Your First Lesson on Linear Functions: What to Cover and How (Ontario, Canada)

The first lesson on linear functions decides how the rest of the unit goes. Get it right and students spend the next three weeks connecting tables, graphs and equations. Get it wrong and they spend those weeks memorizing y equals mx plus b as a shape with no meaning. This is a lesson plan for that opening period in an Ontario classroom, built around one idea: constant rate of change, seen three ways before any formula appears.

The one idea to teach first

Do not open with the equation. Open with a situation that changes at a steady rate. A candle burning down two centimeters an hour. A taxi that charges four dollars to get in and two dollars a kilometer. Students already understand these. What they lack is the vocabulary linking the starting amount to the initial value and the steady change to the rate of change, and the Ontario Curriculum expectations for algebra at this level all run through that connection. If students leave able to say what changes, by how much, and where it started, the slope formula becomes shorthand for something they can already talk about rather than a rule taken on faith.

A four-phase sequence with timings

1. Minutes 0 to 8, the silent table. Put a partly filled table of values on the board with the x column complete and only two y values given. Students copy it and fill in the rest without instruction. Almost every student succeeds. Then ask how they knew. The answers are rate of change in their own words, and that phrasing is worth writing on the board and keeping there.
2. Minutes 8 to 22, three representations of one story. Pairs produce a table, a graph and a sentence rule for the taxi, in that order, on one sheet. Insist that the graph is labeled with units. The purpose is to establish that these are three views of one relationship rather than three tasks, which is what later makes translating between forms feel obvious.
3. Minutes 22 to 38, changing one thing at a time. Now change the flag fall to seven dollars and keep the per-kilometer rate. Students predict what moves before they draw. Then change the rate and keep the flag fall. This isolates the two parameters so that when the symbols arrive, each already has a job.
4. Minutes 38 to 50, naming it. Only now write y equals mx plus b, and map each letter back onto the taxi. Five minutes of notation, then a short practice set matching four equations to four graphs. If the earlier phases went well, this part is fast.

Running this smoothly depends on having varied practice ready at the right moment rather than improvising. The Linear Functions – Teaching Kit with Puzzle, Talk Cards & Practice suits this shape of lesson, with talk prompts for the paired phase and practice pages for the last fifteen minutes, so you are not writing questions on the board while thirty students wait.

The exit check that tells you what to do tomorrow

Two questions on a half sheet, five minutes, collected at the door. First, a table with a constant difference: state the rate of change and the initial value in words. Second, a graph with no equation: describe a real situation it could represent. Nobody writes an equation.

Sort the responses into three piles. Students who name both parameters correctly are ready to move to writing equations tomorrow. Students who find the rate but miss the initial value need one more scenario with a nonzero starting point. Students who describe the graph as going up without quantifying anything repeat the silent table with a partner. The sorting takes six minutes and it is the whole point of the check.

Where the unit goes from here

Lesson two writes equations from tables, lesson three moves between forms, and lesson four introduces systems, which is where students who never internalized rate of change begin to struggle silently. Having the next stage in hand makes pacing easy, and Algebra – Linear Functions and Systems of Equations | Foundations & Problem Solving Workbook | GRADE 7–10 covers that progression with differentiated levels, so a class that splits after the exit check can still work on one idea.

For the strongest students in a Grade 9 or Grade 10 destination course, the stretch is asking what happens when the parameters change systematically, which is the doorway into transformations. Algebra – Higher-Level Functions and Graph Transformations | Foundations for Pre-Calculus Workbook | GRADE 9–12 supports that without pulling the rest of the class forward too early. When the first lesson lands, you hear it two weeks later: a student looks at an unfamiliar graph, says the line climbs three each time and started at minus two, and writes the equation unprompted.

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