1. Copyright © 2005 Pearson Education, Inc. Problem Solving Method Quantitative Literacy

2. Slide 5-2 Copyright © 2005 Pearson Education, Inc. Polya’s Procedure George Polya (1887-1985) developed a general procedure for solving problems.

3. Slide 5-3 Copyright © 2005 Pearson Education, Inc. Guidelines for Problem Solving Understand the Problem. Devise a Plan. Carry Out the Plan. Check the Results.

4. Slide 5-4 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-5 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-6 Copyright © 2005 Pearson Education, Inc. 3. Carrying Out the Plan. Use the plan you devised in step 2 to solve the problem.

7. Slide 5-7 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-8 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-9 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-10 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-11 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-12 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-13 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-14 Copyright © 2005 Pearson Education, Inc. Addition of Integers 1. If the signs of the numbers are the same, add the numbers and the sign remains the same. 2. 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) = –120c) 75 + 98 = 173 b) 270 + (–170) =100d) -183 + -160 = -343

15. Slide 5-15 Copyright © 2005 Pearson Education, Inc. Subtraction of Integers 1. Write the first number with no changes. 2. Change the subtraction symbol to an addition symbol and then replace the second number with its opposite. 3. 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. Slide 5-16 Copyright © 2005 Pearson Education, Inc. Properties Multiplication Property of Zero Division For any a, b, and c where b  0, means that c b = a.

17. Slide 5-17 Copyright © 2005 Pearson Education, Inc. Rules for Multiplication The product of two numbers with like signs (positive  positive or negative  negative) is a positive number. The product of two numbers with unlike signs (positive  negative or negative  positive) is a negative number.

18. Slide 5-18 Copyright © 2005 Pearson Education, Inc. Examples Evaluate: a) (3)(  4)b) (  7)(  5) c) 8 7d) (  5)(8)

19. Slide 5-19 Copyright © 2005 Pearson Education, Inc. Examples Evaluate: a) (3)(  4)b) (  7)(  5) c) 8 7d) (  5)(8) Solution: a) (3)(  4) =  12b) (  7)(  5) = 35 c) 8 7 = 56d) (  5)(8) =  40

20. Slide 5-20 Copyright © 2005 Pearson Education, Inc. Rules for Division The quotient of two numbers with like signs (positive  positive or negative  negative) is a positive number. The quotient of two numbers with unlike signs (positive  negative or negative  positive) is a negative number.

21. Slide 5-21 Copyright © 2005 Pearson Education, Inc. Example Evaluate: a) b) c) d)

22. Slide 5-22 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-23 Copyright © 2005 Pearson Education, Inc. 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. Copyright © 2005 Pearson Education, Inc. Ratio & Proportion 1.3

25. Slide 5-25 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-26 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-27 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-28 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-29 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-30 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-31 Copyright © 2005 Pearson Education, Inc. Multiplication/Division Principle of Equality If a = b, the ac = bc, or For all real numbers a, b, c with c ≠ 0

32. Slide 5-32 Copyright © 2005 Pearson Education, Inc. 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. Slide 5-33 Copyright © 2005 Pearson Education, Inc. Division of Fractions Multiplication of Fractions

34. Slide 5-34 Copyright © 2005 Pearson Education, Inc. Example: Multiplying Fractions Evaluate the following. a) b)

35. Slide 5-35 Copyright © 2005 Pearson Education, Inc. Example: Dividing Fractions Evaluate the following. a) b)

36. Slide 5-36 Copyright © 2005 Pearson Education, Inc. Addition and Subtraction of Fractions

37. Slide 5-37 Copyright © 2005 Pearson Education, Inc. Example: Add or Subtract Fractions Add: Subtract: