Fishtank Learning

8th Grade Math
Unit 8: Bivariate Data
Lesson 3

Objective

Identify and describe associations in scatter plots including linear/nonlinear associations, positive/negative associations, clusters, and outliers.

Warm Up

Sketch a line or curve in the following coordinate grids that show the following:

a. A line with a positive slope

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b. A line with a negative slope

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c. A nonlinear relationship

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Anchor Problems - Problem 1

Your teacher will give you a set of cards. Your job is to use the cards to label the scatter plots based on the types of associations that exist.

For each scatter plot, ask yourself these questions:

Does there appear to be a relationship between the two variables?
If there is a relationship, what happens to one variable as the other changes? For example, as $$x$$ increases, $$y$$…?
If there is a relationship, is it linear or nonlinear?
If there is a linear relationship, is it positive or negative?

Purpose

In this Anchor Problem, students classify scatter plots by identifying whether relationships are linear or nonlinear, positive or negative, or show no association. Through sorting and defending their decisions, students refine their use of precise statistical language and connect visual patterns in data to ideas of slope and direction from prior work with linear functions.

Setup

Print enough copies of the Scatter Plot Relationships Sorting Cards on cardstock, so that each group gets one complete set. Have students cut the cards out in class, or prepare them ahead of time by cutting along the dotted lines.

Facilitation Guidance

Launch the problem by displaying two contrasting scatter plots from the sorting cards (for example, one clearly positive linear and one clearly nonlinear). Ask:
- What’s similar about these? What’s different?
Do not give vocabulary yet. Let students suggest informal language like “goes up,” “straight,” or “pattern.” Then introduce the sorting task and the vocabulary they’ll use from the sorting cards. Frame the task as sense-making, not just matching.
Have students work in groups with one full card set. Their goal is to match description cards to scatter plots and justify their reasoning before finalizing placements.
- As you circulate, listen carefully for how students are defining “linear.” A common overgeneralization is that students expect a linear relationship to mean the points lie in a perfectly straight line. When this arises, validate the thinking — students are drawing on prior experiences with exact lines — and gently shift the focus to overall trend. Ask:
  - If you drew a line to represent the pattern, would it capture the general trend of the points?
  - Does a linear association mean exact, or does it describe the general direction of the data?
- You may also notice students connecting points and concluding that there is no relationship or that the relationship is not linear because it zigzags. Acknowledge that this makes sense procedurally, then redirect attention to the structure of scatter plots. Ask:
  - Are scatter plots meant to be connected point-to-point?
  - When you step back and look at the whole picture, what direction does the data tend to move?
Pause the class once or twice during work time to surface emerging insights. Invite a group that revised their thinking to explain what changed and why. This makes the shift from exact to statistical reasoning visible.
- If students are using absolute language such as “as $$x$$ increases, $$y$$ increases”, press students to refine their claims. MP.6 Attend to precision.
  - What terms can imply that this association is a tendency, not a fact? (Always, tends to, generally, etc.)
  - Do you see any exceptions? Could there be other exceptions, if more data was collected?
Debrief the problem by bringing the class back together and projecting each scatter plot in turn. Invite groups to justify their classifications, focusing on the reasoning.
- What makes a relationship linear instead of nonlinear?
- How can you tell the difference between nonlinear association and no association?
- What evidence supports calling this a positive (or negative) association?
- Why do we say “tends to increase” instead of “increases”?
Make explicit connections to prior work with slope: a positive linear association resembles a line with positive slope; a negative linear association resembles a line with negative slope. Emphasize that scatter plots describe tendencies in data, not exact rules.
Supporting Access for Students

Increase engagement and participation in the discussion by asking students to consistently restate each other’s thinking. This could look like:
- After a student shares an approach, ask the class to restate what they heard to a partner using precise mathematical language.
- Select one or two students to share their restatement with the class, and ask the original speaker if their peer was able to accurately represent their thinking.
- Bring attention to words and phrases that clarify or draw out the meaning of the original explanation.

Notes

In this Anchor Problem, students make the connection between positive linear associations and lines with positive slopes, and negative linear associations and lines with negative slopes. The language and vocabulary is scaffolded for students, as it is given to them in the cards. Throughout the rest of the lesson, listen for students using this language in other examples, and if need be, direct them back to this problem for reminders or guidance. This scaffolded language can also be edited to remove the scaffolding.

Anchor Problems - Problem 2

Pablo is curious if there is a relationship between the amount of time high school students spend on social media and their academic achievement. To explore this, he conducts a survey of 30 randomly selected students from his school, and asks them how many minutes they spent on social media over the weekend and what their most recent grade point average (GPA) was. Pablo summarized his data in the scatter plot below. ]

Scatter plot showing GPA vs. time spent on social media (minutes), with points indicating individual student data.

a. Describe the relationship between the amount of time spent on social media and GPA.

b. Are there any deviations from the pattern? Explain what you see.

c. Are there any clusters of data? What do they mean in context?

Purpose

In this problem, students analyze a contextual scatter plot to describe the direction and strength of an association and to identify clusters and outliers in context. Students refine their ability to interpret structure in data while using precise statistical language and distinguishing between general trends and individual deviations.

Facilitation Guidance

Launch the problem by posing a neutral question before displaying the scatter plot:
- Do you think spending more time on social media necessarily leads to lower grades? Why or why not?
Allow brief partner discussion without evaluating responses. Emphasize that today’s goal is not to prove or disprove opinions, but to examine what the data show.
Display the scatter plot without the questions first and give students quiet time to study it. Give students time to put words to the visible structures that they see MP.7 Look for and make use of structure. . Ask:
- What do you notice right away?
- Where do your eyes go first?
Have students work independently to respond to Parts (a)–(c), then briefly compare answers with a partner. Encourage them to refer to specific features of the graph rather than making general statements. As you circulate, listen for precision in three areas:
- Direction and strength — If students overstate the relationship (“social media lowers GPA”), press them to describe the trend more carefully MP.6 Attend to precision. :
  - Is it exact or does it tend to happen?
  - How strong does the pattern appear?
  - How can you explain that there is an association but not causation?
- Outliers — When students identify a point that stands out, ask:
  - What makes it different? Why does it make sense to call this an outlier?
  - What does that point represent in context?
- Clusters — If students notice a dense region, push interpretation MP.2 Reason abstractly and quantitatively. :
  - What does this concentration suggest about the group of students surveyed?
Debrief the problem by synthesizing around three structural elements: direction, strength, and deviations. Ask:
- How would you describe the overall association?
- Is it strong, moderate, or weak? What evidence supports that?
- Where do you see clusters? What do they represent?
- Are there any outliers? What might explain them?
- Does this scatter plot prove that social media causes lower GPA? MP.2 Reason abstractly and quantitatively.
- What other factors might influence GPA?

Problem Set

View Problem Set

Target Task - Problem 1

Which scatter plot shows a positive nonlinear relationship? Explain how you know.

Two scatter plots with points showing a positive nonlinear relationship in Graph B.

Target Task - Problem 2

Sketch a scatter plot in the coordinate plane below so that the following statements are true about your scatter plot:

There is a negative linear relationship between the variables.
There is a cluster of data.
There are no outliers in the data.

Coordinate grid with axes labeled x and y, ready for plotting data points.

Fishtank Learning

8th Grade MathUnit 8: Bivariate DataLesson 3

Objective

Warm Up

Anchor Problems - Problem 1

Anchor Problems - Problem 2

Problem Set

Target Task - Problem 1

Target Task - Problem 2

8th Grade Math
Unit 8: Bivariate Data
Lesson 3