2. It is not enough to know the skills. It is important to know how to use these skills to solve real-world problems. Problem solving touches every aspect of our lives.

4. Problem solving is an integral part of all mathematics learning. In everyday life and in the workplace, being able to solve problems can lead to great advantages. a b c In the real-world would you be asked to find the length of c or …….

5. Pleasantville Merry TownHoliday City 32 miles 48 miles Amanda needs to go to Holiday City. Since she is in a hurry she needs to take the shortest route possible. What route should she take and how many miles will she save?

6. What steps should we take when solving a word problem? 1. Understand the problem 2. Devise a plan 3. Carry out the plan. 4. Look back

7. What am I asked to find or show? What type of answer do I expect? What units will be used in the answer? Can I give an estimate? What information is given? If there enough or too little information given? Can I restate the problem in your own words? UNDERSTAND THE PROBLEM Ask yourself….

8. Act is out Make a drawing or diagram Look for a pattern Construct a table Identify all possibilities Guess and check Work backward Write an open sentence Solve a simpler or similar problem Change your point of view/logical reasoning DIVISE A PLAN The plan is usually called a strategy Problem-solving strategies include:

9. CARRY OUT THE PLAN Now solve the problem. The original strategy may need to be modified, not every problem will be solved in the first attempt.

10. This is simply checking all steps and calculations. Do not assume the problem is complete once a solution has been found. Instead, examine the problem to ensure that the solution makes sense. LOOK BACK

11. What strategy would you use to solve the following problem? George has written a number pattern that begins with 1, 3, 6, 10, 15. If he continues this pattern, what are the next four numbers in his pattern? What do you need to find? How can you solve the problem? SOLVE: You need to find 4 numbers after 15 that fit the pattern. You can find a pattern. Look at the numbers in the pattern. 3 = 1 + 2 (starting number is 1, add 2 to make 3) 6 = 3 + 3 (starting number is 3, add 3 to make 6) 10 = 6 + 4 (starting number is 6, add 4 to make 10) 15 = 10 + 5 (starting number is 10, add 5 to make 15)

12. Following this pattern… Starting with 15, add 6 to make 21. Starting with 21, you add 7 to make 28 Starting with 28, you add 8 to make 36. So the next four numbers would be 21, 28, 36, and 45 Starting with 36, you add 9 to make 45 Using a pattern made this easy Using a pattern made this easy

13. What strategy would you use to solve this problem? Karen has 3 green chips, 4 blue chips and 1 red chip in her bag of chips. What fractional part of the bag of chips is green? What do you need to find? How can you solve this? SOLVE: Draw 8 chips. G G G G B B B B B B G G B B R R 3 OUT OF THE 8 CHIPS ARE GREEN You need to find how many chips in all. Then you need to find what part of this total is green. Try drawing a picture.

14. What strategy would you use to solve this problem? Steven walked from Coral Springs to Margate. It took 1 hour 25 minutes to walk from Coral Springs to Pompano Beach. Then it took 25 minutes to walk from Pompano Beach to Margate. He arrived in Margate at 2:45 P.M. At what time did he leave Coral Springs? What do you need to find? How can you solve the problem? You need to find what the time was when Steven left Coral Springs. You can work backwards from the time Steven reached Margate. SOLVE: Start at 2:45. This is the time Steven reached Margate. Subtract 25 minutes. This is the time it took to get from Pompano Beach to Margate. The time is 2:00 P.M. Subtract: 1 hour 25 minutes. This is the time to get from Coral Springs to Pompano Beach. Steven left Coral Springs at 12:55 P.M.

15. Another helpful tool when organizing your thoughts and solving problems is to use graphic organizers. There are many different types you can use.

