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BLOCK 20-B Problem solving

BLOCK 20-B Problem solving. 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.

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BLOCK 20-B Problem solving

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  1. BLOCK 20-B Problem solving

  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.

  3. 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. In the real-world would you be asked to find the length of c or ……. c a b

  4. Pleasantville 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? 32 miles Merry Town 48 miles Holiday City

  5. 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

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

  7. DIVISE A PLAN The plan is usually called a strategy Problem-solving strategies include: •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

  8. 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.

  9. LOOK BACK 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.

  10. 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? You need to find 4 numbers after 15 that fit the pattern. What do you need to find? How can you solve the problem? You can find a pattern. SOLVE: 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)

  11. Using a pattern made this easy 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. Starting with 36, you add 9 to make 45 So the next four numbers would be 21, 28, 36, and 45

  12. 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? You need to find how many chips in all . Then you need to find what part of this total is green. How can you solve this? Try drawing a picture. SOLVE: Draw 8 chips. G G G B B B B R 3 OUT OF THE 8 CHIPS ARE GREEN

  13. 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? You need to find what the time was when Steven left Coral Springs. What do you need to find? How can you solve the problem? 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.

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

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

  16. A Simple Hierarchical Graphic Organizer

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

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

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

  20. Venn Diagram

  21. Prime Numbers 5 7 11 13 2 3 Even Numbers 4 6 8 10 Multiples of 3 9 15 21 6 Venn Diagram - example

  22. Multiple Meanings

  23. 3 sides 3 sides 3 angles 3 angles 3 angles = 60° 1 angle = 90° 3 sides 3 angles 3 angles < 90° Multiple Meanings – example Right Equiangular TRI- ANGLES Acute Obtuse 3 sides 3 angles 1 angle > 90°

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

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

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

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

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

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

  30. Problem Set 1 Parallelogram Rhombus Square Quadrilateral Polygon Kite Irregular polygon Trapezoid Isosceles Trapezoid Rectangle

  31. 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

  32. Problem Set 3 Addition Multiplication a + b a times b a plus b a xb sum of a and ba(b) ab Subtraction Division a – b a/b a minus b a divided by b a less b a ÷ b

  33. 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.

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

  35. Possible Solution to PS #2 REAL NUMBERS Irrational Numbers Rational Numbers Integers Counting Numbers Whole Numbers

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

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

  38. 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.

  39. 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.

  40. 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?

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