Just a quickie post today. Lots on my plate: meeting with our principal and instructional coaches, meeting with our local community college to determine new placement criteria based on high school transcripts rather than the unreliable COMPASS test, and evening dinner/talk by our local math council.
In precalculus, we are finished with our study of matrices. And what better way to review so many ideas, processes and applications than with Review Stations. With eight stations, I tried to put all of these topics into eight unique groups.
In one station, “Using Determinants,” I wanted to introduce more ways the determinant could be used besides finding the area of a triangle. One method was using determinants to decide if three points were collinear. One student asked my why it worked, but then immediately thought about it and realized why when he related the work to finding the area of a triangle. Another used determinant to find the equation of a linear function between two points. Again, the initial reaction is to ask “why it works” before thinking it through. I was so proud of the kids when, with a slight nudge, they actually figured it out on their own. I am so lucky to have students who embrace the challenge!! I think giving students new (unseen before) but very related activities builds confidence, resilience and perseverance, especially when they are able to work with their peers to discuss, fine-tune and use these ideas.
In another station, “Solving Systems,” I wanted to add in an activity that celebrated another culture (diversity integration) which is often hard to do in math. In this case, student looked at Chinese counting boards (from 2000 years ago) to solve systems of equations.
As our final application of augmented matrices, we tackled the area of traffic flow solutions. We kept it simple using one way streets and only 3-6 intersections. We did discuss that these problems require computers to handle cities with 100s or 1000s of intersections, but we were using a simplified situation to understand the underlying constructs of the situation. The simple diagram we used is:
Here was our introduction problem situation:
The figure shows the intersections of four one-way streets. As you study the figure, notice that 300 cars per hour want to enter intersection I1 from the north on 27th Avenue. Also, 200 cars per hour want to head east from intersection I2 on Palm Drive. The letters w, x, y, and z stand for the number of cars passing between the intersections.
If the traffic is to keep moving, at each intersection the number of cars entering per hour must equal the number of cars leaving per hour. Use this idea to set up a system of linear equations involving w, x, y, and z.
Write the related augmented matrix. Use your calculator to solve the equation – but be sure to understand how to use row operations to solve as well.
If construction on 27th Avenue limits z to 50 cars per hour, how many cars per hour must pass between the other intersections to keep traffic flowing?
Once again, my students were fascinated and interested in another way matrices can be used in real situations.
Today we looked at how determinants of matrices can actually be of use beyond solving matrix equations. One interesting application is using the determinant of 3 points to find the area of any coordinatized triangle. You can see the two examples we did to use the formula. It was a great way to review the process of finding the determinant of a 3×3 matrix.
However, my students wanted to know WHY this process worked. I was so thrilled with their question and so I gave them a couple of days to think about the “why” of the method. After a night of thinking, we began to explore the problem. I simply scribed for the class as they thought through the relationship. It is a nice way to connect geometry and algebra in an interesting and unexpected way!
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.
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.
- What 2 x 2 matrix results in a reflection over the y-axis?
- 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?
How can one use matrices in real life….let me count the ways!! On Tuesday we explored how matrices can be used to change the contrast of photos, determine ranking of candidates in an election and count paths between objects.
Yesterday we delve into counting paths and raising matrices to powers based on a lesson I found at Engage NY: NYS Common Core Mathematics Curriculum.
Today we looked at Markov chains and transition matrices. Lots of oooo’s and ahhhh’s. Love it when the math’s coolness shines through to the kids!
Today we begin our study of matrices! I so love this chapter! There are so many ways to bring in mathematical topics that aren’t part of the regular prescribed curriculum. There are so many interesting and useful applications of matrices. The beauty and inter-connectedness of mathematical ideas shines through when we use matrices.
My course partner and I decided to try something a little different for this chapter. The mathematical skills involved with matrices are pretty straight forward and we believe our students will pick up the skills relatively quickly…so why “waste” valuable class time demonstrating skills and practicing? We asked ourselves, “should we “flip” this chapter?” I do have my AP Statistics students watch videos pretty regularly to gain skills, but I didn’t think I could find time to create matrix videos for precalculus. What about Khan Academy? These are pre-made videos with support self-quizzes and a way to monitor students completing the required skills. I was hesitant because I find the videos to be very skill-driven and not much pondering or questioning….but how can it be in a non-interactive video. Besides, the purpose of watching the videos is to present skills and have a way to practice the basic process. So why not Khan? There was a good chapter on Matrices and covers all of those “boring” skills in a way that students can control how they receive the information.
Today, we had our students sign up for Khan Academy and then will assign each matrix module in the Khan Precalculus topic as needed while we use class time to dig into the uses of matrices. My students got started: signed up for my Khan Academy class, and assigned the first module-Basic Matrix Operations. Many of my students huddled together to watch a video and then did the prescribed/required 5 correct problems in a row on the video skill. Tomorrow we will begin our adventure into Matrices!