Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Many everyday situations grow or shrink by a steady amount: a phone plan with a monthly fee plus a cost per gigabyte, a candle burning down a fixed length each hour, or a savings account with regular deposits. Students learn to capture such a situation as a linear function y = mx + b, where m is the rate of change and b is the initial value, the amount when the input is zero.
The information can arrive in several ways. A description may state both values directly. Two data points, such as 'after 3 hours the tank holds 140 liters and after 7 hours it holds 60 liters', require computing the rate (60 - 140) ÷ (7 - 3) = -20 and then working back to the starting amount of 200. Tables and graphs supply the same information visually. The final, and most often neglected, step is interpretation: explaining that -20 means the tank loses 20 liters every hour and 200 means it started with 200 liters.
For a $25 fee plus $8 per hour, students write y = 25x + 8. Ask which number is multiplied by the hours: that is the rate.
When an amount decreases over time the rate of change is negative. Writing +20 instead of -20 for a draining tank predicts the tank filling up.
If data starts at x = 3, the first y-value is not b. Students must step back to x = 0 using the rate.
Saying 'the slope is 4' is incomplete. A full interpretation says 4 dollars per ticket or 4 meters per second.
A water tank drains at a steady rate. After 3 hours it holds 140 liters; after 7 hours it holds 60 liters. Write a function for the amount of water w after h hours and interpret it.
Answer: w = -20h + 200. The tank began with 200 liters, drains 20 liters per hour, and empties after 10 hours.
Use situations students care about, such as streaming plans, fundraiser totals or phone battery drain, and have them collect or invent two data points before building the model. Asking 'what happens at zero?' every time keeps the initial value meaningful rather than a number to fill in.
Assessment items often ask students to choose the equation that models a situation and then to interpret the slope or y-intercept. Practice the sentence frames 'The ___ starts at ___' and 'For every additional ___, the ___ increases/decreases by ___'.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: C) c = 6h + 12
The cost per hour (6) is the rate, multiplied by h. The fixed $12 is the initial value, so c = 6h + 12.
Answer: 2 (also accepted: 2 cm)
(11 - 5) ÷ (5 - 2) = 6 ÷ 3 = 2 cm per week.
Answer: 1 (also accepted: 1 cm)
Work back 2 weeks: 5 - 2 × 2 = 1 cm. The model is height = 2w + 1.
Answer: D) The battery loses 8 percentage points each hour
-8 is the rate of change: the charge falls by 8 percentage points every hour. The 100 is the starting charge.
Answer: 14
Solve 25n + 150 = 500. Subtract 150: 25n = 350. Divide: n = 14.
Write a function that describes a relationship between two quantities.
A full lesson with slides, activities and an exit ticket on modeling with linear functions, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.F.B.4 with an answer key, ready in about a minute.
Make a worksheet →Turn modeling with linear functions into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It is the output when the input is zero, the b in y = mx + b. In a story it is usually a starting amount, fee or height.
Yes. A negative rate means the quantity decreases by the same amount for each unit of input, such as fuel used or money spent.