Substitute Guide: Linear Relationships in Four Forms
Learning goal (plain language)
Students practice linear relationships — connecting the same "straight-line" relationship across a table, a graph, an equation (y = mx + b), and a real-world story. They find the slope (the rate, m), the y-intercept (the starting amount, b), and tell whether a relationship is proportional (through the origin) or not. 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 §111.28 (Grade 8 Mathematics). Knowledge & skills 8.4 and 8.5 — proportional and non-proportional linear relationships, slope, and y-intercept.
- Primary — 8.4A: use similar right triangles / rates to develop the idea of slope as a constant rate of change (rise over run).
- Primary — 8.4C: use data to write an equation in the form y = mx + b and interpret slope (m) and y-intercept (b) in context.
- Primary — 8.5I: distinguish proportional (y = kx, through the origin) from non-proportional (y = mx + b, b ≠ 0) linear relationships across tables, graphs, and equations.
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.)
Setup Before class (5 min)
- Count and hand out one student packet per student.
- Confirm each student has a pencil. A basic 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.
Materials What's needed
- Printed student packet (4–6 pages) per student.
- Pencil per student; a basic four-function calculator is optional.
- This guide and the separate answer key (teacher only).
- No technology, no internet, no special supplies.
Timing Suggested pacing (~45–60 min)
| Segment | Time | What students do |
|---|---|---|
| Start — Retrieval warm-up | 5 min | Q1–Q2 (evaluate a rule; read a point off the line) |
| Build — Study the math | 5–10 min | Read reference box, worked example, graph; Q3 |
| Apply — Use what you know | 20–25 min | Q4–Q6 (four forms, error analysis, predict from model) |
| Explain — Justify (CER) | 5–10 min | Q7 claim/evidence/reasoning |
| Close — ACE | 5 min | Articulate / Connect / Extend |
| Continue — early finishers | optional ~15 min | Write a context for y = 8x + 15 |
Script Read aloud to start
"Today's assignment is a paper math packet called Linear Relationships in Four Forms. You will work by yourself with a pencil. A basic calculator is allowed if you have one, but you do not need one. The most important rule: show your work — write down the numbers 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 45 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 y = 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 y) ÷ (change in x), with the y-difference on top.
- 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 Q7 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 Q7 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 (four-function) calculator, quiet self-read-aloud. Not needed / not allowed: phones, internet, graphing/AI calculators, 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 (Q5) and the justification (Q7), which reveal the reversed rise/run slope error and whether students can compare two linear models using both slope and y-intercept. Watch for slope vs. y-intercept swaps on the four-forms item (Q4) and the belief that every line is proportional (Q4d, Q7 check this — only lines through the origin are). The separate answer key lists worked steps, alternate strategies, and a 3-point rubric for Q7.