Line of Best Fit Spiral Lesson
Lesson 5 of 7
Objective: Students will be able to work with a line of best fit.
This is not a test prep unit (although it could be construed as such). My goal is not to simply expose students to test questions and format (although they will get an amazingly wide variety of exposure and practice). I want to help my students look back on the year and fill in the gaps and build on their foundation through repeated practice. All students need a chance to recall the things they have forgotten (or never mastered in the first place). Our curricula often push forward, build up and look ahead. But we can’t forget to spiral back and give students an opportunity to revisit topics from past units. These lessons represent my attempt designing a rewarding and fulfilling review unit for algebra. It is an attempt to prepare students for the summative exams that we use to asses our curriculum design and mark student progress.
The lessons in this unit are built around a simple format:
- A 15-minute start up problem, where we introduce the basic ideas of the concept they are about the review.
- A 45-minute chunk of time devoted to giving students in depth, meaningful practice, where students use very short videos to cover appropriate problems
The work reserved for the final 45 minutes of class is connected to a series of homework assignments. The materials are designed so that whatever students don’t finish in class, they can do at home. If they finish the current assignment, they can move ahead and complete future assignments.
Each day, the lesson starts with some follow up on their practice work. Based on the previous lesson, I will be prepared to ask specific local and global questions about their work. Local questions can be anything specific to a certain aspect of a problem, like "how did you get that number there?" Global questions can be things like, "how does this example connect to the problem before it?"
We do a lot of work with the line of best fit, so I give them a question that spirals back to some of the basics:
Notice that the question is simply asking them to read the graph in a local way. Students are to find a precise x value that matches a precise y value. This is basic language of functional thinking on a graph. Find a corresponding input and output. Of course, they write "how many dollars did Jim earn for working for 5 hours?" Here students need to identify the wage in dollars as the independent variable (input) and the hours of work as the dependent variable (output).
Many students struggle with the precision required to answer this "estimate." They see the word estimate and look at the graph without lining up a ruler to find the most precise answer possible. They need to recognize that the line already represents the acceptable estimate and now our job is to read it precisely.
I then discuss the line of best fit with the class, asking them low inference questions like, "why doesn't it go above/below/through most of the points?" I want students to recall that a line of best fit minimizes the distance to the most points possible. This doesn't mean it has to go through the most points, but that it should use all the data to best represent a general trend.
Before I have them work on the group of problems, I ask them to recreate a line of best fit on their graphing calculator:
This work is split into two groups. The first part consists of work they have already done for me. These six are included in the video links below and match the template. I ask students to do these problems first. The repetition isn't problematic, since it was much earlier in the year and now I want them to do the problems on their own.
There are also new problems online, which can be found here.
Many of these questions can be easily done without a graphing calculator, but I ask them to try and use the graphing calculator on each problem (I know that they may encounter a problem in which linear regression on the graphing calculator would be necessary). Of course, I want to stress the logic of the mathematics, not just the procedure on a calculator. So I look out for their approach to questions like Question 3.
This question fascinates me because students always attempt to use the calculator first when they solve this problem. I see them work as a team to enter all the data points and perform the linear regression via the TI-84 graphing calculator. Then I ask them to step back and use the fact that it is multiple choice to quickly identify the most likely slope and intercept. That moment is the "aha" moment for them. To realize that they key to these types of problems is to think globally as well as locally. In other words, look for general trends, like slope and intercept before you zoom is to the exact slope and intercept of the line of best fit. Then look for specific values that best match the line. Look to see if the slope you have chosen best represents the given line. Do this before you resort to an unnecessary level of precision. Determining the level of precision is part of the fun of these types of problems.
Here is the template I give them for these problems: Line of Best Fit Template