8.SP.A.3Common CoreMathStatistics and ProbabilityGrade 8
8.SP.A.3: Using a Linear Model and Interpreting Its Slope and Intercept
In plain English: 8.SP.A.3 is the Common Core grade 8 math standard that asks students to use the equation of a line of fit to answer questions about measured data. Students make predictions from the equation and explain its slope and intercept in the context, with units, such as 1.5 cm more plant height for each extra hour of daily sunlight. It builds on fitting lines in 8.SP.A.2.
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.
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Investigate patterns of association in bivariate data. Also written as 8.SP.3 · Official standard
In 8.SP.A.2, students fit a straight line to a scatter plot by eye. In this lesson they use the equation of that line, y = mx + b, as a linear model: a rule that describes the pattern in measured data and can be used to make predictions. The data are bivariate measurement data, meaning two measurements from each subject, such as the hours of sunlight a plant gets and its height when fully grown.
Students interpret the two numbers in the equation. The slope m is the predicted change in y for each increase of 1 in x, with units such as cm per hour. The intercept b is the predicted y when x = 0, which only has a real meaning when x = 0 makes sense and is close to the data. Students also solve problems with the model, both ways: find y from x, and find x from y. The lesson works through the official example about plant height and sunlight. All data sets are invented for teaching.
Learning Objectives
By the end of this lesson, students will be able to:
Use the equation of a linear model to predict y from x, and x from y, in the context of measured data
Interpret the slope of a linear model as the predicted change in y for each 1-unit increase in x, with units
Interpret the intercept as the predicted value of y when x = 0, and decide whether it makes sense in the context
Explain why a prediction far outside the data can be unreliable, and why "is associated with" is used instead of "causes"
Prior Knowledge Required
Students should already be comfortable with:
Fitting a line to a scatter plot by eye and judging the fit 8.SP.A.2
Slope and y-intercept in y = mx + b as the rate of change and initial value 8.F.B.4
Solving one-variable linear equations such as 1.5x + 9 = 27 8.EE.C.7
Show Diagram 1 with the side panel covered. A science class grew 12 bean plants, each with a set number of hours of light each day, and measured each plant's height when it stopped growing (invented data). A line of fit has been drawn.
Warm-Up Prompt
"Use the line to estimate the height of a plant that gets 8 hours of light a day and one that gets 9 hours. How much taller is the second plant predicted to be? What would the line say about a plant that gets no light at all? Do you believe it?"
Reading the graph, students find about 21 cm and 22.5 cm, a difference of about 1.5 cm. Following the dashed part of the line to x = 0 gives 9 cm, and many students will say a plant with no light would not grow like that. Keep both ideas on the board: the difference for one extra hour is the slope, and the value at 0 is the intercept.
Direct Instruction20 minutes
Build the key ideas one at a time, and have students copy each sentence frame:
Linear model: the equation y = mx + b of a line of fit. It predicts a typical y for a given x; real data points are near it, not on it.
Slope: the predicted change in y when x increases by 1. Sentence frame: "Each additional [1 unit of x] is associated with [m units of y] more (or less) [y]." The units of the slope are y-units per x-unit.
Intercept: the predicted y when x = 0. Sentence frame: "When [x] is 0, the model predicts [b units of y]." Then ask: does x = 0 make sense, and is it close to the data?
Associated with, not causes: data show that two measurements go together. The words "is associated with" say that without claiming that one causes the other.
Predicting y from x: substitute the x-value into the equation.
Predicting x from y: set the equation equal to the y-value and solve for x.
Stay near the data: a prediction for an x-value far outside the data can be impossible, because the pattern may not continue.
The official example: slope of a plant model (Diagram 1)
In a biology experiment, the line of fit for the bean plants is y = 1.5x + 9, where x is the hours of sunlight each day and y is the mature height in cm. Interpret the slope of 1.5 cm/hr.
Equation: The slope 1.5 cm/hr means that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. It does not mean a plant grows 1.5 cm every hour: the "hr" is an hour of daily sunlight, not an hour of growing time.
Intercept that does not make sense (Diagram 1)
Interpret the intercept 9 of the plant model, and predict the mature height of a plant that gets 10 hours of sunlight each day.
Equation: When x = 0, the model predicts 9 cm. The data only go from 4 to 12 hours, and a bean plant with no light would not grow to 9 cm, so the intercept has no real meaning here. Prediction for 10 hours: 1.5(10) + 9 = 24 cm, inside the data range.
