8.SP.A.1Common CoreMathStatistics and ProbabilityGrade 8
8.SP.A.1: Scatter Plots and Patterns of Association
In plain English: 8.SP.A.1 is the Common Core grade 8 math standard that asks students to construct and interpret scatter plots for bivariate measurement data, pairs of measurements taken from the same people or objects. Students describe the patterns they see: clusters, outliers, positive or negative association, and linear or nonlinear association. It starts the scatter-plot work of Grade 8 Math statistics.
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Investigate patterns of association in bivariate data. Also written as 8.SP.1 · Official standard
Bivariate data are pairs of values: two measurements taken from the same person, object or event, such as the temperature on a night and the number of times a cricket chirps. A scatter plot shows each pair as one point on a coordinate grid, with one measurement on the horizontal axis and the other on the vertical axis. Scatter plots let students see whether the two quantities are connected, which is called an association.
Students first construct scatter plots: they choose which quantity goes on each axis, choose scales that fit the data, and plot every pair. They then describe what they see with the words the standard names. The direction can be positive (both quantities increase together), negative (one increases as the other decreases) or show no association. The form can be linear (the points lie close to a straight line) or nonlinear (they follow a curve). A cluster is a group of points close together, apart from the others, and an outlier is a point far from the overall pattern. All data sets on this page are invented, but they were built to scatter like real measurements.
Learning Objectives
By the end of this lesson, students will be able to:
Construct a scatter plot from a table of bivariate measurement data, with labeled axes and scales that fit the data
Describe the direction of an association as positive, negative or none, and explain what it means in context
Decide whether an association is linear or nonlinear, and explain how the plot shows it
Identify clusters and outliers, and suggest reasons for them in context
Prior Knowledge Required
Students should already be comfortable with:
Plotting ordered pairs (x, y) in the first quadrant to show real-world data 5.G.A.2
Displaying one set of numerical data in dot plots, histograms and box plots 6.SP.B.4
Describing a data distribution by its center, spread and overall shape 6.SP.A.2
"People say you can tell the temperature by counting cricket chirps. One cricket chirped 20 times in 14 seconds on a 60 °F night and 39 times on an 80 °F night. Is that enough to be sure warmer nights mean more chirps? What would you want to collect?"
Students should see that two measurements are not enough: one cricket could be unusual. They need many pairs (temperature, chirps) from many crickets. Tell them each pair is bivariate data, and that today they will learn to show many pairs at once on a scatter plot and to describe the pattern. (For the teacher: a popular rule of thumb says that the number of chirps in 14 seconds plus 40 is about the temperature in °F. It works best for one kind of cricket, the snowy tree cricket, and gives only an estimate.)
Direct Instruction20 minutes
Choose the axes. Put the quantity that might explain or affect the other on the horizontal axis (x), and the other on the vertical axis (y). Temperature might affect chirping, so temperature goes on x.
Choose the scales. Find the smallest and largest value of each quantity and pick equal steps that cover them and fill most of the grid. A scale does not have to start at 0: an axis break, a small zigzag near the origin, shows that part of the axis was skipped.
Plot and label. Plot each pair as one point, and label both axes with the quantity and its unit.
Direction. Positive association: as x increases, y tends to increase (points rise from left to right). Negative association: as x increases, y tends to decrease. No association: the points show no upward or downward trend.
Form. Linear association: the points lie close to a straight line, so y changes by about the same amount for each step in x. Nonlinear association: the points follow a curve, so the change in y grows or shrinks along the plot.
Clusters and outliers. A cluster is a group of points close together and separated from other points. An outlier is a point that lies far from the pattern of the others. Always ask what could explain it, such as a measuring mistake or a different kind of object.
Construct and describe: positive linear with an outlier (Diagram 1)
A class measures 13 crickets (invented data): temperature (°F) and chirps in 14 seconds: (56, 18), (59, 18), (62, 23), (64, 22), (67, 27), (69, 32), (71, 30), (73, 12), (74, 35), (76, 33), (79, 41), (82, 41), (85, 46). Construct a scatter plot and describe it.
