Chapter 6: Expressions and Equations

2017 University of Utah Middle School Math Project, created with the Utah State Office of Education. Shared 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: Turning Word Problems into Equal Number Expressions
• 6.1a Homework: Turning Word Problems into Equal Number Expressions
• 6.1b Class Activity: How Many Expressions Can You Make?
• 6.1b Homework: How Many Expressions Can You Make?
• 6.1c Class Activity: Expressions with Letters (Algebraic Expressions) and Equal Expressions
• 6.1c Homework: Expressions with Letters (Algebraic Expressions) and Equal Expressions
• 6.1d Class Activity: Moving from Number Expressions to Letter Expressions
• 6.1d Homework: Moving from Number Expressions to Letter Expressions
• 6.1e Self-Check: Section 6.1

Section 6.2: Writing, Simplifying, and Solving Algebraic Expressions

• 6.2a Class Activity: Simplifying Letter Expressions, Part 1
• 6.2a Homework: Simplifying Letter Expressions, Part 1
• 6.2b Class Activity: Number Expressions and the Distributive Property (a rule for breaking apart multiplication)
• 6.2b Homework: Number Expressions and the Distributive Property
• 6.2c Class Activity: Simplifying Letter Expressions, Part 2
• 6.2c Homework: Simplifying Letter Expressions, Part 2
• 6.2d Class Activity: Working Backwards with the Distributive Property (Factoring)
• 6.2d Homework: Working Backwards with the Distributive Property (Factoring)
• 6.2e Class Activity: Repeated Multiplication and Exponents (a short way to show repeated multiplying)
• 6.2e Homework: Repeated Multiplication and Exponents
• 6.2f Class Activity: Solving Letter Expressions for a Value
• 6.2f Homework: Solving Letter Expressions for a Value
• 6.2g Class Activity: How Many Expressions Can You Make? Part 2
• 6.2h Class Activity: Writing Letter Expressions for Real-Life Problems
• 6.2h Homework: Writing Letter Expressions for Real-Life Problems
• 6.2i Self-Check: Section 6.2

Section 6.3: Equations and Inequalities with One Unknown

• 6.3a Class Activity: Equations and Their Answers
• 6.3a Homework: Equations and Their Answers
• 6.3b Class Activity: Working Backwards to Solve Equations
• 6.3c Class Activity: Building and Breaking Down Equations
• 6.3c Homework: Building and Breaking Down 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 Fractions and Decimals
• 6.3e Homework: Solving Equations with Fractions and Decimals
• 6.3f Class Activity: Writing Equations to Solve Real-Life Problems
• 6.3f Homework: Writing Equations to Solve Real-Life Problems
• 6.3g Class Activity: Solving Percent Problems with Equations
• 6.3g Homework: Solving Percent Problems with Equations
• 6.3h Class Activity: Understanding the Answer to an Inequality (a math sentence with more-than or less-than)
• 6.3h Homework: Understanding the Answer to an Inequality
• 6.3i Class Activity: Solving Inequalities
• 6.3i Homework: Solving Inequalities
• 6.3j Class Activity: Writing and Solving Inequalities for Real-Life Problems
• 6.3j Homework: Writing and Solving Inequalities for Real-Life Problems
• 6.3k Class Activity: Self-Check: Section 6.3

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What You Will Learn in This Chapter (Utah Core Standards)

• Find the greatest common factor (the largest number that divides evenly into two numbers) for two whole numbers up to 100. Find the least common multiple (the smallest number that both numbers divide into evenly) for two whole numbers up to 12. Use the distributive property to rewrite a sum, like 36 + 8, as a multiple of a smaller sum, like 4 × (9 + 2).

• Write and solve number expressions that use exponents (a short way to show a number multiplied by itself several times).

• Write, read, and solve expressions where letters stand for numbers.
• Write expressions that show math operations using numbers and letters. For example, "Subtract y from 5" becomes 5 – y.
• Name the different parts of an expression using math words like sum, term, product, factor, quotient, and coefficient. Sometimes one part of an expression can be treated as a single piece. For example, in 2 × (8 + 7), you can think of (8 + 7) as one single amount, or as two terms added together.
• Solve expressions when you know what number each letter stands for. This includes expressions from real-life formulas. Follow the correct order of operations (the order in which you do each step of a math problem) when there are no parentheses telling you what to do first. 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 1/2.

• Use math rules to rewrite expressions in different but equal ways. For example, the distributive property changes 3 × (2 + x) into 6 + 3x. It also changes 24x + 18y into 6 × (4x + 3y). Other rules can change y + y + y into 3y.

• Figure out when two expressions are equivalent — meaning they always equal the same number, no matter what number you use for the letter. For example, y + y + y and 3y are equivalent because they always come out equal, no matter what y stands for.

• Understand that solving an equation or inequality means answering a question: which numbers make it true? Try out numbers to see if they make an equation or inequality true.

• Use letters to stand for numbers, and write expressions to solve real-life or math problems. A letter can stand for one unknown number, or for any number in a group of numbers, depending on the problem.

• Solve real-life and math problems by writing and solving equations that look like x + p = q or px = q, where p, q, and x are all positive numbers (including fractions and decimals).

• Write inequalities that look like x > c or x < c to show a real-life or math rule or limit. Understand that these inequalities can have endless numbers of solutions, and show those solutions on a number line.

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Important Words to Know

number expression, equal number expressions, simplify (make simpler), order of operations, grouping symbols (like parentheses, brackets, and fraction bars), the middle dot (∙) used for multiplication, fraction bar as division (a/b means a ÷ b), letter expressions (algebraic expressions), equal letter expressions, solve, sum, difference, product, quotient, simplest form of a letter expression, term, like terms, coefficient (the number in front of a letter), constant (a number that doesn't change), unknown, variable (a letter that stands for a number), and the Commutative Property of Addition and Multiplication (a rule saying you can add or multiply numbers in any order and get the same result).