Intercept that makes sense, and solving for x (Diagram 2)
Phones that started fully charged streamed video, and their battery percent was recorded (invented data). The line of fit is y = -12x + 98, where x is the hours of streaming. Interpret the slope and intercept, and predict when the battery reaches 20%.
Equation: Slope: each additional hour of streaming is associated with 12 fewer percentage points of battery (for example, from 60% down to 48%). Intercept: at 0 hours the model predicts 98%, close to the full charge the phones started with, so it makes sense. Solve -12x + 98 = 20: -12x = -78, so x = 6.5 hours.
Using the slope for a difference
In the bean experiment, one plant gets 3 more hours of sunlight each day than another. How much taller is it predicted to be? How many hours of sunlight does the model link with a mature height of 27 cm?
Equation: Each hour adds 1.5 cm, so 3 hours add 3 × 1.5 = 4.5 cm. For 27 cm, solve 1.5x + 9 = 27: 1.5x = 18, so x = 12 hours.
Crickets and temperature
Students counted cricket chirps in 15 seconds on 10 summer evenings (invented data, 16 to 43 chirps) and recorded the temperature in °F. The line of fit is y = x + 40. Predict the temperature for 30 chirps, and interpret the slope and intercept.
Equation: Prediction: 30 + 40 = 70 °F. Slope: each additional chirp in 15 seconds is associated with a temperature 1 °F higher. Intercept: 0 chirps gives 40 °F, but crickets are mostly quiet in the cold, and 0 chirps is far below the data, so the intercept is not a useful temperature.
Use Diagram 1 to show the slope as a 1-hour step with a 1.5 cm rise, and the dashed part of the line as the region with no data. Use Diagram 2 to show solving for x as reading the graph backward: go across from 20% to the line, then down to 6.5 hours.
Guided Practice15 minutes
A family recorded the average outside temperature and its gas heating bill for 8 winter and spring months (invented data). The line of fit is y = -3x + 215, where x is the average temperature in °F and y is the bill in dollars.
Invented data: average monthly temperature and gas heating bill
Average temperature (°F)
Gas bill ($)
31
124
35
106
40
97
44
81
50
66
55
49
58
40
64
22
Guided practice questions with answers
Question
Answer
Interpret the slope -3.
Each additional 1 °F of average temperature is associated with a bill $3 lower
Interpret the intercept 215. Does it make sense?
At 0 °F the model predicts $215. The data go from 31 °F to 64 °F, so 0 °F is far outside them; treat it as a rough guess at best
Predict the bill for a month that averages 45 °F.
-3(45) + 215 = 80, so about $80
For which average temperature does the model predict a $125 bill?
-3x + 215 = 125, so -3x = -90 and x = 30 °F
One month averages 10 °F warmer than another. How do the predicted bills compare?
10 × 3 = 30, so the warmer month's bill is predicted to be $30 lower
Listen for students who say "the bill goes down $3 each month": x is temperature, not time. Ask them to say the full sentence with both units.
Independent Practice15 minutes
Students work alone, then check with a partner. A swim coach recorded the weekly practice hours of 10 swimmers and their best time for 50 meters (invented data, 2 to 12 hours). The line of fit is y = -0.8x + 42, with y in seconds.
Invented data: weekly practice and 50-meter swim time
Practice (hours per week)
Time (s)
2
40.1
3
39.8
4
38.2
5
38.4
6
37
7
36.1
8
35.9
9
34.4
10
34.3
12
32.2
Independent practice problems with answers
Problem
Answer
Interpret the slope -0.8.
Each additional hour of practice per week is associated with a time 0.8 seconds faster
Interpret the intercept 42. Is it reliable?
A swimmer who does not practice is predicted to swim 50 m in 42 s. 0 hours is outside the data, so it is only a rough guess
Predict the time for a swimmer who practices 7.5 hours a week.
-0.8(7.5) + 42 = 36 seconds
How many hours a week does the model link with a time of 34 seconds?
-0.8x + 42 = 34, so x = 10 hours
What does the model predict for 60 hours a week? Explain.
