HSS.ID.C.7Common CoreMathStatistics and ProbabilityGrades 9-12
HSS.ID.C.7: Interpreting the Slope and Intercept of a Linear Model
In plain English: HSS.ID.C.7 is the Common Core statistics standard that asks students to explain what the slope and the intercept of a linear model mean in the context of real data: the slope as the predicted change in y for each one-unit increase in x, and the intercept as the predicted y when x is 0, including when that value makes no sense. It is usually taught in Algebra I and introductory Statistics.
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
Common Core State Standards for Mathematics · Domain: Interpreting Categorical and Quantitative Data (ID) · Cluster: Interpret linear models Also written as HSS-ID.C.7 or S-ID.7 · Official standard
Students learn to read the two numbers in a fitted linear model, predicted y = a + bx, as statements about the data they came from. The slope b is a rate of change: the predicted change in the response variable for each one-unit increase in the explanatory variable, with units such as dollars per year or inches per shoe size. The intercept a is the constant term: the predicted value of the response when the explanatory variable is 0.
The lesson spends as much time on judging the intercept as on stating it. Sometimes x = 0 is inside or near the data and the intercept is a sensible starting value; often it is far outside the data, or impossible, and the honest interpretation is that it only positions the line. Students practice sentences that name both variables, include units, say "predicted" and avoid claiming that x causes y.
Learning Objectives
By the end of this lesson, students will be able to:
Interpret the slope of a linear model as the predicted change in the response variable per one-unit increase in the explanatory variable, with units, in the context of the data
Use the slope to state the predicted change in y for a change in x of more than one unit
Interpret the intercept of a linear model as the predicted value of the response when the explanatory variable is 0
Decide whether the intercept has a practical meaning by comparing x = 0 with the range of the data and with the context
Correct interpretations that swap the variables, omit units, treat predictions as exact or claim cause and effect
Prior Knowledge Required
Students should already be comfortable with:
Finding and interpreting the rate of change and initial value of a linear function 8.F.B.4
Using the equation of a linear model to solve problems and interpret slope and intercept in bivariate data 8.SP.A.3
Making scatter plots and describing linear association 8.SP.A.1
Post the prompt. Give students three minutes alone, then one minute to compare with a partner.
Warm-Up Prompt
"A tomato seedling is 12 cm tall when it is planted and grows 3 cm each week. Write an equation for its height h after w weeks. Which number tells you where it started, and which tells you how fast it changes? Now imagine you measured 20 real seedlings every week instead. Would the numbers in your equation still be exact?"
Record h = 12 + 3w. Students name 12 as the starting height and 3 as the growth per week (8.F.B.4). The last question sets up the lesson: with real data, points scatter around a line, so a fitted model gives predicted values, and its slope and intercept describe the typical pattern, not every plant.
Direct Instruction20 minutes
Vocabulary. In a linear model predicted y = a + bx, x is the explanatory variable and y is the response. The slope b is the rate of change: predicted change in y per one-unit increase in x, in units of y per unit of x. The intercept a is the constant term: predicted y when x = 0, in units of y. Show the invented data below for a sample of 10 used cars of one model, then fit the least-squares line with technology.
Invented data: 10 used cars of the same model
Car
1
2
3
4
5
6
7
8
9
10
Age (years)
1
2
2
3
4
5
6
7
8
9
Price ($ thousands)
24.1
21.0
23.2
19.0
19.8
15.5
16.4
12.1
13.0
9.3
Name the variables and units: x = age in years, y = price in thousands of dollars.
Slope sentence: "For each additional [unit of x], the predicted [y] increases (or decreases) by [|b| units of y]."
Intercept sentence: "When [x] is 0, the predicted [y] is [a units of y]."
Judge the intercept: is x = 0 possible, and is it inside or close to the range of the data? If not, say that the intercept has no practical meaning and only positions the line.
Check the language: say "predicted" or "on average", keep x and y in the right roles, and do not say that x causes y.
Negative slope, sensible intercept
Technology gives the least-squares line for the used-car table: predicted price = 25.48 - 1.73(age), in thousands of dollars.
