039i - Gradient and Intercept

Lesson 39 — Gradient and Intercept of a Linear Graph Using Digital Tools

Strand: Algebra | Descriptor: AC9M8A02 | Duration: 45 minutes

Learning Intentions

• To calculate the gradient of a linear graph using rise over run.
• To identify the -intercept of a linear graph and connect gradient and intercept to the equation .
• To use a digital graphing tool to investigate how and change a line’s appearance.

Success Criteria

I can:

1. Calculate the gradient between two points using .
2. Classify a gradient as positive, negative, zero or undefined.
3. Identify the -intercept from a graph or from an equation in the form .
4. State the gradient and -intercept directly from an equation without graphing it.
5. Use a digital tool to predict how changing or will affect a graph, before checking.

Warmup

(6 minutes — retrieval, mini whiteboards)

Using the line through and :

1. How much does increase as goes from to ? (This is the “rise.“)
2. How much does increase over the same interval? (This is the “run.“)
3. Divide the rise by the run.

Answers: rise ; run ; .

The hook: That number, , is called the gradient — it measures how steep the line is, and today you’ll learn to read it straight off an equation without any graphing at all.

Activities

Activity 1 — Explicit Instruction: Calculating Gradient (13 min)

Definition.

I do — from two coordinates. Find the gradient through and .

I do — a negative gradient. Find the gradient through and .

Sign convention, made explicit: a line that climbs left-to-right has a positive gradient; a line that falls left-to-right has a negative gradient.

I do — zero and undefined gradients. Through and : — a horizontal line. Through and : — undefined, a vertical line (division by zero is not allowed).

We do: Find the gradient through (a) and (b) and .

You do: Calculate the gradient for each pair of points, and classify it as positive, negative, zero or undefined.

1. and
2. and
3. and
4. and
5. and

(Answers: , positive; , negative; , zero; undefined; , positive.)

Activity 2 — Explicit Instruction: Reading and Straight from an Equation (10 min)

The form . Every linear equation written this way reveals its gradient and intercept instantly: is the gradient; is the -intercept, the point where the line crosses the -axis.

I do: For : gradient , -intercept , so the line crosses the -axis at .

For : gradient , -intercept , crossing at .

I do — rearranging first. For , divide every term by first: , so , .

We do: State and for: ; ; .

You do: State the gradient and -intercept.

(Answers: ; ; ; ; ; .)

Activity 3 — Digital Investigation: what Do and Actually Control? (14 min)

Pairs, using Desmos or GeoGebra (or an equivalent graphing app). Predict, then check, for each stage.

Setup. Graph first. This is the baseline.

Investigation A — the effect of . Add sliders (or type new lines) for , , , , , all on the same axes.

1. Predict, before graphing: which will be steepest? Which will slope downward?
2. Graph them and check your predictions.
3. What happens to the steepness as increases? What controls whether the line slopes up or down?

Investigation B — the effect of . Starting from , graph , , on the same axes.

1. Predict: how will these lines relate to and to each other?
2. Graph them and check.
3. What single feature of each line changed? What stayed exactly the same?

Guiding questions for the write-up:

Prompt	Purpose
Understand: what are you comparing?	A family of lines that share one feature ( or ) and vary the other.
Devise a plan.	Change one variable at a time, keeping the other fixed, so any change you observe can be attributed to a single cause.
Carry out the plan and record observations.	E.g. all lines in Investigation B are parallel — same steepness, different crossing points.
Looking back — can you state a general rule?	Changing rotates the line about a fixed point on the -axis; changing slides the line vertically without changing its steepness.

Answers: Investigation A — larger gives a steeper line; the sign of controls the direction of slope (positive up, negative down); steepest, steepest downward. Investigation B — all lines are parallel (same gradient ), differing only in where they cross the -axis; the gradient stayed fixed while the intercept moved the line vertically.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Find the gradient through and .
2. Classify the gradient of a line through and .
3. State the gradient and -intercept of .
4. Rearrange to state its gradient and -intercept.
5. Reasoning. Two lines are and . Without graphing, explain how you know they are parallel.

Answers: 1. ; 2. undefined (vertical line); 3. ; 4. , so ; 5. both have the same gradient, , and lines with equal gradients never meet — they are parallel, differing only in their -intercepts.

Common Misconceptions

Misconception	How to pre-empt it
Calculating rise over run with the coordinates in mismatched order, e.g. .	Always label points and first, and keep the same order in both numerator and denominator.
Confusing “zero gradient” with “no gradient” (undefined).	Pair a horizontal line () with a vertical line (undefined) every time this distinction is introduced.
Believing a steeper line always has a larger -intercept.	Investigation B shows gradient and intercept are entirely independent features.
Reading and from an equation not yet in the form , e.g. reading directly from .	Insist on dividing through to isolate with a coefficient of before reading off and .
Assuming changing changes the steepness of a line.	Investigation B directly tests and refutes this.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A line passes through and . Find its gradient and -intercept, and write its equation in the form .

Answer

The line passes through , so . Equation: .

E2 (AMC Junior style). A line has gradient and passes through . Find its -intercept.

Answer

Using with : .

E3 (Challenge). Two lines, and , are units apart, measured vertically. If , find the two possible values of .

Answer

or .

E4 (Investigation). Explain why a line with gradient rises more gently than a line with gradient , using the meaning of rise and run.

Answer

Gradient means the line rises just unit for every units moved across — a gentle climb. Gradient means the line rises units for every unit across — a much steeper climb. The larger the gradient’s size, the steeper the line.

Homework

1. Find the gradient through each pair of points: (a) and (b) and (c) and (d) and .
2. State the gradient and -intercept of: (a) (b) (c) .
3. Rearrange each to the form , then state and : (a) (b) (c) .
4. A line passes through and . Find its gradient and write its equation in the form .
5. Reasoning. Explain, using rise and run, why a horizontal line has a gradient of exactly zero.
6. Reasoning. A student says a line with a larger -intercept must be steeper. Use an example to show this is false.
7. Challenge. A line has gradient and passes through . Find its -intercept and write its full equation.
8. Challenge. Two lines pass through the origin with gradients and . Describe how these two lines are related, geometrically, and justify using their gradients.

Answers: Q1 — (a) (b) (c) (d) undefined. Q2 — (a) (b) (c) . Q3 — (a) , (b) , (c) , . Q4 — ; . Q5 — for a horizontal line, never changes, so the rise is always , and for any run. Q6 — e.g. has a large intercept () but a shallow gradient (), while has a small intercept () but a much steeper gradient () — intercept and steepness are independent. Q7 — ; equation . Q8 — they are mirror images of each other in the -axis (or equally steep but sloping in opposite directions), since their gradients are equal in size but opposite in sign.