Welcome back!  Report cards will be issued this Thursday.

1. Welcome back!  Report cards will be issued this Thursday. I have bus duty after school this week at the top of the steps outside the gym from 3:30 to 3:50 p.m.

2. Essential Question: How do we use matrices to solve problems? • How do we apply operations to matrices? Standards: MM3A7: Students will understand and apply matrix representations of vertex-edge graphs. • MM3A7a: Use graphs to represent realistic situations. • MM3A7b: Use matrices to represent graphs, • and solve problems that can be represented by graphs.

3. Core-Plus 2 p. 75 ThinkAboutThis Situation Managing an athletic shoe store is a complicated job. Sales need to be tracked, inventory must be controlled carefully, and changes in the market must be anticipated. In particular, the store manager needs to know which shoes will sell. Think about the shoe store where you bought your last pair of athletic shoes. a) What information might the manager of the store want to know about the kinds of shoes the customers prefer? Make a list.

4. Core-Plus 2 p. 75 ThinkAboutThis Situation Managing an athletic shoe store is a complicated job. Sales need to be tracked, inventory must be controlled carefully, and changes in the market must be anticipated. In particular, the store manager needs to know which shoes will sell. Think about the shoe store where you bought your last pair of athletic shoes. a) What information might the manager of the store what to know about the kinds of shoes the customers prefer? Make a list. b) It is not enough just to have information. The manager needs to organize and manage the information in order to make good decisions. What are some ways the manager might organize the information?

5. Core-Plus 2 p. 75 Investigation 1: There’s No Business Like Shoe Business There are many different brands of athletic shoes, and each band of shoe has many different styles and sizes. Shoe-store managers need to know which shoes their customers prefer so they can have the rights shoes in stock. As you explore shoe data in the investigation, look for answers to this question: How can you construct and use a rectangular array of numbers (a matrix) to organize, display, and analyze information? Work together with the whole class to find out about the brands of athletic shoes preferred by students in your class. a) Make a list of all the different brands of athletic shoes preferred by students in your class. b) How many males prefer each brand? How many females prefer each brand.

6. Core-Plus 2 p. 76 One way to organize and display these data is to use a kind of table. You can do this by listing the brands down one side, writing “Men” and “Women” across the top, and then entering the appropriate numbers. Complete a table like the one below for your class data. Add or remove rows as needed. A rectangular array of numbers like this is called a matrix. Athletic-Shoe Brands Men Women Converse ____ ____ Nike ____ ____ Reebok ____ ____

7. Core-Plus 2 p. 76 One way to organize and display these data is to use a kind of table. You can do this by listing the brands down one side, writing “Men” and “Women” across the top, and then entering the appropriate numbers. Complete a table like the one below for your class data. Add or remove rows as needed. A rectangular array of numbers like this is called a matrix. Athletic-Shoe Brands Men Women Converse ____ ____ Nike ____ ____ Reebok ____ ____ b) The Matrix above has 3 rows and 2 columns. How many rows were needed in the matrix you constructed to display your class data? How many columns?

8. Core-Plus 2 p. 76 One way to organize and display these data is to use a kind of table. You can do this by listing the brands down one side, writing “Men” and “Women” across the top, and then entering the appropriate numbers. Complete a table like the one below for your class data. Add or remove rows as needed. A rectangular array of numbers like this is called a matrix. Athletic-Shoe Brands Men Women Converse ____ ____ Nike ____ ____ Reebok ____ ____ b) The Matrix above has 3 row and 2 columns. How many rows were needed in the matrix you constructed to display your class data? How many columns? Could you organize your class data using a matrix with 2 rows? If so, how many columns would it have?

