8.SP.A.2Common CoreMathStatistics and ProbabilityGrade 8
8.SP.A.2: Fitting a Straight Line to a Scatter Plot and Judging the Fit
In plain English: 8.SP.A.2 is the Common Core grade 8 math standard that asks students to know that straight lines are often used to model how two measured quantities are related. For a scatter plot with a roughly straight-line pattern, students fit a line by eye and judge how well it fits by how close the points are to it. It comes right before using the equation of that line in 8.SP.A.3.
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Investigate patterns of association in bivariate data. Also written as 8.SP.2 · Official standard
In 8.SP.A.1, students made scatter plots: graphs of bivariate data, which means two measurements taken from each subject, such as the age and the price of each car. This lesson asks what to do when the points form a roughly straight band. Students learn that a straight line is a common model for such data: a simple description that follows the pattern closely but not exactly. A line drawn to follow the pattern is called a line of fit (some books say trend line).
Students fit lines informally, which means by eye, with a ruler or a strand of spaghetti, and without formulas. They then judge the fit by looking at how close the points are to the line. They measure each point's vertical distance to the line (straight up or down on the graph) and find the average of those distances to compare two lines. They also meet data where a straight line is a poor model: points that curve, and points with no pattern at all. All data sets on this page are invented for teaching.
Learning Objectives
By the end of this lesson, students will be able to:
Explain why straight lines are often used to model the relationship between two measured quantities
Fit a line to a scatter plot by eye, following the direction of the points with about as many points above the line as below it
Measure the vertical distance from a point to a line of fit and find the average vertical distance
Judge whether a line fits well, and compare two lines for the same data
Recognize a curved pattern or no pattern, where a straight line is a poor model
Prior Knowledge Required
Students should already be comfortable with:
Making scatter plots and describing positive, negative and linear association 8.SP.A.1
Reading and graphing the line y = mx + b, where m is the slope and b is where the line crosses the y-axis 8.F.A.3
Give each pair a printed copy of the car scatter plot from Diagram 1 with no lines on it, and a strand of uncooked spaghetti. The data come from 12 used cars of the same model (invented data).
Invented data: age and price of 12 used cars of one model
Age (years)
Price ($ thousand)
1
22
2
23.5
2
21.5
3
20.5
4
19.5
4
17
5
17.5
6
15
7
14
8
11.5
9
10.5
10
13
Warm-Up Prompt
"Lay your spaghetti on the plot so that it shows the pattern of the points as well as you can. Then use it to guess the price of a 6-year-old car of this model. Compare your strand with your partner's. Are they in the same place?"
The strands should slope down from left to right, and guesses for a 6-year-old car usually land between $15,000 and $17,000. Ask two pairs with different strands to explain their choice. Then ask: "How could we decide which strand is better?" Keep the answers on the board; the lesson gives two tests.
Direct Instruction20 minutes
Define each word as it comes up, and have students copy the steps next to a sketch:
Linear association: the points of a scatter plot cluster around a straight line. Positive if the points rise from left to right, negative if they fall.
Why lines: a line is simple, it shows a steady rate of change, and it lets you estimate values you did not measure. Scientists, businesses and sports analysts use lines this way for car prices, growth, sales and many other measurements.
A model is not exact: real data scatter because of things the model leaves out, such as a car's mileage or condition. A good line of fit usually passes through few data points, or none.
Fitting a line by eye: (1) follow the direction of the band of points; (2) keep about as many points above the line as below it; (3) spread those points along the whole line, not all above on one end; (4) do not let one point far from the others (an outlier) pull the line away from the rest.
Vertical distance: for a point (x, y), find the line's value at the same x. The vertical distance is the difference between the two heights. The point is above the line if its y is larger.
Average vertical distance: add the distances without their signs (their absolute values) and divide by the number of points. A smaller average means the points are closer to the line, so the line fits better.
When a line is a poor model: if the points curve, the distances form a pattern, such as above at both ends and below in the middle. If there is no pattern, no line fits well.
Fitting a line by eye (Diagram 1, Line A)
Fit a line to the 12 used cars in the warm-up table. Line A passes through (0, 24.5) and (10, 9.5).
Equation: Line A falls 15 thousand dollars over 10 years, so y = -1.5x + 24.5. Five points are above it, five are below it and two are on it, and the points above and below are spread from the newest cars to the oldest. It follows the downward band, so it is a reasonable line of fit.