-0.8(60) + 42 = -6 seconds, which is impossible: 60 hours is far outside the data
Closure5-10 minutes
Exit ticket: a bookstore's line of fit for the price of paperback books is y = 0.02x + 4, where x is the number of pages and y is the price in dollars (invented data, 150 to 600 pages). (1) Predict the price of a 300-page book. ($10.) (2) Interpret the slope with units. (Each additional page is associated with a price 2 cents higher.) (3) Does the intercept make sense? (It predicts $4 for a book with 0 pages; that is outside the data, though it could stand for costs such as the cover.)
Differentiation Strategies
For Struggling Students
Give the two sentence frames on a card, with blanks for the units, and have students fill in the units before any numbers
Make a two-column table of x and predicted y for x = 0, 1, 2 and 3, so students see the slope as the step from one row to the next and the intercept as the first row
Use the graph first: read predictions from Diagram 1 or 2, then check them with the equation
For Advanced Students
Ask students to rewrite the plant model with x in minutes of daily sunlight instead of hours, and explain how the slope changes (1.5 cm per 60 minutes, 0.025 cm per minute)
Have students find a data set online or in a science book, fit a line by eye, write its equation and interpret both numbers
Extension (beyond this standard): in high school (HSS.ID.C.7), students interpret the slope and intercept of models computed with technology, and in HSS.ID.C.9 they study why a pattern alone does not prove a cause
Assessment Guidance
What to Look For
A complete interpretation of the slope names both variables, both units, the amount of change and its direction, and uses "is associated with" or "predicted". A complete interpretation of the intercept says what x = 0 means and whether it makes sense and is near the data. Watch for students who read "cm/hr" as growth per hour of time, who switch x and y in the sentence, who treat the intercept as always meaningful, and who trust predictions far outside the data.
02
Classroom Activities
3 Activities
1
Slope and Intercept Match
15 minGroups of 3
Each group gets 12 cards: 4 model cards, 4 slope cards and 4 intercept cards. Groups make 4 sets of three cards and decide whether each intercept makes sense.
The 4 Model Cards (invented data)
Model 1: kittens aged 0 to 8 weeks, y = 100x + 110, where x is age in weeks and y is weight in grams
Model 2: a bath cooling for 30 minutes, y = -0.4x + 40, where x is minutes and y is water temperature in °C
Model 3: bike rides of 2 to 15 km, y = 3.5x + 2, where x is distance in km and y is time in minutes
Model 4: stacks of 2 to 20 paper cups, y = 0.6x + 8.4, where x is the number of cups and y is the height of the stack in cm
The Slope and Intercept Cards (shuffled)
S1: Each additional cup is associated with 0.6 cm more height.
S2: Each additional minute is associated with a temperature 0.4 °C lower.
S3: Each additional week of age is associated with 100 g more weight.
S4: Each additional km is associated with 3.5 more minutes.
I1: At 0 km the model predicts 2 minutes, time spent on things like unlocking the bike.
I2: At birth the model predicts 110 g.
I3: A stack of 0 cups is predicted to be 8.4 cm tall, which makes no sense.
I4: When the bath is first filled, the model predicts 40 °C.
Answer Key
Model 1: S3 and I2 (makes sense: newborn kittens weigh about 100 g)
Model 2: S2 and I4 (makes sense)
Model 3: S4 and I1 (makes sense, as a small start-up time)
Model 4: S1 and I3 (does not make sense: 0 cups is outside the data)
Discussion Questions
Model 4's intercept is close to the height of one cup. Why might that be, even though 0 cups makes no sense?
Model 2 is the only model with a negative slope. What does the negative sign change in the sentence?
2
Bounce Lab
20 minGroups of 4
Groups drop a tennis ball from 7 heights between 50 and 200 cm, measure the height of the first bounce, fit a line by eye and use its equation as a model.
Procedure
Tape a meter stick or tape measure to a wall; one student drops the ball from the chosen height, and another watches the bottom of the ball at the top of its bounce
Drop from each height twice and record the average of the two bounces
Plot drop height (x) and bounce height (y), fit a line by eye, and write its equation from two points on the line
Interpret the slope and intercept, then use the model to predict the bounce from 250 cm and test it
Sample Data and Key (invented)
Drop and bounce heights in cm: (50, 30), (75, 39), (100, 57), (125, 66), (150, 83), (175, 93), (200, 110). One line of fit is y = 0.54x + 2. Slope: each additional cm of drop height is associated with 0.54 cm more bounce, so the ball keeps a bit more than half its height. Intercept: a ball dropped from 0 cm cannot bounce, so the true value is 0; the 2 cm comes from measuring error and from fitting the line by eye. Prediction from 250 cm: 0.54(250) + 2 = 137 cm.
