A First Course in Mathematics Concepts forElementary School Teachers: Theory,
Problems, and Solutions
Marcel B. FinanArkansas Tech UniversitycAll Rights Reserved
February 8, 2006
1 Polyas Problem-Solving Process 3
2 Problem-Solving Strategies 7
3 More Problem-Solving Strategies 20
4 Sets and Operations on Sets 33
5 Numeration Systems 48
6 The Hindu-Arabic System (800 BC) 62
7 Relations and Functions 69
8 Addition and Subtraction of Whole Numbers 83
9 Multiplication and Division of Whole Numbers 93
10 Ordering and Exponents of Whole Numbers 104
11 Whole Numbers: Mental Arithmetic and Estimation 111
12 Algorithms for Addition and Subtraction of Whole Numbers119
13 Algorithms for Multiplication and Division of Whole Num-bers 128
14 Arithmetic Operations in Bases Other Than Ten 136
15 Prime and Composite Numbers 144
16 Tests of Divisibility 152
17 Greatest Common Factors and Least Common Multiples 160
18 Fractions of Whole Numbers 168
19 Addition and Subtraction of Fractions 178
20 Multiplication and Division of Fractions 188
21 Decimals 198
22 Arithmetic Operations on Decimals 209
23 Ratios and Proportions 218
24 Percent 223
25 Solutions to Practice Problems 231
1 Polyas Problem-Solving Process
Problem-solving is the cornerstone of school mathematics. The main reasonof learning mathematics is to be able to solve problems. Mathematics is apowerful tool that can be used to solve a vast variety of problems in technol-ogy, science, business and finance, medecine, and daily life.It is strongly believed that the most efficient way for learning mathemati-cal concepts is through problem solving. This is why the National Councilof Teachers of Mathematics NCTM advocates in Principles and Standardsfor School Mathematics, published in 2000, that mathematics instructionin American schools should emphasize on problem solving and quantitativereasoning. So, the conviction is that children need to learn to think aboutquantitative situations in insightful and imaginative ways, and that merememorization of rules for computation is largely unproductive.Of course, if children are to learn problem solving, their teachers must them-selves be competent problem solvers and teachers of problem solving. Thetechniques discussed in this and the coming sections should help you to be-come a better problem solver and should show you how to help others developtheir problem-solving skills.
Polyas Four-Step ProcessIn his book How to Solve It, George Polya identifies a four-step process thatforms the basis of any serious attempt at problem solving. These steps are:
Step 1. Understand the ProblemObviously if you dont understand a problem, you wont be able to solve it.So it is important to understand what the problem is asking. This requiresthat you read slowly the problem and carefully understand the informationgiven in the problem. In some cases, drawing a picture or a diagram can helpyou understand the problem.
Step 2. Devise a PlanThere are many different types of plans for solving problems. In devisinga plan, think about what information you know, what information you arelooking for, and how to relate these pieces of information. The following arefew common types of plans: Guess and test: make a guess and try it out. Use the results of your guessto guide you.
Use a variable, such as x. Draw a diagram or a picture. Look for a pattern. Solve a simpler problem or problems first- this may help you see a patternyou can use. make a list or a table.
Step 3. Carry Out the PlanThis step is considered to be the hardest step. If you get stuck, modify yourplan or try a new plan. Monitor your own progress: if you are stuck, is itbecause you havent tried hard enough to make your plan work, or is it timeto try a new plan? Dont give up too soon. Students sometimes think thatthey can only solve a problem if theyve seen one just like it before, but thisis not true. Your common sense and natural thinking abilities are powerfultools that will serve you well if you use them. So dont underestimate them!
Step 4. Look BackThis step helps in identifying mistakes, if any. Check see if your answer isplausible. For example, if the problem was to find the height of a telephonepole, then answers such as 2.3 feet or 513 yards are unlikely-it would be wiseto look for a mistake somewhere. Looking back also gives you an opportunityto make connections: Have you seen this type of answer before? What didyou learn from this problem? Could you use these ideas in some other way?Is there another way to solve the problem? Thus, when you look back, youhave an opportunity to learn from your own work.
Solving Applied ProblemsThe term word problem has only negative connotations. Its better tothink of them as applied problems. These problems should be the most in-teresting ones to solve. Sometimes the applied problems dont appear veryrealistic, but thats usually because the corresponding real applied problemsare too hard or complicated to solve at your current level. But at least youget an idea of how the math you are learning can help solve actual real-worldproblems.Many problems in this book will be word problems. To solve such problems,one translates the words into an equivalent problem using mathematical sym-bols, solves this equivalent problem, and then interprets the answer. Thisprocess is summarized in Figure 1.1
Example 1.1In each of the following situations write the equation that describes the sit-uation. Do not solve the equation:
(a) Hermans selling house is x dollars. The real estate agent received 7% ofthe selling price, and Hermans received $84,532. What is the selling priceof the house?(b) The sum of three consecutive integers is 48. Find the integers.