Comparing two lines (Diagram 1, Lines A and B)
Line B joins the first point (1, 22) and the last point (10, 13), so y = -x + 23. Use the average vertical distance to compare Lines A and B.
Equation: Line A: the 12 distances are 1, 2, 0, 0.5, 1, 1.5, 0.5, 0.5, 0, 1, 0.5 and 3.5, for a total of 12 and an average of 12 ÷ 12 = 1.0 thousand dollars. Line B: the total is 17.5, so the average is 17.5 ÷ 12 ≈ 1.46 thousand dollars. Line A fits better. Line B is too flat because the 10-year-old car costs more than the pattern suggests, and seven of the eight cars aged 4 to 10 years are below or on Line B.
Close points and far points (Diagram 2, left and middle)
A runner's 10 training runs (distance in km, time in minutes) are fit with y = 5.5x. The sleep and quiz scores of 10 students are fit with y = 1.5x + 2.
Equation: Runs: the average vertical distance is 0.95 minutes, small next to times of 12 to 55 minutes, so the line fits closely. Sleep: the average vertical distance is 1.55 points on a 20-point quiz, and some points miss by 3. The line shows the upward trend, but a prediction from it could easily be off by 2 or 3 points.
A curved pattern (Diagram 2, right)
A cup of tea cools from 90 °C. Its temperature every 5 minutes for 30 minutes is 90, 70, 56, 46, 39, 33 and 30 °C. A student fits y = -2x + 85.
Equation: The distances are +5, -5, -9, -9, -6, -2 and +5: above the line at both ends and below it in the middle. The temperature drops fast at first and then slowly, so the points curve. A straight line is a poor model here, even though the tea clearly cools over time.
On Diagram 1, have students trace the red segments: each one is a vertical distance to Line A. Point out that Line A passes through only two points and still fits well. On Diagram 2, ask which panel's line they would trust most for a prediction, and why the right panel's line is not a good model even though the tea's temperature always goes down.
Guided Practice15 minutes
Pairs plot the data on grid paper and answer the questions in order. One partner draws, the other records; they switch halfway.
Invented data: noon temperature and visitors at a city pool on 10 summer days
Noon temperature (°F)
Visitors
72
150
75
190
78
205
80
260
83
250
85
310
88
330
90
395
92
370
95
440
Guided practice questions with answers
Question
Answer
Does the scatter plot suggest a linear association? Describe it.
Yes: a positive linear association. Hotter days have more visitors, and the points form a straight band
Fit a line by eye through (75, 180) and (95, 420). Show that its equation is y = 12x - 720.
It rises 240 visitors over 20 °F, 12 per degree. 12 × 75 - 720 = 180 and 12 × 95 - 720 = 420
How many points are above the line and how many are below it?
6 above and 4 below, spread along the line, so the balance is reasonable
Find the average vertical distance.
The distances are 6, 10, 11, 20, 26, 10, 6, 35, 14 and 20, a total of 158, so the average is 15.8 visitors
Is it a good fit? Is 15.8 large or small here?
A good fit: 15.8 visitors is small next to 150 to 440 visitors a day
Some pairs will draw slightly different lines, which is expected. Any line that follows the band with a balance of points gives an average near 15 to 20 visitors. Ask pairs whose average is much larger to look for points that are all on one side.
Independent Practice15 minutes
Students work alone, then compare with a partner. The data give the number of pages and the weight of 8 hardcover books (invented data).
Invented data: pages and weight of 8 hardcover books
Pages
Weight (g)
180
320
220
370
260
455
300
500
340
590
380
610
420
720
480
790
Independent practice problems with answers
Problem
Answer
Make a scatter plot and describe the association.
Positive and linear: books with more pages weigh more, in a narrow straight band
Kai fits y = 1.5x + 40. Count the points above, below and on his line.
6 above, 0 below and 2 on it: Kai's line is too low
Lena fits y = 1.6x + 30. Count the points above and below her line.
4 above and 4 below
Find the average vertical distance for each line. Which line fits better?
Kai: 165 ÷ 8 = 20.625 g. Lena: 103 ÷ 8 = 12.875 g. Lena's line fits better
In one sentence, explain why Kai's line is worse even though it passes through two data points.
