Ngā Pūkaha Raina · Linear Relationships and Graphs
Gradient, y-intercept, and graphing y = mx + c · Years 9–10
Ngā Whāinga Akoranga · Learning Intentions
- Understand that in y = mx + c, m is the gradient (rate of change) and c is the y-intercept.
- Calculate the gradient of a line using the formula m = rise ÷ run = Δy ÷ Δx.
- Plot points from a linear rule using a table of values and draw the straight-line graph.
- Interpret linear graphs in real-world contexts, identifying what the gradient and y-intercept mean.
Paearu Angitu · Success Criteria
- I can identify m and c from an equation in the form y = mx + c and explain what each represents.
- I can calculate the gradient of a line given two points or from a graph.
- I can complete a table of values and plot a linear graph accurately.
- I can explain what the gradient and y-intercept mean in a real-world context (e.g. cost, speed, temperature).
Hononga Marautanga · Curriculum Connections
This activity addresses the following Mathematics curriculum statements (Draft 2025, Phase 4 — Years 9, 10):
- Algebra: In y = mx + c, m and c are constants; x and y are variables. All values of x and y that satisfy the equation lie on the same straight line.
- Algebra: Interpreting rules of the form y = mx + c, using substitution and tables to plot points from the linear graph, connecting the position of points to the rule.
- Algebra: Interpreting and graphing linear equations in the form y = mx + c, using gradient and y-intercept. Calculating gradient and y-intercept of a line given two points or its equation.
- Algebra: The gradient m of a straight line is determined by m = rise ÷ run = Δy ÷ Δx. A vertical line has an infinite gradient.
- Algebra: The constant rate of change of a linear graph is the vertical change divided by the horizontal change between any two points. Multiplying or dividing by a negative number reverses an inequality.
Wāhanga 1 · Understanding y = mx + c
Every straight-line graph can be described by the equation y = mx + c. The two constants tell you everything about how the line behaves.
The steepness and direction of the line.
- Positive m → line goes up left to right
- Negative m → line goes down left to right
- m = 0 → horizontal line
- Larger |m| → steeper line
Where the line crosses the y-axis (when x = 0).
- c = 0 → line passes through origin
- c > 0 → line crosses above origin
- c < 0 → line crosses below origin
For each equation, identify m and c, then describe the line:
| Equation | m (gradient) | c (y-intercept) | Direction (up/down/flat) |
|---|---|---|---|
| y = 3x + 2 | |||
| y = −2x + 5 | |||
| y = 0.5x | |||
| y = −x − 3 |
Wāhanga 2 · Calculating Gradient
The gradient measures the rate of change — how much y changes for every 1-unit increase in x.
Calculate the gradient between each pair of points:
Δy = 9 − 3 =
Δx = 4 − 1 =
m =
Δy =
Δx =
m =
Real world: A car travels at constant speed. After 2 hours it has gone 120 km; after 5 hours, 300 km. What is the gradient, and what does it represent?
Wāhanga 3 · Table of Values and Graphing
Complete the table of values for y = 2x − 1, then plot the points and draw the line:
| x | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y |
Plot your points here — draw axes, label x and y, and draw the line
Where does your line cross the y-axis? Does this match c in your equation? Explain:
Wāhanga 4 · Interpreting Linear Graphs
Linear graphs appear in real-world situations. The gradient always represents a rate — the amount one quantity changes per unit of another. The y-intercept is the starting value.
Write your own real-world linear relationship. Describe what m and c represent in your context:
Aronga Mātauranga Māori · Te Ao Māori Lens
In te ao Māori, the observation of constant relationships across time and space was central to navigation, agriculture, and architecture. Polynesian wayfinders tracked the rising and setting of stars — recognising that their positions changed at predictable, constant rates through the night and across seasons. This is linear thinking in its oldest form.
Maramataka — the Māori lunar calendar — encodes proportional and linear relationships between cycles of the moon, tides, planting seasons, and harvesting times. The pattern thinking behind these systems is mathematically sophisticated, representing centuries of observation and record-keeping passed through oral tradition.
Think of a natural cycle or relationship that changes at a constant rate. How would you express it as a linear equation?
Tuhia ōu whakaaro · Write Your Thoughts
Where do you notice linear relationships in daily life? Think about phone plans, taxi fares, savings accounts, or sports statistics. What does the gradient always tell you in each of these contexts?