Equation: Temperatures run from 56 to 85, so the x-axis goes from 50 to 90 in steps of 5, with an axis break. Chirps run from 12 to 46, so the y-axis goes from 0 to 50 in steps of 10. The points rise from left to right in a narrow, nearly straight band: a positive linear association, about 1 more chirp for each degree. The point (73, 12) is far below the band, so it is an outlier. It could be a different kind of cricket, or a count that was cut short.
Nonlinear association (Diagram 2, left)
Students time 10 swings of a pendulum (a weight hanging on a string) for different string lengths (invented data): length (cm) and time (s): (5, 4.7), (10, 6.6), (20, 8.8), (30, 11.2), (40, 12.5), (50, 14.4), (60, 15.3), (70, 17), (80, 17.8), (90, 19.3), (100, 19.8). Describe the association.
Equation: Longer strings take more time, so the association is positive. From 10 cm to 20 cm the time goes up about 2.2 s, but from 80 cm to 90 cm only about 1.5 s, and the points bend into a curve that flattens out. The association is nonlinear.
Clusters (Diagram 2, right)
Fourteen students (invented data) give their distance from school (km) and travel time (min). Walkers: (0.4, 6), (0.6, 9), (0.8, 11), (1, 13), (1.1, 15), (1.3, 17), (1.5, 20). Bus riders: (4.2, 18), (5, 22), (5.6, 21), (6.3, 25), (7.1, 27), (7.8, 30), (8.5, 31). What pattern stands out?
Equation: The points form two clusters: one between 0.4 and 1.5 km, and one between 4.2 and 8.5 km, with no students in between. The first cluster is students who walk, and the second is students who ride the bus. Within each cluster, farther students take longer, a positive association. Walkers take about 13 minutes per kilometer, much more than bus riders.
Negative linear association
On a morning hike, a student records the elevation (m) and the air temperature (°C) at 8 points (invented data): (1200, 17.9), (1400, 16.4), (1650, 15.3), (1800, 13.8), (2050, 12.6), (2300, 10.7), (2500, 9.9), (2750, 8). Describe the association.
Equation: As elevation increases, temperature decreases: a negative association. Each 250 m or so of climbing lowers the temperature by about 1.5 °C, and the points lie close to a straight line, so the association is linear. There are no clusters or outliers.
Use Diagram 1 to show the parts of a scatter plot: axis labels with units, the axis break, even scales and the outlier. Use Diagram 2 to contrast a curve with a straight band, and one group of points with two groups. Stress that a description has three parts: direction, form, and any clusters or outliers.
Guided Practice15 minutes
Pairs construct the first plot on grid paper, then describe all three data sets (all invented). One partner names the direction and the other names the form; together they look for clusters and outliers.
Guided practice problems with answers
Problem
Answer
Age of a phone (months) and battery capacity (% of new): (0, 100), (4, 98), (8, 93), (12, 91), (16, 86), (20, 85), (24, 80), (28, 78). Construct a scatter plot and describe it.
x-axis 0 to 30 by 5, y-axis 75 to 100 by 5 with an axis break. Negative linear association: the capacity drops about 3 to 4 percentage points every 4 months. No outliers
Diameter (cm) and mass (g) of 7 snowballs: (6, 48), (7, 70), (8, 101), (9, 150), (10, 205), (11, 270), (12, 355). Describe the association.
Positive and nonlinear: each extra centimeter of diameter adds more mass than the one before (22 g from 6 to 7 cm, 85 g from 11 to 12 cm), so the points curve upward
Hours of sun per day and tomatoes picked from 9 plants: (4, 5), (5, 9), (6, 9), (6, 12), (7, 13), (8, 3), (8, 17), (9, 18), (10, 22). Which point does not fit, and what might explain it?
(8, 3) is an outlier: the other plants show a positive linear association, and a plant with 8 hours of sun would be expected to give about 16 tomatoes, like the other 8-hour plant (17). Pests, disease or a missed harvest could explain it
Listen for students who call any rising pattern "linear," and ask them to compare the change in y over two equal steps in x. Also watch for scales with uneven steps, such as 0, 5, 10, 20.
