Operations with Matrices (1 of 2)
Lesson 2 of 10
Objective: SWBAT add and subtract matrices and multiply matrices by a scalar
Over the next two lessons students will determine how to add, subtract and multiply matrices along with multiplying a matrix by a scalar. For the most part, my students understand the concepts covered today intuitively. I approach the day as an opportunity to let students use their reasoning skills to determine the how arithmetic operations work for matrices. We will work together to validate the students' intuitions.
I begin the day with students brainstorming about adding matrices. Students usually think of the obvious when they add. As students are working I am listening to their discussions. The first instinct is to add terms in the same position in the matrices. I let the students know that there is at least one problem that is not possible. By putting Problem d in the list the students will notice that there are not the same number of columns, and, uncover an important criteria that needs to be accounted for in matrix addition.
After students have time to discuss and find the answers, I bring the class together. I ask students to begin by identifying the problem that is not possible. I have students explain why this problem will not work. We then put the other answers on the board to give students the opportunity to confirm their answers.
To move the discussion about matrices forward today, I need to introduce the idea of the dimension of a matrix. Again, I project the definition of a matrix on the board. Today we will focus on the m x n notation. I ask students to look at the definition and determine what m and n represent for a matrix. As students explain that m is the number of rows and n is the number of columns, I write this on the board. I explain the convention that a matrix is sized by its dimensions, m x n. I also let them know that the dimension is sometimes referred to as the order of a matrix.
I ask if students can recall any other objects that are described in this way. Obviously, students are familiar with the dimensions of a room or a container. Many of my students have studied woodworking at school, so they are very comfortable working with dimensions. Next, I give students some problems that ask them to determine the dimension (p. 2) of a matrix.
After determining the dimensions I return to the definition page and we read the paragraph under the definition. This paragraph describes several characteristics that I want students to be familiar with we work with matrices.
- Square Matrices: I will ask, "What would be an example for the dimensions of a square matrix?" I will then put up a square matrix with questions for students to consider. I have groups determine the answers and then share the answers with the class.
- Diagonal of a Matrix: I like to ask students to explain why square matrices are the only matrices with diagonals. I help them to understand that the diagonal of a matrix refers to a vector of numbers where the row and column numbers are equal.
- Equivalent Matrices: The last idea students need to understand before doing more operations is the meaning of equal matrices. Students work on a problem. This problem has students find the value of x and y when they are terms in a matrix. Once students determine the values of x and y. I ask "How did you determine the values? What is meant by equivalent matrices?" As a class, we develop a definition for equivalent matrices.
Computation with matrices
I expect that my students really can do all the operations discussed today without direct instruction. I continue the lesson by asking students to work on a problem involving several operations. For this task, I ask students to work without talking for 3-4 minutes. This gives me a chance to see which students are understanding and problem solving. After about 4 minutes, I ask students to pair-up and compare answers. Once the pairs have time to work together, I will choose different students to put up their answers. Then, we will discuss Question 1 and the idea of scalar multiplication. I explain that a numerical value is also called a scalar, so we are multiplying a number with a matrix.
The last activity of the day asks students to summarize what was learned today. I give students this Summary worksheet. In groups, students work to determine the best explanation for about 5 minutes. I now randomly ask students to share their explanations with the class. After an explanation is shared, I ask the class if anyone would add anything to the explanation that would make it more clear or comprehensive. I also check to see if any students have a completely different explanation.
As class ends students receive Using Matrix Operations to Analyze Data worksheet to attempt. This worksheet gives students the opportunity to practice all three operations covered in today's lesson. The tasks are in context. After finding the sum, difference and a scalar multiplication, students are asked to interpret the matrices in relation to the given situation.