Chapter 6: Expressions and Equations

2017 University of Utah Middle School Math Project, created in partnership with the Utah State Office of Education. Licensed under Creative Commons, CC-BY.

Table of Contents

Chapter 6: Expressions and Equations ......................................................................... 3

• 6.0 Anchor Problem ......................................................................................................... 9
• 6.0 Alternative Anchor Problem ..................................................................................... 11
• 6.0a The Properties of Arithmetic Reference Sheets ..................................................... 13

Section 6.1: The Structure of Numeric and Algebraic Expressions ......................... 18

• 6.1a Class Activity: Translating Contexts to Equivalent Numeric Expressions ............. 19
• 6.1a Homework: Translating Contexts to Equivalent Numeric Expressions .................. 27
• 6.1b Class Activity: How Many Expressions Can You Make? ....................................... 32
• 6.1b Homework: How Many Expressions Can You Make? ............................................ 35
• 6.1c Class Activity: Algebraic Expressions and Equivalence ......................................... 36
• 6.1c Homework: Algebraic Expressions and Equivalence .............................................. 46
• 6.1d Class Activity: Transitioning from Numeric Expressions to Algebraic Expressions ... 54
• 6.1d Homework: Transitioning from Numeric Expressions to Algebraic Expressions ... 59
• 6.1e Self-Assessment: Section 6.1 .................................................................................. 62

Section 6.2: Writing, Simplifying, and Evaluating Algebraic Expressions .............. 69

• 6.2a Class Activity: Simplifying Algebraic Expressions, Part I ...................................... 70
• 6.2a Homework: Simplifying Algebraic Expressions, Part I .......................................... 79
• 6.2b Class Activity: Numeric Expressions and the Distributive Property ...................... 82
• 6.2b Homework: Numeric Expressions and the Distributive Property .......................... 88
• 6.2c Class Activity: Simplifying Algebraic Expressions, Part II ..................................... 91
• 6.2c Homework: Simplifying Algebraic Expressions, Part II ......................................... 98
• 6.2d Class Activity: Modeling Backwards Distribution (Factoring) ............................. 102
• 6.2d Homework: Modeling Backwards Distribution (Factoring) .................................. 104
• 6.2e Class Activity: Repeated Multiplication and Exponents ....................................... 106
• 6.2e Homework: Repeated Multiplication and Exponents ............................................ 115
• 6.2f Class Activity: Evaluating Algebraic Expressions ................................................. 118
• 6.2f Homework: Evaluating Algebraic Expressions ...................................................... 123
• 6.2g Class Activity: How Many Expressions Can You Make, Part II ........................... 126
• 6.2h Class Activity: Writing Algebraic Expressions to Model Real-World Problems ... 127
• 6.2h Homework: Writing Algebraic Expressions to Model Real-World Problems ....... 133
• 6.2i Self-Assessment: Section 6.2 ................................................................................. 136

Section 6.3: Equations and Inequalities in One Variable ......................................... 144

• 6.3a Class Activity: Equations and Their Solutions ...................................................... 145
• 6.3a Homework: Equations and Their Solutions ........................................................... 148
• 6.3b Class Activity: Working Backwards to Solve Equations ....................................... 151
• 6.3c Class Activity: Constructing and Deconstructing Equations ................................. 154
• 6.3c Homework: Constructing and Deconstructing Equations ....................................... 158
• 6.3d Class Activity: Solving Equations with Whole Numbers ....................................... 161
• 6.3d Homework: Solving Equations with Whole Numbers ............................................ 166
• 6.3e Class Activity: Solving Equations with Rational Numbers .................................... 167
• 6.3e Homework: Solving Equations with Rational Numbers ......................................... 171
• 6.3f Class Activity: Writing Equations to Solve Real-World Problems ........................ 172
• 6.3f Homework: Writing Equations to Solve Real-World Problems ............................. 179
• 6.3g Class Activity: Solving Percent Problems with Equations .................................... 182
• 6.3g Homework: Solving Percent Problems with Equations .......................................... 187
• 6.3h Class Activity: Understanding the Solution to an Inequality ................................ 189
• 6.3h Homework: Understanding the Solution to an Inequality ..................................... 194
• 6.3i Class Activity: Solving Inequalities ........................................................................ 197
• 6.3i Homework: Solving Inequalities ............................................................................. 200
• 6.3j Class Activity: Writing and Solving Inequalities to Represent Real-World Problems ... 201
• 6.3j Homework: Writing and Solving Inequalities to Represent Real-World Problems ... 206
• 6.3k Class Activity: Self-Assessment: Section 6.3 ........................................................ 207