Equation: Slope -1.73: for each additional year of age, the predicted price is about $1,730 lower. Intercept 25.48: a car 0 years old has a predicted price of about $25,480. The youngest car is 1 year old, so this is a short extrapolation, but it is a believable price for a new car.
Intercept with no practical meaning
Invented data for 10 adult men with shoe sizes 8 to 13 give predicted height = 53.3 + 1.63(shoe size), in inches (Diagram 2).
Equation: Slope 1.63: each additional shoe size goes with a predicted height about 1.6 inches greater. Intercept 53.3: an adult shoe size of 0 does not exist and is far outside sizes 8-13, so the intercept only positions the line.
Slope for a change of several units
Invented data for 15 households give predicted monthly grocery bill = 180 + 95(number of people), in dollars.
Equation: Slope 95: each additional person goes with a predicted bill $95 higher, so a household with 3 more people has a predicted bill 3 × 95 = $285 higher. Intercept 180: a household of 0 people cannot exist, so 180 is not a meaningful bill.
Intercept as a starting amount
A school raffle records unsold tickets each day for 12 days, starting on day 0. The fitted model is predicted unsold tickets = 500 - 38(days since the sale began).
Equation: Slope -38: the predicted number of unsold tickets falls by 38 per day, so about 38 tickets are sold per day. Intercept 500: on day 0 the model predicts 500 unsold tickets, the amount available at the start, and day 0 is part of the data.
Use Diagram 1 to show the slope as rise over a run of one year, and Diagram 2 to show why an intercept that lies far from the data is an extrapolation.
Guided Practice15-20 minutes
Pairs write a slope sentence and an intercept sentence for each model, then decide whether the intercept is meaningful. Share out after each model. All data are invented.
Screens and sleep: for 40 students, predicted hours of sleep = 8.4 - 0.012(minutes on a phone after 9 p.m.), with minutes from 0 to 180. (Slope: each extra minute goes with 0.012 fewer predicted hours of sleep, so 60 more minutes goes with 0.72 hours, about 43 minutes, less. Intercept: 8.4 predicted hours for a student with no phone time; meaningful, because 0 minutes is in the data.)
Latitude and January temperature: for 30 cities between 25° and 48° north, predicted average January temperature = 139 - 2.75(latitude), in °F. (Slope: each degree farther north goes with a predicted January average 2.75°F colder. Intercept: 139°F at the equator is far outside the data and hotter than any city average, so it has no practical meaning.)
Candle: a class measures a candle every hour for 7 hours: predicted height = 20.3 - 2.4(hours burned), in cm. (Slope: the candle is predicted to lose 2.4 cm per hour. Intercept: 20.3 cm is the predicted height before lighting, which is meaningful because hour 0 was measured.)
Listen for sentences that say "the slope is -2.4" without saying what changes, and for intercept sentences that forget the units.
Independent Practice15 minutes
Students work alone with technology. The invented data record the high temperature and the cups of hot cocoa sold at a school stand on 8 winter days.
Invented data: hot cocoa sales on 8 days
Day
1
2
3
4
5
6
7
8
High temperature (°F)
28
34
37
41
45
48
52
55
Cups sold
92
90
81
79
70
73
60
62
Tasks: (1) Use technology to fit a least-squares line. (Answer: predicted cups = 128.6 - 1.24(temperature).) (2) Interpret the slope in context. (Each 1°F warmer goes with about 1.24 fewer predicted cups sold.) (3) By how much does the predicted number of cups change on a day 10°F warmer? (About 12.4 fewer cups.) (4) Interpret the intercept and decide whether it is meaningful. (At 0°F the model predicts about 129 cups, but 0°F is far below the coldest day in the data, 28°F, so this is an extrapolation and should not be trusted.)
Closure5-10 minutes
Exit ticket: invented data for adults ages 20 to 70 give predicted push-ups in one minute = 52 - 0.45(age in years). (1) Interpret the slope, and state the predicted difference for two adults 10 years apart. (2) Interpret the intercept and explain whether it has a practical meaning. (Answers: each extra year of age goes with 0.45 fewer predicted push-ups, or 4.5 fewer per decade; the intercept is the prediction for age 0, which is outside ages 20-70 and makes no sense for push-ups, so it only positions the line.)