Discussion Questions
Why is it reasonable for this model's intercept to be close to 0?
Would a basketball give the same slope? What would a larger slope mean?
Your prediction from 250 cm was outside your data. How close was your test?
Modification for Distance Learning
Students record a short video of each drop next to a tape measure and pause it at the top of the bounce to read the height.
3
Is It Safe to Predict?
15 minPairs
Pairs get 4 prediction cards. Each card gives a model, the range of its data and two x-values. Pairs make both predictions and label each one safe (inside the data) or risky (outside the data), explaining any impossible result.
The 4 Cards (invented data)
Card A: free-throw percent y = 4x + 45 for 1 to 8 hours of practice a week; predict for 5 hours and for 20 hours
Card B: cups of lemonade sold y = 3x - 120 for days from 50 °F to 90 °F; predict for 70 °F and for 30 °F
Card C: height in cm y = 6x + 78 for children aged 4 to 12; predict for age 8 and for age 25
Card D: phone battery percent y = 1.4x + 6 for 0 to 60 minutes of charging; predict for 40 minutes and for 90 minutes
Answer Key
Card A: 65% (safe); 125% (impossible, a percent of shots made cannot pass 100)
Card B: 90 cups (safe); -30 cups (impossible)
Card C: 126 cm (safe); 228 cm (impossible, since people stop growing in their late teens)
Card D: 62% (safe); 132% (impossible, since the battery stops at 100%)
Discussion Questions
Every risky prediction on these cards was impossible. Can a prediction outside the data be possible but still wrong? Give an example.
For Card C, what happens to real children's growth that the straight line misses?
Challenge Variation
For each card, pairs find the largest x-value for which the prediction is still possible, such as the hours of practice at which Card A reaches 100%.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Plant Model y = 1.5x + 9
Invented data for 12 bean plants, drawn to scale. The line of fit y = 1.5x + 9 is solid over the data (4 to 12 hours) and dashed where there are no data. The red step shows the slope: 1 more hour of daily sunlight goes with 1.5 cm more mature height. The open circle marks the intercept (0, 9), which lies outside the data.
Diagram 2: The Battery Model y = -12x + 98
Invented data for 10 phones that streamed video, drawn to scale. The intercept (0, 98) is close to a full charge, so it makes sense here. The dashed red path reads the model backward: 20% battery left goes with x = 6.5 hours of streaming.
04
Homework Assignment
~30 min
8.SP.A.3 Homework: Using Linear Models
Directions: Each model is a line of fit for invented data. Write every interpretation as a full sentence with both units. Show the equation you solve for each prediction.
Part 1: Interpret the Slope and Intercept (Problems 1-3)
A shop's line of fit for the monthly rainfall and umbrella sales is y = 6x + 25, where x is the rainfall in cm (data from 2 to 20 cm) and y is the number of umbrellas sold. (a) Interpret the slope. (b) Interpret the intercept. (c) Does the intercept make sense? Explain.
In a science fair experiment, the line of fit for a dwarf sunflower variety is y = 4.5x + 60, where x is the hours of sunlight each day (data from 6 to 12 hours) and y is the mature height in cm. (a) Interpret the slope of 4.5 cm/hr in a sentence. (b) Explain why it does not mean that a sunflower grows 4.5 cm every hour. (c) Does the intercept make sense? Explain.
Students measured the temperature at which water boils at 6 places at different elevations. Their line of fit is y = -3.3x + 100, where x is the elevation in km (data from 0 to 4 km) and y is the boiling point in °C. (a) Interpret the slope. (b) Interpret the intercept, and say whether it makes sense. (c) Predict the boiling point in a city 1.6 km above sea level.
Part 2: Solve Problems with the Model (Problems 4-6)
A pot of water on a stove is measured every minute for 9 minutes. The line of fit is y = 8x + 20, where x is minutes and y is the water temperature in °C. (a) Predict the temperature after 5 minutes. (b) When does the model predict 84 °C? (c) What does the model predict after 15 minutes, and why can that not be right?