---

Chapter 6: Expressions and Equations

Utah Core Standards

This chapter is built around the following learning standards:

• Factors and multiples: Find the greatest common factor (the largest number that divides evenly into two numbers) of two whole numbers up to 100, and the least common multiple (the smallest number that both numbers divide evenly into) of two whole numbers up to 12. Use the distributive property—the rule that lets you "break apart" a multiplication problem, such as writing a(b + c) as ab + ac—to rewrite the sum of two whole numbers between 1 and 100 that share a common factor as a multiple of a sum with no common factor. For example, 36 + 8 can be rewritten as 4(9 + 2). (Standard 6.NS.4)

• Exponents: Write and evaluate numerical expressions that include whole-number exponents (a shorthand for repeated multiplication, such as writing 2 × 2 × 2 as 2³). (Standard 6.EE.1)

• Variable expressions: Write, read, and evaluate expressions in which letters stand for numbers. (Standard 6.EE.2)
• a) Write expressions that represent operations using numbers and letters that stand for numbers. For example, "subtract y from 5" becomes 5 – y.
• b) Identify the parts of an expression using correct math vocabulary—sum, term, product, factor, quotient, and coefficient (the number multiplied by a variable)—and recognize that a group of terms can be treated as a single unit. For example, in the expression 2(8 + 7), the whole thing is a product of two factors, while (8 + 7) can be seen both as one single quantity and as a sum of two terms.
• c) Evaluate expressions by substituting specific numbers for the variables, including expressions drawn from real-world formulas. When there are no parentheses to show a different order, perform operations—including those with whole-number exponents—in the standard order of operations. 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 side length s = ½.

• Equivalent expressions: Apply the properties of operations to rewrite expressions in different but equal forms. 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 turns y + y + y into 3y. (Standard 6.EE.3)

• Recognizing equivalence: Determine when two expressions are equivalent—that is, when they produce the same value no matter what number is substituted in. For example, y + y + y and 3y are equivalent because they always give the same result, regardless of the value of y. (Standard 6.EE.4)

• Solving equations and inequalities: Understand that solving an equation or inequality means figuring out which values from a given set (if any) make the statement true. Use substitution—plugging in a number to check—to test whether a specific number makes an equation or inequality true. (Standard 6.EE.5)

• Using variables to model problems: Use variables to represent numbers when writing expressions for real-world or mathematical problems. Understand that a variable can stand for one specific unknown number or, depending on the situation, any number from a particular set. (Standard 6.EE.6)

• Solving real-world equations: Solve real-world and mathematical problems by writing and solving equations in the form x + p = q or px = q, where p, q, and x are all nonnegative rational numbers (numbers that can be written as fractions, including whole numbers and decimals). (Standard 6.EE.7)

• Writing and graphing inequalities: Write an inequality in the form x > c or x < c to represent a limit or condition in a real-world or mathematical problem. Recognize that these inequalities have infinitely many solutions, and represent those solutions on a number line. (Standard 6.EE.8)

Academic Vocabulary

Key terms used throughout this chapter include: numeric expression, equivalent numeric expressions, simplify, order of operations, grouping symbols (parentheses, brackets, and the fraction bar), the middle dot (∙) used for multiplication, the fraction bar as a symbol for division (so a/b can be read as a ÷ b), algebraic expressions, equivalent algebraic expressions, evaluate, 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.