Differentiation Strategies
For Struggling Students
Give sentence frames for the slope and the intercept, with blanks for each variable name and unit, and color-code x and y in the model and in the frame
Start with models whose intercept is clearly a starting amount (the raffle, the candle) before models whose intercept is an extrapolation
Have students compute the prediction at x and at x + 1 and subtract, to see that the difference is always the slope
For Advanced Students
Ask students to rewrite predicted price = 25.48 - 1.73(age) using x = age in months and explain why the slope changes but the intercept does not
Ask students to find a data set in which the intercept is meaningful and one in which it is not, and justify each choice with the data range
Ask why a model with a negative intercept can still give good predictions inside the range of the data
Assessment Guidance
What to Look For
Strong interpretations name both variables in context, give the slope in units of y per unit of x, use "predicted" or "on average", and put the variables in the right roles. For the intercept, look for students who check x = 0 against the data range and the context before interpreting it, and who are willing to say that an intercept has no practical meaning. Watch for sentences that treat a model as exact for every individual, and for slope sentences that describe x changing in response to y.
02
Classroom Activities
3 Activities
1
Model Card Match
15 minPairs
Each pair gets 18 cards: 6 model cards, 6 slope-sentence cards and 6 intercept-sentence cards. Pairs match each model with its two sentences, then mark each intercept card "meaningful" or "no practical meaning". All models come from invented data.
Model Cards
A. Predicted backpack weight (lb) = 3.5 + 1.2(number of textbooks), for 0 to 6 books
B. Predicted rain barrel volume (gallons) = 4 + 11(inches of rain that week), for 0 to 3 inches
C. Predicted resting heart rate (beats per minute) = 78 - 0.6(hours of exercise per week), for 0 to 12 hours
D. Predicted pizza price ($) = 1.40 + 0.95(diameter in inches), for 10- to 18-inch pizzas
E. Predicted texts sent per day = 95 - 3.1(age in years), for ages 14 to 25
F. Predicted minutes to run a mile = 11.2 - 0.3(weeks of training), for 0 to 10 weeks
Key for the Teacher
A: 1.2 lb more per book; an empty backpack is predicted to weigh 3.5 lb (meaningful)
B: 11 more gallons per inch of rain; 4 gallons predicted in a dry week (meaningful)
C: 0.6 fewer beats per minute per weekly hour of exercise; 78 bpm predicted with no exercise (meaningful)
D: $0.95 more per inch of diameter; a 0-inch pizza does not exist (no practical meaning)
E: 3.1 fewer texts per year of age; age 0 is far outside ages 14-25 (no practical meaning)
F: 0.3 minutes faster per week of training; 11.2 minutes predicted before training starts (meaningful)
Modification for Distance Learning
Put the 18 cards on a shared slide and have pairs drag them into six rows, then type the "meaningful" decision next to each intercept card.
2
Stacking Cups Model
25 minGroups of 3-4
Groups measure the height of nested stacks of 1 to 10 identical plastic cups, fit a line with technology, and interpret both numbers physically. The slope turns out to be the height each cup adds (its lip), and the intercept is a height with no stack behind it.
Procedure
Stack cups inside one another and measure the height of stacks of 1, 2, ..., 10 cups to the nearest millimeter
Enter (number of cups, height in cm) into a calculator, Desmos or a spreadsheet and fit a least-squares line
Write a slope sentence and an intercept sentence with units
Measure one cup and its lip, and compare them with the slope and the intercept
Discussion Questions
What part of a cup does the slope measure? Why is it much smaller than the height of one cup?
A stack of 0 cups has height 0. Why is the intercept not 0? What length of a single cup does it match?
Would a different kind of cup change the slope, the intercept, or both?
Challenge Variation
Give each group a second kind of cup. Groups fit a second line, then predict how many cups of each kind make stacks of equal height and test the prediction.
