In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission again comes from my former student Perla Perez. Her topic, from Algebra: adding, subtracting, and multiplying matrices.
How could you as a teacher create an activity or project that involves your topic?
“Cryptography is the study of encoding and decoding messages. Cryptography was first developed to send secret messages in written form.” Cryptography also uses matrices to code and decode these messages by multiplication and the inverse of them. This, however, can be done by using any operations. By using the worksheet below as a foundation for an activity, teachers can have students act like hackers to engage students in computing different operations with matrices. In this activity, prepare the classroom by dividing it into four sections each with one of the phrases separated on the worksheet. Display the message (numerically) that is to be coded. Display the alphabet with corresponding number somewhere visible for students to have references throughout the activity. The instructions given are:
- Students are to get into four groups (more groups can be added for larger classrooms by making the phrase longer).
- Students are given an index card with the matrix [2, 7; 13, 5]
- Students are to add the matrix on each station to the the matrix on the card.
- When completed students must go change the message on the broad with the code.
When the students finish coding the message they can continue developing their skills by having them do this in the beginning of class throughout the lesson plan period. As the lesson progresses the teacher can change the phrase and require different operations to be made to either code or decode or even come up with their own message. With this activity the teacher gets the opportunity to see how the students choose to add the matrices together.
How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.
In today’s society we have access to a plethora of technology that can aid us in our everyday lives. There are so many ways one can learn something with different methods and from different people. The best part about the technology that we have access to is we can be manipulative to fit the needs of our students. When students get to the topic of adding, subtracting and soon multiplying matrices, they should be familiar with what a matrix is, the dimensions of one, and how to solve linear system with them. At this point it is a good a time to bring in a game into play like this one:
In this game the player chooses an operation such as adding, subtracting, multiplying by another matrix or scaler, and its dimensions. When a certain operation is chosen such as multiplication, it only allows the player to choose any size matrix but then spits out one with specific number of rows to multiply it with. The teacher can play this game with their students in any way they sit. The purpose is to get students thinking why and how the operations are working. From there the teacher can introduce the new topic.
How has this topic appeared in high culture (art, classical music, theatre, etc.)?
So many times students don’t understand that what they learn in class is used in everyday life, but teachers can give students the resources and knowledge to see applications of their work. In the video below, it shows different ways matrices can be applied. For instances the operations of matrices are used in a wide variety of way in our culture.
The main one can be in computer programming and computer coding, but they are also seen in another places such as dance and architecture. “In contra dancing, the dancers form groups of four (two couples), and these groups of four line up to produce a long, two-person-wide column” and where each square that is created is a formed by two pairs. Like the video had said, matrices can be used to analyze contra dancing. This can be done by having squares and multiplying them creating different types of configurations. By creating different groups and formations, essentially it is using different operations to create different matrices to.
“Common Topics Covered in Standard Algebra II Textbooks.” Space Math @ NASA. NASA, n.d. Web. 18 Sept. 2015.
Knill, Oliver. “When Was Matrix Multiplication Invented?” When Was Matrix Multiplication Invented? Harvard, 24 July 2014. Web. 18 Sept. 2015.
Smoller, Laura. “The History of Matrices.” The History of Matrices. University of Arkansas at Little Rock, Apr. 2001. Web. 18 Sept. 2015.