Matrices
Lesson 39
Our warm up for lesson 39 has student interrupting a double bar graph. This is a skill they will use by answering questions about the double bar graph. They will also be answer questions about matrices through out this lesson.
Ms. Maldonado surveyed her students to collect some data. She found that out of the 10 girls in her class, 5 liked chocolate ice cream, 2 liked vanilla, and 3 like strawberry. She also found that out of the 12 boys in her class, 4 liked chocolate ice cream, 3 liked vanilla, and 5 liked strawberry. How could this data be organized in a matrix? If we “S” or study the problem we must first underline the question. How could this data be organized in a matrix? Then we ask the question, what is this problem asking me to find. This problem is asking me to find a matrix which organizes the data.
A matrix is a two dimensional system for organizing information. We have matrices which are made of rows and columns, and the dimensions of a matrix actually determine how many rows there are and how many columns there are. As you can see a matrix is somewhat like a table it organize information the lines are just missing. If you look at the matrix we have here it has two rows and 1, 2, 3, 4 columns. We normally write our diminutions rows by columns, for this matrix would be a two by four matrix, each number in the matrix is called an element, so each of these represents something different. And if were to say let’s look at the element 1, what does it represents, it represent, and it represents the number of males who have 2 siblings. We can say that the element in located in row 1 column 3 because you go to the first row and go over to the third column. Matrices help organize data just as a table does. Here we have some sample data from the group, because we don’t exactly what your data will be from your group we will use this data. If we are trying to create a matrix which shows the number males and females who are athletes, and readers and other we can add the data from all the subgroups to create one largest matrix. If we are trying to find our first element, which is in the first column in the first row; we have to look at what our first column first row are in all of the other matrices. Our first row first column element is a 3 here, so that means we are going to have 3 plus 1 plus 2, plus 3, plus 1, plus 7 that’s a whole lot to fit in one small space. We are adding up each of the row one and column one elements to get our total. 3 plus 1, is 4 plus 2 more is 6, plus 3 more is 9, plus 1 more is 10, plus 7 is 17. We also can find our next element by adding the first row second column element, 1 plus 2 is 3, plus 2 is 5, plus 0 is still 5, 5 plus 5 is 10 plus another 0 leaves us with 10. You will continue adding each row and column that is the same together until you find your total matrices.
In problem one we are going to add the two matrices. The each have two rows and two columns, you can only matrices that have the same dimensions. They must have the exact same dimension in order to add. If we look at our first row first column we will have 4 and we will add 2 because it is also in the first row first column. Our first row second column will be 5 plus 8, second row first column will be 3 plus negative 6, second row second column is negative 1 plus 7. When we simplify these we get an answer of 6, 13, negative 3, and 6. In problem 2 we can subtract these two matrices because they have the same dimensions, each of the matrices have 3 rows and 2 columns. We will first show that we have 10 minus 7 because they are in the first row first column, the first row second column will be 8 minus 6, second row first column 3 minus 2, and then 4 minus 4, second row second column, third row 5 minus 10 and 1 minus 0. when we simplify we are left with the matrices of 3, 2, 1, 0, negative 5, 1. Our answer will be a 3 rows by 2 columns matrix. In problem three we will model scalar multiplication, in scalar multiplication you are actually going to multiply the number by every matrices; in other words it will be 2 times 1, 2 times 0, 2 times 3, and 2 times negative e 5. 2 times 1 is 2, 2 times 0 is zero, 2 times 3 is 6, and 2 times negative 5 is negative 10.
We have already “S” the problem so we know this problem is asking me to find, we will now organize the facts. Ms. Maldonado surveyed her students to collect some data; this is a fact; Some students will want to write this fact down, or say this is a necessary fact. And other will say that is unnecessary, I am going to leave it as a necessary fact. But I’m not going to write it down, I don’t think we need to write it. She found out that 10 girls, she has 10 girls in her class, this is a necessary fact; 5 like chocolate, important, 3 like vanilla, important, and 3 like strawberry, important. All of those are necessary fact so we know that we have 10 girls, 10 total girls and out of those 10 girls we know that 5 like chocolate, 2 like vanilla, and 3 like strawberry. If we keep reading in the problem, she also found that the 12 boys in her class, right there is a fact, 12 boys. 4 like chocolate ice cream, 3 like vanilla, and 5 like strawberry, all of these are important facts. So we know that we have 12 boys who like chocolate, 2 like vanilla, and 5 like strawberry. We now need to line up our plan, we are going to choose an operation or operations and write in words what our plan of action will be. Because our question asks to write a matrices, you reallydon’t have an operation taking place. Some of your students may say that our operation is to organize. And when we line up a plan we are going to make a matrices which organizes the data. In “V” verify your plan with action we will first being with an estimate, because we are making a matrices we have 2 categories. Girls and Boys, when we also have chocolate, vanilla, and strawberry. My matrices is going to be a 2 by 3 matrices. Because we are creating something that organizes the boys and the girls we will put them as the 2 of the categories, and then the other 3 categories are chocolate, vanilla, and strawberry. Our first element will be the number of girls who chose chocolate which is 5, in our first row second column we want the number of girls who chose vanilla which is 2, and in our first row third column we want the number of girls who chose strawberry which is 2. Second row we are looking at the data for the boy, and we had 4 boys choose chocolate, 3 choose vanilla, and 5 choose strawberry. In “E” examine our results we have to ask does our answer make sense, the problem was asking me to find the matrices which organizes the date, and that is what we have, is your answer reasonable, we said we would have a 2 by 3 matrices, and we have a 2 by 3 matrices. And is your answer accurate, and some students may want to go back into the problem to make sure they have all of the right data recorded. If we want to write our answer as a complete sentence, Here our matrices represents the data.
