Chapter 6: Expressions and Equations

2017 University of Utah Middle School Math Project, created with the Utah State Office of Education. Licensed under Creative Commons, cc-by.

Table of Contents

Chapter 6: Expressions and Equations

• 6.0 Anchor Problem
• 6.0 Alternative Anchor Problem
• 6.0A The Properties of Arithmetic Reference Sheets

Section 6.1: The Structure of Numeric and Algebraic Expressions

• 6.1a Class Activity: Translating Contexts to Equivalent Numeric Expressions
• 6.1a Homework: Translating Contexts to Equivalent Numeric Expressions
• 6.1b Class Activity: How Many Expressions Can You Make?
• 6.1b Homework: How Many Expressions Can You Make?
• 6.1c Class Activity: Algebraic Expressions and Equivalence
• 6.1c Homework: Algebraic Expressions and Equivalence
• 6.1d Class Activity: Moving from Numeric Expressions to Algebraic Expressions
• 6.1d Homework: Moving from Numeric Expressions to Algebraic Expressions
• 6.1e Self-Assessment: Section 6.1

Section 6.2: Writing, Simplifying, and Evaluating Algebraic Expressions

• 6.2a Class Activity: Simplifying Algebraic Expressions, Part I
• 6.2a Homework: Simplifying Algebraic Expressions, Part I
• 6.2b Class Activity: Numeric Expressions and the Distributive Property
• 6.2b Homework: Numeric Expressions and the Distributive Property
• 6.2c Class Activity: Simplifying Algebraic Expressions, Part II
• 6.2c Homework: Simplifying Algebraic Expressions, Part II
• 6.2d Class Activity: Modeling Backward Distribution (Factoring)
• 6.2d Homework: Modeling Backward Distribution (Factoring)
• 6.2e Class Activity: Repeated Multiplication and Exponents
• 6.2e Homework: Repeated Multiplication and Exponents
• 6.2f Class Activity: Evaluating Algebraic Expressions
• 6.2f Homework: Evaluating Algebraic Expressions
• 6.2g Class Activity: How Many Expressions Can You Make? Part II
• 6.2h Class Activity: Writing Algebraic Expressions to Model Real-World Problems
• 6.2h Homework: Writing Algebraic Expressions to Model Real-World Problems
• 6.2i Self-Assessment: Section 6.2

Section 6.3: Equations and Inequalities in One Variable

• 6.3a Class Activity: Equations and Their Solutions
• 6.3a Homework: Equations and Their Solutions
• 6.3b Class Activity: Working Backward to Solve Equations
• 6.3c Class Activity: Building and Taking Apart Equations
• 6.3c Homework: Building and Taking Apart Equations
• 6.3d Class Activity: Solving Equations with Whole Numbers
• 6.3d Homework: Solving Equations with Whole Numbers
• 6.3e Class Activity: Solving Equations with Rational Numbers
• 6.3e Homework: Solving Equations with Rational Numbers
• 6.3f Class Activity: Writing Equations to Solve Real-World Problems
• 6.3f Homework: Writing Equations to Solve Real-World Problems
• 6.3g Class Activity: Solving Percent Problems with Equations
• 6.3g Homework: Solving Percent Problems with Equations
• 6.3h Class Activity: Understanding the Solution to an Inequality
• 6.3h Homework: Understanding the Solution to an Inequality
• 6.3i Class Activity: Solving Inequalities
• 6.3i Homework: Solving Inequalities
• 6.3j Class Activity: Writing and Solving Inequalities to Represent Real-World Problems
• 6.3j Homework: Writing and Solving Inequalities to Represent Real-World Problems
• 6.3k Class Activity Self-Assessment: Section 6.3

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What You Will Learn in This Chapter

By the end of this chapter, you should be able to do the following things.

Working with factors and multiples
Find the greatest common factor (the largest number that divides evenly into two numbers) of two whole numbers up to 100. Find the least common multiple (the smallest number that both numbers divide into evenly) of two whole numbers up to 12. Use the distributive property — a rule that lets you break a multiplication problem into smaller, easier parts — to rewrite the sum of two whole numbers (from 1 to 100) that share a common factor as that factor multiplied by a new sum. For example, 36 + 8 can be rewritten as 4 × (9 + 2).

Working with exponents
Write and solve numerical expressions that include whole-number exponents (a shortcut for repeated multiplication, like 2³ meaning 2 × 2 × 2).

Writing and understanding expressions with letters
Write, read, and solve expressions where a letter stands in for a number.

• Write expressions that show operations using numbers and letters. For example, "Subtract y from 5" becomes 5 – y.
• Name the different parts of an expression using math vocabulary such as sum, term, product, factor, quotient, and coefficient. You should also be able to treat one part of an expression as a single unit. For example, in 2(8 + 7), you can describe the whole thing as a product of two factors, while also recognizing that (8 + 7) is itself a sum of two terms.
• Solve expressions by plugging in specific numbers for the variables (letters). This includes expressions used in real-world formulas, and expressions with exponents. When there are no parentheses telling you what to do first, follow the order of operations — the agreed-upon sequence for solving a math expression. For example, use the formulas for volume (V = s³) and surface area (A = 6s²) to find the volume and surface area of a cube with a side length of s = 1/2.

Rewriting expressions using math properties
Use the properties of operations (rules like the distributive property) to create equivalent expressions — expressions that always equal the same value even though they look different. For example, the distributive property turns 3(2 + x) into 6 + 3x. It can also turn 24x + 18y into 6(4x + 3y). Similarly, combining repeated terms like y + y + y gives the equivalent expression 3y.

Recognizing equivalent expressions
Tell when two expressions are equivalent — meaning they give the same result no matter what number you substitute in. For example, y + y + y and 3y are equivalent because they always equal the same number, whatever value y stands for.

Solving equations and inequalities
Understand that solving an equation or inequality means answering this question: which values (if any) make the statement true? Use substitution — plugging a number in to check — to test whether a given number makes an equation or inequality true.

Using variables to model problems
Use letters to stand for unknown numbers, and write expressions to help solve real-world or math problems. A variable might represent one specific unknown number, or it might represent any number from a certain group of numbers, depending on the situation.

Solving real-world equations
Write and solve equations shaped like x + p = q or px = q to solve real-world and math problems, where p, q, and x are all nonnegative rational numbers (numbers that can be written as fractions, including whole numbers and decimals, that are zero or greater).

Writing and understanding inequalities
Write an inequality — a statement comparing two values using symbols like > or < — in the form x > c or x < c to describe a limit or condition in a real-world or math problem. Understand that inequalities like these have infinitely many possible solutions, and be able to show those solutions on a number line.

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Key Vocabulary for This Chapter

As you work through this chapter, you'll use the following math terms: numeric expression, equivalent numeric expressions, simplify, order of operations, grouping symbols (such as parentheses, brackets, and the fraction bar), the middle dot (∙) used to show multiplication, and the fraction bar as a way to show division (for example, a/b can be read as a ÷ b).

You'll also work with: algebraic expressions, equivalent algebraic expressions, evaluate (which means to find the value of), sum, difference, product, quotient, the simplified form of an algebraic expression, term, like terms, coefficient, constant, unknown, variable, and the Commutative Property of Addition and Multiplication.