In module 4, students work with numerical and algebraic expressions and equations. First, they learn that exponents represent repeated multiplication, evaluate powers with whole number, fraction, and decimal bases, and use the order of operations to evaluate numerical expressions. Then, students learn why and how to use variables to represent unknown numbers and quantities. They write and evaluate algebraic expressions and use properties of operations to generate equivalent expressions. Students reason about and solve single-variable, one-step equations, and they understand the meaning of a solution to an equation or inequality. At the end of the module, they revisit ratio relationships and write and graph equations in two variables, identifying independent and dependent variables in real-world situations.
Students transition from working with algebraic expressions to working with equations and inequalities in topic D. They use substitution to determine whether a given number makes an equation or inequality a true number sentence, and they develop an understanding of the meaning of a solution. Students graph solutions to inequalities on number lines and interpret solutions graphed on number lines by writing inequalities. Students solve single-variable equations by using tape diagrams and algebraic reasoning, and they apply their understanding of equations to solve geometric problems involving angle measures.
6.C.6: Apply the order of operations and properties of operations (identity, inverse, commutative properties of addition and multiplication, associative properties of addition and multiplication, and distributive property) to evaluate numerical expressions with nonnegative rational numbers, including those using grouping symbols, such as parentheses, and involving whole number exponents.
6.AF.1: Evaluate expressions for specific values of their variables, including expressions with whole-number exponents and those that arise from formulas used in geometry and other real-world problems.
6.AF.4: Understand that solving an equation or inequality is the process of answering the following question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.AF.5: Solve equations of the form x + p = q, x − p = q, px = q, and x/p = q fluently for cases in which p, q and x are all nonnegative rational numbers. Represent real-world problems using equations of these forms and solve such problems.
6.AF.6: Write an inequality of the form x > c, x ≥ c, x < c, or x ≤ c, where c is a rational number, to represent a constraint or condition in a real-world or other mathematical problem. Recognize inequalities have infinitely many solutions and represent solutions on a number line diagram.
I can...
Determine whether a number sentence is true.
Determine whether a number is a solution to an equation by using substitution.
Lesson at a Glance
Through a Whiteboard Exchange, group discussion, and partner work, students compare expressions and number sentences, write number sentences from descriptions, and determine whether number sentences are true or false. Students add to their prior knowledge of the term equation and learn that an equation is a type of number sentence that can be true or false. By substituting values into an equation, students determine whether a number is a solution to the equation. Finally, students analyze equations that model real-world situations. This lesson introduces the terms equation and solution.
I can...
Represent solutions to inequalities on number lines.
Identify whether a number is a solution to an inequality by using substitution.
Lesson at a Glance
A number line activity helps transition students from using numbers to using variables to write inequalities that represent temperatures in various cities. Next, students graph solutions to inequalities on number lines and explain why inequalities can have infinitely many solutions. By substituting a number into an inequality, students then determine whether the number is a solution to the inequality. At the end of the lesson, students write and interpret inequalities that model real-world situations.
This lesson introduces the terms boundary number and boundary point.
I can...Solve addition and subtraction equations by using tape diagrams and algebraic reasoning.
Lesson at a Glance
Students begin the lesson by reasoning about unknown values for a given situation. To identify equations that match the situation, students substitute values for the variable into addition and subtraction equations and determine which values make true number sentences. Students solve addition and subtraction equations, first by using tape diagrams and then algebraically. In pairs, students analyze work to identify and correct mistakes and use substitution to check their solutions. Students progress to solving addition and subtraction equations that have fractions and decimals.
This lesson introduces the term additive identity.
I can...Solve multiplication and division equations by using tape diagrams and algebraic reasoning.
Lesson at a Glance
Students begin the lesson by reasoning about an unknown number when given a description. Students also write the description by using algebraic notation. Next, students solve multiplication and division equations, first by using tape diagrams and then algebraically. In pairs, students analyze work to identify and correct mistakes and use substitution to check their solutions. Students progress to solving multiplication and division equations involving fractions and decimals.
This lesson introduces the terms multiplicative identity and multiplicative inverse.
I can...Solve problems by writing and solving equations.
Lesson at a Glance
In this lesson, students practice writing and solving one-variable equations in the context of angle relationships. Students write descriptions of familiar angle terms and use a protractor to measure angles. Then, in pairs, they use angle relationships to write and solve equations to find unknown angle measures. Students match equations to real-world situations and analyze solution strategies. Realizing they can make an equation simpler by writing the expression on each side of the equal sign with as few terms as possible, students progress to writing equations from real-world situations and solving them to find unknown values.
This lesson introduces the term linear pair.