- 1. Mathematics Reviewer WORD PROBLEMS IN ALGEBRA Solving Word Problems George Polya’s Four-step Problem-Solving 2. Check each step of the plan as you proceed. 5. If the first number is x and another Process This may be intuitive checking or a formal number is k less than the first, then the As part of his work on problem solving, Polya proof of each step. other number is x k . developed a four-step problem-solving process 3. Keep an accurate record of your work. 6. Consecutive integers: If x is the first similar to the following: D. Looking Back integer, then x 1 is the 2nd, x 2 is the A. Understanding the Problem 1. Check the results in the original problem. In 3rd, etc. 1. Can you state the problem in your own some cases, this will require a proof. 7. Consecutive even/odd integers: If x is the words? 2. Interpret the solution in terms of the original first even/odd integer, then x 2 is the 2. What are you trying to find or do? problem. Does your answer make sense? Is 2nd even/odd integer, x 4 is the 2nd 3. What are the unknowns? it reasonable? even/odd integer, etc. 4. What information do you obtain from the 3. Determine whether there is another method 8. Digits. A two digit number can be written problem? of finding the solution. in the from 10T U , where T is the ten’s 5. What information, if any, is missing or not 4. If possible, determine other related or more digit and U is the unit’s digit. A three- needed? general problems for which the techniques digit number can be written in the form B. Devising a Plan will work. 100H 10T U , where H is the The following list of strategies, although not (Source: hundred’s digit. exhaustive, is very useful: http://www.drkhamsi.com/classe/polya.html) N.B. If the digit of a two-digit number 1. Look for a pattern. 10T U is REVERSED, the new number 2. Examine related problems and determine if Solution Strategies & Tips for Particular Word the same technique can be applied. becomes 10U T . Reversing a three-digit Problems 3. Examine a simpler or special case of the number means reading the number backwards; problem to gain insight into the solution of A. Representations for Number Problems i.e., 100H 10T U becomes 100U 10T H . the original problem. 1. If the sum of two numbers is s and x is one 4. Make a table. number, then the other number is s x . B. Age Problems 5. Make a diagram. 2. If the difference between two numbers is d If a is the present age of person A, then a y 6. Write an equation. and x is the smaller number, then the larger is the age of A y years ago, while a y is the 7. Use a guess and check. number is d x . age of A y years from now (or hence). 8. Work backward. 3. If the first number is x and another is k times Age table for age problems: 9. Identify a sub goal. the first, then the other number is kx . Age some C. Carrying out the Plan 4. If the first number is x and another number Persons Age now years ago or 1. Implement the strategy in Step 2 and is k more than the first, then the other from now perform any necessary actions or number is x k . computations.
- 2. C. Mixture/Collection Problems meeting point Note: If B undoes what A does, the “+” becomes a A dA B For mixtures, the general equation is “” between 1/a and 1/b in #s 2, 3, and 4. amount of % concen- amount of dB × = dapart d A d B The situations described above can be extended for solution tration substance 3 or more persons doing a job. while for those involving prices or money, 4. For bodies traveling in air/current: price or cost totol cost quantity × = rate against the rate in still rate of wind F. Some Formulas in Geometry per unit or price 1. Perimeter Formulas wind/current wind/water or current a. square: P 4s, s length of one side Types of No. of Amount/price/percent Total rate with the rate in still rate of wind b. rectangle: P 2l 2w (l = length, w quantities units per unit amount = width wind/current wind/water or current c. triangle: P a b c (a, b, and c are E. Work Problems the sides of the triangle) TOTAL = sum of all entries at last column If person A can finish a job alone in a time units d. circle (circumference): C 2r (r = and B can finish the same job alone in b time radius) D. Motion Problems units, then 2. Area formulas distance rate time 1 a. square: A s 2 General Formula: 1. A can finish of the job in 1 time unit, d rt a b. rectangle: A lw Situations: 1 c. triangle: A 1 bh 2 1. Overtaking = equal distances covered while B can finish of the job in 1 time b d. circle: A r 2 Starting point unit. So after k time units, A and B can 3. Volume Formulas dA A 1 1 a. cube: V s3 finish k and k of the job, a b b. rectangular solid/prism: V lwh d B d A rAt0 B respectively. c. sphere: V 3 r 3 4 d A dB 2. Together, A and B can finish of 1 1 1 d. right circular cylinder: V r 2 h 2. Bodies moving in opposite directions = the a b x e. right circular cone: V 1 r 2 h 3 distance apart is the sum of the distances the job in 1 time unit, where x is the no. of traveled by each body time units that A and B can finish the job together. Anything you can solve in Starting point 3. Suppose that A started working the job five minutes should not A dA B along in the first p time units then B joined be considered a problem. dB for x more time units until they finished the -- George Polya dapart d A d B 1 1 1 job, then p x 1 where 1 a a b 3. Bodies moving toward each other = the sum of represents the whole job. the distance between the origin of 1st body and 4. If A and B were doing the job together for 2nd body to the meeting point is equal to the sum the first q time units until B left, letting A of the distances between the origin of each point finish the job alone in x more time units, © gjnabueg 2 August 2003 1 1 1 Revised 19 July 2009 then q x 1 . All rights reserved a b a