Key Stage 3 maths · Algebra

KS3 coordinates, straight-line graphs and quadratic curves

A graph turns an equation into a picture. In Key Stage 3 children plot points in all four quadrants, draw straight-line graphs and quadratic curves from their equations, learn what the gradient and intercept of y = mx + c tell you, and use graphs to estimate values and find approximate solutions to problems.

Graphs of linear and quadratic functions, explained

The Cartesian plane is the grid with a horizontal x-axis and a vertical y-axis, crossing at the origin (0, 0). Children plot points in the first quadrant from Year 4 and meet all four quadrants in Year 6. At Key Stage 3 they work with negative coordinates routinely, so a point such as (−3, 2) sits to the left of the origin and above it, and they use the grid for graphs rather than just for positions. The order is always x first, then y, along the corridor and then up the stairs. Children also choose scales so a graph fits the page, perhaps counting in 2s, 5s or 10s on an axis rather than in 1s.

A linear function has a graph that is a straight line. A child builds a table of values by putting in a few values of x and calculating y, plots the points and joins them with a ruler. In the form y = mx + c, the number m is the gradient, which tells you how steep the line is: how far y changes for each 1 step across. The number c is the intercept, the point where the line crosses the y-axis. So y = 2x + 3 climbs 2 for every 1 across and starts at height 3. A negative gradient, as in y = −2x + 4, means the line slopes downwards. Children are also given equations that are not yet in this form, such as 3y + 6x = 12, and rearrange them into y = mx + c to read off the gradient and intercept. A horizontal line has the equation y = a number, and a vertical line has x = a number. In a real context the gradient and intercept have meanings: on a graph of a taxi fare against miles, the gradient is the cost per mile and the intercept is the fixed charge.

Quadratic functions such as y = x² − 2x − 3 contain an x². Their graphs are smooth, symmetrical curves, called parabolas, rather than straight lines. A table of values is still the way in, but children have to be careful with negative numbers when squaring, and they join the points with a smooth freehand curve, not with a ruler. The curve either dips and rises like a bowl or arches like a hill, and has a turning point at its lowest or highest value.

The reason for drawing graphs is to use them. Children read off a value of y for a given x, or work backwards to find x for a given y, and are told when an answer is only approximate. Two straight lines drawn on the same axes cross at the point that makes both equations true, so reading off the crossing point gives an approximate solution to simultaneous equations. Children also read graphs of real situations, including piece-wise linear graphs made of connected straight sections (such as a mobile tariff that changes after a data limit), exponential graphs that climb faster and faster (such as a doubling population) and reciprocal graphs that fall away (such as journey time against speed). For these the programme of study asks children to find approximate answers from graphs they are given, rather than to draw them from an equation.

Words your child will hear

Quadrant: One of the four regions of the coordinate grid, divided by the x-axis and y-axis.
Origin: The point (0, 0) where the axes cross.
Gradient: How steep a line is: the change in y divided by the change in x.
Intercept: Where a graph crosses an axis. The y-intercept is the c in y = mx + c.
Linear: Giving a straight-line graph. y = 3x − 1 is linear.
Quadratic: Containing an x² term and giving a curved, symmetrical graph.
Table of values: A list of x-values with the matching y-values, used to plot a graph.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Work with coordinates in all 4 quadrants
Recognise, sketch and produce graphs of linear and quadratic functions of 1 variable with appropriate scaling, using equations in x and y and the Cartesian plane
Reduce a given linear equation in 2 variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically
Use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations
Find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

A line has the equation 3y + 6x = 12. Write it in the form y = mx + c, state the gradient and y-intercept, and find where it crosses the x-axis.

Get the y term on its own. Subtract 6x from both sides.
```
3y = −6x + 12
```
Divide every term by 3.
```
y = −2x + 4
```
Compare with y = mx + c. The gradient m is −2, so the line slopes downwards, falling 2 for each 1 across. The y-intercept c is 4, so the line crosses the y-axis at (0, 4).
The line crosses the x-axis where y = 0. Substitute and solve.
```
0 = −2x + 4
2x = 4
x = 2
```
Check with a third point. When x = −1, y = −2 × (−1) + 4 = 6, and 3 × 6 + 6 × (−1) = 18 − 6 = 12, which matches the original equation.

y = −2x + 4: gradient −2, y-intercept (0, 4), crossing the x-axis at (2, 0).

Common mistakes and how to help

Writing coordinates the wrong way round, especially with negatives

Children read (−3, 2) as three up and two across, or lose track of which way negative numbers go.

What helps: Remember “along the corridor, then up the stairs”: x moves left or right, y moves up or down. Pick any point on a map or board game and describe how to reach it from the origin.

Getting squares of negative numbers wrong in the table of values

For y = x² − 2x − 3 at x = −2, a child squares −2 and gets −4 instead of 4. The curve then comes out lopsided, with a point far away from the others.

What helps: Write (−2)² = (−2) × (−2) = 4 and put brackets round negative numbers when substituting. The graph of a quadratic should be symmetrical, so a point that breaks the pattern is the one to recalculate.

Joining quadratic points with straight ruler lines

Linear graphs are drawn with a ruler, so the same habit carries over to curves and produces a zig-zag with sharp corners.

What helps: Plot extra points near the turning point to see the shape, then draw a smooth curve freehand, turning gently, without lifting the pencil more than needed.

Mixing up gradient and intercept, or ignoring the sign of the gradient

In y = −2x + 4 children say the gradient is 2 or 4, or that the line goes up. Rearranging 3y + 6x = 12, some divide the x term by 3 but forget the number, giving y = −2x + 12 instead of y = −2x + 4.

What helps: Say m is the multiplier of x and c is the number on its own. A negative gradient is a line that falls as you move right. Check by plotting two points and seeing which way the line goes.

How to help at home

Hide treasure on a grid. Draw a grid on paper with the origin in the middle and the axes running from −5 to 5. Take turns to hide a “treasure” at a point and give clues as (x, y) pairs, including negative ones, until the other person finds it.
Draw the story of a journey. After a car or train trip, sketch a distance-time graph together using straight sections, which makes it a piece-wise linear graph. Flat sections are stops, steep sections are fast stretches. Ask your child to read off how far you were after an hour and when you reached the halfway point.
Predict before you plot. Give an equation like y = 3x − 2 and ask where it starts and which way it slopes before any points are plotted. Then plot a few and see whether the prediction was right.
Check the symmetry. When your child draws a quadratic from a table, ask them to fold the page along the vertical line of symmetry through the turning point. Both halves should match. If they don’t, a value in the table has probably been miscalculated.

Builds on

Year 6: Introduction to algebra

Leads on to

Key Stage 4: Factorising, indices and identities
Key Stage 4: Straight-line and quadratic graphs
Key Stage 4: Quadratics, simultaneous equations, inequalities
Key Stage 4: Special sequences and the nth term
Key Stage 4: Algebraic fractions, proof and functions
Key Stage 4: Exponential, trig and circle graphs
Key Stage 4: Quadratic formula, iteration, inequalities
Key Stage 4: Quadratic sequences and surd ratios