Using the arithmetic calculations in Problems 1 and 2, you can generalize with a symbolic representation, utilizing a variable.
3. Consider the situation in which a 24-inch piece of wire is cut into two parts. Suppose you have no measuring device available, so you will denote the length of the first part by the symbol x, representing the variable length.
a. Using Problems 1 and 2 as a guide, represent the length of the second part symboli-cally in terms of x.
b. What are the reasonable replacement values for the variable x?
Notice that your symbolic representation in Problem 3 provides a general method for calcu-lating the length of the second part given the length x of the first part. The method is all about the operation, subtracting the length of the first part from 24. This symbolic representation,
, is often called an algebraic expression in the variable x.
24 - x
Now suppose you take the two parts of the snipped wire and bend each to form a square as shown in the following figure.
Wire I
Square I
Wire II
Square II
Remember, the area of a square equals the length of a side squared. The perimeter of a square is four times the length of one side.
4. Refer back to Problem 1, in which one piece of wire measuring 8 inches is cut from a 24-inch piece of wire.
a. Explain in words how you can use the known length of the wire to determine the area of the first square. What is this area?
b. Write down the sequence of arithmetic operation(s) you used to determine your answer in part a.
c. Explain in words how to determine the area of the second square.
d. Write down the sequence of arithmetic operation(s) you used to determine your answer in part c.
5. Refer back to Problem 3, in which one piece of wire cut from a 24-inch piece measured x inches.
a. Using Problem 4b as a guide, represent the area of the first square symbolically using x.
b. Similarly, represent the area of the second square symbolically using x.
c. Replace x by 8 in your algebraic expressions in parts a and b to calculate the areas of the two squares formed when a length of 8 inches is cut from a 24-inch piece of wire. Compare your results with the answers in Problem 4a and c.
When a specific number (an input value) replaces a variable in an algebraic expression, as in Problem 5c, the resulting calculation produces a single output value. This process is often called evaluating an algebraic expression.
Here are some additional problems to explore arithmetically and algebraically.
6. Suppose you have 8 dimes and 6 quarters in a desk drawer.
a. Explain how you can use this information to determine the total value of these coins.
What is the value?
b. Write down and describe in words the sequence of arithmetic operations you used to determine your answer.
c. Suppose you have d dimes and q quarters in a desk drawer. Represent the total value of these coins symbolically in terms of d and q.
d. Evaluate your algebraic expression in part c when there are 14 dimes and 21 quarters.
The algebraic expression in Problem 6c contained a variable multiplied by a constant num-ber. In such a situation, it is not necessary to include a multiplication symbol. The arithmetic operation between a constant and a variable is always understood to be multiplication when no symbol is present.
Some special terminology is often used to describe important features of an algebraic expres-sion. The terms of an algebraic expression are the parts of the expression that are added or subtracted. For example, the expression in Problem 6c has two terms, namely 0.10d and 0.25q. A numerical constant that is multiplied by a variable is called the coefficient of the variable. For example, in the expression in Problem 6c, the coefficient of the variable d is 0.10 and the coefficient of the variable q is 0.25.
7. a. Suppose a rectangle has length 8 inches and width 5 inches. Determine its perimeter and its area.
b. If the length of the rectangle is represented by l and its width is represented by w, write an expression that represents its perimeter and its area.
c. Suppose the length and width are each increased by 3 inches. Determine the new length, width, perimeter, and area of the expanded rectangle.
d. Represent the new length and width of the expanded rectangle in part c symbolically using the original dimensions l and w.
e. Represent the perimeter and area of the expanded rectangle symbolically in terms of the original length l and width w.
f. Evaluate the expressions in part e for the dimensions of the original rectangle. How do they compare to your results in part c?
g. Suppose the length and width of the original rectangle are doubled and tripled, respectively. Determine the new length, width, perimeter and area of the expanded rectangle.
h. Represent the new length and width symbolically using the original dimensions l and w.
i. Represent the new perimeter and new area symbolically using the original dimensions w and l. Is your representation consistent with your answer to part g?
8. In Activity 2.3 (College Expenses, Problem 4), you purchased a meal ticket at the be-ginning of the semester for $150. The daily lunch special costs $4 in the college cafete-ria. You want to be able to determine the balance on your meal ticket after you have purchased a given number of lunch specials.
a. Identify the input and output variables in the meal ticket situation.
b. Write a verbal description of the sequence of operations that you must perform on the input value to obtain the corresponding output value.
c. Translate the verbal description in part b to a symbolic expression using x to repre-sent the input.
d. Suppose you have purchased 16 lunch specials so far in the semester. Use the alge-braic expression from part c to determine the remaining balance.
An algebraic expression gives us a very powerful symbolic tool to represent relationships be-tween an input variable and output variable. This symbolic rule generalizes the many possi-ble numerical pairs, as might be represented in a tapossi-ble or graph. The remainder of Chapter 2 will further your fluency in this language of algebra.