Solution.(a) The equation describing this situation is
x 0.07x = 84, 532.
(b) If x is the first integer then x + 1 and x + 2 are the remaining integers.Thus,
x+ (x+ 1) + (x+ 2) = 48.
In each of the following problems write the equation that describes eachsituation. Do not solve the equation.
Problem 1.1Two numbers differ by 5 and have a product of 8. What are the two numbers?
Problem 1.2Jeremy paid for his breakfest with 36 coins consisting of nickels and dimes.If the bill was $3.50, then how many of each type of coin did he use?
Problem 1.3The sum of three consecutive odd integers is 27. Find the three integers.
Problem 1.4At an 8% sales tax rate, the sales tax Peters new Ford Taurus was $1,200.What was the price of the car?
Problem 1.5After getting a 20% discount, Robert paid $320 for a Pioneer CD player forhis car. What was the original price of the CD?
Problem 1.6The length of a rectangular piece of property is 1 foot less than twice thewidth. The perimeter of the property is 748 feet. Find the length and thewidth.
Problem 1.7Sarah is selling her house through a real estate agent whose commission rateis 7%. What should the selling price be so that Sarah can get the $83,700she needs to pay off the mortgage?
Problem 1.8Ralph got a 12% discount when he bought his new 1999 Corvette Coupe. Ifthe amount of his discount was $4,584, then what was the original price?
Problem 1.9Julia framed an oil painting that her uncle gave her. The painting was 4inches longer than it was wide, and it took 176 inches of frame molding.What were the dimensions of the picture?
Problem 1.10If the perimeter of a tennis court is 228 feet and the length is 6 feet longerthan twice the width, then what are the length and the width?
2 Problem-Solving Strategies
Strategies are tools that might be used to discover or construct the meansto achieve a goal. They are essential parts of the devising a plan step, thesecond step of Polyas procedure which is considered the most difficult step.Elementary school children now learn strategies that they can use to solve avariety of problems. In this section, we discuss three strategies of problemsolving: guessing and checking, using a variable, and drawing a picture or adiagram.
Problem-Solving Strategy 1: Guess and CheckThe guessing-and-checking strategy requires you to start by making a guessand then checking how close your answer is. Next, on the basis of this result,you revise your guess and try again. This strategy can be regarded as a formof trial and error, where the information about the error helps us choose whattrial to make next.This strategy may be appropriate when: there is a limited number of possibleanswers to test; you want to gain a better understanding of the problem; youcan systematically try possible answers.This strategy is often used when a student does not know how to solve aproblem more efficiently of if the student does not have the tools to solve theproblem in a facter way.
Example 2.1In Figure 2.1 the numbers in the big circles are found by adding the numbersin the two adjacent smaller circles. Complete the second diagram so that thepattern holds.
Understand the problemIn this example, we must find three numbers a, b, and c such that
a+ b = 16,a+ c = 11,b+ c = 15.
See Figure 2.2.
Devise a planWe will try the guess and check strategy.
Carry out the planWe start by guessing a value for a. Suppose a = 10. Since a + b = 16 thenb = 6. Since b+ c = 15 then c = 9. But then a+ c is 19 instead of 11 as it issupposed to be. This does not check.Since the value of a = 10 yields a large a + c then we will reduce our guessfor a. Take a = 5. As above, we find b = 11 and c = 4. This gives a + c = 9which is closer to 11 than 19. So our next guess is a = 6. This implies thatb = 10 and c = 5. Now a+ c = 11 as desired. See Figure 2.3.
Look backIs there an easier solution? Looking carefully at the initial example and thecompleted solution to the problem we notice that if we divide the sum ofthe numbers in the larger circles by 2 we obtain the sum of the numbers inthe smaller circle. From this we can devise an easier solution. Looking atFigure 2.1, and the above discussion we have a+ b+ c = 21 and a+ b = 16.This gives, 16 + c = 21 or c = 5. Since a + c = 11 then a = 6. Finally, sinceb+ c = 15 then b = 10.
Example 2.2Leah has $4.05 in dimes and quarters. If she has 5 more quarters than dimes,how many of each does she have?
