What are Matrices?
Lesson 1 of 10
Objective: SWBAT represent data in a matrix and determine the dimensions of a matrix.
In recent years, my school has not taught matrices in Algebra 1 or Algebra 2 Matrices appear on many college entrance exams, so this class provides students with the opportunity to understand and develop skills in the use of matrices.
Today, the work introduces how matrices can be used to analyze data. The bell work ask students to consider different ways to organize data. After a few minutes of thinking and discussing the bell work students share their ideas. Students usually think of tables and spreadsheets. Some may just make a list. Giving students time to think about different ways to organize the data helps me to determine what commonly comes to mind for students. I may get a student that says a matrix but that is rare.
What is a matrix?
I explain this unit develops how to use a common mathematical tool to organize data for analysis. The tool is called a matrix. I now give students the definition of a matrix to read. Today the students read the definition individually and write down what they do not understand.
After a couple of minutes I have the students discuss the definition with their groups. I move around and listen to what the students are discussing about the definition. The groups need to try and answer the questions members have jotted down. If no one can answer the question then the question will be brought to the class to determine the answer. Students often struggle with the term array. Students may also confused by the notation for matrices.
Once groups have talked for 3-4 minutes, I bring the class back together. I randomly pick a group to see if they have any questions they could not determine as a group. Since matrices are new, someone in the group will always ask a question. I will restate the question to make sure the questions is clear to everyone including myself. l then ask the other groups to answer the question. As I have listened to the discussion so far, I may know which group has a good answer. I may ask that group to explain what was decided in their group. Once the students have given their thoughts on the question I organize the information and make sure the question has been answered correctly. The class continues to ask and answer questions until most questions are answered.
I now want students to make a matrix to describe the data in the bell work. I expect 2 different matrices for the data. We will put both answers on the board. I ask the students to determine whether the matrices are correct. It is generally the case that someone will comment, "So there will be different ways to write the matrices? How can that be?" I try to help the class understand that depending on what we want the rows and columns to represent, we can write the matrix in more than one way. The rows could be the type of shoes or it could be the cities. This is also true for the columns. I ask if there are any other ways to write the matrix. Some students may say we could make a matrix for each city which is correct, but I ask, "Is that a single matrix?"
Now that students see how to write a matrix I want students to understand how they can be used to analyze data. I give students a matrix to analyze. I have groups work together to answer the questions on the worksheet. After a few minutes, different groups put their answers on the board. Once the answers are on the board, I ask a different group to explain how the answers were found. Using this method of sharing gives me a chance to assess multiple groups on understanding of the question.
Students are now given Using Matrices to Analyze Data worksheet. This worksheet has 2 different real world situations where matrices can be used to analyze data. I like the examples that are not what students would think of as big mathematical careers (archeology and sociology). The questions give students the opportunity to see how the matrices are used in these careers. I give students time to work on the worksheet in their groups. I listen to the discussions and answer any questions.
As class ends I ask students to fill out an exit slip. Today the students will answer the following question:
If the principal stops you in the hall and asks what you learned in Precalculus today what would you tell him?
I let students know that they need to be very precise in their answer. Asking this question gives students a chance to think about the learning objective for today. I want answers such as "I learned how to use matrices to answer questions." Some students will say I learned about matrices. These are the students that are either rushing to finish the exit slip or they did not understand the use of matrices.