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Gradient and Intercept: Free Printable Worksheet for Year 9
- ForYear 9
- StageKey Stage 3
- CostFree
- FormatPrintable PDF
Gradient and Intercept is a free printable maths worksheet for Year 9 (ages 13 to 14), in Key Stage 3. Students find the gradient of a line from two points, read the gradient and y intercept from an equation, and write a line as y = mx + c. It takes about 20 minutes and needs only a pencil.
It comes from the Mini Minds Hub, the free library of printables from Jaques of London, the British games and toy company that has been making games for families since 1795. Print both pages, give page 1 to the student and keep page 2 for the grown-up. It has notes, help, a stretch idea and the answers. Black and white printing works well.
What is on the sheet
Students start by finding a gradient from two points on a line. Some of the points have negative coordinates, so signs need care. Next they read the gradient and y intercept straight from an equation in the form y = mx + c, including equations written in a different order. The sheet ends by asking them to build an equation from two points. There is space to show working under each question.
- Gradient of the line through (0, 1) and (2, 5)
- Gradient through (1, 2) and (3, 8)
- Gradient through (0, 6) and (3, 0)
- Gradient through (2, 1) and (6, 3)
- Gradient through (-3, -1) and (1, 3)
- y = 4x - 3: gradient ___ and y-intercept ___
- y = 5 - 2x: gradient ___ and y-intercept ___
- Line through (0, 2) and (4, 10): write it as y = mx + c
Worked example: gradient from two points
Find the gradient of the line through (7, 9) and (9, 21).
This example uses different numbers from the sheet. It shows two ways to reach the same gradient.
Method 1: change in y over change in x- Find the change in y: 21 minus 9 = 12.
- Find the change in x in the same order: 9 minus 7 = 2.
- Divide the change in y by the change in x: 12 ÷ 2 = 6.
- Mark both points on squared paper and join them with a ruler.
- Start at (7, 9). Count 2 across to reach x = 9, then 12 up to reach y = 21.
- 2 across gives 12 up, so 1 across gives 6 up.
How to find a gradient in four steps
Pick two points
Choose any two points on the line. Label one as the first point and one as the second, and keep that order.
Find the change in y
Take the first y value away from the second y value. This is the rise. It can be negative.
Find the change in x
Take the first x value away from the second x value, in the same order. This is the run.
Divide and check
Divide the rise by the run. Then sketch the line quickly. A line that goes up from left to right should not have a negative answer.
What gradient and intercept tell us
The gradient of a line measures how steep it is. It tells you how much y changes when x goes up by 1. On a straight line this is the same everywhere along it. A line that rises from left to right has a positive gradient. A line that falls has a negative gradient. A flat line has a gradient of zero.
In y = mx + c, the letter m is the gradient and c is the y intercept, the point where the line crosses the y axis. Together they describe a line completely. This is part of the Key Stage 3 work on graphs of linear functions. Equations are not always written in the same order, so read each number with its sign.
Did you know?
- The gradient of a straight line is the same wherever you measure it along the line.
- Gradient is also the word used for how steep a hill, road or railway is.
- In the UK, road signs warn drivers about steep hills, and the steepness is shown as a percentage.
- The y intercept is the point where a line crosses the y axis. At that point, x is always 0.
Skills and curriculum link
- Finding a gradient from two points
- Reading y = mx + c
- Working with negative numbers
- Writing an equation of a line
- Sketching straight line graphs
Ideas to help you support
- Squared paperPlotting points makes a gradient visible. Students can count across and up, then check the number they calculated.
- Colour the stepsUse one colour for the change in y and another for the change in x. This helps stop the two being swapped.
- Keep the orderSubtract in the same order on the top and the bottom. Ask students to say which point they called first.
- Sketch to checkA quick sketch shows whether a line should slope up or down. That is a fast check on the sign of an answer.
Key words
- GradientA number that tells you how steep a line is. It is also called the slope.
- Y interceptThe point where a line crosses the y axis. Here x is 0.
- CoordinatesA pair of numbers, such as (3, 5), that gives a position on a grid. The x value comes first.
- RiseHow far a line goes up or down between two points. Going down gives a negative rise.
- RunHow far a line goes across between two points, from left to right.
- Equation of a lineA rule linking x and y that is true for every point on the line.
How to use it
- Print both pages. Page 1 is for the student. Keep page 2, which has the notes and answers.