A driver records the distance driven since filling up and the fuel left. The line of fit is y = -0.07x + 50, where x is km driven (data from 0 to 600 km) and y is liters of fuel. (a) Interpret the slope and the intercept. (b) After how many km does the model predict 15 liters left? (c) How much fuel does the model predict is used on a 200 km trip?
Two classes grew bean sprouts. Class A's line of fit is y = 1.1x + 3 and Class B's is y = 0.8x + 5, where x is days after sprouting (data from 0 to 14 days) and y is height in cm. (a) Which class's sprouts grew faster? How do you know? (b) Predict each class's height on day 10. (c) On which day do the two models predict the same height?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Slope Sentences
Names both variables and units, the amount and the direction
Missing units or direction
Missing or wrong
Intercept Sentences
States the prediction at x = 0 and judges whether it makes sense
States the value without judging it
Missing or wrong
Predictions
Correct equation and answer with units, both directions
One computing error
Missing or wrong method
Reasonableness
Explains every impossible or risky prediction
Notices a problem but does not explain it
Accepts impossible results
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. All data sets and models are invented. Your score updates as you answer, 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
An ice cream shop's line of fit is y = 8x - 380, where x is the day's high temperature in °F (data from 70 °F to 95 °F) and y is the number of cones sold. What does the slope 8 mean?
Answer: C
The slope is the change in y (cones) for each increase of 1 in x (°F), so each extra degree is associated with about 8 more cones. Choice A switches the two variables. Choice B describes an intercept, and the intercept here is -380. Choice D treats x as days, but x is temperature.
Question 2 of 20 · Multiple Choice
A line of fit for laptops is y = -0.6x + 9, where x is the laptop's age in years (data from 0 to 6 years) and y is its battery life in hours. What does the intercept 9 mean?
Answer: A
At x = 0 (a new laptop) the model predicts 9 hours, and age 0 is in the data, so the intercept makes sense. Choice B mixes up the intercept and the slope, which is -0.6 hours per year. Choice C reads 9 as an x-value: the model gives 3.6 hours at 9 years, not 0.
Question 3 of 20 · Multiple Choice
Use the laptop model y = -0.6x + 9. What battery life does it predict for a 4-year-old laptop?
Answer: D
-0.6(4) + 9 = -2.4 + 9 = 6.6 hours. Choice A adds 2.4 instead of subtracting it. Choice B subtracts the age 4 without multiplying by the slope. Choice C subtracts the slope only once, as if the laptop were 1 year old.
Question 4 of 20 · Multiple Choice
A line of fit for children aged 6 to 16 is y = -0.25x + 12.5, where x is age in years and y is hours of sleep per night. What does the slope mean?
Answer: B
The slope -0.25 means 0.25 hours less sleep for each year, and 0.25 hours is 15 minutes. Choice A reads 0.25 hours as 25 minutes. Choice C drops the negative sign. Choice D uses the intercept as the slope.
Question 5 of 20 · Multiple Choice
Use the ice cream model y = 8x - 380. For which high temperature does it predict 260 cones sold?
Answer: A
Solve 8x - 380 = 260: 8x = 640, so x = 80 °F, inside the data. Choice B divides 260 by 8 and ignores the -380. Choice C subtracts 380 instead of adding it: (260 - 380) ÷ 8 = -15. Choice D substitutes 260 for x instead of for y.
Question 6 of 20 · Multiple Choice
The ice cream model y = 8x - 380 has an intercept of -380. How should it be interpreted?
Answer: B
The intercept is the prediction at 0 °F: -380 cones, which is impossible, and 0 °F is far below the 70 to 95 °F data. So it has no real meaning. Choice C drops the sign and treats 0 °F as a real day in the data. Choice D confuses the intercept with the slope.
Question 7 of 20 · Multiple Choice
In another biology experiment, a linear model for tomato plants has a slope of 2.4 cm/hr, where x is the hours of sunlight each day and y is the mature height in cm. Which interpretation is correct?
Answer: D
The slope compares plants: one more hour of sunlight each day is associated with about 2.4 cm more mature height. Choice A reads cm/hr as a growth speed over time, a common misreading of this kind of slope. Choice B describes an intercept. Choice C turns the ratio upside down.
Question 8 of 20 · Multiple Choice
Use the tomato model with slope 2.4 cm/hr. One plant gets 4 more hours of sunlight each day than another. How much taller is it predicted to be?