Independent Practice15 minutes
Students sketch each plot quickly on a small grid, describe it in one or two sentences, and then compare with a partner. All data are invented.
Independent practice problems with answers
Problem
Answer
Minutes of jumping rope and heart rate (beats per minute): (0, 72), (1, 98), (2, 118), (3, 131), (4, 140), (5, 146), (6, 150).
Positive nonlinear: the heart rate rises fast in the first minutes (26 beats in the first minute) and then levels off (4 beats in the last minute)
Age of a hiking boot (months) and tread depth (mm): (0, 6), (3, 5.4), (6, 4.9), (9, 4.1), (12, 3.7), (15, 2.9), (18, 2.5).
Negative linear: the tread wears down about 0.2 mm per month, and the points lie close to a straight line
Forearm length (cm) and minutes to walk to school for 8 students: (22, 14), (23, 6), (24, 19), (25, 9), (26, 16), (27, 5), (28, 12), (29, 18).
No association: the points are scattered with no upward or downward trend. Forearm length does not tell you anything about walking time
Mass (kg) and distance per liter of fuel (km) for 9 vehicles. Motorcycles: (190, 31), (210, 28), (230, 33), (250, 26). Cars: (1250, 16), (1400, 14), (1550, 13), (1700, 11), (1800, 12).
Two clusters: the light motorcycles with high fuel distances, and the heavy cars with lower ones. Overall, heavier vehicles go fewer kilometers per liter, a negative association
Screen size (in) and price ($) of 8 TVs in one store: (32, 180), (40, 240), (43, 260), (50, 330), (55, 380), (55, 1200), (65, 520), (75, 700).
Positive association: bigger screens cost more. (55, 1200) is an outlier, far above the other 55-inch TV at $380; it could be a newer or higher-quality model
Closure5-10 minutes
Exit ticket: (1) Sketch a scatter plot with a negative linear association and one outlier, and label the outlier. (2) A plot of the ages of 10 trees (years) and their trunk circumference (cm) rises from left to right in a nearly straight band. Name the association. (Positive linear.) (3) Give one reason a data set can have an outlier.
Differentiation Strategies
For Struggling Students
Give pre-drawn axes with the scales already marked, so students focus on plotting and describing
Use a word bank card with a tiny sketch next to each term: positive, negative, no association, linear, nonlinear, cluster, outlier
Have students cover the plot with a sheet of paper and slide it from left to right, saying whether the points go up or down
For Advanced Students
Ask students to find a real data set (for example, from a sports page) and describe its association in a short paragraph
Have students explain how the pattern in Diagram 2 (right) would look if the walkers and bus riders were drawn with the same symbol, and why the colors help
Challenge (beyond this standard): lay a strand of spaghetti along the band of points in Diagram 1 and use it to predict the chirps at 70 °F. Fitting lines is the next standard, 8.SP.A.2
Assessment Guidance
What to Look For
A complete scatter plot has both axes labeled with quantities and units, even scales that fit all the data, an axis break when a scale does not start at 0, and every pair plotted once. A complete description names the direction, the form, and any clusters or outliers, and says what the pattern means in context ("warmer nights, more chirps"). Watch for students who plot the two variables as separate dot plots, who call every rising pattern linear, and who delete outliers without a reason.
02
Classroom Activities
3 Activities
1
Pendulum Lab
20 minPairs
Pairs build a pendulum from string and a few metal washers, time 10 swings for different string lengths, and construct a scatter plot of their own measurements.
Procedure
Tie 3 washers to a string and tape the other end to the edge of a desk
Set the length (from the tape to the middle of the washers) to 10, 20, 40, 60, 80 and 100 cm
Pull the washers a little to one side and let go. Time 10 full swings (over and back counts as one)
Record each pair (length, time) in a table and plot the pairs on grid paper: length on the x-axis, time on the y-axis
Backup Data (invented, for pairs who cannot finish)
Length (cm) and time for 10 swings (s): (5, 4.7), (10, 6.6), (20, 8.8), (30, 11.2), (40, 12.5), (50, 14.4), (60, 15.3), (70, 17), (80, 17.8), (90, 19.3), (100, 19.8). This is the data set in Diagram 2 (left).
