The Mastermind Project, Day 2: Choosing the Best Representation
Lesson 3 of 11
Objective: SWBAT construct histograms on the real number line, and they will consider the relative strengths of dot plots, box plots, and histograms.
Opener: Why not dot plots?
Today's opener is somewhat of a trick question: I ask students to create a dot plot for the Greed data, in hopes that they will see that a dot plot is not the best representation for this particular data set. Students should have this data set in their notebooks, but it's also on the side board.
Part of using tools strategically (MP5) is knowing which one is the best for a given situation, especially when there's nothing stopping us from using one that's less appropriate. There are two reasons students may recognize a dot plot as inappropriate for this data. The superficial one is that it's just annoying to make the plot. When the data is so spread out, and nothing repeats, trying to maintain precision when placing each of these dots becomes an arduous task. Once the plot is made, the richer understanding I'm looking for is that a dot plot of this data doesn't really tell us anything. It's hard to draw any conclusions from looking at all these dots just sprinkled along a number line. It's not too hard to find the middle of the data, and we can get a basic feel for how the data is spread out or concentrated in a certain place, but there must be a better way.
I give students a little more than 5 minutes to play with this task and come to their own conclusions, before leading a brief conversation about the merits and weaknesses of using a dot plot in this situation. I ask students to compare the dot plot to the box plot we made on the first day of class. I ask them to compare this dot plot to the one we made for the Mastermind data during the previous class. Then I say that today we're going to look at a third representation of data on the number line.
The purpose of the opener was to show students that although they are easy to make, dot plots don't always help us to make sense of a data set. So we have a thing called a histogram. When dot plots are just a set of individual dots on a number line, histograms can better reveal behavior of data within ranges.
I start by reviewing the learning target (slide #3 of Greed MM and Histograms), and introducing the idea that today we're going to look at histograms. I make a show of the fact that a histogram is not a bar graph, and I explain that the width of the bucket actually tells us something. A superficial difference between the histogram and the bar graph is that while there are spaces between the bars of a bar graph, there are no spaces in between the bins of a histogram. More importantly is what this represents: a histogram is a accounting for an entire range of a quantitative data, while a bar graph can represent the counts of pretty much anything.
Then, as I share the notes on slides #7 through 13 with the class, I work through the example with students. I explicitly make the point that the problem with making dot plot for the Greed data is that it's quite spread out. If we want to be able to group data into ranges of results, we can use a histogram. These ranges can be called "bins". If we make a number line counting by 100's, then each bin will contain all values within the range 0 to 100, 100 to 200, etc. One key detail is that the upper bound of each bin is the lower bound of the next bin, and it's the lower bound that is included in each bin, not the upper. I more elegantly make this point to students by showing them the example on slide #9.
Once we've decided on the width of each bin, we can use a frequency table to tabulate the results. Because the data set is still on sticky notes, I also move the stickies around to show what it means for each data point to be in a bin.
After we have the frequency of data points within each bin, we plot the data by using the height of each bin to represent frequency. Then, after we complete the first histogram as a group, I tell students to practice by making another histogram, this time with a bin width of 50, for the same data. This gives me time to circulate and see what each of my students understand.
Finally, I mention that there are some histograms on Problem Set 1 (from survey), and anyone who finishes ahead of the class should take a look at that.
Students have a choice of two record sheet prompts today.
See also slide #18 Greed MM and Histograms for the prompts.