Mathematics • Year 8 • Unit 2 • Lesson 9

Finding Gradient from Two Points

Build fluency with the gradient formula m = (y₂ − y₁) / (x₂ − x₁). One worked example, one guided example with blanks, then eight independent problems including negatives and special cases.

Build · I Do / We Do / You Do

1. I do, fully worked example

Given two coordinates, you can find the gradient without ever drawing the line.

Problem. Find the gradient of the line through (2, 3) and (6, 7).

Gradient = rise ÷ run = 4 ÷ 4 = 1, equal rise and run give a 45° line.

Step 1, Label your points.

(x₁, y₁) = (2, 3) (x₂, y₂) = (6, 7)

Reason: pick either point as "1", just be consistent for both subtractions.

Step 2, Calculate the rise (y₂ − y₁).

Rise = y₂ − y₁ = 7 − 3 = 4

Reason: "y on top" of the gradient fraction.

Step 3, Calculate the run (x₂ − x₁).

Run = x₂ − x₁ = 6 − 2 = 4

Reason: "x on bottom".

Step 4, Compute m.

m = rise / run = 4 / 4 = 1

Answer: m = 1 (positive, line slopes uphill at 45°).

Stuck? Revisit lesson § "The Gradient Formula", "y on top, x on bottom".

2. We do, fill in the missing steps

Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks

Problem. Find the gradient through (1, 2) and (5, 14).

Step 1, Label points: (x₁, y₁) = (______, ______) and (x₂, y₂) = (______, ______).

Step 2, Rise = y₂ − y₁ = ______ − ______ = ______.

Step 3, Run = x₂ − x₁ = ______ − ______ = ______.

Step 4, m = rise / run = ______ / ______ = ______.

Sign check: y went up as x went up, so m should be ____________ (positive / negative).

Stuck? Subtract in the SAME order for both: if (5,14) is (x₂,y₂), then both subtract the (1,2) version.

3. You do, independent practice

Show your working under each problem. First four are foundation, next two are standard, last two are extension.

Foundation, apply the formula

3.1 Find m through (1, 3) and (4, 9). 1 mark

3.2 Find m through (0, 0) and (5, 10). 1 mark

3.3 Find m through (2, 7) and (5, 1). 1 mark

3.4 Find m through (3, 5) and (7, 5). State the gradient type. 1 mark

Standard, including negatives

3.5 Find m through (0, 0) and (−2, 6). 2 marks

3.6 Find m through (−3, −2) and (1, 4). 2 marks

Extension, simplify and decide

3.7 Find m through (−1, 2) and (3, −4). Give your answer as both a simplified fraction and a decimal. 2 marks

3.8 Show that the gradient through (1, 3) and (4, 9) is the same whether you label (1,3) as (x₁,y₁) or as (x₂,y₂). 2 marks

Stuck on 3.8? Try both labellings. As long as you subtract in the same order top and bottom, the negative signs cancel.

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What I'll revisit before next class:

Answers, Do not peek before attempting

Section 2, We do

(x₁, y₁) = (1, 2), (x₂, y₂) = (5, 14). Rise = 14 − 2 = 12. Run = 5 − 1 = 4. m = 12/4 = 3. Sign check: positive.

3.1, (1, 3) and (4, 9)

m = (9 − 3)/(4 − 1) = 6/3 = 2.

3.2, (0, 0) and (5, 10)

m = (10 − 0)/(5 − 0) = 10/5 = 2.

3.3, (2, 7) and (5, 1)

m = (1 − 7)/(5 − 2) = −6/3 = −2.

3.4, (3, 5) and (7, 5)

m = (5 − 5)/(7 − 3) = 0/4 = 0. Gradient type: zero (horizontal line).

3.5, (0, 0) and (−2, 6)

m = (6 − 0)/(−2 − 0) = 6/(−2) = −3.

3.6, (−3, −2) and (1, 4)

m = (4 − (−2))/(1 − (−3)) = 6/4 = 3/2 (= 1.5).

3.7, (−1, 2) and (3, −4)

m = (−4 − 2)/(3 − (−1)) = −6/4 = −3/2 = −1.5.

3.8, Order independence

If (x₁,y₁) = (1,3) and (x₂,y₂) = (4,9): m = (9−3)/(4−1) = 6/3 = 2.
If (x₁,y₁) = (4,9) and (x₂,y₂) = (1,3): m = (3−9)/(1−4) = −6/−3 = 2.
Both give m = 2. The order doesn't matter as long as you subtract consistently top and bottom.