Answer: B
Each extra hour adds 2.4 cm, so 4 hours add 4 × 2.4 = 9.6 cm. Choice A uses a single hour. Choice C adds 4 and 2.4 instead of multiplying. Choice D divides 2.4 by 4.
Question 9 of 20 · Multiple Choice
When does the intercept of a linear model have a real meaning in the context?
Answer: C
The intercept is a prediction at x = 0, so it is only meaningful if x = 0 is a real situation near the data, such as a phone at 0 hours of streaming. Choice A fails for the plant model, whose intercept 9 is positive but meaningless. Choice D confuses a point on the graph with a meaning in the context.
Question 10 of 20 · Multiple Choice
What does the laptop model y = -0.6x + 9 predict for a 20-year-old laptop, and what does that show?
Answer: B
-0.6(20) + 9 = -12 + 9 = -3 hours, which is impossible. The data only cover 0 to 6 years, and the pattern does not continue that far. Choice A drops the negative sign. Choice D gives -0.6 × 20 = -12 without adding the intercept 9. Choice C adds 12 to 9.
Question 11 of 20 · Multiple Choice
A line of fit for a car is y = -0.4x + 58, where x is the highway speed in mph (data from 50 to 80 mph) and y is the fuel economy in miles per gallon. What does the slope mean?
Answer: A
The slope -0.4 is the change in miles per gallon for each 1 mph increase in speed, and the negative sign means it goes down. Choice B drops the sign. Choice C switches x and y. Choice D uses the intercept instead of the slope.
Question 12 of 20 · Multiple Choice
A line of fit for weekly grocery bills is y = 55x + 40, where x is the number of people in a household (data from 1 to 6) and y is the bill in dollars. Predict the bill for a household of 4.
Answer: D
55(4) + 40 = 220 + 40 = $260. Choice A forgets to add the intercept 40. Choice B multiplies both numbers by 4. Choice C adds 55, 40 and 4.
Question 13 of 20 · Multiple Choice
Why do statisticians say the slope "is associated with" a change instead of saying one variable "causes" the change?
Answer: D
A scatter plot shows that two measurements go together; something else may explain the pattern, so the careful wording avoids claiming a cause. Choice A is true of many slopes but is not the reason for the wording. Choice C is false: neither number proves a cause.
Question 14 of 20 · Multiple Choice
After a snowstorm, on a warm sunny day, a line of fit for the snow depth passes through (2, 40) and (6, 28), where x is hours after the storm ended and y is depth in cm. What is the intercept of this model?
Answer: C
The slope is (28 - 40) ÷ (6 - 2) = -3 cm per hour. Going back 2 hours from (2, 40) adds 2 × 3 = 6 cm, so the intercept is 46 cm: the predicted depth when the storm ended. Choice A subtracts the 6 cm instead of adding it. Choice B uses the first point, which is at 2 hours, not 0. Choice D gives the slope.
Question 15 of 20 · Short Answer
A vet's line of fit for 15 dogs is y = 0.6x - 14, where x is the height at the shoulder in cm (data from 30 to 70 cm) and y is the weight in kg (invented data). (a) Predict the weight of a dog 50 cm tall. (b) Interpret the slope. (c) Interpret the intercept and say whether it makes sense.
(a) 0.6(50) - 14 = 30 - 14 = 16 kg. (b) Each additional cm of shoulder height is associated with 0.6 kg more weight. (c) A dog 0 cm tall is predicted to weigh -14 kg. That is impossible, and 0 cm is far outside the data, so the intercept has no real meaning.
Question 16 of 20 · Short Answer
A line of fit for apartments is y = -45x + 2,400, where x is the distance from the city center in km (data from 1 to 20 km) and y is the monthly rent in dollars (invented data). (a) Interpret the slope. (b) At what distance does the model predict a rent of $1,950?
(a) Each additional km from the center is associated with a rent $45 lower per month. (b) Solve -45x + 2,400 = 1,950: -45x = -450, so x = 10 km.
Question 17 of 20 · Short Answer
A hiker's line of fit is y = -0.18x + 2, where x is the km hiked and y is the liters of water left in her bottles (invented data). (a) Interpret the intercept. Does it make sense? (b) After how many km does the model predict 0.38 liters left?
(a) At 0 km the model predicts 2 liters, the water she starts with. It makes sense, because 0 km is the start of the hike. (b) Solve -0.18x + 2 = 0.38: -0.18x = -1.62, so x = 9 km.
