• Throughout the lesson, it is helpful to emphasize how remarkably strong the association is between attendance and test scores. The strength of the association, combined with its positive direction, will naturally entice students into viewing the relationship as causal. Then the discussion question’s result – that raising attendance did not significantly improve test scores – becomes even more surprising. The surprise sticks with students and serves as an anchor moment for distinguishing correlation from causation throughout the unit.
• This lesson introduces the language and tools students will use to describe relationships between two quantitative variables. As students learn to identify form, direction, strength, and unusual features, it can be helpful to continually comment on these features in the scatterplots they analyze throughout the unit. This practice supports later work in the unit with correlation, regression equations, and the coefficient of determination (r²).

First, download this lesson's slide deck and handout key to see the prompt and sample responses for the Lesson Starter. Then, check out the additional background notes below.

Instructional routine: Ten-Minute Talk. The lesson provides space for students to jot down their thoughts on the prompt, before discussing with a partner and engaging in whole group discussion. In this case, they are asked to tell a story. It may make sense to offer more time for the independent thinking (writing) portion for this prompt and to only use a few examples for discussion (reducing whole group discussion time). You can find more background on implementing a Ten-Minute talk here.

Purpose & Background: This Lesson Starter is designed to get students thinking about the stories of scatterplots. It also provides an opportunity to formatively assess what students may already know about associations between two variables. Throughout the lesson, students will explore ways to determine and describe relationships identified through scatterplots, including direction and strength. In the Lesson Synthesis, students are invited to refine the story they created in the Lesson Starter by replacing their informal descriptions with the more precise language from the lesson.

First, download this lesson's handout key and read through its Discussion Question section. Then, check out our model discussion norms and the additional background notes below.

• The diagram below may be helpful when discussing the statement from the handout key: “Students experiencing poverty might not only face attendance barriers. They might also experience hunger or have less study time (due to working a job or taking care of siblings). Because of these confounding factors, focusing solely on attendance may not provide students with all the resources they need to be successful.”



• There are additional possible confounding variables that students may identify during the discussion. Common examples include access to tutoring, housing stability, and school resources. The key idea is that these factors may simultaneously influence both attendance and academic achievement.

• The attendance context provides a useful example of why strong associations should be interpreted cautiously. The lesson intentionally presents a remarkably strong relationship between attendance and achievement, before introducing evidence that attendance interventions did not substantially improve test scores. This creates a natural opportunity to discuss the distinction between association and causation.
• Research on chronic absenteeism has found that attendance is associated with a wide range of student outcomes, particularly among students experiencing poverty. At the same time, researchers have identified many factors that influence attendance, including health, transportation, housing stability, family circumstances, and school conditions. These factors provide concrete examples of potential confounding variables that may influence both attendance and academic achievement.

• There are three common correlation / causation fallacies. High School Statistics primarily focuses on the first, but it is helpful to understand all three:
• Confounding variable fallacy: A lurking variable influences both the explanatory and response variables simultaneously, creating the observed association. This is the primary focus of the lesson’s attendance example, and students may be able to think back to numerous other examples from the course.
• Directional fallacy: Occurs when we believe X causes Y, but in reality, Y causes X. For example, we might assume that higher attendance causes stronger academic performance, when in reality stronger academic performance increases positive feelings about school, which then leads to higher attendance rates. More examples and explanations of this fallacy can be found here.
• Spurious correlation fallacy: Occurs when two variables are correlated purely by chance, despite having no meaningful connection. For striking examples of these, check out this great collection of spurious correlations from Tyler Vigen. One such example is the surprisingly strong association between the number of Disney movies released per year and the yearly motor vehicle theft rate across the United States.
• The terms explanatory variable and response variable are preferred over independent variable and dependent variable because they imply fewer assumptions about the nature of the relationship. In many observational studies, one variable may help explain or predict another, without implying a causal relationship. Using explanatory and response language helps reinforce this distinction, rather than using “dependent” (which could imply causation). In addition, the word “independent” already has a specific meaning elsewhere in statistics and probability, so this terminology helps avoid unnecessary ambiguity.
• A large-magnitude correlation coefficient does not guarantee that a linear model is appropriate. Anscombe’s Quartet provides a striking example: four data sets with identical correlation coefficients can exhibit dramatically different patterns, including non-linear relationships and influential outliers. Reinforce that scatterplots should always be examined before interpreting r.
• Outliers in a bivariate setting differ from outliers in a univariate setting. A point is not necessarily unusual simply because it is far from the other observations. Instead, an outlier is a point that does not follow the overall pattern seen in the data. This idea will be revisited later through residuals and regression.