9. Core-Plus 2 p. 75 One way to organize and display these data is to use a kind of table. You can do this by listing the brands down one side, writing “Men” and “Women” across the top, and then entering the appropriate numbers. Complete a table like the one below for your class data. Add or remove rows as needed. A rectangular array of numbers like this is called a matrix. Athletic-Shoe Brands Men Women Converse ____ ____ Nike ____ ____ Reebok ____ ____ b) The Matrix above has 3 row and 2 columns. How many rows were needed in the matrix you constructed to display your class data? How many columns? Could you organize your class data using a matrix with 2 rows? If so, how many columns would it have? The size of a matrix is written as m x n, where m is the number of rows and n is the number of columns. Thus, the sample matrix in Part a is a 3 x 2 matrix (which is read “3 by 2”). What is the size of the matrix you constructed to display your class data?

10. Core-Plus 2 p. 77 Suppose you are a manager of a local FleetFeet shoe store. Data on monthly sales of Converse, Nike, and Reebok shoes are shown in the matrix below. Each entry represents the number of pairs of shoes sold. Monthly Sales____________________________________________ J F M A M J J A S O N D Converse 40 35 50 55 70 60 40 70 40 35 30 80 Nike 55 55 75 70 70 65 60 75 60 55 50 75 Reebok 50 30 60 80 70 50 10 75 40 35 40 70

11. Core-Plus 2 p. 77 4. Monthly Sales____________________________________________ J F M A M J J A S O N D Converse 40 35 50 55 70 60 40 70 40 35 30 80 Nike 55 55 75 70 70 65 60 75 60 55 50 75 Reebok 50 30 60 80 70 50 10 75 40 35 40 70 Describe any patterns you see in the data.

12. Core-Plus 2 p. 77 4. Monthly Sales____________________________________________ J F M A M J J A S O N D Converse 40 35 50 55 70 60 40 70 40 35 30 80 Nike 55 55 75 70 70 65 60 75 60 55 50 75 Reebok 50 30 60 80 70 50 10 75 40 30 40 70 Describe any patterns you see in the data. Describe any general trends over time that you observe. Which months have the highest sales? What could be a reason for the high sales?

13. Core-Plus 2 p. 77 4. Monthly Sales____________________________________________ J F M A M J J A S O N D Converse 40 35 50 55 70 60 40 70 40 35 30 80 Nike 55 55 75 70 70 65 60 75 60 55 50 75 Reebok 50 30 60 80 70 50 10 75 40 30 40 70 Describe any patterns you see in the data. Describe any general trends over time that you observe. Which months have the highest sales? What could be a reason for the high sales? Are there any outliers in the data? If so, explain why you think they could have occurred.

14. Core-Plus 2 p. 77 4. Monthly Sales____________________________________________ J F M A M J J A S O N D Converse 40 35 50 55 70 60 40 70 40 35 30 80 Nike 55 55 75 70 70 65 60 75 60 55 50 75 Reebok 50 30 60 80 70 50 10 75 40 30 40 70 Describe any patterns you see in the data. Describe any general trends over time that you observe. Which months have the highest sales? What could be a reason for the high sales? Are there any outliers in the data? If so, explain why you think they could have occurred. How many pairs of Nikes were sold over the year?

15. Core-Plus 2 p. 77 4. Monthly Sales____________________________________________ J F M A M J J A S O N D Converse 40 35 50 55 70 60 40 70 40 35 30 80 Nike 55 55 75 70 70 65 60 75 60 55 50 75 Reebok 50 30 60 80 70 50 10 75 40 30 40 70 Describe any patterns you see in the data. Describe any general trends over time that you observe. Which months have the highest sales? What could be a reason for the high sales? Are there any outliers in the data? If so, explain why you think they could have occurred. How many pairs of Nikes were sold over the year? How many pairs of all there brands together were sold in February?

