How to Find Gradient from a Table of Values at KS3

What is gradient?

Gradient is defined as:

Gradient = change in y ÷ change in x

This is sometimes written using the Greek letter delta (Δ) as Δy ÷ Δx, or described as "rise over run" — how much the graph rises (or falls) for every unit you move to the right.

• A positive gradient means the line goes up from left to right.
• A negative gradient means the line goes down from left to right.
• A zero gradient means the line is horizontal.

For a straight-line graph, the gradient is the same everywhere along the line. For a curved graph, the gradient changes from point to point.

How do you find gradient from a table of values?

1. Choose any two rows from the table. For a linear (straight-line) relationship, any two rows will give the same gradient.
2. Find the change in y. Subtract the y-value of the first row from the y-value of the second row: Δy = y₂ − y₁.
3. Find the change in x. Subtract the x-value of the first row from the x-value of the second row: Δx = x₂ − x₁.
4. Divide. Gradient = Δy ÷ Δx.
5. Check with another pair of rows. For a linear relationship, the gradient must be the same for every pair of points. If it is not, the relationship is non-linear.

Worked example 1: a linear table

The table shows values for a straight-line graph.

Step 2 — Change in y (using x = 1 and x = 3):

Δy = 11 − 5 = 6

Step 3 — Change in x:

Δx = 3 − 1 = 2

Step 4 — Gradient:

Gradient = 6 ÷ 2 = 3

Step 5 — Check using x = 2 and x = 5:

Δy = 17 − 8 = 9, Δx = 5 − 2 = 3, gradient = 9 ÷ 3 = 3 ✓

The gradient is 3, meaning y increases by 3 for every 1 unit increase in x. The equation of this line is y = 3x + c. Substituting x = 1, y = 5: 5 = 3(1) + c, so c = 2. The full equation is y = 3x + 2.

Worked example 2: a non-linear table

Now consider this table, which comes from a quadratic curve.

Gradient between x = 1 and x = 2:

Δy = 4 − 1 = 3, Δx = 1, gradient = 3 ÷ 1 = 3

Gradient between x = 2 and x = 3:

Δy = 9 − 4 = 5, Δx = 1, gradient = 5 ÷ 1 = 5

Gradient between x = 3 and x = 4:

Δy = 16 − 9 = 7, Δx = 1, gradient = 7 ÷ 1 = 7

The gradient is not constant — it increases each time. This tells you the relationship is non-linear. The table in fact shows y = x², a quadratic curve whose steepness increases as x increases.

The differences between the y-values (3, 5, 7, …) are themselves increasing — a sign of a quadratic relationship, as you will learn when studying sequences at GCSE.

How does this connect to plotting straight-line graphs?

When you plot the (x, y) pairs from a linear table, all the points lie on a straight line. The gradient you calculated from the table is the same as the gradient m in the equation y = mx + c.

To find c from the table, use any row: c = y − m × x.

Example: Using the first worked example with gradient 3 and the row (2, 8): c = 8 − 3 × 2 = 8 − 6 = 2, giving y = 3x + 2.

What does gradient represent in real-life contexts?

Gradient from a table of values has a practical meaning depending on the context:

• In a distance–time table, gradient = distance ÷ time = speed.
• In a cost–quantity table, gradient = change in cost ÷ change in quantity = cost per item.
• In a temperature–time table, gradient = change in temperature ÷ change in time = rate of cooling or heating.

Always state the units of the gradient. If distance is in metres and time in seconds, gradient has units of m/s (metres per second).

Common mistakes to watch out for

• Dividing the wrong way: Always divide Δy by Δx, not Δx by Δy.
• Not checking for linearity: Calculate gradient between at least two different pairs of points before assuming the relationship is linear.
• Using negative differences incorrectly: If y decreases as x increases, Δy is negative and the gradient is negative — a falling line.
• Forgetting to subtract in the same direction: If you calculate Δy = y₂ − y₁, you must also calculate Δx = x₂ − x₁ using the same order.

Frequently asked questions

Q: Can I use any two rows from a linear table to find gradient?

Yes — for a linear relationship the gradient is the same between every pair of points. Using rows further apart (a larger Δx) reduces the chance of arithmetic errors, so choosing the first and last row is often the most efficient approach.

Q: What if the x-values in the table are not evenly spaced?

The formula Δy ÷ Δx still applies. The x-values do not need to be equally spaced — you just subtract the two x-values you have chosen. Be careful when the x-values are decimal or fractional, and check your division.

Q: How is gradient from a table different from gradient from a graph?

From a graph, you draw a right-angled triangle on the line and read off the rise and run. From a table, you use the numbers directly. Both methods use the same formula (Δy ÷ Δx) and should give the same answer — use whichever format the question provides.

Q: What if two x-values are the same but y is different?

That situation cannot arise in a function (where each x maps to exactly one y), but if it did it would mean the "line" is vertical, with an undefined gradient. You will not see this in standard KS3 problems.

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