Lesson type: Connected representations — one linear relationship seen five ways.
Stepping Stone: connect linear representations — move between a story, a table, coordinates, an equation and a graph of the same line.
NEO Cornerstones: Connection · Movement · Reflection · Creativity · Rest · Nutrition
45-minute shape: Connect 15 · Move 8 · Reflect 10 · Create 5 · Rest/Nourish 4 · Evidence 3
Choose only the support that helps now. Every route returns you to this lesson.
Every graph in this unit sits on a grid of coordinates: a way of pairing numbers with position on a page. That idea was shaped by the philosopher and mathematician René Descartes (1596–1650) — which is why we call it the Cartesian plane. Here is his story, in about three minutes.
Captions are on their way. If you would rather skip the film, that is completely fine — scroll on to begin.
Linear relationship — a relationship whose graph is a straight line: equal steps in x give equal steps in y.
Representation — a way of showing the relationship: in words, a table, coordinates, an equation, or a graph.
Equation y = mx + c — the line's rule: c is the y-intercept (the value of y when x = 0), and m is the step — how much y changes each time x goes up by 1.
What to do: drag the m and c sliders and watch all five representations change together — the story, the table, the coordinates, the equation and the graph.
What you are trying to notice: these are not five different things — they are five costumes on one relationship. Change the line and every view updates at once.
Graph & coordinates
Carry this forward: a straight line is one relationship in five costumes. Everything else in this unit is a closer look at its two numbers — m and c.
What to do: press Next step to move along the line one x at a time. Watch the dot climb the same amount at every step, and the running total grow.
What you are trying to notice: the line is a constant step — each time x goes up by 1, y goes up by the same amount, m. That steady step is what makes it straight.
Carry this forward: equal steps in x give equal steps in y. That equal step has a name you will meet soon: the gradient.
What to do: each card shows a line in one costume — a table or a story. Choose the equation that matches it, then check.
What you are trying to notice: you can move from any one representation to another by finding the two numbers: where it starts (c) and how much it steps (m).
Carry this forward: to match representations, read off the start and the step — that is c and m, the whole of y = mx + c.
What to do: pick a real situation, then set the m and c sliders so the line tells its story. Check how close you are.
What you are trying to notice: a straight line can model a real situation — the starting amount is c and the steady change is m.
Make it your own: on the Scratchpad, invent your own straight-line story and sketch its graph, table and equation.
Holding one relationship in five forms at once asks a lot of your attention. Take a breath before moving on.
Notice one thing: which costume of a line feels most natural to you — the story, the table, the equation or the graph? There is no right answer — just what your own attention noticed.
You can jot it on the Scratchpad, say it aloud, or simply hold it in mind. You can pause here and return whenever you are ready.
Mode: contextual. Straight-line relationships are everywhere a fixed amount meets a steady rate.
Why it matters: a phone tariff (a monthly fee plus a cost per gigabyte), a taxi fare (a flag price plus a rate per mile), wages (a bonus plus an hourly rate), a savings plan, a currency conversion — each is a line with an intercept c (the fixed amount, when x = 0) and a steady rate m. People read these lines every day to compare deals and predict costs. Being able to move between the story, the numbers and the graph is what lets you choose well.
Next time you see a "fixed fee plus a rate", picture the line — what is its fixed amount (c), and how steep is it (m)?
Your task: show one line in five costumes.
Submission choices: an annotated Scratchpad image, a written solution, a labelled poster, or a spoken/video walkthrough.
Filename pattern: NEO_MATH_SL_A_LINE_AS_A_RELATIONSHIP_FirstName_Date
| Criterion | Approaching Proficiency | Developing Proficiency | Proficiency | Refining Proficiency |
|---|---|---|---|---|
| Understanding | Knows a line can be shown as a graph and an equation. | Links two representations with support. | Shows one relationship in several representations. | Moves fluently between all five and states the shared relationship. |
| Application | Reads a value from a graph or table. | Builds a table from a rule with prompting. | Generates a table, coordinates and graph from an equation. | Translates confidently in any direction, including from a story. |
| Communication | Records a representation with limited labelling. | Labels a table or graph. | Sets out each representation clearly. | Explains so another learner could rebuild the line. |
| Reflection | Names one thing noticed. | Spots that the views describe one line. | Explains how c and m show up across representations. | Connects representations to real linear situations. |
Use support in this order: 🤔 Socratic prompt → 💡 hint → ✅ check → 👀 worked solution. The worked solution unlocks separately for each question, once you have used the prompt, used the hint and checked two answers.
Use the Scratchpad to predict, sketch, test and revise.
You can use this board, paper or another way of recording your thinking. The Scratchpad is optional.
Angles are in degrees. Examples: 9 × sin(37) = 5.42; sin⁻¹(0.5) = 30