates are plotted: (1, 3), (2, 4) and (3, 5). Someone asks: what is the rule?

Which idea is closest to what you think right now?
How sure are you?

Maths · Algebra

One point, two rules

Test each coordinate in both rules. Tick every rule it follows, or tick neither.

  • A Follows y = x + 2
  • B Follows y = 2x
  1. (2, 4)
  2. (1, 3)
  3. (3, 6)
  4. (5, 5)
  5. (4, 6)
  6. (0, 0)

Maths · Algebra

What shape is the rule?

Sort each rule by the kind of line its coordinates make when plotted. Ask yourself what the coordinates share.

Still to sort

Vertical line (0)

Every coordinate has the same x value.

Horizontal line (0)

Every coordinate has the same y value.

Diagonal line (0)

x and y both change as you move along.

6 of 6 still to sort.

Predict, then check

Four coordinates: (1, 2), (2, 4), (3, 6) and (4, 7). Plot them in your head: the first three line up, the last one wanders off.

Could ONE rule give all four of these coordinates?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

  • A coordinate follows a rule when the relationship between its x and its y works inside that one coordinate.
  • A rule can be said in words or written as an equation, such as y = 5x, y = ½x, y = x + 2 or y = x − 10.
  • You make coordinates by choosing an x, substituting it into the rule and working out y. Plot them and they make a line.
  • A point is on the line exactly when its x and y values fit the rule, and a rule must work for every coordinate you have.

The big picture

A rule such as y = x + 2 is a test that a coordinate passes when its y is its x add 2. You can make coordinates by substituting x values, and the coordinates that pass line up into a straight line that holds every answer to the rule.

Key points

1A rule is the relationship between x and y inside each coordinate. The coordinate (3, 6) follows 'the y coordinate is the x coordinate add 3' because 3 plus 3 gives 6.
2In an equation, y stands for the y coordinate and x for the x coordinate. 'The y coordinate is 5 lots of the x coordinate' is y = 5x, and half the x coordinate is y = ½x or y = x/2.
3Substitution means putting a value in place of a letter. Choose an x, substitute it into the rule, and you get the matching y. You can also choose a y and work the other way to find x.
4To draw the line, plot coordinates that fit the rule, check they lie in a straight line, then draw it with a ruler all the way across the grid. A line holds every coordinate that follows the rule, including non-whole-number ones, and unless you are given restrictions it is infinitely long.
5Any point on the line satisfies the rule and points off the line do not, so test a point by checking whether its x and y values satisfy the rule.
6A single coordinate may fit several rules, so check a rule against two, three or more coordinates. If no single rule works for every coordinate, the points do not make a straight line.
7Technology such as Desmos can plot coordinates, draw the line from a rule you type in, and make it easy to spot coordinates that do not follow the rule.

Worked example

Problem

Draw the line y = 2x − 1.

⚠ Watch out

Stopping the line at the last point you plotted. The line is every coordinate that follows the rule, so draw it with a ruler all the way across the grid, not just between your points.

🧠

Memory hook

A rule is a doorkeeper: each coordinate either gets in or is turned away, and the line is everyone who got in. (Unlike a real guest list, this one never ends, because the line goes on forever.)

✓

Check yourself

Is (4, 12) on y = 3x? Is (4, 7)? Put x = 4 into the rule each time. Answer: 3 × 4 = 12, so (4, 12) is on it; (4, 7) is not.

Flashcards

(13)
What does it mean for a coordinate to follow a rule?
The relationship between its x and its y works. For example (3, 6) follows 'y is x add 3' because 3 + 3 = 6.
Write as equations: (a) the y coordinate is 5 lots of the x coordinate; (b) half the x coordinate.
(a) y = 5x. (b) y = ½x, or y = x/2.
What does substitution mean?
Putting in place of another. In algebra you replace a letter with a value, such as x = 3 in y = 4x − 5.
How do you make coordinates from a rule?
Choose x values and apply the rule to find each y. Or choose y values and work the other way to find x.
Why is 'add 1 each time' not the rule for (1, 3), (2, 4), (3, 5)?
It describes the pattern between coordinates. A rule is the relationship between x and y inside each coordinate: here y = x + 2.
(2, 4) fits both y = x + 2 and y = 2x. What does that tell you?
A single coordinate may fit several rules, so look at two, three or more coordinates together and check the rule works for all of them.
What line do coordinates with the same x value make? Give an example rule.
A vertical line, such as x = 4.
What line do coordinates with the same y value make? What is special about y = 0?
A horizontal line, such as y = 2. The line y = 0 is the x-axis.
What kind of line do y = 2x, y = x + 2 and y = ½x make?
Diagonal lines.
What does the line for a rule represent?
All the coordinates that follow the rule, including non-integer ones. Unless you are given restrictions, it is infinitely long.
How can you test whether a point is on a line?
Check whether its x and y values satisfy the rule. If they do, it is on the line. If not, it is off it.
Coordinates follow no single rule. What do they look like plotted?
They do not make a straight line, because no rule works for every coordinate.
What can technology such as Desmos do with rules and coordinates?
Plot coordinates, generate coordinates, draw the line from a rule, and make it easy to spot coordinates that do not follow the rule.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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