1. Aim: Problem Solving Course: Math Literacy Do Now: Aim: Why is problem solving sooooo problematic??!! A Melissa lives at the YWCA (A) and works at Macy’s (B). She walks to work. How many different routes can she take? B Sutter St. Post St. Geary St. O’Farrell St. Stockton St.Powell St.Mason St.Taylor St.

2. Aim: Problem Solving Course: Math Literacy Guidelines for Problem Solving FirstYou have to understand the problem SecondDevise a plan. Find the connection between the data and the unknown. Look for patterns, related to previously solved problem or a know formula, or simplify the given info to give you an easier problem. ThirdCarry out the plan FourthLook back and examine the solution obtained

3. Aim: Problem Solving Course: Math Literacy Guidelines for Problem Solving FirstYou have to understand the problem What facts are given? What does the problem tell me? Do I know what all the words and phrase mean? Did I read the problem carefully? What is the problem asking me?

4. Aim: Problem Solving Course: Math Literacy Guidelines for Problem Solving SecondDevise a plan. What am I trying to find out? What facts do I need to know? Should I make a chart, a table or a diagram? Can I do the problem in my head or do I need paper and pencil or a calculator? What steps should I follow?

5. Aim: Problem Solving Course: Math Literacy Guidelines for Problem Solving ThirdCarry out the plan Review the steps and formulate/calculate the answer. FourthLook back and examine the solution obtained Reread the problem and ask yourself: Does the answer make sense.

6. Aim: Problem Solving Course: Math Literacy Guidelines for Problem Solving Read Plan Solve Reflect Step 1 Step 2 Step 3 Step 4

7. Aim: Problem Solving Course: Math Literacy Model Problem A library has 2890 science books. The science books are classified into three categories: life, earth, and physical science. There are 190 more books in the earth category than in each other category. How many books are in each category. Understand the problem Devise a plan Carry out the plan Look back

8. Aim: Problem Solving Course: Math Literacy Do Now Problem A B 1 1 1 2 1 3 4 3 6 10 4 20 1 1 Understand the problem Devise a plan Carry out the plan Look back no backtracking; no cutting thru backyards one way to arrive at this point start small Each # of ways to a point is found by adding the two numbers from the upper/left vertices that connect to that point.

9. Aim: Problem Solving Course: Math Literacy Guidelines for Problem Solving A B 1 1 1 2 1 3 4 3 6 10 4 20 1 1 Each vertex in the rows is found by adding the two numbers from the above row that connect to it A B 1 1 1 2 1 3 4 3 6 10 4 20 1 1

10. Aim: Problem Solving Course: Math Literacy Pascal’s Triangle 1 11 121 1331 14641 1510 51 1615201561 172135 2171 The first & last numbers in each row are 1 Every other number in each row is formed by adding the two numbers above the number.

11. Aim: Problem Solving Course: Math Literacy Pascal’s Triangle & Expansion of (x + y) n (x + y) 0 = 1 (x + y) 1 = 1x + 1y (x + y) 2 = 1x 2 + 2xy + 1y 2 (x + y) 3 = 1x 3 + 3x 2 y + 3xy 2 + 1y 3 (x + y) 4 = 1x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + 1y 4 (x + y) 5 = 1x 5 + 5x 4 y + 10x 3 y 2 + 10x 2 y 3 + 5xy 4 + 1y 5 In each expansion there is n + 1 terms. In each expansion the x and y have symmetric roles. The sum of the powers of each term is n. The coefficients increase & decrease symmetrically. 5 5 4 4 expansion of (x + y) n zero row 1st row

12. Aim: Problem Solving Course: Math Literacy Melissa’s Trip 1 11 121 1331 14641 1510 51 1615201561 172135 2171 A 3 blocks down 3 blocks over

13. Aim: Problem Solving Course: Math Literacy Extension How many different ways could Melissa get from the YWCA to the YMCA (D)? A D

14. Aim: Problem Solving Course: Math Literacy Model Problem A jokester tells you that he has a group of cows and chickens and that he counted 13 heads and 36 feet. How many cows and chickens does he have? Understand the problem Devise a plan Carry out the plan Look back No. of chickens No. of cows No. of Heads No. of Feet 0 13 13 52 1 12 13 50 2 11 13 48 Look for patterns need 36 feet 12 feet gotta go each additional chicken reduces No. of feet by 2 need additional 6 chickens 8 5 13 36

15. Aim: Problem Solving Course: Math Literacy Model Problem If a family has 5 children, in how many different birth orders could the parents have a 3-boy, 2-girl family? BBBGG BBGGB BGGBB etc. Understand the problem Devise a plan Carry out the plan Look back

16. Aim: Problem Solving Course: Math Literacy Model Problem If a family has 5 children, in how many different birth orders could the parents have a 3-boy, 2-girl family? B G B G B G B G B G B G B G B G B G B G B G B G B G B G B G BBBBB BBBBG GGGGG B G.....