Every other point is above it, so the line sits below the band instead of through its middle
Closure5-10 minutes
Exit ticket: the points (1, 8), (2, 11), (3, 13), (4, 17), (5, 18), (6, 22) show a linear association. (1) Ravi fits the line y = 3x + 8. How many points are above it and how many below? (0 above and 6 below.) (2) What should Ravi change? (Move the line down, about 3 or 4 units, keeping the same steepness.) (3) In one sentence, say what the average vertical distance tells you about a line of fit.
Differentiation Strategies
For Struggling Students
Give pre-printed scatter plots on a large grid, so every point sits on a grid line and vertical distances can be counted in squares
Use a clear plastic ruler with one straight line drawn on it, so students can slide the line and see the points on both sides
Give a checklist to tick for each line: follows the direction, points on both sides, sides balanced along the whole line
For Advanced Students
Ask students to find a line that makes the average vertical distance for the car data smaller than 1.0, and to explain why small changes matter less than the direction
Have students sketch a data set where two very different lines have the same average vertical distance, and discuss what else they would look at
Extension (beyond this standard): in high school (HSS.ID.B.6), students use technology to find the least-squares regression line, which makes the sum of the squared vertical distances as small as possible. Students can compare it with their own line
Assessment Guidance
What to Look For
A well-placed line follows the direction of the points, has roughly equal numbers of points above and below, and has those points spread along its whole length. Students should measure distances straight up or down, not along a slanted path to the line, and should drop the signs before averaging. Watch for students who join the first and last points, who force the line through as many points as possible, who say a line fits well because it passes through the origin, and who fit a straight line to a clearly curved pattern without comment.
02
Classroom Activities
3 Activities
1
Spaghetti Lines
15 minPairs
Each pair gets a printed scatter plot of the foot length and height of 14 students (invented data) and two strands of spaghetti. Each partner places a strand as a line of fit, then the pair decides which strand fits better.
Each partner lays a strand without talking, then tapes it down
For each strand, count the points above it and below it
Choose 4 points and measure their vertical distances to each strand, in grid squares, with a ruler
Mark two points on the better strand and read their coordinates, so the line can be drawn again
Answer Key
Lines vary. One reasonable line passes through (150, 22.5) and (180, 27.3), about y = 0.16x - 1.5. It has 8 points above it, 5 below and 1 on it, and an average vertical distance of about 0.42 cm. So a foot length predicted from this line is usually within about half a centimeter.
Discussion Questions
Did your two strands have the same steepness? Which mattered more for the fit: the steepness or the height of the strand?
Your line passes through at most one or two points. Why is it still a good model?
Would you use your line to predict the foot length of a 120 cm child? Why might that be risky?
Modification for Distance Learning
Students plot the data in an online graphing tool and drag a line with two movable points instead of placing spaghetti.
2
Measure and Model
20 minGroups of 4
The class collects real data: each student measures their height and arm span (fingertip to fingertip, arms stretched out) with a tape measure. Groups plot the class data, fit a line by eye and judge its fit.
Procedure
Partners measure each other in centimeters, standing against a wall, and write both numbers on a sticky note
Each group copies all the class notes into a table and makes a scatter plot, height on the horizontal axis
Each group fits a line by eye, writes two points on it, and finds the average vertical distance for 6 points of its choice
Groups post their lines and compare them
Sample Data for Teachers (invented)
If a class cannot measure, use these 10 students: height and arm span in cm (152, 149), (155, 157), (158, 159), (161, 158), (163, 166), (166, 164), (168, 171), (171, 169), (174, 172), (177, 181). The line y = x (arm span equals height) has 5 points above it and 5 below, with an average vertical distance of 2.5 cm.
Discussion Questions
Many people's arm span is close to their height. Does your class data support that?
Which student is farthest from your line? Could that be a measuring mistake?
Is a vertical distance of 2.5 cm large or small for arm spans of about 150 to 180 cm?
3
Line Judge
15 minGroups of 3
Each group gets 4 line cards. Each card has a small data table and a line someone proposed. Groups plot the points, draw the line, and give a verdict: good fit, line in the wrong place, or no straight line fits.