We will close our lesson by looking at the essential questions. Number 1 how could your school use matrices? Your school could use matrices to organize what classes students take, their grade levels, the number of student in a grade, the grades that students make, what they eat for lunch, there are many ways that they could use a matrix. How a business could use matrices, question number 2. They could organize and types of items they have, they can organize the money earn, or the money they spend, matrices can be used to organize many different things.
In lesson 39 we are going to use our graphing calculator to add, subtract, and multiply matrices. First we must learn how to enter a matrix before we can add or subtract. We are going to enter the 2 matrices here, the first step is to press second X inverse in order to get to the matrices, from here you are going to arrow over twice to edit and we going to enter our first matrices in matrix A by pressing enter. It’s already a 2 by 2 which is what we need so we can arrow down, but we have to change the entries, if we look at out problem, we see that our matrices has 4, 5, 3, and negative 1. Your students have to make their screen match this matrix, looking at the screen, we will enter 4, 5, 3, and negative 1. To enter our second matrix we will go back to second X inverse, which is matrix, we will arrow to the right on to edit and go down to matrix B and press enter. If we look back at our problem, we can see that our matrix is 2, 8, negative 6, and 7. Looking at our screen we are going to have a 2 by 2 matrix, so we have to change that 1 to a 2, and we will enter 2, 8, negative 6, 7. Your students must look at their problem and look at their screen and make sure that they match. From here we can quit. We have entered our matrices as A and B. To add these two matrices we will press second matrix we have entered our first matrix in A so we will press enter on A and we are going to add it to B by pressing second X inverse arrowing down to B and pressing enter. So now we have A plus B and if you press enter, it will give you your matrix. This is the sum 6, 13, negative 3, and 6. Looking at our problem for 4 plus 2 is 6, 5 plus 8 is 13, 3 plus negative 6 is negative 3, and negative 1 plus 7 is 6. In lesson 39 we can also subtract matrices, we must first enter the 2 matrices we have in our original problem, we will do this by pressing second inverse, arrowing over to edit, and pressing enter. We will enter our first matrices which is a 3 row by 2 column, so we have to change the matrix to say 3 by 2. Looking at our screen, we see we have a 3 by 2 matrix and it lines up on the screen exactly as your problem does. So from here your students have to pay attention to how they enter there numbers correctly. Once they have entered the matrices correctly, they have to enter the second matrix, so they will hit second Y inverse arrow to the right to edit, and then arrow down to enter the second matrix in B. It is also a 3 by 2 so they must change the dimensions, then using their arrow keys and enter they will enter the second matrices. It is very important that hey pay attention to their problem and what they are entering into their calculator to make sure they match. From here we have entered the 2 matrices so you can hit second quit to get back to your regular screen. We will hit second X inverse for matrix and press enter on matrix A so that we have A and we want to subtract matrix B; so we hit second matrix arrow down and press enter on B. We have A minus B, and when we press enter it gives us our 3 by 2 difference, which is 3, 2, 1, zero, negative 5, 1. You can check your answer because 10 minus 7 is 3, 8 minus 6 is 2, 3 minus 2 is 1, 4 minus 4 is zero, 5 minis 10 is negative 5, and 1 minus zero is 1. In lesson 39 we also do scalar multiplication, we first have to enter our matrices 1, zero, 3, negative 5, which is a 2 by 2 matrices, we will do this by pressings second X inverse and going to our matrix A, arrow to the right twice where edit is highlighted and press enter on A. It is not a 3 by 2 it is a 2 by 2, and when your student change the dimension they will see they have 4 places, or 4 entries for their number, and if they pay attention they enter matrix A. From here they are using scalar multiplication so they will do 2 and they can either press times or just leave it as 2 second matrix and press enter on that matrix A, so they have 2 times A, and press enter. This give us an answer of 2, zero, 6, negative 10. We can check our answer because 2 times 1 is 2 and 2 times 0 is 0 and 2 times 3 is 6, and 2 times negative 5 is negative 10.