Day 145: Transforming Geometric Figures with Matrices!

Another tantalizing use of matrices.  On Tuesday (no class yesterday), my precalc kiddos used the TI Nspired Activity: Matrix Transformations to explore the effects of various matrices on polygons.

Using their experience, I asked the groups to determine the polygon matrix and then the transformation matrices for various situations.  We looked at the graph and the point-rule to deepen their grasp of how to determine the transformation matrix.  By hand we cranked out the elements of the resulting matrix to see what was really happening.

 

Once we debriefed and analyzed HOW the transformation matrices actually worked and WHY they worked (via point-rules), they began a second Nspire Activity: Linear Transformations.

Day 144: Nspire to the Rescue

Seniors in our school had their senior “skip” day on Monday.  ARGH!!  We had just started linear regression inference on Friday, and Monday was discussing the conditions for inference for the population slope.  I knew if I simply plowed along, I would plow most of the “skippers” under.  So how do I not condone the “skip” (and reteach the lesson as if no one had attended) but also not penalize the skippers academically (while also validating and rewarding those that did not skip)?

Quite the dilemma, but while my students took their Chi-Square test, I was able to put together an Nspire companion document to this problem:

Alternate Example 12.1a: Fresh flowers?

For their second-semester project, two AP Statistics students decided to investigate the effect of sugar on the life of cut flowers. They went to the local grocery store and randomly selected 12 carnations. All the carnations seemed equally healthy when they were selected. When the students got home, they prepared 12 identical vases with exactly the same amount of water in each vase. They put one tablespoon of sugar in 3 vases, two tablespoons of sugar in 3 vases, and three tablespoons of sugar in 3 vases. In the remaining 3 vases, they put no sugar. After the vases were prepared and placed in the same location, the students randomly assigned one flower to each vase and observed how many hours each flower continued to look fresh. Here are the data and computer output….

Here are some of the screens:

Because my students were already logged in to the Navigator, I could send it right away.  They didn’t have to enter the data, and by using a split screen, I could re-introduce (re-enforce) the conditions and visually show what needed to be done to check the conditions.  Those students who were there on Monday found the document to be “awesome,” in their own words.  They said it actually helped them solidify their understanding better.  And we were able to cover the topic in 10 minutes.

Addendum 4/23: in their opener today, they could articulate the mnemonic, what it stood for AND how to check the 5 conditions!!  Whoot-whoot!

Day 143: Matrices and Geometry

It is time to explore another, different, interesting use of matrices….this time in the field of transformational geometry!  A few years ago, when I first decided I wanted, no needed!, to have a unit on Matrices in precalculus (since the topic was not part of our Algebra 1 nor Algebra 2 curriculum), I hunted for interesting and engaging uses of matrices.  I found a pre-made activity with student and teacher handouts on the TI Math Nspired website called Matrix Transformations

The handout needed almost not tweaking!  Always a plus since I am forever changing activities to do just exactly what I want it to do.  An example of an exploration question is:

Grab and move the sliders for each element of the multiplication matrix until the polygon in Quadrant I is a reflection of the polygon in Quadrant II.

1. What 2 x 2 matrix results in a reflection over the y-axis?
2. Why does this matrix multiplication result in a reflection over the y-axis?

Here is a progression of motions for the question:

I like how the activity is a nice marriage of Action-Consequence theory (moving sliders to see what happens) and multiple representations of the mathematical idea (numeric-matrix, visual-graph, and abstract-matrix calculation in green).  Here are my “giggling boys” exploring, giggling, discussing, giggling, solidifying their observations, giggling…. You get the idea!

How do you give your students experiences where they make sense of the mathematics by trying things and adjusting their thinking as they go along?

 

Day 136: Do The Really Intersect?

Its the day before Spring Break.  We finished the Midterm yesterday and they did brilliantly!  Not a great time to introduce new material.  An easy way out is to “show a movie” but to waste precious time with students with an experience that has no real academic purpose goes against everything I believe.

So, how do I keep my precalculus students engaged and interested?  I wanted something that reviewed an old, but not critical, learning but added an intellectual twist that will catch and keep their attention.  And I remembered a workshop I attended about 5 years ago (I’m sorry but I just can’t find the name of who presented and would love to give you credit…just comment below with your name and any other info you’d like to share) using the Nspire to look at the relationship between polar and rectangular equations in graph form.  The presenter shared 3 Nspire documents that had nice animations.

For this experience, though, I used Geogebra because the kids could manipulate easily with their iPads.

It was the 4th document which inspired me to create an investigation around a system of two polar equations.  Luckily the document was created, but I wanted my students to reflect more while using the Nspire document.

 

For instance, I had the students do this:

Move to 2.1. Without talking to your group, watch the animation alone!  You must watch carefully because you cannot re-graph again.  Try to observe how many times these graphs’ paths intersect.

1. What did YOU notice?
2. Once everyone has completed the task alone, talk with your group. What did other people in your group notice? Try to paraphrase your discussion.

Then later the real thinking started.  My students  used their Nspire to do the following:

Move to 4.1. You will see the results of graphing the two polar graphs in the rectangular plane.  Does this affect your answer?  Be specific!

1. Use the rectangular graphs to give the coordinates of the points (in polar form please) of intersection of the limacon, r₁ = 3 + 2 cosθ, and the four-leaved rose, r₂ = 5 sin(2θ). Label the intersection points on both the rectangular graph and the polar graph.  The first point is labeled P₁ on both graphs.

