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Problem Solving Method

Quantitative Literacy. Problem Solving Method. Polya’s Procedure. George Polya (1887-1985) developed a general procedure for solving problems. Guidelines for Problem Solving. Understand the Problem. Devise a Plan. Carry Out the Plan. Check the Results. 1. Understand the Problem.

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Problem Solving Method

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  1. Quantitative Literacy Problem Solving Method

  2. Polya’s Procedure • George Polya (1887-1985) developed a general procedure for solving problems.

  3. Guidelines for Problem Solving • Understand the Problem. • Devise a Plan. • Carry Out the Plan. • Check the Results.

  4. 1. Understand the Problem. • Read the problem carefully, at least twice. • Try to make a sketch of the problem. Label the given information given. • Make a list of the given facts that are pertinent to the problem. • Decide if you have enough information to solve the problem.

  5. 2. Devise a Plan to Solve the Problem. • Can you relate this problem to a previous problem that you’ve worked before? • Can you express the problem in terms of an algebraic equation? • Look for patterns or relationships. • Simplify the problem, if possible. • Use a table to list information to help solve. • Can you make an educated guess at the solution?

  6. 3. Carrying Out the Plan. • Use the plan you devised in step 2 to solve the problem.

  7. 4. Check the Results. • Ask yourself, “Does the answer make sense?” and “Is it reasonable?” • If the answer is not reasonable, recheck your method and calculations. • Check the solution using the original statement, if possible. • Is there a different method to arrive at the same conclusion? • Can the results of this problem be used to solve other problems?

  8. Example: Selling a House The Sharlow’s are planning to sell their home. They want to be left with $139,500 after paying commission to the realtor. • If a realtor receives 7% of the selling price, how much must they sell the house for? • If a realtor receives a flat $5000 and then 3% of the selling price, how much must they sell their house for?

  9. Solution • Translate the problem in algebraic terms as follows: selling price less commission is amount left. x - 0.07x = $139,500 Thus, the Sharlow’s need to sell their home for $150,000.

  10. Solution continued • If the realtor receives a flat fee of $5000, then 3% commission, first add 3% to the amount they wish to be left with. $139,500 + 3% $143,814. Then add $5000 to this total. $143,814 + $5000 = $148,814.

  11. Example: Taxi Rates In Mexico, a taxi ride costs $4.80 plus $1.68 for each mile traveled. Diego and Juanita budgeted $25 for a taxi ride (excluding tip). • How far can they travel on their $25 budget? • If they include a $2 tip, then how far can they travel?

  12. Solution • We know that the initial charge plus the mileage charge can equal $25. So, if we let x equal the distance, in miles, driven by the taxi for $25, then • If they wish to give a $2 tip, we solve the same way only allowing only $23 to be the budget.

  13. Integers • The set of integers consists of 0, the natural numbers, and the negative natural numbers. • Integers = {…-4,-3,-2,-1,0,1,2,3,4,…} • On a number line, the positive numbers extend to the right from zero; the negative numbers extend to the left from zero.

  14. Addition of Integers • If the signs of the numbers are the same, add the numbers and the sign remains the same. • If the signs of the numbers are different, subtract the larger value minus the smaller value. The result inherits the sign of the value farthest from zero on the number line. Evaluate: a) 20 + (–140) = –120 c) 75 + 98 = 173 b) 270 + (–170) =100 d) -183 + -160 = -343

  15. Subtraction of Integers • Write the first number with no changes. • Change the subtraction symbol to an addition symbol and then replace the second number with its opposite. • Carry out the rules for the addition of signed numbers. a – b = a + (b) = solution Evaluate: a) –7 – 3 = –7 + (–3) = –10 b) –7 – (–3) = –7 + 3 = –4

  16. Multiplication Property of Zero Division For any a, b, and c where b 0, means that c• b = a. Properties

  17. Rules for Multiplication • The product of two numbers with likesigns (positive  positive or negative  negative) is a positivenumber. • The product of two numbers with unlikesigns (positive  negative or negative  positive) is a negative number.

  18. Examples • Evaluate: • a) (3)(4) b) (7)(5) • c) 8 • 7 d) (5)(8)

  19. Examples • Evaluate: • a) (3)(4) b) (7)(5) • c) 8 • 7 d) (5)(8) • Solution: • a) (3)(4) = 12 b) (7)(5) = 35 • c) 8 • 7 = 56 d) (5)(8) = 40

  20. Rules for Division • The quotient of two numbers with likesigns (positive  positive or negative  negative) is a positivenumber. • The quotient of two numbers with unlikesigns (positive  negative or negative  positive) is a negative number.

  21. Example • Evaluate: • a) b) • c) d)

  22. Order of Operations • The order in which we carry out operations is important! • Different methods result in different solutions. • We must agree upon only one solution for every problem.

  23. Order of Operations • Rank 1: Grouping Symbols • (Parentheses, Brackets, etc. innermost first) • Rank 2: Exponents (Powers) & Roots (from left to right) • Rank 3: Multiplication & Division (from left to right) • Rank 4: Addition & Subtraction (from left to right) • Follow these in order of operation and you always arrive at the standard solution for a problem.

  24. Ratio & Proportion 1.3

  25. The Rational Numbers • The set of rational numbers, denoted by Q, is the set of all numbers of the form p/q, where p and q are integers and q 0.

  26. Fractions • Fractions are numbers such as: • The numerator is the number above the fraction line. • The denominator is the number below the fraction line.

  27. Reducing Fractions • In order to reduce a fraction, we divide both the numerator and denominator by the greatest common divisor. • Example: Reduce to its lowest terms. • Solution:

  28. Mixed Numbers • A mixed number consists of an integer and a fraction. For example, 3 ½ is a mixed number. • 3 ½ is read “three and one half” and means “3 + ½”.

  29. Improper Fractions • Rational numbers greater than 1 or less than -1 that are not integers may be written as mixed numbers, or as improper fractions. • An improper fraction is a fraction whose numerator is greater than its denominator. An example of an improper fraction is 12/5.

  30. Fundamental Property of Proportions • The ratios , form the proportion If and only if ad = bc (cross products are equal) and b ≠ 0, d ≠ 0

  31. Multiplication/Division Principle of Equality • If a = b, the ac = bc, or For all real numbers a, b, c with c ≠ 0

  32. Proportion Models • Percent Problems • Geometric Problems • Diluted Mixture Problems • DS = “Desired Strength” of solution to be made (Percent) • AS = “Available Strength” of solution to be used (Percent) • AU = “Amount” of available solution “Used” • AM = “Amount” of desired solution “Made”

  33. Multiplication of Fractions • Division of Fractions

  34. Evaluate the following. a) b) Example: Multiplying Fractions

  35. Evaluate the following. a) b) Example: Dividing Fractions

  36. Addition and Subtraction of Fractions

  37. Add: Subtract: Example: Add or Subtract Fractions

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