Linear Relationships in Four Forms
Overview
This is a self-contained, low-tech substitute packet in which students practice linear relationships. Working alone with a pencil (a basic calculator is allowed but not required), students evaluate a rule and read a point off a line, study a worked example that shows one relationship as a table, a graph, and the equation y = mx + b, and then apply the ideas: matching the four forms, interpreting slope and y-intercept in context, fixing a mis-found slope, and predicting a value from the model. They finish by justifying which of two plans costs less with a claim, evidence, and reasoning. A key idea throughout is telling proportional lines (through the origin) apart from non-proportional lines (with a starting amount).
At a glance
Grade: 8
Subject: Mathematics
Time: about 45 minutes core plus a 15-minute optional extension
Materials: printed packet and a pencil; a basic calculator is allowed but not required (no computer or internet)
Work mode: independent
Standards (provisional): 19 TAC §111.28 (Grade 8 Mathematics), 8.4A (slope as a constant rate of change), 8.4C (write y = mx + b from data and interpret slope and y-intercept), and 8.5I (distinguish proportional from non-proportional linear relationships). Provisional — pending educator verification against the current official TAC source. Standards are paraphrased, not quoted.
Accessible version of the student activity
The full student activity is reproduced below in plain, screen-reader-friendly HTML. It reflows on phones and at 200% zoom. Write your answers on the printed packet, and show your work.
Start (5 minutes) — Retrieval warm-up
- Evaluate a rule: the rule is y = 2x + 5. Find y when x = 4. Show the multiplication and addition.
- Read a point off a line: on the graph in the Build section (Figure 1), find the point where x = 2 and write it as an ordered pair (x, y).
Build (5–10 minutes) — Study the math
A linear relationship makes a straight line and changes by the same amount each step. The rate of change, or slope m, is how much y changes for each 1 that x increases: m = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁). The y-intercept b is the value of y when x = 0 — the point (0, b) where the line crosses the vertical axis. Together they give the slope-intercept form: y = mx + b. The same relationship can appear as a table (a fixed step in y for each fixed step in x; the x = 0 row gives b), a graph (a straight line crossing at (0, b)), an equation (y = mx + b), or a story (a starting amount b plus a steady rate m per one). A linear relationship is proportional only when b = 0 — the line passes through the origin (0, 0) and y = mx. If b is not 0, the line is linear but not proportional.
Worked example — one relationship, four forms
A pool starts at 4 inches of water, and a hose adds 3 inches every minute. Because there is a starting amount of 4, this is linear but NOT proportional. The slope is m = 3 inches per minute and the y-intercept is b = 4 inches, so the equation is y = 3x + 4 (x = minutes, y = depth in inches). Checking the slope from rows (1, 7) and (2, 10): m = (10 − 7) ÷ (2 − 1) = 3. The line crosses the y-axis at (0, 4), not the origin.
| Time x (minutes) | Depth y (inches) | Change in y |
|---|---|---|
| 0 | 4 | — (start) |
| 1 | 7 | +3 |
| 2 | 10 | +3 |
| 3 | 13 | +3 |
| 4 | 16 | +3 |
- Using y = 3x + 4, how deep would the water be at 6 minutes? Show your work.
Apply (20–25 minutes) — Use what you know
Table B (gym membership): 0 visits cost $20; 2 visits cost $30; 4 visits cost $40; 6 visits cost $50. Story: a gym charges a one-time $20 sign-up fee plus $5 for each visit.
| Visits x | Total cost y (dollars) |
|---|---|
| 0 | 20 |
| 2 | 30 |
| 4 | 40 |
| 6 | 50 |
- Match the four forms using Table B and the story:
- 4a. Find the slope m (dollars per visit); show m = (change in y) ÷ (change in x) using two rows.
- 4b. Find the y-intercept b (cost at 0 visits), then write the equation y = mx + b.
- 4c. Interpret in context: what does the slope mean, and what does the y-intercept mean (use "per visit" and "sign-up fee")?
- 4d. Is this relationship proportional or not proportional? Explain using the y-intercept.
- Error analysis (a mis-found slope): a student used rows (2, 30) and (4, 40) and wrote "m = run ÷ rise = (4 − 2) ÷ (40 − 30) = 2 ÷ 10 = 0.2." Find the mistake, explain it, and give the correct slope.
- Multi-step application using y = 5x + 20:
- 6a. Predict the total cost of 12 visits; show y = 5(12) + 20.
- 6b. A member paid $75 total. How many visits? Set up 5x + 20 = 75 and solve for x, showing your steps.
Explain (5–10 minutes) — Justify your strategy (Claim, Evidence, Reasoning)
Question 7. Two bike-rental plans: Plan A is y = 4x + 10 ($10 sign-up plus $4 per hour); Plan B is y = 6x ($6 per hour, no sign-up). For a 3-hour rental, which plan costs less, and why? Also state which plan is proportional. Write a Claim (your answer), Evidence (the numbers you calculated), and Reasoning (why your strategy works, using slope and y-intercept).
Sentence stems you may use: "For 3 hours, Plan ___ costs less because…"; "I found Plan A's cost by… and Plan B's cost by…"; "Plan B is proportional because its y-intercept is…"; "The slope tells me… and the y-intercept tells me…".
Close (5 minutes) — ACE
- Articulate: explain in your own words how to find the slope from two rows of a table using rise over run.
- Connect: name one packet item where you used y = mx + b.
- Extend: give a new real-life non-proportional linear relationship (a starting amount plus a steady rate) not used in this packet.
Continue (optional, ~15 minutes) — Early finisher
Write a context for the equation y = 8x + 15: invent a real-world situation the equation could describe (say what x and y stand for), explain what the slope 8 and the y-intercept 15 mean in your story, and use the model to predict y when x = 5. Make sure your story is non-proportional (it must have a starting amount).
Turn in
Hand in the whole packet with your name, class period, and date, with questions 1–7 and the ACE box answered and your work shown. Include the optional challenge if you did it.