1. Why could the apparent intersection point Q₁ be called a “false” intersection point? What aspect(s) of a polar graph make it appear to be a point of intersection?  Label the other false intersection points on the polar graphs in the same manner.
2. Show on the rectangular graphs above that the second-quadrant angle θ for point Q₁ corresponds to a point on the limacon but not to a point on the rose. What are the coordinates of point on the rose that correspond to the location Q₁?  Is there a mathematical relationship between the point on the limacon related and the point on the rose which correspond to the location Q₁?

I was so delighted with how engaged my students were…to see their heads down, fingers posed over their iPads and some conversations, too.  The Math Practices were everywhere today! In fact, they didn’t even realize class was over, and that’s a biggie for a 7th period just before Spring Break.

Happy Spring Break!  Happy Easter!  I’ll be back in a week.

Day 135: Rock, Paper, Scissors Anyone?

Do you remember playing Rock, Paper, Scissors as a kid? As an adult? Against a computer?  I’ve seen various lesson plans around RPS and probability of winning, but none seemed to fit the level and sophistication I was looking to use with my AP Stats class.  Well the New York Times science section had a great article Rock Paper Scissors: You vs. the Computer.  If you haven’t played the computer, its a kick and I highly recommend it!

Then I remembered reading Bob Lochel’s post Rock-Paper-Scissors and Two-Way Tables.  And found his second post, Chi-Square Tests: Rock Paper Scissors.   I liked the idea, but the gold came in the response sections of both posts.  Doug Page  shared a worksheet he has developed for using the Rock, Paper, Scissors applet and also a Google Form for having students submit results.  I stole blatantly! 
Since my students use the iPad, they can’t access Flash animations, so I had to assign the play as homework.  I hate to do this because so many kids just don’t follow-up and complete the assignment – and we don’t get the data we could if we collected in the classroom.  And the same happened with this class, but at least we had enough to continue with the introduction to Chi-Square test of Homogeneity.  I think next year, they will use their own data rather than the class data…or compare their results to the class results.  Will have to think more about this next year.

I required the students to calculate the components by hand – they need to know where the components come from and how they are related to the final χ² statistic.  Once we finished this problem, we tackled another problem using the calculator.  Because of the hand-calculating of the components earlier, they then understood what the expected values matrix meant and the components matrix.

 

How do you incorporate electronic experiences to develop engagement?

 

Day 66: Oh What We Can Do With Trig Functions!

Today in Precalculus, I had my students explore what happens when they encounter things like f(x) = |sin x|, g(x) = 2x + 3 cos 6x, and h(x) = (2x + 3)cos 6x.  

Their job today was to work with various situations, make conjectures, test them and eventually make a generalization about what they are seeing.  Eventually I would like them to connect the equation and what is actually happening graphically.

Day 42: Blocking Trees Activity

We are an 1 to 1 iPad school and I have been trying to use the use them effectively.  Can’t say I’ve been particularly successful in using them to enhance mathematical understanding, but I have had good success with using them to help students become “better” students….but I digress from the point of this post: Blocking Trees activity.

   

I am so lucky this year as my AP Stats students are really interested in learning statistics and are willing to “play” when I have activities for them to pursue.  I’ve used the Blocking Activity: Can you see the trees for the forest? which gives students a hands-on experience with the 2001 AP Statistics Examination Free Response #4 “fruit tree” problem on which students performed miserably. This activity was originally presented at the 2001 NCSSM Statistical Leadership Institute and I find it is very successful in getting at the idea that blocking can reduce variability. In this activity, students use simulation (yay…and introduction to simulation before we actually study it!) to model a completely randomized design and two different block designs for the fruit tree setting.  Above are screenshots of the Nspire supported activity using a Quick Poll to collect the data from each student and then to send compiled data immediately back to students.  Notice the outlier at 60…we found out the student forgot to find the average.  Block A is the “correct” way to block for the forest preexisting condition and we talked about the two things we look for in our sampling results: low bias (center should represent the population parameter) and low variability (each sample should not vary far from the true value).

Day 38: Sampling Vocabulary

 
This activity ROCKS!!  It is from the TI Math Nspired site for Statistics and I’ve found that these activities ALWAYS work great in the classroom.  In this simple activity, they practice probability sample methods such as simple random sampling, stratified random sampling (and what disproportionate samples are), cluster sampling and systematic sampling.  Students actually experience what the sampling vocabulary means through this activity…I guess it is quasi-kinesthetic.  I heard great discussion around the idea of SRS versus the other three methods.  It was awesome since we spend about 10 minutes talking about the two requirements of an SRS and through their math talk, they could practice and solidify their understanding of a simple random sample.  How do you use technology to help students aquire vocabulary?

Day 37: Corny Sampling in AP Statistics

  

Today in AP Stats block we did the famous An Exercise inSampling : Rolling Down the River activity.  Students randomly selected their four samples, determined the yield and average yield with each method and sent their results in a Quick Poll document using the Nspire Navigator.  We then looked at our results and talked about the advantages and disadvantages of each method.  Loved using the colors to highlight each sample type.

Day 34: Nspired Rational Functions

Another great day in Precalculus.  I am so blessed with students who are really interested in understanding the mathematics rather than just memorizing for the test!  I am in “teacher heaven.” Today I gave my students the conditions for a rational function: Given the following information, write the rational function that has zeros at x = 2, x = 3, and vertical asymptote at x = 5.  It has a removable discontinuity  at x = -1.    It has a horizontal  asymptote of y = -3.  I then had them submit their answers via a QuickPoll with the TI Navigator system.  Then we compared and listed all the answers that were unique.  And then they were to look at each one and talk about what they thought need to be “fixed.”  Great discourse and they didn’t even need me to call on them…lots of cross-class discussion.  Learn so much from “mistakes.”