Discussion Questions
In the backup data, the time for a 40 cm string (12.5 s) is less than double the time for a 20 cm string (8.8 s). What does that tell you about the form of the pattern?
About how many times as long must the string be to double the time? (About 4 times: 10 cm takes 6.6 s and 40 cm takes 12.5 s. For small swings, physics predicts exactly 4 times; real measurements come close.)
Did any of your points fall off the pattern? What could have caused it?
Modification for Distance Learning
Students use a key ring or a small toy on a shoelace at home, record three lengths, and add their points to a shared class plot.
2
Pattern Card Sort
20 minGroups of 3
Each group gets 8 cards, each with a small invented data set of 6 pairs. Groups sketch each scatter plot on a mini grid and sort the cards into groups: positive linear, positive nonlinear, negative linear, negative nonlinear, clusters, outlier and no association.
The 8 Cards
Card 1: Minutes on a treadmill and calories burned: (5, 52), (10, 91), (15, 146), (20, 184), (25, 240), (30, 279)
Card 2: Hours after noon and length of a flagpole's shadow (m): (0, 1.6), (1, 2.1), (2, 3.1), (3, 4.6), (4, 7.2), (5, 12.5)
Card 3: Average outdoor temperature (°C) and monthly heating bill ($): (-5, 176), (0, 158), (5, 124), (10, 101), (15, 66), (20, 47)
Card 4: Height (cm) and day of the month of birth: (150, 12), (162, 3), (155, 28), (170, 17), (158, 22), (166, 7)
Card 5: Length (cm) and mass (g) of fruit in a bowl: (1.8, 4), (2, 5), (2.3, 6), (5.2, 62), (5.5, 70), (6, 80)
Card 7: Students painting a mural and hours it takes: (1, 24), (2, 12.5), (3, 8), (4, 6.2), (6, 4.1), (8, 3.2)
Card 8: Minutes in the shower and liters of water used: (4, 41), (6, 52), (8, 77), (10, 88), (12, 115), (15, 133)
Answer Key
Positive linear: Cards 1 and 8
Positive nonlinear: Card 2 (the shadow grows faster and faster)
Negative linear: Card 3
Negative nonlinear: Card 7 (the time drops fast at first, then slowly)
Two clusters: Card 5 (grapes and plums)
Outlier: Card 6, the point (50, 18); the other points are positive linear
No association: Card 4
Discussion Questions
Card 6 is the only card with an outlier. What might explain the point (50, 18)?
On Card 7, going from 1 to 2 students cuts the time by more than 11 hours, but going from 6 to 8 students saves less than 1 hour. How does that show a nonlinear pattern?
Card 5 shows a positive association overall. Why is "two clusters" a more useful description?
3
Alphabet Race
15 minWhole class, then individuals
Every student times how long it takes to write the letters A to Z, first with the hand they usually write with and then with the other hand. The class pools the pairs, and each student constructs and describes the scatter plot.
Procedure
Partners time each other with a stopwatch, to the nearest second
Each student writes the pair (usual hand, other hand) on a sticky note and posts it on the board
Students choose scales and plot every pair on grid paper, with usual-hand time on the x-axis
Each student writes a three-part description: direction, form, clusters or outliers
Backup Data (invented, 12 students)
Seconds with the usual hand and with the other hand: (12, 31), (14, 38), (15, 35), (16, 17), (16, 44), (17, 41), (18, 49), (19, 46), (20, 55), (21, 52), (23, 60), (24, 65).
Discussion Questions
In the backup data, which point is an outlier, and what kind of student might it describe? ((16, 17): someone who writes almost as fast with either hand.)
Is the association positive or negative? What does it say about fast and slow writers?
Challenge Variation
Challenge (beyond this standard): students lay a strand of spaghetti through the middle of the points and use it to predict the other-hand time of a student who takes 22 seconds with the usual hand. This previews 8.SP.A.2.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Positive Linear Association With an Outlier
Invented data for 13 crickets: temperature (°F) and chirps in 14 seconds, drawn to scale. The zigzag on the horizontal axis is an axis break: the scale starts at 50 °F instead of 0. Twelve points rise from left to right in a nearly straight band, a positive linear association. The red point (73, 12) is far below the band, so it is an outlier.