16. Investigation 2: Analyzing Matrices Matrices can be used to organize all sorts of data, not just sales data. In this investigation, you will analyze some situations in archeology, sociology, and sports. As you explore these different situations, look for answers to this question: How can you interpret and operate on a matrix to help understand and analyze data? Archeology Archeologists study ancient civilizations and their cultures. One way they study these cultures is by exploring sites where the people once lived and analyzing objects that they made. Archeologists use matrices to classify and then compare the objects they find at various archeological sites. For example, suppose that pieces of pottery are found at five different sites. The pottery pieces have certain characteristics: they are either glazed or not glazed, ornamental or not, colored or natural, thin or thick. Core-Plus 2 p. 78

17. Information about the characteristics of the pottery pieces at all five sites is organized in the matrix below. A “1” means the pottery piece has the characteristic and a “0” means it does not have the characteristic. Pottery Characteristics Glaze Orn Color Thin Site A 0 1 0 0 Site B 1 0 0 0 Site C 1 0 1 0 Site D 1 1 1 1 Site E 0 1 1 1 What does it mean for pottery to be “glazed”? “Ornamental”? Core-Plus 2 p. 79

18. Information about the characteristics of the pottery pieces at all five sites is organized in the matrix below. A “1” means the pottery piece has the characteristic and a “0” means it does not have the characteristic. Pottery Characteristics Glaze Orn Color Thin Site A 0 1 0 0 Site B 1 0 0 0 Site C 1 0 1 0 Site D 1 1 1 1 Site E 0 1 1 1 What does it mean for pottery to be “glazed”? “Ornamental”? What does the “1” in the third row and the first column mean? Core-Plus 2 p. 79

19. Information about the characteristics of the pottery pieces at all five sites is organized in the matrix below. A “1” means the pottery piece has the characteristic and a “0” means it does not have the characteristic. Pottery Characteristics Glaze Orn Color Thin Site A 0 1 0 0 Site B 1 0 0 0 Site C 1 0 1 0 Site D 1 1 1 1 Site E 0 1 1 1 What does it mean for pottery to be “glazed”? “Ornamental”? What does the “1” in the third row and the first column mean? Is the pottery at site E thick or thin? Core-Plus 2 p. 79

20. Information about the characteristics of the pottery pieces at all five sites is organized in the matrix below. A “1” means the pottery piece has the characteristic and a “0” means it does not have the characteristic. Pottery Characteristics Glaze Orn Color Thin Site A 0 1 0 0 Site B 1 0 0 0 Site C 1 0 1 0 Site D 1 1 1 1 Site E 0 1 1 1 What does it mean for pottery to be “glazed”? “Ornamental”? What does the “1” in the third row and the first column mean? Is the pottery at site E thick or thin? Which site has pottery pieces that are glazed and thick but are not ornamented or colored? Core-Plus 2 p. 79

21. Information about the characteristics of the pottery pieces at all five sites is organized in the matrix below. A “1” means the pottery piece has the characteristic and a “0” means it does not have the characteristic. Pottery Characteristics Glaze Orn Color Thin Site A 0 1 0 0 Site B 1 0 0 0 Site C 1 0 1 0 Site D 1 1 1 1 Site E 0 1 1 1 What does it mean for pottery to be “glazed”? “Ornamental”? What does the “1” in the third row and the first column mean? Is the pottery at site E thick or thin? Which site has pottery pieces that are glazed and thick but are not ornamented or colored? How many of the sites have glazed pottery? Explain how you used the rows and columns of the matrix to answer the question. Core-Plus 2 p. 79

22. Pottery Characteristics Glaze Orn Color Thin Site A 0 1 0 0 Site B 1 0 0 0 Site C 1 0 1 0 Site D 1 1 1 1 Site E 0 1 1 1 You can use the matrix in Problem 1 to determine how much the pottery pieces differ between sites. For example, the pieces found at sites A and B differ on exactly two characteristics—glazes and ornamentation. So you can say that the degree of difference between the pottery pieces at sites A and B is 2. a. Explain why the degree of difference between pottery pieces at sites A and C is 3. Core-Plus 2 p. 79

