5.19 - Variables

The student will

a) investigate and describe the concept of variable;

b) write an equation to represent a given mathematical relationship, using a variable;

c) use an expression with a variable to represent a given verbal expression involving one operation; and

d) create a problem situation based on a given equation, using a single variable and one operation.

BIG IDEAS

• So that I can make a story problem when given an equation
• So that I can figure out the missing piece of a problem (i.e How much gas can I get if I have \$25 and it costs \$2.09 a gallon?)
• So that I have a foundation of knowledge for a career in engineering, mathematics, chemistry, physics, and finance.
• So that I can make good financial decisions (i.e comparing offers from two phone companies when signing a long term contract; comparing the price of similar items in the grocery store based on price per unit)
• So that I can improve my critical thinking and problem solving skills
• So that I can win at game of Yahtzee, Math Boggle, Battleship, or Chess

UNDERSTANDING THE STANDARD

• A variable is a symbol that can stand for an unknown number (e.g., a + 4 = 6) or for a quantity that changes (e.g., the rule or generalization for the pattern for an input/output table such as x + 2 = y).
• An algebraic expression, an expression with a variable, is like a phrase; a phrase does not have a verb, so an expression does not have an equal symbol (=).
• A verbal expression describing a relationship involving one operation can be represented by an expression with a variable that mathematically describes the relationship. Numbers are used when quantities are known; variables are used when the quantities are unknown. For example, when b stands for the number of cookies in one full box, “the number of cookies in a full box and four extra” can be represented by b + 4; “three full boxes of cookies” by 3b; “the number of cookies each person would receive if a full box of cookies were shared among four people” by b/4 .
• An equation is a statement that represents the relationship between two expressions of equal value (e.g., 12 × 3 = 72 ÷ 2).
• A problem situation about two quantities that are equal can be expressed as an equation.
• An equation may contain a variable and an equal symbol (=). For example, the sentence, “A full box of cookies and four extra equal 24 cookies.” can be written as b + 4 = 24, where b stands for the number of cookies in one full box. “Three full boxes of cookies contain a total of 60 cookies” can be written as 3b = 60.
• Another example of an equation is b + 3 = 23 and represents the answer to the word problem, “How many cookies are in a box if the box plus three more equals 23 cookies?” where b stands for the number of cookies in the box?
• Teachers should consider varying the letters used (in addition to x) to represent variables.  The symbol x is often used to represent multiplication and can be confused with the variable x.  In addition to varying the use of letters as variables, this confusion can be minimized by using parentheses [e.g., 4(x) = 20 or 4x = 20] or a small dot raised off the line to represent multiplication  [4 • x = 20].
• By using story problems and numerical sentences, students begin to explore forming equations and representing quantities using variables.
• An equation containing a variable is neither true nor false until the variable is replaced with a number and the value of the expressions on both sides are compared.

ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

• Describe the concept of a variable (presented as boxes, letters, or other symbols) as a representation of an unknown quantity. (a)
• Write an equation with addition, subtraction, multiplication, or division, using a variable to represent an unknown quantity. (b)
• Use an expression with a variable to represent a given verbal expression involving one operation (e.g., “5 more than a number” can be represented by y + 5). (c)
• Create and write a word problem to match a given equation with a single variable and one operation. (d)

KEY VOCABULARY

Updated: May 29, 2019