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.
In topic A, students write, interpret, and evaluate numerical expressions. Students learn that exponents represent repeated multiplication, and they evaluate powers with whole number, fraction, and decimal bases. Through an exploration of the hierarchy of operations, students learn the conventional order of operations and then apply it to evaluate numerical expressions.
Students transition from numerical expressions to algebraic expressions in topic B as they learn why and how to use variables to represent unknown numbers and quantities. Students learn to precisely define variables as values with units and to identify constraints on variables. When representing mathematical and real-world descriptions with algebraic expressions, students use common conventions and recognize how subtle variations in wording indicate different meanings.
In topic C, students write and identify equivalent algebraic expressions. Students use the distributive property to write products as sums or differences, to factor algebraic expressions, and to combine like terms.
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.
In topic E, students write and graph two-variable equations and identify independent and dependent variables. After representing ratio relationships with graphs and two-variable equations, students transition to representing relationships of the form
y = x + b and y = x - b.
Students interpret the meanings of points on graphs and also interpret the meanings of coefficients, variables, operators, and constants in equations that represent real-world situations. At the end of the topic, students analyze and model relationships between two quantities when they complete a modeling task about climbing the steps of the Statue of Liberty.