Question 18 of 20 · Short Answer
In a biology experiment, a linear model for the mature height of pea plants is y = 2.5x + 30, where x is the grams of fertilizer given each week (data from 2 to 10 grams) and y is the height in cm. Interpret the slope and the intercept.
Slope: each additional gram of fertilizer per week is associated with 2.5 cm more mature height. Intercept: with no fertilizer, the model predicts a height of 30 cm. This one can make sense, since plants do grow without fertilizer and 0 grams is not far below the data, but it is still a prediction outside the data.
Question 19 of 20 · Short Answer
A line of fit for students' vertical jumps passes through (10, 44) and (30, 56), where x is the hours of training in a month and y is the jump height in cm (invented data, 8 to 32 hours). Find the slope and intercept and interpret both.
Slope: (56 - 44) ÷ (30 - 10) = 12 ÷ 20 = 0.6 cm per hour: each additional hour of training a month is associated with a jump 0.6 cm higher. Intercept: 44 - 0.6(10) = 38 cm, the predicted jump with no training. It is a rough estimate, since 0 hours is a little outside the data. The model is y = 0.6x + 38.
Question 20 of 20 · Short Answer
A line of fit for a young oak tree's height is y = 0.9x + 0.4, where x is the age in years (data from 1 to 8 years) and y is the height in meters. Use it to predict the height at 100 years, and explain whether you trust the result.
0.9(100) + 0.4 = 90.4 m. This is not trustworthy: 100 years is far outside the 1 to 8 years of data, and trees grow more slowly as they age, so an oak would be far shorter than 90 m. A linear model should only be used near its data.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.SP.A.3 mean?
8.SP.A.3 means students use the equation of a line of fit to solve problems about measured data and explain what its slope and intercept mean in the situation. For example, they predict a plant's height from its hours of sunlight and explain the slope as the extra height linked with each extra hour.
What is a linear model in 8th grade math?
A linear model is the equation y = mx + b of a line that fits a scatter plot. It describes the typical relationship between two measurements and is used to make predictions. The data points are near the line, not exactly on it.
How do you interpret the slope in 8.SP.A.3?
Say how much y is predicted to change when x goes up by 1, with both units. A useful frame is: "Each additional [unit of x] is associated with [m units of y] more (or less) [y]." A negative slope means y is predicted to go down.
How do you interpret the y-intercept of a line of fit?
The intercept is the predicted y when x = 0. Say what x = 0 means in the situation, then decide whether it makes sense. It is meaningful when x = 0 is a real situation near the data, such as a phone at the start of streaming, and not meaningful when x = 0 is impossible or far from the data.
What does the official 8.SP.A.3 example about plants mean?
The standard's example says a slope of 1.5 cm/hr means an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. The "hr" is an hour of daily sunlight, so the slope compares plants grown with different amounts of light. It does not say how fast a plant grows each hour.
Why say "is associated with" instead of "causes"?
Because a scatter plot shows only that two measurements go together. Another factor could explain the pattern, or the data could come from observation rather than an experiment. "Is associated with" describes the pattern without claiming a cause.
Can you use a linear model to predict far outside the data?
It is risky. The pattern may change outside the data, and the model can give impossible answers such as a negative time or a battery above 100%. Predictions are most trustworthy for x-values inside the range of the data.
Is 8.SP.A.3 the same as 8.F.B.4?
No, but they are close. 8.F.B.4 is about building a linear function and reading its rate of change and initial value. 8.SP.A.3 applies the same ideas to a line of fit for scattered data, so students also judge whether the intercept makes sense and whether a prediction is trustworthy.
What mistakes do students make with 8.SP.A.3?
Common ones are leaving out units, switching x and y in the slope sentence, reading a slope like cm/hr as a speed over time, treating every intercept as meaningful, substituting a y-value for x, and trusting predictions far outside the data.
How does 8.SP.A.3 prepare students for high school?
In high school statistics (HSS.ID.C.7), students interpret the slope and intercept of linear models again, for models found with technology. They also study correlation and why a pattern does not prove a cause (HSS.ID.C.9). The sentence frames from 8.SP.A.3 carry over directly.
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Related Standards
5 standards
These standards connect to 8.SP.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.SP.A.2Prerequisite
Fit a straight line to a scatter plot informally and judge how well it fits