3
Fix the Interpretation
15 minGroups of 3
Groups get one model and six flawed interpretations. They name the error in each statement and rewrite it correctly. The model: for invented data on 24 weekend days with highs from 55°F to 90°F, predicted bikes rented at a rental shop = -38 + 2.2(high temperature in °F).
Flawed Statements
"Each bike rented raises the temperature by 2.2 degrees." (variables swapped)
"Every 1°F increase makes exactly 2.2 more people rent bikes." (treats the model as exact and causal)
"At 0°F the shop rents -38 bikes." (an impossible extrapolation, far below 55°F)
"The slope of the model is -38." (confuses the intercept with the slope)
"On a day 10°F warmer, predicted rentals go up by 2.2." (forgets to multiply: 10 × 2.2 = 22)
"The intercept means 38 of the shop's bikes are broken." (invents a meaning the data cannot support)
Procedure
Each student fixes two statements, then passes the sheet to the next student to check
The group writes one final slope sentence and one final intercept sentence on chart paper
Groups post their chart paper and compare wording in a short gallery walk
Modification for Distance Learning
Post the six statements in a shared document. Each student comments a fix under one statement, and the group votes on the clearest rewrite.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Slope and Intercept of a Fitted Line
Invented prices of 10 used cars, drawn to scale with the least-squares line predicted price = 25.48 - 1.73(age). The triangle shows the slope: one more year of age, 1.73 thousand dollars lower predicted price. The open circle at (0, 25.48) is the intercept, the predicted price of a car 0 years old.
Diagram 2: When the Intercept Is an Extrapolation
Invented heights of 10 adult men with shoe sizes 8 to 13 and the line predicted height = 53.3 + 1.63(shoe size), drawn to scale. The solid part covers the data. The dashed part runs back to shoe size 0, which no adult wears, so the intercept of 53.3 inches has no practical meaning.
04
Homework Assignment
~30 min
HSS.ID.C.7 Homework: Slope and Intercept in Context
Directions: Write every interpretation as a full sentence that names both variables, gives units and uses the word "predicted". For each intercept, say whether it has a practical meaning and why. Use a graphing calculator, Desmos or a spreadsheet for Problem 3. All data are invented.
Part 1: Interpreting Slope and Intercept (Problems 1-3)
A model for 20 apartments from 450 to 1,400 square feet is predicted monthly rent = 410 + 1.35(square feet), in dollars. (a) Interpret the slope. (b) How much higher is the predicted rent of an apartment 200 square feet larger than another? (c) Interpret the intercept and decide whether it has a practical meaning.
For 25 vehicles weighing 2,500 to 5,500 pounds, predicted fuel efficiency = 44.8 - 0.0052(weight in pounds), in miles per gallon. (a) Interpret the slope per pound and per 1,000 pounds. (b) What is the predicted difference in fuel efficiency between a 3,000-pound car and a 4,500-pound SUV?
Six students recorded their weekly typing practice and their typing speed: (1 hour, 28 words per minute), (2, 34), (4, 37), (5, 44), (7, 46), (9, 55). (a) Use technology to fit a least-squares line, rounding the slope to two decimal places and the intercept to one. (b) Interpret the slope. (c) Interpret the intercept and decide whether it is meaningful.
Part 2: Intercepts, Errors and Comparisons (Problems 4-6)
For the years 2010 to 2023, a model of a theater chain's average ticket price is predicted price = 7.90 + 0.21t, in dollars, where t is years since 2010. (a) Interpret 7.90 and 0.21. (b) Rewrite the model using x = years since 2020 and interpret the new intercept.
For 40 coffee shops on days from 60°F to 95°F, predicted iced coffees sold = -45 + 3.2(temperature in °F). Jordan writes: "The slope means an iced coffee costs $3.20 more for each degree, and the intercept means the shops lose 45 coffees at 0°F." Explain both errors and write correct interpretations.