16. Types of Graphic Organizers Hierarchical diagramming Sequence charts Compare and contrast charts

17. A Simple Hierarchical Graphic Organizer

18. A Simple Hierarchical Graphic Organizer - example Algebra Calculus Trigonometry Geometry MATH

19. Category What is it? Illustration/Example What are some examples? Properties/Attributes What is it like? Subcategory Irregular set Compare and Contrast

20. Positive Integers Numbers What is it? Illustration/Example What are some examples? Properties/Attributes What is it like? Fractions Compare and Contrast - example Whole Numbers Negative Integers Zero -3, -8, -4000 6, 17, 25, 100 0

21. Venn Diagram

22. Venn Diagram - example Prime Numbers 57 11 13 Even Numbers 4 6 810 Multiples of 3 9 15 21 3 2 6

23. Multiple Meanings

24. Multiple Meanings – example TRI- ANGLES RightEquiangular AcuteObtuse 3 sides 3 angles 1 angle = 90° 3 sides 3 angles 3 angles < 90° 3 sides 3 angles 3 angles = 60° 3 sides 3 angles 1 angle > 90°

25. Series of Definitions Word=Category +Attribute = + Definitions: ______________________ ________________________________

26. Series of Definitions – example Word=Category +Attribute = + Definition: A four-sided figure with four equal sides and four right angles. Square Quadrilateral 4 equal sides & 4 equal angles (90°)

27. Four-Square Graphic Organizer 1. Word: 2. Example: 3. Non-example:4. Definition

28. Four-Square Graphic Organizer – example 1. Word: semicircle 2. Example: 3. Non-example:4. Definition A semicircle is half of a circle.

29. Matching Activity Divide into groups Match the problem sets with the appropriate graphic organizer

30. Matching Activity Which graphic organizer would be most suitable for showing these relationships? Why did you choose this type? Are there alternative choices?

31. Problem Set 1 Parallelogram Rhombus Square Quadrilateral Polygon Kite Irregular polygon Trapezoid Isosceles TrapezoidRectangle

32. Problem Set 2 Counting Numbers: 1, 2, 3, 4, 5, 6,... Whole Numbers: 0, 1, 2, 3, 4,... Integers:... -3, -2, -1, 0, 1, 2, 3, 4... Rationals: 0, …1/10, …1/5, …1/4,... 33, …1/2, …1 Reals: all numbers Irrationals: π, non-repeating decimal

33. Problem Set 3 Addition Multiplication a + b a times b a plus b a x b sum of a and b a(b) ab SubtractionDivision a – b a/b a minus b a divided by b a less b a ÷ b

34. Problem Set 4 Use the following words to organize into categories and subcategories of Mathematics: NUMBERS, OPERATIONS, Postulates, RULE, Triangles, GEOMETRIC FIGURES, SYMBOLS, corollaries, squares, rational, prime, Integers, addition, hexagon, irrational, {1, 2, 3…}, multiplication, composite, m || n, whole, quadrilateral, subtraction, division.

35. Possible Solution to PS #1 POLYGON Parallelogram: has 2 pairs of parallel sides Kite Square, rectangle, rhombus Kite: has 0 sets of parallel sides Irregular: 4 sides w/irregular shape Quadrilateral Trapezoid: has 1 set of parallel sides Trapezoid, isosceles trapezoid Kite

36. Possible Solution to PS #2 REAL NUMBERS

37. Possible Solution PS #3 Operations Subtraction Multiplication Division Addition ____a + b____ ___a plus b___ Sum of a and b ____a - b_____ __a minus b___ ___a less b____ ____a / b_____ _a divided by b_ _____a  b_____ ___a times b___ ____a x b_____ _____a(b)_____ _____ab______

38. Possible Solution to PS #4 Numbers Operations RulesSymbols Geometric Figures Mathematics Triangle Quadrilateral Hexagon Integer Prime Rational Irrational Whole Composite Addition Subtraction Multiplication Division Corollary Postulatem║nm║n √4 {1,2,3…}

39. Mike, Juliana, Diane, and Dakota are entered in a 4- person relay race. In how many orders can they run the relay, if Mike must run list? List them.

40. Mrs. Stevens earns $18.00 an hour at her job. She had $171.00 after paying $9.00 for subway fare. Find how many hours Mrs. Stevens worked. Try solving this problem by working backwards.

41. Use the work backwards strategy to solve this problem. A number is multiplied by -3. Then 6 is subtracted from the product. After adding -7, the result is -25. What is the number?