Month: July 2013

Categories
Algebra Middle School Statistics Technology

4 Engaging Ideas From Twitter Math Camp

This past week, over 100 math teachers descended upon the Drexel University campus for Twitter Math Camp 2013.  It was a fantastic opportunity to meet people I had communicated with via Twitter for some time, make new friends, and share math ideas.  It’s a real rush to hang out with colleagues who share similar ideals on math instruction, and a commitment to improve our practices.  Check out the hashtag #tmc13 on Twitter to look back on some of the action and reactions, and find new math folks to follow.

While there’s so much to share from TMC13, I know there are many math friends who couldn’t attend who are looking forward to hearing about the goings-on, so in this post I share 4 ideas from this year’s Twitter Math Camp I am eager to try in my classroom right away.

EliELI’S BALLOONS – Followers of the blog know that I am a big fan of the Desmos online graphing calculator.  The highlight of the week for me, and I suspect for many, was having Desmos founder Eli Luberoff model a lesson using his creation.  Eli’s enthusiasm for sharing Desmos, and his sincere desire to work with teachers to improve the interface, are infectuous.  There were many “oooh” and “aah” moments from the assembled group, and a loud cheer for the “nthroot” command…yes, it’s a pretty geeky group!  (thanks to @jreulbach for tweeeting out the great picture of Eli showing off Desmos’ position when you Google “graphing calculator”)

Eli’s lesson idea has a simple and engaging premise:

  • Hand out balloons
  • Blow up the balloon.  For each breath, have a partner record the girth of the balloon
  • Consider the data set

Balloons

That’s it.  No worksheet.  No convoluted instructions.  Eli walked us through an exploration of the data set using Desmos, using the table to record the data, and considered various function models: is a square root model?  Is it logarithmic?  The group eventually settled upon a cube root as the proper model – and how often in class do we encounter data best modeled by a cube root?!  Since the explanatory (air entering the balloon) is volume, and the response variable (girth) is linear, the cubic model makes perfect sense. Fun stuff.  But wait…there’s more!  Eli then analyzed the fit of curve by looking at the squares of the residuals.  Click the graph below to check out my best-shot recreation of Eli’s presentation, and play around with the fit of the curve by toying with the “a” slider.

More great new additions to Desmos are coming.  Thanks to Eli for letting us preview some of them!  Was a pleasure meeting you and hearing about your fascinating story.

GLENN’S PROBLEM POSING – Glenn Waddell is a colleague I feel I have a lot in common with, in that we have both experienced the frustrations of trying to “spread the word” to colleagues of the great new ideas, and strong need, for inquiry-based mathematics.  In this session, Glenn presented a framework for problem posing in mathemtics which can be employed equally-well with real-life problems (see the “meatball” example in Glenn’s Powerpoint, which was adapted from a Dan Meyer “math makeover” problem) or with a garden-variety drill problem.

The framework asks that teachers lead students in a discussion that goes beyond just the problem in front of us.  Think about the many attributes of a problem, list them, consider changes to them and their consequences, and generalize results.  Glenn suggests the book “The Art of Problem Posing” by Brown and Walter as a resource for getting started, which employs the problem posing framework.

Framework

Glenn led the group through an exploration of a quadratic equation, where we started by listing its many attributes.

Glenn Problem

Now we consider changes to attributes:

  • What would happen if there were a “less than” sign, rather than equals?
  • What would happen if the last sign were minus?
  • What is it were an x-cubed, rather than x-squared?

There’s no limit to the depth or number of adaptations, and that’s why I like this method of problem posing for all levels of courses.

Download Glen’s presentation on the TMC wiki, and explore the wiki to get the flavor of many of the sessions.

A PAIR OF LESSONS FROM MATHALICIOUS

Logo

If you have never visited Mathalicious, go now….take a look at some of the free preview lessons, and you will become lost in the great ideas for hours.  THEN, make sure you sign up and get access to all of the engaging lessons.  Here is a company that is doing it right: lessons come with a video or visual hook, data which naturally lead to a discussion of tghe underlying mathematics, and just the right amount of structure to encourage students to contribute their thoughts and ideas.  At TMC, Mathalicious founder Karim Kai Ani led the group through two lessons.  A brief summary is given here, but I encourage you to check out the site and subscribe….you’ll be glad you did.

The “Romance Cone” – What is the appropriate age difference between two romantic partners?  Is there a general rule?  A fun lesson, “Datelines” on Mathalicious, where students explore a function and its inverse, without using those scary-looking terms.  I have been looking for an opening activity for our Algebra 2 course, which brings back ideas of function, inverses and relationships, and looking forward to trying this as a my first-day hook.  Also a great activity for Algebra 1.

PRISM = PRISN? – I have led my probability students through an exploration of false positives in medical testing for many years, and I like how this activity puts a new twist, and some great new conceptual ideas, on the theme.  “Ripped from the headlines”, this lesson challenges students to consider government snooping, and the flagging of perhaps innocent citizens.  If a citizen is flagged, what is the probability they are dangerous?  How often are we missing potentially dangerous folks in our snooping?  What I really liked here was the inclusion of Venn Diagrams, with sets representing “Flagged” and “Dangerous” people, where the group was asked to describe and compare the diagrams.  Fascinating discussions, and a good segue into Type I and Type II error for AP Stats if you want to take it that far.

This lesson does not appear to be available on the Mathalicious site yet, (update from Mathalicious – will be released in the Fall) but will be using it when it is completed!  Later that day, the TMC teachers broke into smaller groups to gain behind-the-scenes access to the Mathalicious writing formula.  Thanks to Kate and Chris for sharing, listening, and giving us all the opportunity to contribute ideas.

Categories
Algebra

Completing the Literal Square

An interesting post comparing polynomial division methods on the More Than a Geek blog reminded me of my own experience teaching completing the sqaure.  There were times in my career where I absolutely dreaded teaching this method, and tried my best to dance around it.  Now, my attitudes have changed, as mastering this method and looking at it in different ways provides so many interesting avenues for analyzing a quadratic function.

The post asks if the “box” method for teaching muliplication of binomials could be extended to completing the square, and I am happy to report that it can!  In fact, using binomial boxes may reach more visual learners and let them complete a square in a more literal sense.

First, some reminder of the “box” method for multiplying binomials (Note, don’t EVER call it FOIL!!!).  This method allows a more visual approach of the double-distributive property, and a visual organizer for students who require this level of structure:

Mult Binomials

USING THE BOX METHOD FOR COMPLETING THE SQUARE

We start with a quadratic, which we are interested in converting to vertex form:

STEP 1: shove that -9 out of the way, and set up a binomal multiplication box:

Step 1

STEP 2:  Fill in the x-squared box, and put half of the b term in each of the x-term boxes:

Step 2

STEP 3:  Now, quite literally, we need to complete the square by filling in the last box.  Also, since we add 25 to right side, we must also subtract 25 from the same side.

Step 3

STEP 4: A little housecleaning, and we have our quadratic in vertex form.

Step 4

My ideas for this are hardly unique. Check out these great blogs for more completing the square action:

Finally, enjoy a quick video where I walk through the box method, with a few stickier examples.

Categories
Algebra Statistics

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:

Math Makeover

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:

Axes1

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?
Linear

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

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?
Quadratic

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:

Data

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.

 Excel

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