- Read the first question together. Ask the student to say what a gradient is in their own words.
- Let them work in pencil, with working out in the lines. Squared paper is useful for plotting points.
- Check against page 2. Ask them to explain how they dealt with any negative numbers.
- Come back another day to redo the trickier questions. Repeating a familiar sheet builds confidence.
Help, stretch and game ideas
If it feels too hard
Plot each pair of points on squared paper and join them with a ruler. Count how far the line goes across, then how far it goes up or down. Write the rise over the run. Do the first few questions together before working alone.
To stretch confident children
Choose your own gradient and y intercept and write the equation. Work out two points on that line, then give only the points to a friend. Can they find your equation? Check whether they match yours.
Make it a game
Play equation match. One player writes a line as y = mx + c. The other sketches it on squared paper. Swap roles, then compare sketches. Score a point each time the gradient and the y intercept are right.
Mistakes to watch for
Watch forThe student subtracts the y values one way round and the x values the other, so the sign of the answer is wrong.
Try thisName the first and second point before starting. Use that same order for both the top and the bottom of the fraction.
Watch forThe student divides the change in x by the change in y, so the gradient is upside down.
Try thisRemember that gradient is how far the line goes up for each step across. Check with a sketch of the line.
Watch forWith an equation such as y = 11 minus 6x, the student says the gradient is 11 because it is written first.
Try thisThe gradient is the number that multiplies x, with its sign. The y intercept is the number on its own. Rewrite it as y = negative 6x plus 11.
Talk about it
- What does a steep line tell you? What does a flat line tell you?
- How can you tell from two points whether a line will go up or down?
- Why does it not matter which two points on a line you choose?
- What does the y intercept tell you about a line?
- How would you explain to a friend what m and c mean?
- Where have you seen steepness measured in real life?
Take it off the page
Ramp experiment
Make a ramp from a board and a pile of books. Measure the height and the distance along the floor. Divide them, then change the pile and see what happens.
Hill spotting
On a walk, look for steep and gentle slopes. Rank them from steepest to flattest and sketch what you think the lines would look like.
Grid drawing
On squared paper, draw lines with different gradients through the same point. Label each one as steeper, gentler or going downhill.
Who it is for
Parents
Sit with your child for the first question and let them explain their thinking. Squared paper and a ruler help a lot. Ten to fifteen minutes at a time works well, and you can finish it another day.
Teachers
Use it as a starter, a plenary, homework or a cover task. It follows teaching of straight line graphs and gives quick practice with two points and y = mx + c. Page 2 has support, stretch and answers.
Home educators
Pair it with graph paper, a ruler and a real slope to measure. Students can plot their own lines, check their answers and explain each method aloud. Print it again for revision a few weeks later.
Where this skill fits
Before this sheet, students plot coordinates in all four quadrants, work with negative numbers and meet ideas of ratio and proportion. These give the base for understanding steepness as a rate of change.
After it, students move on to sketching and comparing straight line graphs, finding equations from graphs and solving problems with linear functions. These ideas continue into Key Stage 4 algebra and graphs.
Questions parents and teachers ask
- What age is Gradient and Intercept for?
It is written for Year 9 (ages 13 to 14), in Key Stage 3. It also works as revision for older students who need practice with straight line graphs.
- What will students practise?
They find a gradient from two points, read the gradient and y intercept from y = mx + c, and write the equation of a line from two points. Negative slopes are included.
- How long does it take?
It takes about 20 minutes. Students who sketch each line may take a little longer, and confident students may finish sooner and move on to the stretch idea.
- Are the answers included?
Yes. Page 2 is for the grown-up and includes the answers, notes on how to use the sheet, help for students who find it hard and a stretch idea.
- Do students need squared paper?
No, but it helps. Plotting each pair of points and counting across and up makes the gradient easier to see and gives students a way to check their working.
- What if my child gets the signs wrong?
Ask them to sketch the line first. A line that falls from left to right has a negative gradient. Then check that they subtracted in the same order on the top and bottom.
- Is it free, and can I use it in class?
Yes. It is free to print for home and classroom use. Teachers can use it as a starter, plenary, homework or table task. It needs no preparation beyond printing.
Preview the full PDF
Made by Jaques of London, making games for families since 1795. A printable supports classroom learning and does not replace a teacher.