Solution.Understand the problemWhat are we asked to determine? We need to find how many dimes and howmany quarters Leah has.What is the total amount of money? $4.05.What else do we know? There are five more quarters than dimes.
Devise a planPick a number, try it, and adjust the estimate.
Carry out the planTry 5 dimes. That would mean 10 quarters.
5 $0.10 + 10 $0.25 = $3.00.
Increase the number of dimes to 7.
7 $0.10 + 12 $0.25 = $3.70.
Try again. This time use 8 dimes.
8 $0.10 + 13 $0.25 = $4.05
Leah has 8 dimes and 13 quarters.
Look backDid we answer the question asked, and does our answer seem reasonable?Yes.
Problem 2.1Susan made $2.80 at her lemonade stand. She has 18 coins. What combina-tion of coins does she have?
Problem 2.2A rectangular garden is 4 feet longer than it is wide. Along the edge of thegarden on all sides, there is a 2-foot gravel path. How wide is the garden ifthe perimeter of the garden is 28 feet? (Hint: Draw a diagram and use theguess and check strategy.)
Problem 2.3There are two two-digit numbers that satisfy the following conditions:
(1) Each number has the same digits,(2) the sum of digits in each number is 10,(3) the difference between the two numbers is 54.What are the two numbers?
Understanding the problemThe numbers 58 and 85 are two-digit numbers which have the same digits,and the sum of the digits is 13. Find two two-digit numbers such that thesum of the digits is 10 and both numbers have the same digits.Devise a planSince there are only nine two-digit numbers whose digits have a sum of 10,the problem can be easily solved by guessing. What is the difference of yourtwo two-digit numbers from part (a)? If this difference is not 54, it can pro-vide information about your next guess.Carry out the planContinue to guess and check. Which numbers has a difference of 54?Looking backThis problem can be extended by changing the requirement that the sum ofthe two digits equal 10. Solve the problem for the case in which the digitshave a sum of 12.
Problem 2.4John is thinking of a number. If you divide it by 2 and add 16, you get 28.What number is John thinking of?
Problem 2.5Place the digits 1, 2, 3, 4, 5, 6 in the circles in Figure 2.4 so that the sum ofthe three numbers on each side of the triangle is 12.
Problem 2.6Carmela opened her piggy bank and found she had $15.30. If she had onlynickels, dimes, quarters, and half-dollars and an equal number of coins ofeach kind, how many coins in all did she have?
Problem 2.7When two numbers are multiplied, their product is 759; but when one is sub-tracted from the other, their difference is 10. What are those two numbers?
Problem 2.8Sandy bought 18 pieces of fruit (oranges and grapefruits), which cost $4.62.If an orange costs $0.19 and a grapefruit costs $0.29, how many of each didshe buy?
Problem 2.9A farmer has a daughter who needs more practice in mathematics. Onemorning, the farmer looks out in the barnyard and sees a number of pigs andchickens. The farmer says to her daughter, I count 24 heads and 80 feet.How many pigs and how many chickens are out there?
Problem 2.10At a benefit concert 600 tickets were sold and $1,500 was raised. If therewere $2 and $5 tickets, how many of each were sold?
Problem 2.11At a bicycle store, there were a bunch of bicycles and tricycles. If there are 32seats and 72 wheels, how many bicyles and how many tricycles were there?
Problem 2.12If you have a bunch of 10 cents and 5 cents stamps, and you know that thereare 20 stamps and their total value is $1.50, how many of each do you have?
Problem-Solving Strategy 2: Use a variableOften a problem requires that a number be determined. Represent the num-ber by a variable, and use the conditions of the problem to set up an equationthat can be solved to ascertain the desired number.This strategy is most appropriate when: a problem suggests an equation;there is an unknown quantity related to known quantities; you are trying todevelop a general formula.
Example 2.3Find the sum of the whole numbers from 1 to 1000.
Solution.Understand the problemWe understand that we are to find the sum of the first 1000 nonzero wholenumbers.
Devise a planWe will apply the use of variable strategy. Let s denote the sum, i.e.
s = 1 + 2 + 3 + + 1001 (1)
Carry out the planRewrite the sum in s in reverse order to obtain
s = 1000 + 999 + 998 + + 1 (2)
Adding ( 1) - ( 2) to obtain
2s = 1001 + 1001 + 1001 + + 1001 = 1000 1001.
Dividing both sides by 2 to obtain
s =1000 1001
Look backIs it true that the process above apply to the sum of the first n whole integers?The answer is yes.
Example 2.4Lindsey has a total of $82.00, consisting of an equal number of pennies,nickels, dimes and quarters. How many coins does she have in...