# Thinking Outside the Box – Working Forward and Working Backwards with Box and Whisker Plots

Box and whisker plots baffle most middle school students, at least initially.   I think they get lost in the language and acronyms associated with the plot long before they ever try to make sense of the representation.    To address that, several years ago, one of my colleagues and I decided to break the instruction into multiple lessons.    We decided to start by simply focusing on the Five Number Summary the first day in order to build background knowledge.    This would give students the chance to become familiar with the vocabulary and to work with the vocabulary before asking them to apply the concepts inherent in the vocabulary to create a data representation.    Then, after students had a solid grasp of the Five Number Summary, we would move on to creating and analyzing box and whisker plots.

Day One – The Five Number Summary

We started with an NCTM lesson, “It’s All In the Cards”.   Each student starts out with a set of twelve number cards (1-12).   They put them in increasing order and then find the median.    Next, they find the median of the lower half of the data and the median of the upper half of the data.   As they do this, I introduce the Lower Quartile (LQ) and Upper Quartile (UQ) vocabulary.    We repeat the process with 11 cards so that they have the opportunity to work with an odd number of data points.   Next, we repeat the process with 10 and then with 9 cards.  (Initially, I use the numbers in cardinal order.   As we progress, I choose random sets of numbers from the cards.)  At this stage, we start to talk about the parts created by the median, the LQ, and the UQ.    How big are they?   What percentage of the data is in this section of the cards?   As students begin to realize that the data is being broken into quarters, I introduce the idea of Q1 and Q3 as alternate names for the LQ and UQ.    I also ask them where Q2 would be.

Its All In The Cards Cards 1 to 12

For homework, I give students a handout in which they need to find the Five Number Summary for three different data sets.    As I reflected on this first day of instruction, I felt that students had a pretty strong grasp of the concepts.   However, I felt like there was enough time and space in the lesson to increase the cognitive demand.   I didn’t want to jump forward to the box and whisker plot because I wanted them to live with the ideas in the lesson for a little bit of time first.   However, I did want more than I had with just the “find the five number summary” that I had in place.   I decided I would like to try adding several problems both to the lesson and to the homework handout in which I ask the students to essentially work backwards.   I would give them a five number summary and then ask them to create a data set to fit it.   I felt this would require students to think in a different way, to recognize multiple solutions, and provide the opportunity for some rich discourse about why a data set would or would not fit and how to alter a data set to make it fit if needed.   Here is the updated version of the file.   Five Number Summary Version 2 for Higher Cog Demand

Day Two – Making and Analyzing Box and Whisker Plots

I began the lesson by having students create a data set by measuring the time needed to find Waldo in a “Where’s Waldo” picture.    To do this, I took apart a Where’s Waldo book and laminated the pages.   I had students work in pairs: one student searched for Waldo while the other tracked the time and then they reversed roles.    Students then used the times to create a class data set for the time to find Waldo. They also created the Five Number Summary for the class data.

I used the Where’s Waldo Five Number Summary as the basis for instruction on how to make a Box and Whisker Plot.   After creating the box and whisker plot, students worked on a foldable to summarize the main ideas and practice the concept.  (I got the initial version of this foldable from my colleague, Laura, who does not have a blog but should.   I took her version and made a few modifications to fit my needs).

First, students had to summarize the main ideas associated with box and whisker plots.    This forced them to crystalize their thinking.   It also provided them with a resource they could use to study for the upcoming test.

Next, they used a data set provided in the foldable to find the five number summary, to find the range and IQR, and to create a box and whisker plot.  This was the “Working Forward” thinking.

Once they could “Work Forward”, the students had a new twist on the task, “Working Backward”.   They were given a box and whisker plot.   From this, they had to find the five number summary, range and IQR, and determine which of several data sets corresponded to the plot.

Students had one additional opportunity to practice working both forward and backward in the foldable.

Box and Whisker Foldable Version 2

Moving forward, I decided to modify the second “Moving Backward” problem.   I am still providing the box and whisker plot and requiring students to find the five number summary, range and IQR.  However, rather than asking students to select the correct data set, I am asking them to create a data set to match the box and whisker plot.   I could control the level of difficulty by specifying the number of items in the data set.

Homework for Day 2 was the Migraine Medicine handout from the NCTM Navigating Through Data Analysis in Grades 6-8

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