17. Aim: Problem Solving Course: Math Literacy Model Problem If a family has 5 children, in how many different birth orders could the parents have a 3-boy, 2-girl family? Start with smaller family and look for pattern 1 child: B one way G one way 2 children: BB one way BG GB GG one way two ways 3 children: BBB one way BBG BGB GBB BBG GBG GGB GGG one way three ways

18. Aim: Problem Solving Course: Math Literacy Model Problem If a family has 5 children, in how many different birth orders could the parents have a 3-boy, 2-girl family? 2 children:1BB21GG two ways for 1 B and 1 G 1 child: 1B 1G 3 children:1BBB 3 3 1GGG 3 ways for 2B & 1G 3 ways for 1B & 2G 1 11 121 1331 14641 1510 51 1615201561 172135 2171 Pascal’s triangle binomial expansion

19. Aim: Problem Solving Course: Math Literacy Model Problem If a family has 5 children, in how many different birth orders could the parents have a 3-boy, 2-girl family? 1 11 121 1331 14641 1510 51 1615201561 172135 2171 5B4B1G3B2G2B3G1B4G5G 1 child 2 child 3 child 4 child Ten different ways

20. Aim: Problem Solving Course: Math Literacy Do Now: Aim: Why is problem solving sooooo problematic??!! As I was going to St. Ives, I met a man with seven wives. Every wife had seven sacks. Every sack had seven cats. Every cat had seven kits. Kits, cats, sacks and wives, How many were going to St. Ives

21. Aim: Problem Solving Course: Math Literacy Gauss When the famous German mathematician Karl Gauss was a child, his teacher required students to find the sum of the first 100 natural numbers. The teacher expected this problem to keep the class occupied for some time. Gauss answer almost immediately. 1 + 2 + 3 + · · · · + 50 + 51 + · · · · · + 98 + 99 + 100 101 x 50 pairs of numbers = 5050

22. Aim: Problem Solving Course: Math Literacy Model Problem Find the sum of natural numbers from 1 to 50. 1 + 2 + 3 + · · · · + 25 + 26 + · · · · · + 48 + 49 + 50 51 x 25 pairs of numbers = 1275 Will this method for finding the sum of the natural numbers always work? Is there a formula?

23. Aim: Problem Solving Course: Math Literacy Model Problem 1 + 2 + 3 + · · · · + · · · · + · · · · · · · (n-2) + (n-1) + n Will this method for finding the sum of the natural numbers always work? Is there a formula? n + 1 n + 1 sum of pairs

24. Aim: Problem Solving Course: Math Literacy Examine a Related Problem Ryan was building matchstick square sequences, as shown below. He use 67 matchsticks to form the last figure in his sequence. How many match sticks did he use for the entire project? 47 10 Every added box requires three more matches

25. Aim: Problem Solving Course: Math Literacy Examine a Related Problem 4 + 7 + 10 + · · · + · · · · + · · · · · · · · 61 + 64 + 67 71 even # of numbers? If so, how many pairs? sums equal 71 for successive outer pairings

26. Aim: Problem Solving Course: Math Literacy Examine a Related Problem 4 + 7 + 10 + · · · + · · · · + · · · · · · · · 61 + 64 + 67 71 +3 # of TermTerm 14 27 = 4 + 3 310 = 4 + 3 + 3 413 = 4 + 3 + 3 + 3 n 11 pairs 71 x 11 = 781 n = 22 3n + 1= 67

27. Aim: Problem Solving Course: Math Literacy Draw a Diagram On the first day of math class, 20 people are present in the room. To become acquainted with one another, each person shakes hands just once with everyone else. How many handshakes take place? Understand the problem Devise a plan Carry out the plan Look back John shakes Mary’s hand and Mary shakes John’s hand counts only as one handshake, not two.

28. Aim: Problem Solving Course: Math Literacy Draw a Diagram On the first day of math class, 20 people are present in the room. To become acquainted with one another, each person shakes hands just once with everyone else. How many handshakes take place? three people four people 3 handshakes 6 handshakes 5 people10 handshakes

29. Aim: Problem Solving Course: Math Literacy Draw a Diagram On the first day of math class, 20 people are present in the room. To become acquainted with one another, each person shakes hands just once with everyone else. How many handshakes take place? three peoplefour people 3 handshakes6 handshakes 5 people 10 handshakes each member shakes the hand of two other people 3 x 2 = 6 John shakes Mary’s hand and Mary shakes John’s hand counts only as one handshake, not two. each member shakes the hand of 3 other people 4 x 3 = 12 each member shakes the hand of 4 other people 5 x 4 = 20 6/2 = 3 12/2 = 620/2 = 10 (20 x 19)/2 = 190 total handshakes

30. Aim: Problem Solving Course: Math Literacy Model Problems How much dirt is in a a hole 2 feet long, 3 feet wide and 2 feet deep? Two US coins have a total value of 55 cents. One coin is not a nickel. What are the two coins? Walter had a dozen apples in his office. He ate all be 4. How many are left? Sal owns 20 blue and 20 brown socks, which he keeps in a drawer in complete disorder. What is the minimum number of socks that he must pull out of the drawer on a dark morning to be sure he has a matching pair?

31. Aim: Problem Solving Course: Math Literacy Types of Problems In your own wordsAsks u to discuss or rephrase main ideas or procedures using your own words. Level 1Mechanical and drill Level 2require understanding of concepts and related to past examples Level 3extension of past problems requiring creative thought Problem Solvingoriginal thinking; not based on past examples