Diagram 2: Nonlinear Association and Clusters
Left: invented pendulum data, string length (cm) and time for 10 swings (s). The points rise but bend into a curve that flattens, a positive nonlinear association. Right: invented data for 14 students, distance from school (km) and travel time (min). The points form two clusters, walkers (green circles) and bus riders (blue squares).
04
Homework Assignment
~30 min
8.SP.A.1 Homework: Scatter Plots and Association
Directions: All data sets are invented. Construct every scatter plot on grid paper with labeled axes, units and even scales (use an axis break if a scale does not start at 0). Describe each pattern by its direction, its form, and any clusters or outliers.
Part 1: Construct Scatter Plots (Problems 1-3)
Number of pages and mass (g) of 10 books: (120, 146), (160, 170), (180, 420), (200, 221), (224, 236), (256, 280), (288, 298), (320, 342), (352, 355), (400, 418). (a) Construct a scatter plot. (b) Describe the association. (c) One book is a hardcover and the rest are paperbacks. Which point is it, and how does the plot show it?
Distance from a speaker (m) and sound level in decibels (dB, a unit that measures loudness), measured with a phone app: (1, 88), (2, 83), (3, 78), (4, 76), (6, 73), (8, 70), (10, 67), (12, 67), (16, 64). (a) Construct a scatter plot. (b) Describe the association. (c) Compare the drop in sound level from 1 m to 4 m with the drop from 4 m to 16 m.
A coach records the standing long jump (cm) and the 40 m sprint time (s) of 10 players: (150, 7.3), (162, 7), (170, 7.1), (178, 6.8), (185, 6.7), (192, 6.5), (200, 6.6), (208, 6.3), (216, 6.2), (228, 5.9). (a) Construct a scatter plot with the jump on the x-axis. (b) Describe the association. (c) Explain in words what the pattern says about the players.
Part 2: Interpret Patterns (Problems 4-6)
A student records minutes of screen time and minutes of reading on 14 days. School days: (60, 35), (65, 30), (68, 26), (70, 28), (72, 31), (75, 33), (78, 24), (80, 25), (85, 27), (90, 22). Weekend and holiday days: (150, 15), (165, 11), (175, 9), (190, 6). (a) Construct a scatter plot, using one symbol for both kinds of days. (b) Describe any clusters and explain them. (c) Describe the overall association.
Write a real-world pair of measurements that you would expect to show each pattern, and sketch a small scatter plot for each: (a) positive linear association, (b) negative linear association, (c) nonlinear association, (d) two clusters.
Age of 10 used cars of the same model (years) and their value (thousands of dollars): (1, 26.5), (2, 23), (3, 20.5), (4, 17.5), (5, 15.5), (6, 13), (8, 10.2), (9, 8.8), (10, 7.8), (12, 6.2). (a) Find the average drop in value per year from year 1 to year 3, and from year 9 to year 12. (b) Use your answers to decide whether the association is linear or nonlinear. (c) Describe the association fully.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Axes and Scales
Both axes labeled with units; even scales fit all data; axis break where needed
One label, unit or scale problem
Axes missing or scales uneven
Plotting
Every pair plotted once, in the right place
One or two points misplaced
Many points wrong or missing
Describing Patterns
Names direction, form, and clusters or outliers correctly
One part missing or wrong
No description, or mostly wrong
Interpreting in Context
Explains what the pattern means for the real quantities
Explanation vague
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
All data sets in this quiz are invented. Sketch a quick scatter plot on scrap paper when a question gives data. 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
Which data set could be shown on a scatter plot?
Answer: B
A scatter plot shows bivariate measurement data: two measurements from each person or object, one on each axis, so each student becomes one point (height, arm span). Choice A is a category, not a measurement. Choice C is only one measurement per student, which fits a dot plot or histogram. Choice D gives two categories, not measurements, so they cannot be placed on number-line axes.
