Teaching objectives
Studying functions by hand is tedious — drawing a single graph can take fifteen minutes. The goal of this lab is to help students experiment with the parameters of the linear equation so they can visualise them directly. Why does a line go up, go down, or cross the axis exactly there? In 12 rooms we explore the full anatomy of y = m·x + n, from watching a line take shape in real time to deducing its equation from just two points. It is designed for secondary-school students encountering linear functions rigorously for the first time.
What you'll learn
Students travel in two directions: from equation to graph and from graph to equation, until both readings become automatic. The journey alternates free exploration with increasingly demanding identification challenges.
- Build y = m·x + n with sliders and read the table of values generated automatically (room 1).
- Calculate points for a specific function and discover that the difference between consecutive heights is always m — "the step height" (rooms 2 and 3).
- Identify the slope m directly from five different graphs (room 4).
- Understand the y-intercept n as the point where the line crosses the y-axis, and locate it visually with a fixed m (rooms 5 and 6).
- Read m and n from a graph and write the full expression; then do the reverse by choosing the correct graph from four very similar ones (rooms 7 and 8).
- Explore families of lines: lines with the same n all pass through (0, n); lines with the same m are parallel (room 9).
- Calculate slope using the formula m = Δy/Δx by dragging two points, and solve step-height challenges (room 10).
- Deduce m and n from two given points: first find the slope, then solve for n (room 11).
- Find the intercepts with the axes Ox and Oy by setting y = 0 and x = 0 respectively (room 12).
Key mathematical ideas
- The linear function y = m·x + n is a straight line in the Cartesian plane; each parameter has a precise geometric role.
- The slope m measures the vertical change per unit of horizontal change: m = Δy / Δx.
- The y-intercept n is the value of y when x = 0; it shifts the line vertically without changing its inclination.
- Two lines with the same m are parallel; two lines with the same n meet at (0, n).
- The x-intercept is found by setting y = 0 → x = −n/m; the y-intercept is always (0, n).
- Given two points, the line is fully determined: compute m from the ratio of differences and solve for n.
Room-by-room contents
Room 1 · Build y = m·x + n
The student uses two sliders (m and n) and observes in real time how the value table and the line on the graph change. This is the interactive introduction to the form y = m·x + n.
Student tasks
- Move the m and n sliders.
- Observe how the table fills in and how the line shifts and rotates.
Room 2 · The height of the step
The fixed function y = 3·x − 4 is used. The student calculates y for four values of x and then observes the highlighted "staircase" on the graph, seeing that each step always rises by the same amount.
Student tasks
- Calculate y for four values of x in y = 3·x − 4.
- Note how much y rises from one step to the next and relate it to m = 3.
Room 3 · The slope m
Slope is explored with a slider: an animation shows the reading "1 step to the right, m steps up (or down)". The geometric meaning of m as the step height is consolidated.
Student tasks
- Move the m slider and observe the step animation.
- Check what happens when m is negative (the line goes downward).
Room 4 · Identify the slope
A quiz with five sub-questions: a line is shown and the student chooses the correct slope from several options. Practice reading m directly from the graph.
Student tasks
- Analyse each line and estimate its slope.
- Select the correct value of m in the five mini-quizzes.
Room 5 · The y-intercept n
With m fixed at 3, the student moves n with a slider and sees how the line shifts up and down in parallel without changing its inclination. This introduces n as the "starting point" on the y-axis.
Student tasks
- Move the n slider with m = 3 fixed.
- Describe what changes in the line as n is raised or lowered.
Room 6 · Find n
The slope m is given and the line is shown; the student must discover n by reading on the graph where the line crosses the y-axis (the y-intercept).
Student tasks
- Visually locate where the line crosses the y-axis.
- Write the correct value of n.
Room 7 · From graph to equation
Complete reverse challenge: a line is shown and the student must write both m and n to obtain the equation y = m·x + n. This integrates the skills from rooms 4 and 6.
Student tasks
- Read the slope m from the graph.
- Read the y-intercept n and write the complete equation.
Room 8 · Recognise the graph
The opposite direction to room 7: an equation is given and the student chooses which of four very similar graphs corresponds to it. Develops visual discrimination between lines with similar m and n.
Student tasks
- Read the given equation.
- Identify and select the correct graph from the four options.
Room 9 · Families of lines
Two families are explored: lines with the same n (concurrent at (0, n)) and lines with the same m (parallel). The student switches modes and verifies both properties with animation.
Student tasks
- Activate "same n" mode and verify that all lines pass through (0, n).
- Switch to "same m" mode and see that the lines are parallel.
Room 10 · Slope as a step
Two draggable points on the graph; moving them visualises the step Δy/Δx. Then, two numerical exercises in which the student calculates Δy, Δx and m.
Student tasks
- Drag the two points and observe the step they form.
- Calculate Δy, Δx and the slope m in the two proposed exercises.
Room 11 · The line through two points
Given two specific points, the student derives the complete equation: first calculating m using the step formula (m = Δy/Δx), then solving for n by substituting a point into y = m·x + n.
Student tasks
- Calculate the slope m using the step formula.
- Substitute a point to derive n and write the final equation.
Room 12 · Intersections with the axes
Given y = m·x + n, the student finds both intercepts: with the y-axis (immediately, it is n) and with the x-axis (by setting y = 0 and solving x = −n/m). The route closes with a direct application of the equation.
Student tasks
- Identify the y-axis intercept without calculating (it is n).
- Set y = 0 to find the x-axis intercept and verify it on the graph.
Rooms to project
The most striking ones to show and discuss in class.