Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
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
A line of best fit is more than a drawing once it has an equation. Suppose a model for a group of apple trees is y = 0.8x + 3, where x is the tree's age in years and y is its height in meters. The slope, 0.8, says that each additional year of age is associated with about 0.8 meters of extra height. The intercept, 3, is the model's height for a tree of age 0, which may or may not make physical sense.
This standard asks students to use such equations to answer questions in context: predict the height of a 6-year-old tree (0.8 × 6 + 3 = 7.8 m), find the age at which the model predicts 9 m, and explain what the slope and intercept mean in words with units. Students also learn to be cautious. Predictions inside the range of the data (interpolation) are more reliable than predictions far outside it (extrapolation), and an intercept can be meaningless when an input of zero lies far from the data, such as a person aged 0 in a study of adult runners.
A slope of 1.5 cm per hour does not mean the plant is 1.5 cm tall. It is the change in height for each extra hour.
Model predictions are estimates. Real data points scatter around the line, so actual values will differ somewhat.
A model of children's heights predicts a 40-year-old would be over 3 meters tall. Models only hold near the data used to build them.
In a model of car value against age, the intercept may be sensible, but in a model of adult foot length against height it is not. Students should say when it makes no sense.
A linear model for weekly hours of practice (x) and points scored per game (y) on a basketball team is y = 1.5x + 4. Interpret the slope and intercept, and predict the points for a player who practices 8 hours a week.
Answer: The model predicts about 16 points per game; the slope means 1.5 points per extra weekly hour of practice.
Build models from data the class has already plotted in 8.SP.A.1 and 8.SP.A.2, then ask students to write two sentences: one about the slope and one about the intercept, each naming both variables and units. Following up with an absurd extrapolation, such as predicting a 100-year-old's reaction time, makes the limits of models memorable.
State test items commonly give a model and ask what the slope represents or what value the model predicts. Students should practice substituting accurately and phrasing interpretations as associations rather than guarantees.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 90
y = 3(80) - 150 = 240 - 150 = 90 cups.
Answer: C) The puppy gains about 2.5 pounds per week
The slope is the change in mass for each additional week, about 2.5 pounds.
Answer: 4
Solve -60x + 420 = 180. Subtract 420: -60x = -240. Divide: x = 4 years.
Answer: A) A person of height 0 cm would have arm span -2 cm, which has no real meaning
The intercept is the predicted y when x = 0. A height of 0 cm is far outside the data, so the intercept has no meaningful interpretation here.
Answer: B) Age 30
Age 30 is far outside the 5 to 12 range used to build the model, so predicting there is risky extrapolation.
Answer: 7.8 (also accepted: 7.8 m)
y = 0.8 × 6 + 3 = 4.8 + 3 = 7.8 meters.
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
A full lesson with slides, activities and an exit ticket on using linear models with data, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.SP.A.3 with an answer key, ready in about a minute.
Make a worksheet →Turn using linear models with data into a quiz students answer online that marks itself, with a class summary for you.
Build a test →As the change in the y-variable associated with a one-unit increase in the x-variable, stated with both units, for example 1.5 cm of growth per extra hour of sunlight.
No. It only makes sense when an x-value of zero is realistic and close to the data. Students should say when it is not.