Question 2 of 20 · Multiple Choice
A scatter plot shows hours of practice (x) and free throws made out of 20 (y). The points rise from left to right. What does this show?
Answer: C
Points that rise from left to right mean that larger x-values go with larger y-values, a positive association. Choice A describes points that fall from left to right. Choice B would need points with no upward or downward trend. Choice D would need two separate groups of points.
Question 3 of 20 · Multiple Choice
A candle burns for 6 hours. Its height (cm) after each hour is: (0, 20), (1, 18.4), (2, 17.3), (3, 15.6), (4, 14.5), (5, 12.7), (6, 11.6) (invented data). Which describes the scatter plot of (hours, height)?
Answer: B
The height goes down as the hours go up, a negative association, and it drops by about 1.4 cm each hour, so the points lie close to a straight line: linear. Choice A has the direction backward. Choice C is wrong because the drop stays about the same each hour instead of speeding up or slowing down.
Question 4 of 20 · Multiple Choice
Eight bean plants (invented data) give height (cm) and number of leaves: (10, 6), (12, 8), (14, 21), (15, 8), (18, 11), (20, 11), (22, 14), (25, 15). Which point is an outlier?
Answer: D
The other plants show a positive linear pattern, about 0.6 more leaves per centimeter, so a 14 cm plant would have about 8 leaves. (14, 21) has far more, so it is the outlier. Choices A and B are the largest and smallest plants, but they fit the pattern: an extreme value is not the same as an outlier. Choice C also fits the pattern.
Question 5 of 20 · Multiple Choice
At a dog park, 10 dogs are measured (invented data): shoulder height (cm) and mass (kg): (25, 5), (28, 6), (30, 7), (33, 9), (35, 10), (55, 27), (58, 30), (60, 32), (62, 34), (65, 38). Which best describes the scatter plot?
Answer: A
Five dogs are 25-35 cm tall and 5-10 kg, and five are 55-65 cm tall and 27-38 kg, with nothing in between: two clusters. Taller dogs are heavier, a positive association. Choice C is wrong: splitting into clusters does not remove the association. Choices B and D have the direction backward.
Question 6 of 20 · Multiple Choice
A sunflower's height (cm) is measured every 10 days (invented data): (10, 8), (20, 25), (30, 60), (40, 120), (50, 175), (60, 210), (70, 230), (80, 238). Which describes the scatter plot of (days, height)?
Answer: C
The height always increases, so the association is positive. But the growth per 10 days changes a lot: 17 cm, then 35, 60, 55, 35, 20 and 8. The points follow an S-shaped curve, so the association is nonlinear. Choice A would need about the same growth every 10 days. Choice B has the direction backward.
Question 7 of 20 · Multiple Choice
Eight students (invented data) give the number of letters in their last name and their 100 m running time (s): (4, 15.8), (5, 14.2), (6, 16.9), (7, 14.8), (8, 16.1), (9, 15.1), (10, 16.5), (11, 14.6). Which describes the scatter plot?
Answer: D
The times go up and down with no trend as the names get longer, so there is no association. Choices A and B would need the points to rise or fall from left to right. Choice C would need separate groups of points.
Question 8 of 20 · Multiple Choice
Kai wants to know whether the time students spend reading each week is connected to their vocabulary quiz scores. How should he set up the scatter plot?
Answer: A
A scatter plot needs one measurement on each axis. Reading time might affect the score, so it goes on the horizontal axis and the score on the vertical axis. Choices B, C and D put names on an axis, which is not a measurement, so they would not show an association.
Question 9 of 20 · Multiple Choice
Heights in a data set run from 142 cm to 188 cm. The vertical axis has 10 grid squares. Which scale fits all the data and uses the most of the grid?
Answer: B
Starting at 140 with 5 cm per square covers 140 to 190, which fits 142 to 188, and the data fill 46 of the 50 cm shown. Choice C only reaches 160, so the taller heights do not fit. Choice A fits but squeezes the data into about a quarter of the grid. Choice D fits but uses less than half of the grid.
Question 10 of 20 · Multiple Choice
A scatter plot shows the age of used bikes in years (x) and their selling price in dollars (y). What does the point (3, 140) mean?
