# 1.2 – Variables and Problem Solving

## Section 1 (Apex Pg 1 – 20)

Objectives

• Translate everyday language into mathematical sentences.
• Convert mathematical sentences into variables and expressions.
• Interpret the different parts of mathematical sentences.

Key Words

• Variable – an unknown or changeable quantity
• Constant – a value in a formula that does not change (unlike a variable).
• Expression – a combination of numbers, variables, and operations that does not contain an equals sign or inequality sign. An expression that includes variables is often called an algebraic expression.
• Coefficient – a number that multiplies a variable.
• Term – part of algebraic expressions. Terms are separated by + or – signs. A term may include a number, a variable, or both. The variable may have an exponent.
• Operator – a symbol that represents a mathematical process, such as addition, subtraction, multiplication, or division.
• Substitution – the process of replacing a variable with a number or expression.
• Variable Expression – a math expression that contains one or more variables. An expression that does not contain any variables is called a numerical expression.

Notes

• Examples of Variable Phrases & Letters
• Half the height of the building: 1/2 * h
• The number of buildings: b
• The length of the rope: l
• The number of ropes: r
• Half the speed of the car: 1/2 * s
• The number of cars: c
• The number of different colors on the page: c
• The number of pages: p

• Writing Variables Mathematically
1. Change the variable (word) to a letter
2. Decide which math word should be used (see chart below)
• ex. The sum of 35 and the number of carrots: 35 + c
• ex. The sum of cups of coffee and 18 donuts: c + 18
• Difference: Subtract
• ex. The difference of \$25 and the price of a movie ticket: 25 – m
• ex. The difference of pizza slices and the 3 you ate: p – 3
• Product: Multiply
• Write products of variables with numbers, then letters: ex. 3x, 45t, 500m, etc.
• ex. The product of 50 and the number of employees: 50e
• ex. The product of 40 and distance to the finish line: 40d
• Quotient: Divide
• ex. The quotient of \$30 and your siblings: 30 / s
• ex. The quotient of chocolate left and the 4 people who ate it: c / 4

Math Code Words

• sum
• plus
• in all
• are
• altogether / together
• total / in total of
• increased by
• spend / spent
• combined
• joined
• both
• and (between 2 #s)
• more / more than
• how many

Subtraction

• minus
• different / difference
• decreased by
• are not
• left / left over
• less / less than
• from
• fewer
• remain
• subtract
• off
• “er”words
• spent
• how many more
• how many did not have
• how much more
• take away … from
• taller / shorter

Multiplication

• times
• by / multiplied by
• double / triple
• twice
• product / product of
• of / multiple of
• each (before the question)
• every (before the question)
• apiece (before the question)
• per (before the question)

Division

• divide
• quotient
• separate
• equal / equally
• divvy up / cut up
• dish out
• half / third (and other fractions)
• parts
• share
• split
• each (in the question)
• every (in the question)
• apiece (in the question)
• per (in the question)

Examples of Mathematical Phrases

• z increased by 8: z + 8
• 5 less than y: y – 5
• The number of plates on the table minus the 4 saved for dessert: p – 4

Simplify the Numerical Parts of Phrases

2. Rewrite the phrase mathematically
• ex. k + 31 – 11 – 5: add 31 – 11 – 5 following integer rules.
• 31 – 11 = 20.
• 20 – 5 = 15.
• So, k + 15 is the answer.
• ex. j + 16 – 48 + 4: combine 16 and -48 following integer rules.
• 16 – 48 = -32.
• -32 + 4 = -28.
• So, j – 28 is the answer.
• ex. k – 24 / 4: start with 24 / 4.
• 24 / 4 = 6.
• So, k – 6 is the answer.
• ex. 7 * 5 * n: multiply 7 * 5 first.
• 7 * 5 = 35.
• So, 35n (remember that you remove the times sign when you are writing the product of a number and a variable).

## Section 2 (Apex Pg 21 – 28)

Objectives

• Express answers to problems as solution sets.
• Construct a step-by-step strategy for problem solving by asking yourself questions.

Key Terms

• Equation – a mathematical statement that says two expressions are equal.
• Inequality – a mathematical sentence that has two or more expressions separated by inequality signs (<, >, ≤, ≥, or ≠).
• Mathematical Sentence – comparisons of two mathematical expressions. The sentence is called an equation if the two expressions are equal.

Mathematical Sentences MUST have equality or inequality symbols

• ex. The number of apples in the box plus 5 more adds up to 16 apples: m + 5 = 16
• ex. The original price of a computer was reduced by \$250. The new discounted price is \$1499: p – 250 = 1499
• ex. A high school had 140 students at the start of the year, but then some students left the school. After that, the school had 126 students: 140 – s = 126
• ex. Pedro charged \$933 to his credit card. Even after the cost of his next purchase, he was still under his credit limit of \$1500: 933 + c < 1500
• ex. Diana has 235 pokemon cards. If Becca were to double the number of pokemon cards that she has, she would still have fewer than Diana has: 2b < 235
• ex. Ms. Haze’s algebra class has too many students in it. If the class were split exactly in half to form 2 new classes, each class would have no more than 17 students in it: s ÷ 2 ≤ 17
• ex. Cathy checks the temperature at 10:00 p.m. She notices if the temperature falls by 5 more degrees, it will break the record-low temperature for that date, which is 46 degrees: t – 5 < 46