In Uint 7, Expressions, students extend their arithmetic work to include using letters to represent numbers in order to understand that letters are simply "stand-ins" for numbers and that arithmetic is carried out exactly as it is with numbers. Students explore operations in terms of verbal expressions and determine that arithmetic properties hold true with expressions because nothing has changed—they are still doing arithmetic with numbers. Students determine that letters are used to represent specific but unknown numbers and are used to make statements or identities that are true for all numbers or a range of numbers. They understand the relationships of operations and use them to generate equivalent expressions, ultimately extending arithmetic properties from manipulating numbers to manipulating expressions. Students read, write and evaluate expressions in order to develop and evaluate formulas. From there, they move to the study of true and false number sentences, where students conclude that solving an equation is the process of determining the number(s) that, when substituted for the variable, result in a true sentence. They conclude the unit using arithmetic properties, identities, bar models, and finally algebra to solve one-step, two-step, and multi-step equations.
6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for..
Topic C: Replacing Letters and Numbers (6.EE.A.2c, 6.EE.A.4)
Lesson 7: Replacing Letters with Numbers
Lesson 8: Replacing Numbers with Letters
Topic D: Expanding, Factoring, and Distributing Expressions (6.EE.A.2a, 6.EE.A.2b, 6.EE.A.3, 6.EE.A.4)
Lesson 9: Writing Addition and Subtraction Expressions
Lesson 10: Writing and Expanding Multiplication Expressions
Lesson 11: Factoring Expressions
Lesson 12: Distributing Expressions
Lessons 13–14: Writing Division Expressions
Topic E: Expressing Operations in Algebraic Form (6.EE.A.2a, 6.EE.A.2b)
Lesson 15: Read Expressions in Which Letters Stand for Numbers
Lessons 16–17: Write Expressions in Which Letters Stand for Numbers
Topic F: Writing and Evaluating Expressions and Formulas (6.EE.A.2a, 6.EE.A.2c, 6.EE.B.6)
Lesson 18: Writing and Evaluating Expressions—Addition and Subtraction
Lesson 19: Substituting to Evaluate Addition and Subtraction Expressions
Lesson 20: Writing and Evaluating Expressions—Multiplication and Division
Lesson 21: Writing and Evaluating Expressions—Multiplication and Addition
Lesson 22: Writing and Evaluating Expressions—Exponents
Equation (An equation is a statement of equality between two expressions.)
Equivalent Expressions (Two expressions are equivalent if both expressions evaluate to the same number for every substitution of numbers into all the variables in both expressions.)
Exponential Notation for Whole Number Exponents (Let 𝑚 be a nonzero whole number. For any number 𝑎, the expression 𝑎𝑚 is the product of 𝑚 factors of 𝑎 (i.e., 𝑎𝑚 = 𝑎 ∙𝑎 ∙ ⋅⋅⋅ ∙𝑎 ⏟ 𝑚 times ). The number 𝑎 is called the base, and 𝑚 is called the exponent or power of 𝑎.)
Expression (An expression is a numerical expression, or it is the result of replacing some (or all) of the numbers in a numerical expression with variables.)
Linear Expression (A linear expression is an expression that is equivalent to the sum/difference of one or more expressions where each expression is either a number, a variable, or a product of a number and a variable.)
Number Sentence (A number sentence is a statement of equality between two numerical expressions.)
Numerical Expression (A numerical expression is a number, or it is any combination of sums, differences, products, or divisions of numbers that evaluates to a number.)
Solution of an Equation (A solution to an equation with one variable is a number such that the number sentence resulting from substituting the number for all instances of the variable in both expressions is a true number sentence. If an equation has more than one variable, then a solution is an ordered tuple of numbers such that the number sentence resulting from substituting each number from the tuple into all instances of its corresponding variable is a true number sentence.)
Truth Values of a Number Sentence (A number sentence is said to be true if both numerical expressions evaluate to the same number; it is said to be false otherwise. True and false are called truth values.)
Value of a Numerical Expression (The value of a numerical expression is the number found by evaluating the expression.)
Variable (A variable is a symbol (such as a letter) that is a placeholder for a number.)