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## Math Makeover – Let’s Talk About Skid Marks!

This week’s Math Makeover from Dan Meyer features an interesting data set about skid marks, and a scary-looking predicting function:

Later this week, I am off to Twitter Math Camp, where Max Ray will lead an algebra 2 group.  As preparation, we’ve been asked to think about the unique challenges teaching Algebra 2 presents, and make some comparisons to Algebra 1 and Pre-Calc.  This data set is nice for looking at some of those differences.  Here’s my “short and snappy” summary of the differences:

• Algebra 1 is almost exclusively about mastering linear functions.  Let’s get comfortable with moving through the data – graph – function triangle, and look at different forms.  We may look at some quadratics at the end of the course, but primarily, Alg 1 = linear functions.
• In Algebra 2, we are exposed to a bank of functions (rational, radical, quadratic, cubic, exponential) and should be challenged to make decisions.  How do these functions behave?  How are their transformations similar?  Which function is approproate for a given data set?

The data set in the problem above can be used to start exploring the differences between linear and non-linear functions.  I’m thinking this would be a good opener to an Algebra 2 course, where we begin having discussions about the possibile behavior of data.

HERE’S WHAT I AM GOING TO DO WITH THIS PROBLEM

First, we need a visual to encourage discussion of the data set to come:

For a short video, there’s a lot going on here.  In a discussion, here’s what I want to get from my students:

• What is this video about?
• What’s measurable?  What are the variables?
• Is there anything else you’d like to see / know?  Hoping that students recognize that we would like to see more videos, with different car speeds.

Before we look at any data, let’s talk about the variables here and what our gut says about how they behave.  We have two vairables (car speed and skid mark length, which we have hopefully drawn out through conversation).  Which is independent?  Which is dependent?  I think many students would argue that skid mark length depends on how fast you are going, so we would set up our axes like this:

We don’t have any data yet, but how do we think the data will behave.  Certainly, longer skid marks will be associated with higher speeds, so we can predict a data set would reflect that.

Is it reasonable to expect that speed and skid length will share a linear relationship?  In other words, does each additional mph of speed increase the skid mark length by a fixed amount?  If not, how else might the relationship be portrayed?

Perhaps skid marks stabilize after a certain speed?  Is this reasonable? Is there a point where perhaps speed does not matter?

What would this graph imply about the relationship between speed and skid length? Do we feel that speed will have a bigger influence over skid length, the faster you go?

In any case, I want my students to make an argument about the nature of the relationship, and be able to summarize their thoughts, before we start looking at data!

After our discussion, prehaps the next day, let’s take a look at that data set:

I have two issues to address here first:

1. Which variable is x, and which is y.  As an algebra teacher, does it matter?  As a stats teacher as well, it matters a lot.  I’m much more comfortable here with making speed x, and mark length y, but the data is clearly formatted in the other direction.  I’m sticking with speed an the independent variable.
2. Do I care about the point (0,0)?  Is it germane to this discussion?  Can we ignore it?  There’s probably a domain / range discussion to be had here.  Since (0,0) was not an observed value here (in my mind), I am leaving it out and only considering the other 4 points.

So, here is the task for my students.  With a group, defend whether the data suggest (or do NOT suggest) a linear relationship between speed and skid length.

That’s it.  I’m hoping the students will make a poster or some visual, look at a line-of-best-fit, see how well the points fit their line, and defend their ideas.

Looks pretty linear, but looking at the slopes between consecutive points reveals that the slope is growing with increased speed.  I’m in no rush here to establish the true nature of the relationship.  This will be a data set to come back to later in Algebra 2 when we have our tool box of functions more clearly fleshed out.

And that nasty radical function? We will come back to it during our radical functions unit, but will attempt to verify its validity, rather than just take it for gospel.  The file below contains some nice information on skid marks, speeds, and the effects of surfaces as well.

Determining Vehicle Speeds From Skid Marks

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## Problem Makeover – Carpeting

This week’s problem makeover from Dan Meyer features a room in need of carpeting:

There’s a lot of stuff going on in this problem:

• Overlaying rectangular carpet over an irregular shape.  We’re going to need to worry about waste at some point, perhaps.
• The double-sided tape.  We’ll have to look at the perimeters of the individual carpet pieces and add them up later.
• Costs.  We definitely don’t want to waste carpet or tape, or at least minimize the amount of waste.
• Meters vs Yards vs Feet.  Yeah, this aspect of the problem is just mean  Maybe if Lesa had her fill of coffee before measuring the room, she would have used feet.

I think the piece of this problem which may be easy for kids to overlook, and least comprehend, are those seams.  Seams are bad, and add to cost.  This problem needs something tactile to get kids thinking about the seams and coming up with their own ways to  think about them and minimize them.

HERE’S WHAT I AM GOING TO DO

Groups of students will be given a bunch of 4×4 inch sqaures and a bigger shape to tile.

The instructions are simple:

Use the squares to cover the triangle in the most efficient manner possible.

As groups complete the task, they will then meet with other groups and defend why they feel their coverage method is the most efficient.

I’ve left the phrase “most efficient” open for interpretation here.  I’m hoping that a strong definition of efficiency will emerge from group discussions.  As groups continue discussions and start to reach agreement on the characteristics of efficiency, have members of teams approach the front board, and start a class-wide list of these characteristics.  Here’s what I am hoping will emerge:

• Using the least number of squares as possible is important.
• Having squares overlap, if it can be avoided, is preferable.
• Seams are ugly, but necessary.  When we are able, we want to minimize these seams.

This last bullet may be the toughest to coax from kids.  But this is a good time to bring in carpeting as a nice analogy.  Does the carpet at your house have seams?  Where are they?  If you were laying carpet, where would you try to put them?

Once we have a stronger definition of efficiency, it’s time to try a second tiling:

This time have students write up their solution and defend their efficiency.  There’s a nice opportunity for differentiation here.  Students with a strong geometry background could use rules regarding secants and chords to generalize the optimal solution, or even use software to model the scenario.  Middle school students, or those with less exposure to geometry, could simply measure the seams.

Now for the main event.  Students will be asked to tile one last shape:

This time, we add a new wrinkle to the problem.  To get students thinking harder about efficiency, we add some costs to the mix.

• Each square used costs \$10.
• Every inch of seam in the tiling will cost \$2 per inch.

It’s Saturday morning during the summer, so I’m not sure if these are “good” costs for the task, but we can adjust them.  Teams will calculate their costs, defend their solutions, and be assessed on how well they minimized the costs.

Geometry isn’t usually my “thing”, but this challenge appealed to me, because the original task was so filled with stuff that I know many students would shut down.  And I feel there is a natural opporunity here for students to assess their solutions and communicate.  In the end, I want my students to:

1. Communicate their problem-solving process
2. Defend their process, and evaluate the process of other teams
3. Utilize their findings in different situations.