The 4 Cards (invented data)
Card 1, hours a candle has burned and its height (cm): (0, 20), (1, 18.4), (2, 16.1), (3, 14.6), (4, 12.5), (5, 11). Proposed line: y = -1.8x + 20
Card 2, distance from school (km) and bus ride time (min): (1, 6), (2, 9), (3, 13), (4, 15), (5, 19), (6, 22). Proposed line: y = 3.2x + 8
Card 3, bike speed (km/h) and stopping distance (m): (10, 3), (15, 5), (20, 9), (25, 13), (30, 18), (35, 24). Proposed line: y = 0.8x - 6
Card 1: good fit. The points are within 0.3 cm of the line, an average vertical distance of 0.8 ÷ 6 ≈ 0.13 cm
Card 2: wrong place. All 6 points are about 5 minutes below the line; y = 3.2x + 3 fits with an average vertical distance of 2 ÷ 6 ≈ 0.33 min
Card 3: no straight line fits well. The points are above the line at both ends and below it in the middle, because stopping distance grows faster at higher speeds
Card 4: no line fits. Height does not go up or down with birth month; the points scatter with no pattern
Discussion Questions
Card 2's line has the right steepness. What is wrong with it, and how would you fix it without changing the steepness?
Card 3's line has an average vertical distance of 1 meter, which seems small. Why is it still a poor model?
Card 4 is the only card with no association. How can you tell from the plot before drawing any line?
Challenge Variation
Groups make their own card with a data set and a deliberately flawed line, and trade with another group to judge.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Two Lines of Fit for the Used Car Data
Invented data for 12 used cars of one model, drawn to scale. Line A (solid, y = -1.5x + 24.5) was fit by eye; the red segments are its vertical distances, with an average of 1.0 thousand dollars. Line B (dashed, y = -x + 23) joins the first and last points and has an average vertical distance of about 1.46 thousand dollars, so Line A fits better.
Diagram 2: Close Fit, Loose Fit and a Curved Pattern
Three invented data sets, each with a line fit by eye and its vertical distances in red. Left: 10 training runs, fit by y = 5.5x, with points close to the line. Middle: sleep and quiz scores of 10 students, fit by y = 1.5x + 2, with points far from the line. Right: a cooling cup of tea, fit by y = -2x + 85; the points curve, so a straight line is a poor model.
04
Homework Assignment
~30 min
8.SP.A.2 Homework: Fitting and Judging Lines of Fit
Directions: Make every scatter plot on grid paper with a ruler. All data are invented. When you fit a line by eye, write two points on your line so someone else could draw it. Show every distance you use.
Part 1: Fit a Line by Eye (Problems 1-3)
The age (years) and trunk width (cm) of 10 red maple trees in a park are (5, 7), (7, 9), (8, 11), (10, 12), (12, 16), (13, 15), (15, 19), (17, 20), (18, 23), (20, 24). (a) Make a scatter plot and describe the association. (b) Fit a line by eye and name two points on it. (c) How many points are above your line and how many are below it?
At a football game, a snack stand records the temperature (°F) and the number of hot drinks sold at 8 games: (35, 215), (40, 193), (45, 184), (50, 158), (55, 155), (60, 128), (65, 120), (70, 99). Mateo fits the line y = -3.2x + 330. (a) Count the points above and below his line. (b) Explain what is wrong with Mateo's line. (c) Draw a better line and name two points on it.
A ball is dropped from 150 cm. Its bounce number and the height of that bounce (cm) are (0, 150), (1, 98), (2, 62), (3, 41), (4, 26), (5, 17), (6, 11). (a) Make a scatter plot. (b) A classmate fits y = -22x + 125. Find the vertical distance of each point to this line, with its sign. (c) Is a straight line a good model for these data? Explain using the pattern of the distances.
Part 2: Judge the Fit (Problems 4-6)
A phone charges from a low battery. The minutes of charging and the battery percent are (10, 22), (20, 38), (30, 51), (40, 68), (50, 80), (60, 95). (a) For the line y = 1.5x + 7, find the vertical distance of each point. (b) Find the average vertical distance. (c) Is the line a good fit? Explain.
The ages (months) and heights (cm) of 8 toddlers are (12, 75), (15, 78), (18, 82), (21, 84), (24, 87), (27, 89), (30, 92), (36, 96). Line P is y = 0.9x + 65 and Line Q is y = 0.8x + 66. (a) Find the average vertical distance for each line. (b) Which line fits better? (c) What does the pattern of Line Q's distances tell you about where it sits?