Answer: A
Each point stands for one bike: x is its age and y is its price. Choice B reads x as a count of bikes. Choice C swaps the two coordinates. Choice D treats one point as a summary of all the data.
Question 11 of 20 · Multiple Choice
On a scatter plot, each time x increases by 1, y increases by about 5, across the whole plot. How would you describe the association?
Answer: C
y increases as x increases, so the association is positive, and it grows by about the same amount for each step in x, so it is linear. Choice B would need the increase to change along the plot. Choice A has the direction backward.
Question 12 of 20 · Multiple Choice
Priya plots hours studied and test scores for her class. One point, (5, 20), is far below the others. She checks her notes and finds that this student really scored 70: she typed 20 by mistake. What should she do?
Answer: C
Checking an outlier can show that it was recorded wrongly. Here the true value is known, so Priya corrects the typing mistake and plots (5, 70). Choice A is wrong: a real outlier should be kept and explained, not removed. Choice B keeps a value that she knows is wrong. Choice D invents a value to fit the pattern, which is never acceptable.
Question 13 of 20 · Multiple Choice
A cup of hot cocoa cools on a counter. Its temperature (°C) every 5 minutes is: (0, 80), (5, 62), (10, 50), (15, 41), (20, 35), (25, 31), (30, 28) (invented data). Which describes the scatter plot of (minutes, temperature)?
Answer: D
The temperature goes down as the minutes go up, so the association is negative. The drops every 5 minutes are 18, 12, 9, 6, 4 and 3: they shrink, so the points follow a curve that flattens out, which is nonlinear. Choice B would need about the same drop every 5 minutes. Choices A and C have the direction backward.
Question 14 of 20 · Multiple Choice
Which set of points shows a linear association?
Answer: A
In choice A, y goes up by about 2 for each step (2.1, 1.8, 2.3), so the points lie close to a straight line. In choice B, y doubles each time, so the steps grow (2, 4, 8). In choice C, the drops shrink (5, 1.6, 0.9). In choice D, the steps grow (3, 5, 7). Those three are nonlinear.
Question 15 of 20 · Short Answer
Ages (years) and heights (cm) of 8 children (invented data): (4, 101), (5, 112), (6, 114), (7, 124), (8, 125), (9, 136), (10, 137), (11, 146). Describe how you would set up the axes and scales for a scatter plot, then describe the association.
Put age on the x-axis from 3 to 12 in steps of 1 (with an axis break, or start at 0), and height on the y-axis from 100 to 150 in steps of 5 with an axis break. The points rise in a nearly straight band: a positive linear association, about 6 cm of growth per year. There are no clusters or outliers.
Question 16 of 20 · Short Answer
Nine students (invented data) give their hours of sleep and their reaction time (in milliseconds, thousandths of a second) on a computer test: (5, 334), (5.5, 312), (6, 309), (6.5, 296), (7, 293), (7.5, 276), (8, 410), (8.5, 270), (9, 255). Describe the association, name the outlier, and suggest a reason for it.
The other eight points show a negative linear association: more sleep goes with faster (smaller) reaction times, about 15-20 ms less per extra hour. The outlier is (8, 410), much slower than students with similar sleep. That student may have been distracted, missed the start, or had a problem with the mouse.
Question 17 of 20 · Short Answer
Nine students in a typing class are tested after different numbers of weeks (invented data): weeks and typing speed (words per minute): (0, 18), (1, 25), (2, 31), (3, 35), (4, 38), (5, 41), (6, 42), (7, 44), (8, 45). Describe the association and explain how you know its form.
The association is positive and nonlinear. Speed rises every week, but the weekly gains shrink: 7 words per minute in the first week and only about 1 per week at the end. The points bend into a curve that flattens out.
Question 18 of 20 · Short Answer
Nine eggs from a farm (invented data) give length (cm) and mass (g): (3.1, 10), (3.2, 10.5), (3.3, 11), (3.4, 12), (5.5, 52), (5.7, 55), (5.8, 57), (6, 61), (6.2, 64). The farm keeps chickens and quails. Describe the pattern in a scatter plot of these eggs.
