Taxi Fares: MAPs Activity- Cutting out what you don’t need

I really enjoy using MAPs tasks in class, but with my recent “cutting out what you don’t need” outlook, I am finding myself looking twice even at this resource.

Consider the following graph:

Taxi Cab Graph

What do you notice?  What patterns do you see?

Depending on the level of student, you can get a variety of answers.  It has a positive correlation, or that it’s an increasing function that is not continuous.  You can get responses about statistics: the range, min/mas, mode, median, etc.  All of those observations are great for what the original worksheet has students do.  I have even considered cutting out the last sentence in the directions, hiding the axis labels- having students provide them as well.

One question a student asked: “Why would he graph his fares?”  A good question- where would this problem be placed in Dan Meyer’s (Real World,Fake Work) quadrant?
I answered: “That is a good question,  I think he did that so he could look for a pattern.”
Student: “Do I need to use the graph?”
Me: “How do you want to look at it?”
Student: “Didn’t it say he wrote down the times and distances?  I’ll use his list.”  Well, that is a problem since I didn’t create one and part of why I am having students work on this problem is to gather that information from graphs.
Me: “I don’t have it with me, is there a way you can re-create his list?”
Student: “Yes, I can use the graph.”
The student then starts working to make his list.

My point is: students can connect a lot of background experience to the graph, and it does the class a world of good to talk about them.  After you get those ideas floating around in their heads you hit em with the real math:

David needs to make $30 per hour.  Should he charge by hour or mile?  How much should he charge?  Explain your reasoning.

I can walk around and look at student work, asking questions here and there to understand their approach and thinking.  Students would solve this question in whatever manner makes sense to them.

This is what the problem actually looks like from MAPs:

Taxi Problem

I tried this with one class, and they blindly followed the directions for 1 without considering why 4 hours was chosen.  The bad thing about that is that they assumed that the work they did for 1, they were supposed to extend to 2.  They came up with 14min 40 sec for a 4mile trip, which they then approximated to 4 trips per hour, and 16 miles total.  They then got an answer of $1.88, which was close.  They did all of this work without thinking about what or why they used 4 mile trips, so when the second handout came- they were lost because it did not resemble their work.  We then backtracked and re-examined their work (I knew this was coming so I didn’t intervene right away), and they did the process using all of David’s trips.  Many students struggled with this, they did not have the academic stamina to redo a problem that was solved (in their minds).  To make this less painful for my students, I asked students to write why the sheet had them examine a 4 mile trip.  Once they saw a point to re-working the problem, they went back to the graph and measures of central tendency.

I plan on hitting students with this same type of problem next week, cutting out what I don’t need, and see how students progress.


Human Histogram

Since it looks like I may have an alteration to my teaching assignment next year, I feel it is time to dust this off and pump it up.


Human Height Histogram


I started using this activity when I was teaching 7th grade.  There is a lot of physical growth that happens during the middle school years, and this is a great way for students to look at data that is meaningful- data about themselves.

We start by making predictions about which gender has the greatest average height at the beginning of the year.  Typically in 7th grade it was the girls, and we devise a way to determine it.  Students come up with a way to determine the mean height of both groups, which usually involves measurement since students are not accurate with their actual height.  We spend the day measuring all the students in the grade, and then look for the measures of central tendency.  Students will tell me the range, median, mode and mean of each gender, and they also are curious how that stacks up for the group as a whole.  I ask them how we could display this visually and they immediately tell me to graph it.  The type of graph will vary however, many 7th graders are overly “precise” in their measurement of height since they are trying to be taller than their friend.  The number one graph of choice for my students has been a bar graph, so I allow them to try this method.  After attempting to draw a bar for each student, they decide to use a different graph.  This normally results in a scatter plot where we use different colors for boys and girls.  My 7th graders are not particularly adept with reading scatter plots yet, and will complain about the representation, saying that it is more confusing than looking at a list of numbers.  So we brainstorm for a new idea.

After students struggle with other graph types, I ask them if they would want to go back to a bar graph.  I usually get a unanimous response of “YES!”  We talk about how bar graphs display data, and I ask students if instead of graphing each individual student, if we looked at graphing heights instead.  Students think on this a bit and then get into how this could help or be more complicated.  They also start arguing about the heights that we recorded, saying that there is too big of a variation of heights to graph.  I suggest “grouping” heights together.  Students really latch onto this idea, and we brainstorm on how to group the heights.  Typically students either decide on 1″ or 1/2″ intervals.  They then get to work graphing the heights and how many occurrences there are.

One thing I do during the measurement process is take a picture of each student as they are being measured.  I then print out everyone’s picture (normally a head shot, and typically multiple ones- some on colored paper) and when we as a class believe we have a good graph, allow students to “graph themselves” on the wall outside my door.  This creates a visual of the class overall.  We then create a bimodal graph slightly below it that represents each gender.  (A sample is below, but I no longer have pictures of my student ones so this is shown generically)


Students typically do not like this because some of their faces are covered.  So we represent it in another way with a vertical histogram, females on the left side and males on the right (stole this idea from Stem and Leaf plots).  This give students a great visual of the height distribution by gender.

Double histogram hieght

This is what I want students to analyze, I even take them outside and take a picture of them in this format.  They make conjectures on why this distribution occurs, what it may have looked like in earlier grades, and what it will look like when they graduate.  I then ask students to make predictions on how much they will grow over the school year, and we create one last vertical histogram from that.  Then we wait.

At the end of the year, we go through the measuring process once again.  We determine the measures of central tendency and graph our results.  Students then compare their predictions to the actual results.  When I ask students to compare the two, I get all sorts of humorous reasons why things may or may not line up.  “I drank coffee every morning this year because I was tired all the time, Mom told me coffee stunts my growth,” has to be one of my favorite student comments.

Now that I will be teaching high school as well, this will be a fun multiple year project, I may have issues storing graphs for a few years until I measure my students again, but I think it will be a great extension to this.  When I dust these off for the seniors, I can’t wait to hear what their comments are, what the remembered about it, and what kind of estimations they now come up with for their height distribution.