Mathematical Modeling: Best Night's Sleep

I got a Jawbone band for Christmas this year and I have loved seeing how well I do in achieving my sleep goal and steps goal.  I am finding that I am actually a really good sleeper and that my attempt to take 10,000 steps a day is hit or miss.  As a teacher, I do better with my steps on the weekend and days when my students are testing are my worst.  I thought that my students could benefit from taking a look at some real-life data and apply what they've learned from our studies of ratio, similarity, and proportions.

With almost 200 students this year, I did not want to grade that many tests for the end of the chapter, so I made this a group assessment.  My students sit in groups of four as it is, so the groups were already chosen.  As you can see from the write-up below, each student had a different responsibility in the group.  Students dove in to the data before choosing roles that spoke to them.  I was impressed to see how many different methods there were that students latched on to.  Some chose percentages, some chose to represent their data in bar graphs, other pie charts; it was interesting to see what they were drawn to.

Here are the three pieces of data that I showed the students:

I actually cut off the bottom portions that showed the amount of sound sleep, deep sleep, how long it took to fall asleep, and how long each person was in bed for.  Students were given the bar graphs as well as the amount of time each person slept and what percent of their goal was met.

I started class by showing the three graphs through the projector so that they could see them with color.  Students determined what the data showed them, what additional information they needed to know, and asked other questions that came up.  I chose to answer some and leave some for them to answer because the data was available to do so.

At this time, I gave students the following handout for them to read and decide what role they were interested in.  

__________________________________________________________________________________________________________________________________________________________________

Name: ____________________________________________

Geometry 2013-2014

Date: _____________________________ Block: ________

Chapter 7 Similarity Test: Who Got the Best Night's Sleep?

            This is a group test, but each student is responsible for submitting his or her own portion of the test.  Please put a check next to the role that you had in your group in the list below.  If you are a group of 4, each student should take on one role.  If you are a group of 3, each student will take on their own role and then share the role of “Processor”.  When your group is finished, please staple all of the parts together with this page as a “cover page” for each section.  Please staple the pages in the order as they appear below.

________ Group Member 1: Graphics

            When considering who got the best night's sleep, it’s important to compare “apples to apples”.  You may notice that in the three sets of data that I provided you, all three of them use different scales because they went to bed and woke up at different times.  The job of the Graphic team member is to create a visual that uses the same scale for all three sets of data and create this graphic in a way that helps support your argument for who got the best night sleep.  You will be graded on accuracy, neatness, and quality of your graphic.

________ Group Member 2: Data

            Looking at the three sets of data, you can see that there is some “number crunching” to do.  In order to compare the three sets of data accurately, you need to use the same scale or the same units of measure.  It is your responsibility to help your group make sense of the numbers so that you can compare them accurately.  You will be graded on the accuracy, neatness and quality of your calculations.

________ Group Member 3: Argument Writer

            I purposefully chose data that didn’t have a “clear winner” so that you would have to defend your answer to “Who got the best night's sleep?”  Think about what you value in a “good” sleep.  Is it length of time spent sleeping?  Most deep sleep?  There is no right answer here, so you need to convince me that your answer is correct.  Imagine that you are convincing a skeptic, or someone who got a different answer than you.  You will be graded on the clarity of your argument, how well you connect it to your data and graphic, and the quality of your writing.

________ Group Member 4: Processor

            I may have some questions as to how you arrived at your final answer, but that’s why we have the Processor.  I would like to know how your group worked together to come to your final answer.  This should be a descriptive account of what your group did, but more importantly, WHY?  Because this is a chapter test on similarity, please make sure to mention HOW you used similarity, ratios, and/or proportions.  You will be graded on the clarity of your process description, how well you connect it to the other three group members, and the quality of your writing.

_________________________________________________________________________________

_________________________________________________________________________________

I found that students really struggled to understand how they could translate this data into a common measurement.  Lots of students used percent, which I thought was great, but if they didn't think to use percentage, they struggled to come up with an alternate way.

I have to say that I was please with the level of engagement and focus that my students exhibited.  They were interested to know who got the most sleep and what the data meant.  I felt that this was a good exercise in mathematical modeling because students did not feel the pressure of arriving at the "right" answer and they were rewarded for their process as opposed to their answer.  Also, the use of real-world data was beneficial and useful because it made things more applicable and gave the activity context...everybody has slept before. :)

Transitioning to Transformations

Last week I gave my students a task from Illustrative Mathematics called Similar Triangles as an introduction to the concept of Angle-Angle Similarity.  Here's how it played out...

A brief disclaimer is that although I have been teaching Geometry all year, I have not made the complete transition to teaching it through Transformations.  I have done a few things here and there, but more than anything it was to try tasks and strategies out for myself and to challenge my students.  They have had some experience with transformations and each time we do something with them, the students seem to grasp the concepts quite well.  When doing the AA task, I encouraged my students to use patty paper as a tool for doing transformations.  Also, this task was their first introduction to dilations.

I started by drawing 2 equiangular triangles on the board (see picture below) and asked the students if the 2 triangles were similar.  A lot of them said NO because there wasn't a "nice" number to multiply 3 by to get 8. :) So, I wrote the equations from the picture and we came to the agreement that 8/3 or 3/8 would have done the trick, depending on which way we were scaling/dilating.  They seemed to find this pretty magical...this should be an indication of the level of number sense my students are working with. :)

So, as they worked on the AA task, they did great with the translating and rotating.  I wrote up a list of requirements (see picture below) and they were able to articulate them quite well.  

They did great with the "vagueness" that I had been struggling with.  So, when it came to dilating Triangle ABC (I didn't worry too much about the prime notation for this task), they knew that they had to make it bigger, but they didn't necessarily know by how much.  This is where I wrote the new equation from the picture below (AB x ? = DE) and they were actually able to connect it back to the 3 and 8 sided triangles.  The magic (or madness for some of them) continued.  Ultimately, they were able to say that you needed to multiply it by DE/AB in order to dilate the smaller triangle to the larger triangle, and I was pleasantly surprised.

Another issue that I'm having is, Where do I go from here?  I guess I'm still having difficulty wrapping my head around the fact that THIS is the new definition of similarity, but I'm getting there.  Pulling it all together and not seeing these tasks as individual exercises is something that I'm still working on.  I appreciate your feedback.  Please forgive my vagueness or lack of precision in notation, but I think that mostly comes from trying to meet my students where they are, and let's be honest...where I am, too.

Thanks for reading!