23. Pottery Characteristics Glaze Orn Color Thin Site A 0 1 0 0 Site B 1 0 0 0 Site C 1 0 1 0 Site D 1 1 1 1 Site E 0 1 1 1 You can use the matrix in Problem 1 to determine how much the pottery pieces differ between sites. For example, the pieces found at sites A and B differ on exactly two characteristics—glazes and ornamentation. So you can say that the degree of difference between the pottery pieces at sites A and B is 2. a. Explain why the degree of difference between pottery pieces at sites A and C is 3. b. Find the degree of difference between the pottery pieces at sites D and E. Core-Plus 2 p. 79

24. 3. You can construct a new matrix that summarizes the degree of difference information. Degree of Difference A B C D E Glaze Orn Color Thin A __ 2 3 __ __ Site A 0 1 0 0 B 2 __ __ __ __ Site B 1 0 0 0 C 3 __ __ __ __ Site C 1 0 1 0 D __ __ __ __ __ Site D 1 1 1 1 E __ __ __ __ __ Site E 0 1 1 1 What number would best describe the difference between site A and site A? Core-Plus 2 p. 79

25. 3. You can construct a new matrix that summarizes the degree of difference information. Degree of Difference A B C D E Glaze Orn Color Thin A __ 2 3 __ __ Site A 0 1 0 0 B 2 __ __ __ __ Site B 1 0 0 0 C 3 __ __ __ __ Site C 1 0 1 0 D __ __ __ __ __ Site D 1 1 1 1 E __ __ __ __ __ Site E 0 1 1 1 What number would best describe the difference between site A and site A? What number should be placed in the third row, fourth column? What does this number tell you about the pottery at these two sites? Core-Plus 2 p. 79

26. 3. You can construct a new matrix that summarizes the degree of difference information. Degree of Difference A B C D E Glaze Orn Color Thin A __ 2 3 __ __ Site A 0 1 0 0 B 2 __ __ __ __ Site B 1 0 0 0 C 3 __ __ __ __ Site C 1 0 1 0 D __ __ __ __ __ Site D 1 1 1 1 E __ __ __ __ __ Site E 0 1 1 1 What number would best describe the difference between site A and site A? What number should be placed in the third row, fourth column? What does this number tell you about the pottery at these two sites? Complete a copy of the degree of difference matrix shown above. Core-Plus 2 p. 79

27. 3. You can construct a new matrix that summarizes the degree of difference information. Degree of Difference A B C D E Glaze Orn Color Thin A __ 2 3 __ __ Site A 0 1 0 0 B 2 __ __ __ __ Site B 1 0 0 0 C 3 __ __ __ __ Site C 1 0 1 0 D __ __ __ __ __ Site D 1 1 1 1 E __ __ __ __ __ Site E 0 1 1 1 What number would best describe the difference between site A and site A? What number should be placed in the third row, fourth column? What does this number tell you about the pottery at these two sites? Complete a copy of the degree of difference matrix shown above. Describe one of the two patterns you see in the degree of difference matrix. Core-Plus 2 p. 79

28. Complete problems 1 and 2 in the Central High Booster Club Learning Task on your own paper with a partner. Put your name on the top of your paper and your partner’s name on the bottom of your paper.

29. Central High School Booster Club Learning Task: In order to raise money for the school, the Central High School Booster Club offered spirit items prepared by members for sale at the school store and at games. They sold stuffed teddy bears dressed in school colors, tote bags and tee shirts with specially sewn and decorated school insignias. The teddy bears, tote bags, and tee shirts were purchased from wholesale suppliers and decorations were cut, sewn and painted, and attached to the items by booster club parents. The wholesale cost for each teddy bear was $4.00, each tote bag was $3.50 and each tee shirt was $3.25. Materials for the decorations cost $1.25 for the bears, $0.90 for the tote bags and $1.05 for the tee shirts. Parents estimated the time necessary to complete a bear was 15 minutes to cut out the clothes, 20 minutes to sew the outfits, and 5 minutes to dress the bears. A tote bag required 10 minutes to cut the materials, 15 minutes to sew and 10 minutes to glue the designs on the bag. Tee shirts were made using computer generated transfer designs for each sport which took 5 minutes to print out, 6 minutes to iron on the shirts, and 20 minutes to paint on extra detailing.

