Grade 8 · Mathematics · 30 minutes
Unit 2 · Functions and linear systems · Lesson 1 of 2
Linear Functions in a Bike-Rental Model
Connect a table, graph and equation while distinguishing linear from nonlinear change.
Mr. S Teaches · Original starter mini-lesson · MS8-MATH-02Learning objective
Construct a linear rule from two data points and interpret its rate and initial value.
Before you begin
- Grade 7 equations
- Coordinate graphing
Gather your materials
- Paper and pencil
- Ruler and graph paper
Language to know
- function
- A rule assigning exactly one output to each input.
- slope
- Change in output divided by change in input on a line.
- initial value
- The output when the input is zero.
Understand the concept
A table represents a function when one input does not receive two different outputs. A linear relationship changes at a constant rate. In y = mx + b, m gives change per input unit and b is the output at zero. A quadratic-looking pattern such as square area does not have equal output changes for equal input steps.
Teach it step by step
- Read a fictional bike-rental table: hours 1, 3, 5; costs $11, $21, $31. Calculate change in cost divided by change in hours.
- Use the constant $5/hour rate to work backward from 1 hour to 0 hours.
- Write C = 5h + 6, plot (0,6), (1,11), (3,21) and (5,31), and connect the modeled values.
- Check that each allowed input has one output; explain that the real rental policy may restrict possible hours.
- Compare square side/area pairs (1,1), (2,4), (3,9) and explain why those changes are not linear.
Make the thinking visible
Worked example or model response
(21 − 11)/(3 − 1) = $5/hour. At zero hours the rule gives 11 − 5 = $6, so C = 5h + 6. A cost of $36 gives 5h + 6 = 36 and h = 6 hours.
Guided practice
Build a rule from points (2,13) and (4,19). Find the output at zero and at input 5, then plot the points.
Independent practice
1. Is {(1,2),(1,3),(2,4)} a function? 2. A service table has (0,4),(2,10),(4,16). Write its linear equation and explain both constants.
Quick assessment
How can a linear function fail to be proportional?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: Rate (19−13)/(4−2) = 3; initial value 7; y = 3x + 7, so y = 22 at x = 5.
- Independent 1: No; input 1 has outputs 2 and 3. 2: y = 3x + 4; rate 3 output units per input unit and initial value 4.
- Quick check: A nonzero initial value produces a line that does not pass through (0,0), so the ratio y/x is not constant.
Look for: Look for an explanation of the choices, not only a final answer. Require units and a check against the context.
Make it work for your learner
More support
Color-code input and output columns and label the units on a slope triangle.
Extension challenge
Construct two functions with equal outputs at x = 4 but different rates, and predict which is larger afterward.
Selected contributions · Official sources
Standards connections
Contributes to Common Core standard
8.F.A.1Common Core · Grade 8
Decide whether a set of input-output pairs represents a function.
Read the official standard ↗This lesson contributes selected skills; it does not fully address the standard.
Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.
Contributes to Common Core standard
8.F.A.3Common Core · Grade 8
Contrast constant-rate linear and nonconstant-rate examples.
Read the official standard ↗This lesson contributes selected skills; it does not fully address the standard.
Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.
Contributes to Common Core standard
8.F.B.4Common Core · Grade 8
Construct a linear rule and interpret its rate and starting value.
Read the official standard ↗This lesson contributes selected skills; it does not fully address the standard.
Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.