The line y = 0.5x models the minutes played and the points scored by the players of two basketball teams. Team A: (10, 6), (15, 8), (20, 10), (24, 11), (28, 15), (32, 16). Team B: (10, 2), (15, 11), (20, 6), (24, 17), (28, 9), (32, 20). (a) Find the average vertical distance for each team. (b) For which team does the line give more trustworthy predictions? (c) Could the line still show the trend for the other team? Explain.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Scatter Plots
Axes labeled with units, points plotted accurately
Minor plotting errors or missing labels
Missing or wrong
Line of Fit
Follows the band, balanced points above and below, two points named
Reasonable line but unbalanced or points not named
No line, or the line ignores the pattern
Vertical Distances
All distances and averages correct, signs dropped before averaging
One or two computing errors
Distances missing or measured the wrong way
Judging the Fit
Verdict supported by the average and the pattern of the distances
Verdict with a partial reason
No verdict or no reason
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. All data sets 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
Why are straight lines so often used to model the data in a scatter plot?
Answer: B
A line is easy to draw and use, and when the points form a straight band its steady rate of change describes the pattern well. Choice A is false: a line of fit usually misses most points. Choice C confuses a pattern with a cause. Choice D is false: curved data need a curve.
Question 2 of 20 · Multiple Choice
For which pair of measurements would a straight line most likely be a good model?
Answer: C
Apples of one kind weigh about the same, so each extra apple adds about the same weight: a straight band. Choice A curves, since the area grows faster and faster as the side gets longer. Choice B has no association. Choice D rises and then levels off in the teenage years, so the points curve.
Question 3 of 20 · Multiple Choice
The points (1, 4), (2, 7), (3, 7), (4, 10), (5, 12), (6, 13) show a linear association. Which line fits them best?
Answer: D
y = 2x + 2 passes through three points and misses the others by 1, an average vertical distance of 3 ÷ 6 = 0.5. Choice A has the right steepness but is too high: all 6 points are below it. Choice B is too flat (average 10 ÷ 6 ≈ 1.67), and Choice C is too steep (average 14 ÷ 6 ≈ 2.33).
Question 4 of 20 · Multiple Choice
Ana fits a line to a scatter plot of 11 points. Nine points are above her line and two are below it. What should she do?
Answer: B
A line of fit should have about as many points above it as below it, so Ana's line is too low and should move up. Choice C makes the problem worse. Choice A ignores the imbalance. Choice D changes the steepness for no reason: a line of fit does not need to pass through the origin.
Question 5 of 20 · Multiple Choice
A line of fit is y = 3x + 4. What is the vertical distance from the point (8, 31) to the line?
Answer: A
At x = 8 the line is at 3 × 8 + 4 = 28, and 31 - 28 = 3, so the point is 3 units above the line. Choice B gets the side wrong: the point is higher than the line. Choice C forgets the + 4 and computes 31 - 24. Choice D gives the height of the line, 28, instead of the distance.
Question 6 of 20 · Multiple Choice
Two students fit lines to the same scatter plot. Line P has an average vertical distance of 2.4 points and Line Q has an average vertical distance of 6.1 points. Which statement is best?
Answer: C
A smaller average vertical distance means the points are closer to the line on average, so Line P fits better. Choice A reverses the meaning of the average. Choice D is not needed: the averages were found from the same data, so they can be compared directly.
Question 7 of 20 · Multiple Choice
On a scatter plot, the points are above the line of fit at both ends and below it in the middle. What does this tell you?
Answer: B
Points above at both ends and below in the middle follow a curve that bends away from the line, so no straight line will fit well. Choice C only counts sides and misses the pattern. Choice A would put even more points above the line at the ends. Choice D is wrong: a curve is a pattern.
Question 8 of 20 · Multiple Choice
A line of fit misses 6 points by these vertical distances: 2, -1, 0.5, -3, 1.5 and -2.5 (a negative distance means the point is below the line). What is the average vertical distance?
Answer: A
Drop the signs and add: 2 + 1 + 0.5 + 3 + 1.5 + 2.5 = 10.5. Then divide by 6 points: 10.5 ÷ 6 = 1.75. Choice B keeps the signs, so distances above and below cancel. Choice C is the total, not divided by 6. Choice D is the largest single distance.