30. The booster club parents made spirit items at three different work meetings and produced 30 bears, 30 tote bags, and 45 tee shirts at the first session. Fifteen bears, 25 tote bags, and 30 tee shirts were made during the second meeting; and, 30 bears, 35 tote bags and 75 tee shirts were made at the third session. They sold the bears for $12.00 each, the tote bags for $10.00 each and the tee shirts for $10.00 each. In the first month of school, 10 bears, 15 tote bags, and 50 tee shirts were sold at the bookstore. During the same time period, Booster Club members sold 50 bears, 20 tote bags, and 100 tee shirts at the games. The following is a matrix, a rectangular array of values, showing the wholesale cost of each item as well as the cost of decorations. "wholesale" and "decorations" are labels for the matrix rows and "bears", "totes", and "shirts" are labels for the matrix columns. The dimensions of this matrix called A are 2 rows and 3 columns and matrix A is referred to as a [2 x 3] matrix. Each number in the matrix is called an entry. It is sometimes convenient to write matrices (plural of matrix) in a simplified format without labels for the rows and columns. Matrix A can be written as an array where the values can be identified by their positions. In this system, the entry a22 = .90, which is the cost of decorations for tote bags.

31. Write and label matrices for the information given on the Central High School Booster Club's spirit project. a. Let matrix B show the information given on the time necessary to complete each task for each item. Labels for making the items should be cut/print, sew/iron, and dress/decorate. b. Find matrix C to show the numbers of bears, totes, and shirts produced at each of the three meetings. c. Matrix D should contain the information on items sold at the bookstore and at the game. d. Let matrix E show the sales prices of the three items. Matrices are called square matrices when the number of rows = the number of columns. A matrix with only one row or only one column is called a row matrix or a column matrix. Are any of the matrices from 1. square matrices or row matrices or column matrices? If so, identify them.

32. Since matrices are arrays containing sets of discrete data with dimensions, they have a particular set of rules, or algebra, governing operations such as addition, subtraction, and multiplication. In order to add two matrices, the matrices must have the same dimensions. Also, if the matrices have row and column labels, these labels must also match. Consider the following problem and matrices: Several local companies wish to donate spirit items which can be sold along with the items made by the Booster Club at games help raise money for Central High School. J J's Sporting Goods store donates 100 caps and 100 pennants in September and 125 caps and 75 pennants in October. Friendly Fred's Food store donates 105 caps and 125 pennants in September and 110 caps and 100 pennants in October. How many items are available each month from both sources? To add two matrices, add corresponding entries. Sept Oct Sept Oct Let J = caps  100 125 and F = caps 105 110 pennants 100 75 pennants 125 100 Sept Oct Sept Octthen J + F = caps 205 235 = caps 100 + 105 125 + 110   pennants 225 175 pennants 100 + 125 75 + 100 Subtraction is handled like addition by subtracting corresponding entries.

33. Summarize the Mathematics In this investigation, you explored how matrices can be used to organize and display data. The Shoe Outlet sells women’s shoes, sizes 5 to 11, and men’s shoes, sizes 6 to 13. The manager would like to have an organized display of the number of pairs sold this year for each shoe size. Describe a matrix that could be used to organize these data. What is the size of your matrix? What are some advantages of using matrices to organize and display data? What are some disadvantages? Explain how the same information can be displayed in a matrix in different ways. Be prepared to share your explanations and thinking with the entire class. Core-Plus 2 p. 77

34. Math 3 Preview and Acceleration Unit 1, Lesson 1, Day 1 (Previewing Day 2) Do you think these operations can be done? What conditions do you think might be necessary? How do you think the operations would be completed? 8 7 6 4 1 2 + 3 4 5 7 5 8 4 1 1 3 -1 2 7 2 4 + 5 8 -2 10 -3 6 3 5 7 9 2 4 -6