While this support activity is designed for a face-to-face, synchronous delivery, it should be noted that supporting text and interactive examples have been embedded in the digital assignment page to assist asynchronous or hybrid course delivery and to be made more accessible to students performing make-up work.

Notes for synchronous active-learning delivery

Use this corequisite support activity to prepare students for finding and interpreting the coefficient of determination, [latex]R^{2}[/latex]. This support activity’s main aim is to develop students’ intuitive understanding of [latex]R^{2}[/latex] and support number sense around its value.

A large component of this support activity is exploration using the DCMP Data Analysis Tools. The tools perform fairly well on smartphones, so small group work may be possible if at least one person in each group has a phone they are willing to use. If not, then you may work through the exploratory questions as a whole class.

Questions 12 and 13 exemplify the exploratory approach. To wrap up the class period, you could pose these questions as a challenge to the whole class and indicate that you will discuss all the groups’ plots at the end of the period. If all groups or students have access to their own computers, then they could take screenshots of their scatterplots along with the [latex]R^{2}[/latex] values and insert them into shared, editable Google documents for the whole class. If not every group has their own computer or phone, you will again have to rely on the classroom’s computer and projector to conduct these parts of the support activity. Here are some ideas for executing the exploratory components as a large group:

• Groups can sketch their ideas on paper and submit the sketches to the instructor, who can then reproduce the sketches on the Explore Linear Regression tool.
• If the projector is projecting the image on the screen onto a whiteboard or smartboard, have students indicate with a pen on the board where the instructor should place data points.
• Have each group (or a subset of the groups) come up to the computer to make their plots themselves.

After all groups have presented their scatterplots, discuss as a class the similarities and differences among the different scatterplots. What ultimately led to [latex]R^{2}[/latex] being so small in all these cases?

Highlight the last question, and if you have time, ask what other information we might need besides [latex]r[/latex] and [latex]R^{2}[/latex] to make decisions about the appropriateness of fitting a line.

In this support activity, there are two skills that support number sense surrounding [latex]R^{2}[/latex]: conversion of decimals to and from percentages and squaring. Consider relating these operations explicitly to the numbers students saw in the exploratory phase.

Possible student confusions or pitfalls:

• In Question 3, students may answer that the coefficient of determination is always less than the correlation coefficient based on their exploration. Once you have finished the squaring section of the lesson, explicitly revisit this question and highlight the fact that the coefficient of determination may be equal to the correlation coefficient in the case of 0 or 1.
• Be prepared to address that due to rounding in the tool, a correlation coefficient of (approximately) 1 could lead to a coefficient of determination that is less than 1 (see sample answer to Question 2, Part A, for example). This could be a time to highlight the fact that [latex]R^{2}[/latex] is more sensitive than [latex]r[/latex].