Question 9 of 20 · Multiple Choice
Which is NOT a good guideline for fitting a line by eye?
Answer: C
The first and last points may be unusual, so a line through them can miss the pattern, as Line B does in Diagram 1. Choices A, B and D are all guidelines from the lesson.
Question 10 of 20 · Multiple Choice
One of 15 points on a scatter plot is far away from the others. How should it affect a line fit by eye?
Answer: D
A single outlier should not pull the line away from the main pattern, so fit the line to the band and mention the outlier, which may be a mistake or a special case. Choice A lets one point control the line. Choice C throws away a useful model because of one point.
Question 11 of 20 · Multiple Choice
The line y = 2x + 1 is fit to the points (2, 5), (4, 8), (6, 14), (8, 16), (10, 21). How many of the points are above the line?
Answer: A
At x = 2, 4, 6, 8 and 10 the line is at 5, 9, 13, 17 and 21. Only (6, 14) is above it. Choice B counts the points below the line, (4, 8) and (8, 16). Choice C counts the points that are not below, including the two on the line. Choice D counts every point.
Question 12 of 20 · Multiple Choice
What does it mean to call a line of fit a model of the data?
Answer: A
A model describes the pattern simply and closely but not exactly, so it can be used to estimate. Choices B and C describe an exact rule, which real measurements rarely follow. Choice D is false: many good lines of fit do not pass through (0, 0).
Question 13 of 20 · Multiple Choice
The points (1, 3), (2, 6), (3, 6), (4, 9), (5, 11) are fit by two lines: Line A is y = 2x + 1 and Line B is y = 2.5x. Which line fits better, and what is its average vertical distance?
Answer: D
Line A misses by 0, 1, 1, 0 and 0, so its average is 2 ÷ 5 = 0.4. Line B misses by 0.5, 1, 1.5, 1 and 1.5, an average of 5.5 ÷ 5 = 1.1, so Line A fits better. Choice A picks the larger average. Choice B gives Line B's total, not divided by 5. Choice C gives Line A's total, not divided by 5.
Question 14 of 20 · Multiple Choice
Which description best matches a well-placed line of fit?
Answer: D
A line of fit follows the direction of the points and keeps them balanced on both sides along its length. Choice A can force the line through a few points and away from the rest. Choice B joins two points that may not follow the pattern. Choice C describes a line that is too high or too low.
Question 15 of 20 · Short Answer
A student practices typing for 8 weeks. The week number and typing speed (words per minute) are (1, 18), (2, 21), (3, 25), (4, 26), (5, 31), (6, 33), (7, 35), (8, 40) (invented data). Fit a line by eye, name two points on it, and count the points above, below and on your line.
Lines vary. One reasonable line passes through (0, 15) and (8, 39), so it is y = 3x + 15. It has 3 points above, 2 below and 3 on it, spread along the line, and an average vertical distance of 5 ÷ 8 ≈ 0.63 words per minute. Any line that rises about 3 words per minute each week and keeps the points balanced is acceptable.
Question 16 of 20 · Short Answer
A student walks for different lengths of time and records minutes and steps: (10, 1050), (20, 2100), (30, 2950), (40, 4100), (50, 5000) (invented data). Line 1 is y = 100x and Line 2 is y = 110x - 100. Find the average vertical distance for each line and say which fits better.
Line 1 misses by 50, 100, 50, 100 and 0 steps, so its average is 300 ÷ 5 = 60 steps. Line 2 misses by 50, 0, 250, 200 and 400, so its average is 900 ÷ 5 = 180 steps. Line 1 fits better. Line 2 is too steep: the points fall further below it as the minutes grow.
Question 17 of 20 · Short Answer
The age (months) and weight (kg) of a puppy are (2, 5), (4, 11), (6, 16), (8, 19), (10, 21), (12, 22) (invented data). A student fits y = 1.6x + 5. Find the signed vertical distances and decide whether a straight line is a good model.
The line gives 8.2, 11.4, 14.6, 17.8, 21 and 24.2, so the distances are -3.2, -0.4, +1.4, +1.2, 0 and -2.2. The points are below the line at both ends and above it in the middle, because the puppy grows fast and then slows down. The pattern is curved, so a straight line is a poor model, even though the average distance is only 8.4 ÷ 6 = 1.4 kg.