A nursery tracks two tree species for 1 to 10 years after planting. Species A: predicted height = 4.5 + 1.8(years); Species B: predicted height = 7.0 + 1.1(years), in feet. (a) Interpret each slope and intercept. (b) Which species is predicted to grow faster, and by how much per year? (c) After how many years do the two models predict the same height? Round to the nearest tenth.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Slope Interpretation
Predicted change in y per one unit of x, with units, variables in the right roles
Correct number but units or context missing
Variables swapped or slope misidentified
Intercept Interpretation
Predicted y at x = 0 in context, judged against the data range
Stated correctly but not judged
Missing or confused with the slope
Computation
Fitted line, multi-unit changes and predictions correct
Choose an answer or write your own, then read the explanation. All data sets are invented. The score updates as you go, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
For invented data on 25 bakeries, predicted loaves sold per day = 310 - 18p, where p is the price of a loaf in dollars. What does -18 mean?
Answer: A
The slope is the predicted change in the response (loaves sold) for each one-unit increase in the explanatory variable (price in dollars). Choice D swaps the roles of the variables. Choice B describes x = 0, which is the intercept's job, and uses the wrong number.
Question 2 of 20 · Multiple Choice
For 30 puppies of one breed weighed from birth to 12 weeks (invented data), predicted weight = 2.1 + 1.4w, in pounds, where w is age in weeks. What does 2.1 mean?
Answer: B
2.1 is the intercept, the predicted weight at w = 0, which is birth. Week 0 is part of the data, so the value is meaningful. Choice A describes a slope, and the slope here is 1.4, not 2.1. Choice D would be 2.1 + 1.4 = 3.5 pounds.
Question 3 of 20 · Multiple Choice
For invented data on 22 runners, predicted 5K time = 34.5 - 0.9(weeks of training), in minutes. How does the predicted time change over 6 more weeks of training?
Answer: C
Each week changes the predicted time by -0.9 minutes, so 6 weeks change it by 6 × (-0.9) = -5.4 minutes. Choice B gives the change for only one week. Choice D uses 29.1, which is the predicted time after 6 weeks (34.5 - 5.4), not the change.
Question 4 of 20 · Multiple Choice
Which model has an intercept with a sensible interpretation in context? (All data are invented.)
Answer: D
In D, 0 hours is the start of the hike and is part of the data, so 3.0 liters is the predicted amount of water at the start. In A a house with 0 square feet does not exist, in B no newborn runs a marathon, and in C an engine of 0 liters is impossible: each of those intercepts is an extrapolation with no practical meaning.
Question 5 of 20 · Multiple Choice
For invented data on 18 bike commuters, predicted commute time = 4 + 3.6d, where time is in minutes and d is distance in miles. What are the units of the slope 3.6?
Answer: A
The slope is a rate: change in y (minutes) per one-unit change in x (miles), so it is 3.6 minutes per mile. Choice B inverts the rate. Choice C gives the units of the intercept, 4 minutes, which is the part of the trip that does not depend on distance.
Question 6 of 20 · Multiple Choice
For invented data on summer days from 70°F to 100°F, predicted visitors at a city pool = -210 + 9.5(temperature in °F). Which statement about the intercept is correct?
Answer: B
The intercept is the prediction at 0°F, which is -210 visitors. That is impossible, and 0°F is 70 degrees below the coolest day in the data, so it only positions the line. Choices C and D treat -210 as a rate of change; the rate of change is 9.5 visitors per degree. Choice A drops the negative sign.
Question 7 of 20 · Multiple Choice
For invented data on 12 used laptops, predicted battery life = 9.8 - 0.75(years of use), in hours. Which interpretation of the slope is best?
Answer: D
The slope gives the predicted change per one-unit increase in x, and 0.75 hours is 45 minutes. Choice A is close but claims that every laptop changes by exactly that amount; real laptops scatter around the line. Choice B uses the intercept as the slope.
Question 8 of 20 · Short Answer
For invented data on apple trees 3 to 15 years old in one orchard, predicted apples per tree = 40 + 22(age in years). Interpret the slope and the intercept, and say whether the intercept has a practical meaning.
Slope: each additional year of age goes with about 22 more predicted apples per tree. Intercept: the model predicts 40 apples for a tree of age 0. That has no practical meaning: age 0 is outside the data (3 to 15 years), and a newly planted tree does not bear 40 apples.
