1-1 Using VariablesNCSCOS: 1.02 – Use formulas and algebraic expressions, including iterative and recursive forms, to model and solve problems.

Obj. 1: Modeling Relationships with Variables If you earn an hourly wage of $6.50

your pay is the number of hours you work multiplied by 6.5.

In the table at the bottom left, the variable h stands for the number of hours you worked. A variable is a symbol, usually a letter, that represents one of more numbers. The expression 6.50h is an algebraic expression. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Algebraic expressions are sometimes called variable expressions.

Hours Worked Pay (dollars)

1 6.50 x 1

2 6.50 x 2

3 6.50 x 3

h 6.50 x h

Ex. 1: Writing an Algebraic Expression Write an algebraic expression for each

phrase: Seven more than n

“More than” indicates addition. Add the first number 7 to the second number n.

The difference of n and seven “Difference” indicates subtraction. Begin with

the first number n. Then subtract the second number 7

The product of seven and n “Product” indicates multiplication. Multiply the

first number 7 by the second number n. The quotient of n and seven

“Quotient” indicates division. Divide the first number n by the second number 7.

𝒏+𝟕

𝒏−𝟕

𝟕𝒏

𝒏𝟕

Your turn to write an algebraic expression:

The quotient of 4.2 and c

t minus 15

Remember: To translate an English phrase into an algebraic expression, you may need to define one or more variables first.

𝟒 .𝟐𝒄

𝒕−𝟏𝟓

Ex. 2: Writing an Algebraic Expression Define a variable and write an algebraic

expression for each phrase.

Two times a number plus 5 Relate: two times a number plus 5 Define: Let n = the number. Write:2 x n + 5

7 less than three times a number  Relate: 7 less than three times a number Define: Let a = the number. Write:3 x a - 7

Your turn to define a variable and write an algebraic expression for each phrase: 9 less than a number

The sum of twice a number and 31

The product of one half of a number and one third of the same number

Obj. 2: Modeling Relationships with Equations and Formulas

You can use algebraic expressions to write an equation. An equation is a mathematical sentence that uses an equal sign. If the equation is true, then the two expressions on either side of the equal sign represent the same value. An equation that contains one or more variables in an open sentence. In everyday language, the word “is” often suggests an equal sign in the associated equation.

Ex. 3: Writing an Equation Track One Media sells all CDs for $12 each. Write

an equation for the total cost of given number of CDs. Relate: The total cost is 12 times the

number of CDs bought. Define: Let n = the number of CDs bought.

Let c = the total cost. Write: c = 12 x

n Suppose the manager at Track One Media raises

the price of each CD to $15. Write an equation to find the cost of n CDs.

Suppose the manager at Track One Media uses the equation c = 10.99n. What could this mean?

Ex. 4: Real-World Problem Solving Write an equation for the data in the table below:

Relate: Change equals $20.00 minus cost of purchase.

Define: Let c = cost of item purchased.Let a = amount of change.

Write: a = $20.00 - c

Cost of Purchase Change from $20

$20.00 $0$19.00 $1.00

$17.50 $2.50

$11.59 $8.41

Exercises: Practice and Problem Solving

Ex. 1: Practice by Example: Write an algebraic expression for each phrase.

1. 4 more than p2. y minus 123. 12 minus m4. The product of c and 155. The quotient of n and 86. The quotient of 17 and k7. 23 less than x8. The sum of v and 3

Ex. 2: Define a variable and write an expression for each phrase.

9. 2 more than twice a number.10. A number minus 1111. 9 minus a number12. A number divided by 8213. The product of 5 and a number14. The sum of 13 and twice a number15. The quotient of a number and 616. The quotient of 11 and a number

Exercises: Practice and Problem Solving Continued

Ex. 3: Define variables and write an equation to model each situation.

17. The total cost is the number of cans times $0.70.

18. The perimeter of a square equals 4 times the length of a side.

19. The total length of rope, in feet, used to put up tents is 60 times the number of tents.

20. What is the number of slices of pizza left from an 8-slice pizza after you have eaten some slices?

Exercises: Practice and Problem Solving Continued

Exercises: Practice and Problem Solving Continued Ex. 4: Define variables and write an

equation to model the relationship in each table.Number of

WorkersNumber of

Radios Built

1 132 263 394 52

Number of Tapes

Cost

1 $8.502 $17.003 $25.504 $34.00

Number of Sales

Total Earnings

5 $2.0010 $4.0015 $6.0020 $8.00

Number of Hours

Total Pay

4 $326 $488 $64

10 $80

21.

23.

22.

24.

Apply Your Skills: Write an expression for each phrase.25. The sum of 9 and k minus 1726. 6.7 more than 5 times n27. 9.85 less than the product of t and 3728. The quotient of 3b and 4.529. 15 plus the quotient of 60 and w30. 7 minus the product of v and 331. The product of m and 5, minus the quotient of t

and 732. The sum of the quotient of p and 14 and the

quotient of q and 333. 8 minus the product of 9 and r

Write a phrase for each expression.34. q + 5

35. 3 – t

36. 9n + 1

37. 7hb

Define variables and write an equation to model the relationship in each table.

Number of Days Change in Height (meters)

1 0.1652 0.3303 0.4954 0.660

Time (months) Length (inches)

1 4.12 8.23 12.34 16.4

39. 40.

Use the table below:

Does each statement fit the data in the table? Explain. Hours worked = lawns mowed x 2 Hours worked = lawns mowed + 3

Lawns Mowed Hours

1  

2  

3 6

Challenge:

Suppose you drop the ball from a window 20 ft. above the ground. Predict how high the ball will bounce.

Drop Height (ft.)

Height of First Bounce (ft.)

1

2 13

4 25

The table at the right shows the height of the first bounce when a ball is dropped from different heights.

Write an equation to describe the relationship between the height of the first bounce and the drop height.

Open-Ended Describe a real-world situation that each

equation could represent. Include a definition for each variable.

d = 5t

a = b + 3

c =

Standardized Test Prep1. Which is an algebraic

expression for “six less than k”?

a. 6 – kb. k – 6

2. Which is an algebraic expression for “the product of a and 10”?

a. a + 10b. a – 10c. 10a

3. Which is an algebraic expression for “9 more than v”?

a. v + 9b. v– 9c. 9 – vd. 9v

4. A container of milk contains 64 ounces. Which equation models the number n of ounces remaining after you have drunk m ounces?

5. Which equation models the relationship in the table if r represents the row number and t represents the number of tulips?

6. Which is an algebraic expression for “the quotient of r + 5 and b”?

Standardized Test Prep

Row Number Number of Tulips

1 32 63 94 12

Add, subtract, multiply, or divide.7. 0.2 + 0.78. 0.13 + 0.919. 0.6 + 0.7510. 1.09 + 0.3711. 0.9 x 0.0712. 0.58 – 0.49

13. 0.8 – 0.6614. 1.32 – 0.3915. 2 x 0.516. 0.69 317. 0.6 0.218. 1.21 11

19. List four prime numbers between 20 and 50.