Question 18 of 20 · Short Answer
A line of fit for 8 points has these vertical distances: 4, 5, 3, 6, 4, 2, 5 and 3, and every point is above the line. Find the average vertical distance. What is wrong with the line, and how should it change?
The average is 32 ÷ 8 = 4 units. Every point is above the line, so the line is too low. Keep its steepness and move it up about 4 units, so points fall on both sides of it.
Question 19 of 20 · Short Answer
Jin's line of fit passes through 4 of the 10 points on a scatter plot, and the other 6 points are all above it by 3 to 5 units. Mia's line passes through none of the points, but 5 are above it and 5 are below it, each within 1 unit. Whose line is the better model? Explain.
Mia's line. Its points are balanced on both sides and all within 1 unit, so its average vertical distance is at most 1. Jin's line has 6 points on one side, missing by 3 to 5 units, so its average is at least 1.8 (18 ÷ 10) and it sits below the band. Passing through points does not make a line a good fit.
Question 20 of 20 · Short Answer
A student reads Line A in Diagram 1 and writes: "Every 1-year-old car of this model costs $23,000." What is right about the reading, and what is wrong with the statement?
The reading is right: at x = 1, Line A gives -1.5(1) + 24.5 = 23 thousand dollars. The statement is wrong because a line of fit is a model: it gives a typical price, not the price of every car. The 1-year-old car in the data sold for $22,000, and the cars miss Line A by about $1,000 on average, so a price near $23,000, give or take about $1,000, is a better statement.
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Frequently Asked Questions
10 Questions
What does 8.SP.A.2 mean?
8.SP.A.2 means students know that straight lines are often used to model two related measurements, fit a line to a scatter plot by eye, and judge the fit by how close the points are to the line. Everything is informal: students use a ruler, a strand of spaghetti or a graphing tool, not a formula.
What is a line of fit?
A line of fit is a straight line drawn on a scatter plot to show the pattern of the points. It is also called a trend line. It is a model: it describes the data simply and closely, but it usually passes through few of the points, or none.
How do you draw a line of fit by eye?
Follow the direction of the band of points, and place the line so about as many points are above it as below it. Check that the points on each side are spread along the whole line, not bunched at one end, and do not let a single outlier pull the line away from the rest.
Does a line of fit have to go through the origin or through data points?
No. A line of fit only needs to follow the pattern. Forcing it through the origin, or through the first and last points, often moves it away from most of the data. In Diagram 1, the line through the first and last cars fits worse than a line that passes through only two points in the middle.
How do you know if a line of fit is good?
Look at the vertical distances from the points to the line. The fit is good when the distances are small compared with the data values and show no pattern. To compare two lines for the same data, find each line's average vertical distance: the smaller one fits better.
Is 8.SP.A.2 the same as linear regression?
No. In 8.SP.A.2, students fit lines informally, by eye. The least-squares regression line, computed with technology, comes in high school statistics (HSS.ID.B.6). Grade 8 students can still compare their own line with a computed one as an extension.
What if the scatter plot is curved?
Then a straight line is a poor model, even if it seems close. The sign of the distances shows it: points sit above the line in some places and below it in others, in a pattern. Say that the data are not linear, and describe the curve in words, such as "falls fast at first, then slowly".
How is 8.SP.A.2 different from 8.SP.A.1 and 8.SP.A.3?
In 8.SP.A.1, students make scatter plots and describe patterns. In 8.SP.A.2, they fit a line to a linear pattern and judge the fit. In 8.SP.A.3, they write the equation of that line and use its slope and intercept to answer questions about the context.
What mistakes do students make with 8.SP.A.2?
Common ones are joining the first and last points, forcing the line through as many points as possible, measuring distances at a slant instead of straight up or down, keeping the signs when averaging distances, and fitting a straight line to curved data without noticing the pattern.
How can parents help with 8.SP.A.2 at home?
Measure something together at home, such as the height of family members and their arm span, and plot the pairs on grid paper. Ask your child to lay a straight edge where it fits best and to explain why. Then ask how far the farthest point is from the line, and whether that is a lot or a little.
07
Related Standards
5 standards
These standards connect to 8.SP.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.SP.A.1Prerequisite
Construct and interpret scatter plots and describe patterns of association