Question 9 of 20 · Multiple Choice
For the years 2015 to 2024, a community college's predicted enrollment = 8,400 + 260t, where t is years since 2015. What does 8,400 represent?
Answer: A
The intercept is the predicted value when t = 0, and t = 0 means the year 2015, which is in the data. Choice C forgets that t counts years since 2015. Choice B describes the slope, 260 students per year.
Question 10 of 20 · Multiple Choice
Using the model predicted enrollment = 8,400 + 260t (t = years since 2015), by how much is predicted enrollment expected to grow from 2018 to 2023?
Answer: D
From 2018 to 2023 is 5 years, so predicted enrollment grows by 5 × 260 = 1,300 students. Choice B counts 6 years. Choice C, 8,400 + 1,300 = 9,700, is a predicted enrollment (for 2020), not a growth amount.
Question 11 of 20 · Multiple Choice
For a class, predicted final exam score = 20 + 0.8(midterm score). A student writes: "The slope means that each extra point on the final adds 0.8 points to the midterm." What is wrong?
Answer: C
The explanatory variable is the midterm score and the response is the predicted final score, so the rate is 0.8 final points per midterm point. The student read the model backward. Choice B confuses the slope with the intercept, which is the prediction for a midterm score of 0.
Question 12 of 20 · Short Answer
A lab group heats water on a hot plate and records its temperature every minute from t = 0 to t = 20. The fitted model is predicted temperature = 21.5 + 3.2t, in °C. Interpret the intercept and say whether it is meaningful.
The intercept 21.5 is the predicted temperature at t = 0, when heating starts: about 21.5°C, close to room temperature. It is meaningful because t = 0 is a measured time in the data.
Question 13 of 20 · Multiple Choice
Two new stores track weekly sales. Store P: predicted units sold = 200 + 15w. Store Q: predicted units sold = 350 + 9w, where w is weeks since opening (invented data for weeks 1 to 20). Which statement is correct?
Answer: B
The slopes are the growth rates: 15 units per week for P and 9 for Q, a difference of 15 - 9 = 6 units per week. Choice A compares intercepts, which describe predicted sales at opening, not growth. Choice D uses the slope as the intercept.
Question 14 of 20 · Short Answer
For invented data from 16 points along a mountain trail, predicted temperature = 72 - 0.0035(elevation in feet), in °F. Interpret the slope per foot and per 1,000 feet of elevation.
Each additional foot of elevation goes with a predicted temperature 0.0035°F lower. For 1,000 more feet, the predicted temperature is 1,000 × 0.0035 = 3.5°F lower.
Question 15 of 20 · Multiple Choice
For invented data on 50 adults with heights from 150 cm to 185 cm, predicted arm span = -6 + 1.03(height), in cm. What does the intercept -6 tell you?
Answer: D
The intercept is the prediction at x = 0: an arm span of -6 cm for a height of 0 cm, far outside 150-185 cm and impossible. Choice A misreads the intercept as a difference between arm span and height; for a 170 cm adult the model predicts 169.1 cm, not 164. Choice C mixes the slope with the intercept.
Question 16 of 20 · Multiple Choice
Invented data for 5 homes give (miles from the city center, price in $ thousands): (2, 620), (5, 560), (8, 515), (12, 440), (15, 395). Use technology to fit a least-squares line. Which interpretation of its slope is correct?
Answer: A
Technology gives predicted price ≈ 651.15 - 17.28(miles), in thousands of dollars, so the slope is about -17.28 thousand dollars, or $17,280, per mile. Choice B ignores that prices are in thousands. Choice C interprets the intercept, not the slope. Choice D uses the mean price.
Question 17 of 20 · Short Answer
For invented data, predicted phone battery remaining = 100 - 8.5h, in percent, where h is hours of video streaming since a full charge (h from 0 to 10). Interpret the slope and the intercept.
Slope: each additional hour of streaming goes with 8.5 percentage points less predicted battery. Intercept: at h = 0, right after a full charge, the predicted battery level is 100%, which is meaningful because h = 0 is in the data.
