Students will be able to construct and deconstruct the box and whisker plot.

Students can understand a box and whisker plot by constructing it.

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?"

15 minutes

I start off today's lesson by reviewing a fairly sophisticated box and whisker question, something like: Box and Whisker Plot 3.

I give students a few moments to deal with the problem, then we compare results. For example, if 25% of the data is below 5 degrees, we could interpret this to mean that the first quartile is 5 and that the average of B and C needs to be 5, since the first quartile would be between those data points. So one discussion is around how we can quickly find a value for C that averages with B to get 5. This conversation usually starts with something basic, like “we know that B is 2 below 5 and that C needs to be two above 5, so we add 2 and 5 and get 7.”

I usually return with a low inference question, “how can you check this?” Many of my students will recognize that we can take the average of B and C. When they report this I will ask them to reason out loud. “And what do you get?” They will say something like, “3 + 7 = 10 and 10/2 = 5.”

Now, I want to turn up the questions a notch, “what if the quartile wasn’t so close to 3, do we need to count up in the same way? Or is there a more *efficient method*?” I am hoping that my students will say something like, “Can we use algebra?” If so, I will scaffold their work a little it by saying:

I like your thinking, lets construct this equation. We are given the value for B as 3 and know the average of two numbers needs to be 5. Can we reconstruct the steps of finding the average by using a variable?

From here I expect that the conversation will go something like this Sample Dialogue.

We will proceed to solve for x and establish the algebraic method for finding the average and evaluate specific questions like, “do we need parenthesis around 3 + x?” I usually resolve those question by plugging in specific values for x to see how that could alter the answer.

After, we will use a similar technique to construct the other quartiles and discuss the *variety* of correct answers (students can use many different numbers and still reach the same value for the quartiles) we move onto the second part of the question.

I want to make sure to cover the Interquartile Range (Q3 – Q1) and the procedure for determining outliers. I want to help students to build their intuition around the idea that if a max is above the sum of the third quartile and 1.5 times the interquartile range or minis below the difference of the first quartile and 1.5 times the interquartile range, then those numbers are outliers.

45 minutes

For practice I give students a set of box and whisker plots and a helpful Template to follow along. The templates are set up to help me recognize when a student needs help. They rate how they feel about each problem as they finish and I look at these numbers to figure out if they feel comfortable with the material. I try and keep this rating system simple. Something like, give yourself a 4 if you really understood the question and give yourself a 1 if you felt really overwhelmed.

Practice Problems

- Box and Whisker 1
- Box and Whisker 2
- Box and Whisker 3
- Box and Whisker 4
- Box and Whisker 5
- Box and Whisker 6
- Box and Whisker 7
- Box and Whisker 8
- Box and Whisker 9

Since all of the work is accessible via video on YouTube and/or internet archive (which is not blocked at schools and can run on *very* slow networks), students start a video, pause it, try the problem and check the worked example. These videos are more worked example than instruction, but they seem to really help.

I use this time to conference with groups of students on issues they might have had during the start up or questions they might have as they attempt the problems through video. If I see a common issue, I might give them an exit ticket to bring home and return the next day (it would be very similar to the start up problem but with different numbers).

Previous Lesson