Substitute Guide: Linear Functions in Four Forms
Learning goal (plain language)
Students practice linear functions — connecting the same "straight-line" function across a table, a graph, an equation in function notation f(x) = mx + b, and a real-world context. They find the slope (the rate of change, m), the y-intercept (the starting amount, b), write equations from contexts, and solve equations such as f(x) = k. You do not need a math background to run this. Every formula and a worked example are printed in the student packet. Students show their work; you do not need to teach the method — the packet does the teaching. Your job is to hand it out, read the start script, keep the room on task, and collect the packets.
Standards alignment
Framework: 19 TAC §111.39 (Algebra I). Knowledge & skills for linear functions — slope as a rate of change, writing linear equations, solving linear equations, and interpreting linear functions in context.
- Provisional — A.2(C): write linear equations in slope-intercept form (y = mx + b or f(x) = mx + b) given a table, a graph, or a verbal description.
- Provisional — A.3(A): determine the slope of a line as a rate of change given two points, a table, or a graph.
- Provisional — A.3(B): interpret the meaning of slope and intercepts of a linear function in the context of a real-world situation.
- Provisional — A.5(A): solve linear equations in one variable, including equations of the form f(x) = k.
Provisional — pending educator verification against the current official TAC source.
(Standards are paraphrased here, not quoted. Please confirm codes and wording against your official source. This packet is not claimed to be "aligned to the TEKS" until reviewed.)
Setup Before class (5 min)
- Count and hand out one student packet per student.
- Confirm each student has a pencil. A basic or graphing calculator is allowed but not required; students who have one may use it.
- Write the date on the board and remind students to fill in the name line.
- Keep the answer key with you — it is a separate file and is not in the student packet.
- Standard vs. block: on a ~50-minute period, students do items 1–6, 8, and ACE (item 7 is optional). On a ~90-minute block, students also complete item 7, the extra applied modeling task.
Materials What's needed
- Printed student packet (5–7 pages) per student.
- Pencil per student; a basic four-function or graphing calculator is optional.
- This guide and the separate answer key (teacher only).
- No technology, no internet, no special supplies.
Timing Suggested pacing (standard ~50 min · block ~90 min)
| Segment | Standard (~50 min) | Block (~90 min) | What students do |
|---|---|---|---|
| Start — Retrieval warm-up | 6 min | 8 min | Q1–Q2 (evaluate f(x); slope from two points) |
| Build — Study the math | 8–12 min | 12–15 min | Read reference box, worked example, graph; Q3 |
| Apply — Use what you know | 20–26 min | 28–34 min | Q4–Q6 (four forms + solve, write from context, error analysis) |
| Block — Extra applied task | optional | 12–15 min | Q7 (draining-tank modeling; x-intercept) |
| Explain — Justify (CER) | 6–10 min | 8–12 min | Q8 claim/evidence/reasoning |
| Close — ACE | 5 min | 5–7 min | Articulate / Connect / Extend |
| Continue — early finishers | optional | optional | Write a context for f(x) = 7x + 20 |
Script Read aloud to start
"Today's assignment is a paper math packet called Linear Functions in Four Forms. You will work by yourself with a pencil. A basic or graphing calculator is allowed if you have one, but you do not need one. The most important rule: show your work — write down the numbers and steps you use, not just a final answer. Everything you need is printed in the packet, including a worked example, a graph, and all the formulas. Start with the warm-up, then read the Build section before the Apply questions. If a question is tricky, skip it and come back. Put your name, class period, and date at the top now. You have about [50 / 90] minutes; raise your hand if you need help reading a question."
Support When students ask for help
- Students show their work; you do not need to teach the method. Point them back to the printed reference box and the worked example — the method is right there.
- You may read any part aloud to a student or the class.
- If a student is stuck, prompt with "What does the worked example do?" or "In f(x) = mx + b, which number is the rate and which is the starting amount?" rather than giving the answer.
- For slope, remind them the packet shows: slope = rise ÷ run = (change in output) ÷ (change in x), with the output-difference on top.
- Remind students that f(4) means the output at x = 4 — it is not "f times 4."
- It is fine if a student cannot finish the optional early-finisher challenge.
Access Accommodations & language support
- Read-aloud of any question is allowed for all students.
- Point students to the worked example, the formulas in the reference box, and the sentence stems for Q8 and ACE.
- A calculator is allowed for any student who wants one — this removes the arithmetic barrier so students can focus on the reasoning.
- Extended time is fine; the core can stop after Q8 if time runs short.
- Students may show work with numbers and short phrases if writing full sentences is a barrier, as long as the reasoning is clear.
- The packet is grayscale-safe — the graph uses black lines, labeled points, and text, so it prints and reads clearly in black and white.
Tools Allowed & not allowed
Allowed: pencil, the printed packet, a basic or graphing calculator, quiet self-read-aloud. Not needed / not allowed: phones for messaging, internet, AI tools, or getting answers from another student.
Teacher follow-up (for the returning teacher)
The highest-value items to review next class are the error analysis (Q6) and the justification (Q8), which reveal the reversed rise/run slope error and whether students can compare two linear functions using both slope and y-intercept. Watch for slope vs. y-intercept swaps on the four-forms item (Q4), the f(x)-as-multiplication misconception, and the belief that every line is proportional (Q4e, Q8 check this — only lines through the origin are). The solve items (Q4d, Q7c) and the ACE Articulate prompt show who can move from evaluating a function to solving f(x) = k. The separate answer key lists worked steps, alternate strategies, misconceptions, and a 3-point rubric for Q8.