MATH 021 Introductory Algebra

Unit 1: Mathematical Expressions 1.1 Problem Solving and Numerical Representations Introduction Everyone must have felt at least once in his or her life how wonderful it would be if we could easily solve a problem using one method that works every time. Problem solving can be challenging, and there are no universal approaches one can take to solving problems. Basically one must explore possible methods until recognizing the path to a solution. Guessing can help, and algebraic thinking can be involved. In general, as one gains experience in solving problems, one develops one's own techniques and strategies. Thus the guessing is not arbitrary. Rather, it is educated. In this section we are going to learn a framework for problem solving and get a glimpse of strategies that are often used by experts. The strategies are based on the work of George Polya. For further study, read: G. Polya, How to Solve It, A New Aspect of Mathematical Method, Second Ed., Princeton University Press, Princeton, NJ, 1985. A Framework for Problem Solving The following four phases can be identified in the process of solving problems: (1) Understanding the problem (2) Making a plan of solution (3) Carrying out the plan (4) Looking back i.e. verifying Before we apply this process, it is important to introduce some algebraic vocabulary. Definitions and Key Concepts Variable: A variable, usually represented by a letter or symbol, can be defined as:  A quantity that may change within the context of a mathematical problem.  A placeholder for a specific value. Download for free at 1.1 Problem Solving and Numerical Representations 12 Definitions and Key Concepts Input-Output table: An input-output table is the numerical representation of a rule. This table has at least two columns. The first column (input) is for values that you know; the second column (output) lists the answer that you get for each input value, after applying an appropriate rule. The rule for an input-output table is the symbolic representation. Rules are equations that begin with: y = . A rule can also be called a formula. Example 1: When the Mathematics Department moved to its current location, faculty considered various ways to place cubicles in the large, open space. Each cubicle contains a desk for one professor, and uses three partitions for sides. One suggestion for how to place the cubicles, was to make one long row. Figure 1 (below) shows this arrangement for five faculty. How many partitions will be needed to arrange the cubicles in one long row for 30 faculty? 1 2 3 4 5 Figure 1. Solution: (1) Understanding the problem – This is a “find it” type of problem. If we use the cubicle layout in Figure 1 above, we need to find how many partitions will be needed for 30 cubicles – one cubicle for each professor. (2) Making a plan of solution – Since a pattern seems to be involved in this problem (each new cubicle requires an additional two partitions), it may be helpful to create an input-output table that lists the number of cubicles, and the number of partitions that would be needed. Creating a table is known as a numerical representation. (3) Carrying out the plan – The first cubicle needs three partitions, but the second cubicle only needs two more partitions since one partition is shared. Each new cubicle needs two additional partitions. Download for free at: 1.1 Problem Solving and Numerical Representations 13 To read the table, read across each row individually. For example, if you only wanted to make three cubicles, then you would need 7 partitions. Input, x Number of Cubicles Output, y Number of Partitions 1 3 2 5 3 7 4 9 5 11 Following this pattern, it would be cumbersome to make a table that has 30 rows to represent 30 cubicles, so this is where algebra can be helpful. Using the variables x (to represent the number of cubicles) and y (to represent the number of partitions), the rule (or formula) to find the number of partitions is y = 2x + 1. For now, you will be given the rule when you work with tables. Eventually, you will learn how to come up with the rule yourself. Some of you may even be able to guess the rule now! (4) Looking back i.e. verifying – Let’s see if the rule works: for 4 cubicles, we replace x with 4: Look back at Figure 1. For four cubicles, were there 9 partitions? YES! The rule works, so now we can answer the original question and apply it for 30 faculty, or x = 30: y = 2(30) + 1, y = 61. Answer: 61 partitions will be needed to make cubicles in one long row for 30 faculty. Practice Using Input-Output Tables Example 2: American Fitness charges a one-time registration fee of $50, plus $25 per month of membership at its gym. Answer the following questions using the rule y = 25x + 50, where x represents the number of months of membership and y represents the total cost of membership for that number of months. a) How much will it cost a student to become a member for two years? b) How much will it cost a student to become a member for thirty months?