The points form two clusters: four small eggs about 3.1-3.4 cm long and 10-12 g (quail eggs), and five large eggs about 5.5-6.2 cm long and 52-64 g (chicken eggs), with nothing in between. Longer eggs are heavier, a positive association.
Question 19 of 20 · Short Answer
A ball rolls down a long, gentle ramp. Maya records the time (s) and the distance it has rolled (cm): (1, 5), (2, 21), (3, 44), (4, 81), (5, 124) (invented data). She says the association is linear because the distance always increases. Is she right? Explain using the changes in distance.
No. The association is positive, but it is nonlinear. In the four 1-second steps the distance goes up by 16, 23, 37 and 43 cm. The steps keep growing, so the points curve upward instead of lying close to a straight line. "Always increases" describes the direction (positive), not the form.
Question 20 of 20 · Short Answer
A driver tests a car at different steady speeds (invented data): speed (mph) and fuel efficiency (miles per gallon): (30, 28), (40, 31), (50, 32), (60, 30), (70, 26), (80, 22). Describe the association. Is it positive or negative?
It is nonlinear, and it is neither positive nor negative overall. Fuel efficiency rises from 30 to 50 mph (a positive trend), peaks at about 50 mph, then falls from 50 to 80 mph (a negative trend). The points form a curve that goes up and then down.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.SP.A.1 mean?
8.SP.A.1 means students make scatter plots from pairs of measurements and describe the patterns they show. The patterns named in the standard are clustering, outliers, positive or negative association, linear association and nonlinear association. Students both build the plots and explain what they mean in context.
What is bivariate measurement data?
It is data with two measurements for each person, object or event. For example, the height and mass of each dog, or the temperature and the chirps of each cricket. The word "bivariate" means "two variables." Each pair becomes one point on a scatter plot.
How do you make a scatter plot?
Choose which quantity goes on each axis, pick even scales that cover the smallest and largest values, and plot each pair as one point. Label both axes with the quantity and the unit. If a scale does not start at 0, draw an axis break (a small zigzag) near the origin.
What is the difference between positive and negative association?
In a positive association, both quantities tend to increase together, so the points rise from left to right. In a negative association, one tends to decrease as the other increases, so the points fall from left to right. If there is no upward or downward trend, there is no association.
What is the difference between linear and nonlinear association in 8.SP.A.1?
A linear association has points close to a straight line, and a nonlinear association has points along a curve. A quick test: compare how much y changes over two equal steps in x, one near the left and one near the right. About the same change suggests linear; a very different change suggests nonlinear.
What is an outlier on a scatter plot, and should you remove it?
An outlier is a point far from the pattern of the other points. It should not be removed just because it looks odd. First check whether it was measured or recorded wrongly. If it is real, it often tells you something interesting, such as a different kind of object in the data.
What does clustering mean on a scatter plot?
Clustering means the points form separate groups with empty space between them. Clusters usually mean the data mix two or more kinds of things, such as walkers and bus riders, or cars and motorcycles. Describing the clusters, and what makes them different, is part of interpreting the plot.
Does an association mean one quantity causes the other?
No. An association only says the two quantities tend to change together. Sometimes one does affect the other, but sometimes a third factor affects both, or the pattern is a coincidence. High school statistics studies this difference more carefully.
What mistakes do students make with scatter plots?
A common mistake is using uneven steps on an axis, which bends the pattern. Others include swapping the coordinates of a pair, calling every rising pattern linear, treating an extreme but well-fitting point as an outlier, and describing the direction without the form.
How does 8.SP.A.1 connect to 8.SP.A.2 and high school statistics?
It comes first. In 8.SP.A.2, students fit a straight line to scatter plots that show a linear association, and in 8.SP.A.3 they use the line's equation to make predictions. In high school (HSS.ID.B.6), students fit functions to data and judge how well they fit.
07
Related Standards
6 standards
These standards connect to 8.SP.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.G.A.2Prerequisite
Graph points in the first quadrant to represent real-world problems
Lesson coming soon
6.SP.B.4Prerequisite
Display numerical data in dot plots, histograms and box plots
