Lesson One - Notation Development (whole class discussion)
In order for students to be successful in solving word problems, they must become comfortable in working with numerical expressions as well as algebraic expressions. They should evaluate complex numerical expressions, especially those that will be similar in word problems. We want them to be able to write and verbalize expressions. Numerical and algebraic expressions are used throughout algebra. Numerical expressions contain all numbers whereas algebraic expressions contain numbers and variables.
Reading and translating the meaning of algebraic notations are therefore essential skills. Students should be able to convert verbal descriptions given in word problems into numerical expressions, and be able to read and evaluate numerical expressions including parentheses and nested parentheses. In this unit, we only intend to develop linear equations, so issues of exponents will not arise.
Numerical Expressions
Basic:
4+ 4
2*3
12-9
-11-4
24/6
Complex:
(2+4)5 - 7(6-1)
18 (2 + (12 /3))
(24 /3-2) + 4(12x - 3)
4(2x (7-3) + 5x) - 5 + 3/4
Ask the students to provide more examples.
Algebraic Expressions
As stated earlier, concise representation is one of the ways mathematics saves work. It uses letters and numbers instead of words. An algebraic expression is a mathematical phrase that consists of one or more numbers and variables along with one or more arithmetic operations. In the algebraic expression 6x, the letter x is called a variable and 6 is the constant. It is also called the coefficient of x. Variables are symbols used to represent unspecified numbers or values. Any letter may be used as a variable. (This example is to show students how algebraic expressions look. At this point, students may not know how to convert verbal descriptions to expressions and know how to read expressions. These skills will be developed). The focus is recognizing algebraic expressions versus numerical expressions.
Basic:
5x
3t-9
6 + m/n
m * 5n 4ab / 3g
Complex:
x(90 + 30)
2(81x + 9x -10)
2(3x - 4) -3(x - 5)
2(3(2x-4)-5) + ½ (x+7) ñ 3/2
Ask students to provide more examples.
Any time two or more numbers, letters, or combination of letters and numbers are juxtaposed, it indicates multiplication. The number is called the coefficient of the variable. In algebraic expressions, a raised dot or parentheses are often used to indicate multiplication. Using "x" to indicate multiplication is avoided because it can be easily mistaken for the letter "x" (that is used as a variable). The following notations will be discussed in detail.
xy x ?y x(y) (x)y (x)(y)
In each expression, the quantities being multiplied are called factors, and the result is called the product. The above examples of algebraic expressions will now be verbalized.
Construct Algebraic Expression/Equations
It is important in the lesson to formulate the questions clearly and completely, and especially, to define the variables carefully and completely. The teacher should do this at the start, and gradually transfer the responsibility to the class, and then to each student. Therefore, although not shown in the unit due to space limitations, for each word problem, we will translate verbal descriptions into a verbal model and translate verbal model into a mathematical model or algebraic equation.
Verbal Description —> Verbal Model —> Algebraic Expression/Equation
Example 1:
Verbal Description: The sale price of a basketball is $18. If the sale price is $7 less than the original price, what is the original price?
Verbal Model: Sale Price = Original Price - Discount
Algebraic Equation: $18 = Original Price - $7
Example 2:
Verbal Description: The total income that an employee received in 1992 was $21,550. Of that $750 represented a bonus given at the end of the year. How much was the employee paid each week? Assume that each weekly paycheck contained the same amount, and that the year consisted of 52 weeks.
Verbal Model: Income for year = 52 times weekly pay + Bonus
Labels: Income for year = $21,550
Weekly pay = x (in dollars)
Bonus = $750
Algebraic Equation: 21,550 = 52x + 750
Problems
1) You are paid $6 an hour. How much will you earn for working a certain number of hours? Let x represents hours. (6x)
2) A person is paid 3 cents for each aluminum soda can, and 2 cents for each steel soda can collected. What is the total amount the person will earn for the cans collected? (.03x + .02y)
3) A person is paid 3 cents for aluminum soda can, and 2 cents for each steel soda can collected, and $45 a week for collecting other kinds of trash in the city park. (.03x + .02y + 45)
Write a verbal description for each of the following
1) 7x - 12 (Twelve less than the product of seven and a number)
2) 7(x-12) (Twelve is subtracted from a number and result is multiplied by seven).
It would be a good idea to have the students compare these two expressions. The second is equal to 72 les than the first.
3) (5+x)/2 (The sum of five and a number, all divided by two)
One of the primary goals of this unit is learning to read word problems and figure out what expressions and relations are needed to solve them. We need to emphasize to students that most word problems do not contain verbal expressions that clearly identify the arithmetic operations involved and that we need to sometimes rely on common sense and our experiences. This is an important skill, both for mathematics, and for making sense of many non-mathematical situations.
Example:
1) A cash register contains x quarters. Write an expression for this amount of money in dollars. (.25x)
2) A cash register contains n nickels and d dimes. Write an expression for this amount of money in cents. (5n + 10d)
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