Question 18 of 20 · Multiple Choice
For invented data on toddlers aged 18 to 36 months, predicted number of words known = -310 + 25(age in months). A parent says the model shows that a newborn knows -310 words. What is the best response?
Answer: B
Age 0 is far outside the data, and a negative word count is impossible, so the intercept has no practical meaning. Inside the data range the model can still be useful. Choice D goes too far: a meaningless intercept does not make the predictions for ages 18 to 36 months wrong.
Question 19 of 20 · Short Answer
A student measured a plant each week after transplanting it: (week 0, 6.0 cm), (1, 8.9), (2, 12.2), (3, 14.8), (4, 18.1), (5, 20.6). Use technology to fit a least-squares line, then interpret its slope and intercept.
Technology gives predicted height ≈ 6.06 + 2.95w. Slope: the plant is predicted to grow about 2.95 cm per week. Intercept: the predicted height at transplanting (week 0) is about 6.06 cm, which is meaningful because week 0 was measured (6.0 cm).
Question 20 of 20 · Multiple Choice
For invented data, predicted hours of sleep = 9.1 - 0.4(hours of homework). Two students' homework times differ by 2.5 hours. How far apart are their predicted hours of sleep?
Answer: C
The predicted difference is 2.5 × 0.4 = 1.0 hour, with less predicted sleep for the student who does more homework. Choice A is the difference for only one hour of homework. Choice B, 9.1 - 1.0, is a predicted amount of sleep (for 2.5 hours of homework), not a difference.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSS.ID.C.7 mean?
It means students can explain the slope and the intercept of a linear model in words that fit the data. The slope is the predicted change in y for each one-unit increase in x, and the intercept is the predicted y when x is 0, which students must judge for sense before they use it.
Is HSS.ID.C.7 taught in Algebra 1 or in Statistics?
Both. It is usually taught in Algebra I, alongside fitting lines to scatter plots, and again in an introductory Statistics course, where the models come from least-squares regression output.
How do you write a slope interpretation in context?
Use a frame such as: "For each additional [unit of x], the predicted [y] increases (or decreases) by [slope with units of y]." For example, "For each additional year of age, the predicted price of the car is about $1,730 lower." Name both variables, give units, and say "predicted" or "on average".
When does the y-intercept not make sense?
When x = 0 is impossible in context or lies far outside the range of the data. A shoe size of 0, a pizza with a 0-inch diameter or a bicycle with 0 gears are all examples. In those cases say that the intercept is an extrapolation with no practical meaning and that it only fixes where the line sits.
Why should interpretations say "predicted"?
Because a fitted model describes the overall pattern, not each individual. The data points scatter around the line, so the slope is the change in the predicted value, and a particular car or student can be above or below it.
Does a slope show that x causes y?
No. A slope describes how the predicted y changes with x in the data, not why. With observational data, a lurking variable may drive both. Deciding when a relationship is causal is the subject of HSS.ID.C.9, so interpretations under HSS.ID.C.7 should say "is associated with" or "goes with" rather than "causes".
What are the units of a slope?
Units of y per unit of x, for example degrees per hour, liters per day or inches per shoe size. The intercept has the units of y alone. Writing the units is a quick check that the variables are in the right roles.
How is HSS.ID.C.7 different from 8.SP.A.3?
It builds on it. In grade 8 students use a given linear model and interpret its slope and intercept. In high school the models come from data fitted with technology, the contexts are more varied, and students must decide when the intercept has no practical meaning and when a prediction is an extrapolation.
Is interpreting slope and intercept on the SAT?
Yes, it is a real connection. The digital SAT asks what the slope or intercept of a linear model means in context, in the Algebra domain and, for lines of best fit on scatter plots, in Problem-Solving and Data Analysis.
What happens to the intercept if x is measured as years since 2000?
The intercept becomes the predicted value in 2000, because that is when x = 0. Shifting the starting year changes the intercept but not the slope. This is why a model in "years since" form usually has a meaningful intercept, while a model using the calendar year directly does not.
07
Related Standards
6 standards
These standards connect to HSS.ID.C.7: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.B.4Prerequisite
Construct a linear function and interpret its rate of change and initial value
