In this section, the key understandings are that an equation with a variable represents a relationship that can be true or false, and a solution to the equation is a value for the variable that makes the equation true. Relationships are limited to those that can be expressed with or .

First, students revisit tape diagrams and equations as tools for representing relationships and learn about variables and solutions. To find solutions to equations, students reason about the numbers and relationships—with or without tape diagrams, and use substitution to determine if a given value makes the equation true. They also build on their understanding that equations are not always true.

Next, hanger diagrams are used to help students understand the concept of keeping equations balanced. Students use repeated calculations and make use of structure to generalize an algebraic method for solving equations of the form and . They then choose the best strategies to solve equations of that form as well as real-world problems that can be represented by such equations. In the last lesson of the section, students discover that equations of the form can be used to represent and solve percentage problems.

In this section, students apply their understanding of algebraic expressions and equations to represent two quantities that change in relationship to one another. Students explore these relationships through multiple representations, such as tables, graphs, and equations. They also delve into the concepts of independent and dependent variables.

First, students create and analyze tables and graphs that represent how one quantity affects another. The choice of independent and dependent variables is driven by the context or the modeler. For example, when mixing two colors of paint, the independent variable could be either color depending on what we know and want to know.

Next, the context of traveling at a constant rate is used to create representations of the relationship between time and distance. Students explore how switching independent and dependent variables affects the different representations, as shown in the following graphs.

Finally, an optional lesson invites students to explore additional contexts in which two quantities vary together, such as in the relationship of side lengths of a rectangle with a constant area and in a doubling relationship, and allows for student choice in representation methods.

A graph of 10 points plotted in the coordinate plane with the origin labeled "O". The horizontal t axis is labeled "time in hours". The numbers 0 through 10, in increments of 2, are indicated, and there are vertical gridlines midway between. The vertical axis is labeled "distance traveled in miles". The numbers 0 through 250, in increments of 25, are indicated, and there are horizontal gridlines midway between. The data are as follows: 1 comma 25. 2 comma 50. 3 comma 75. 4 comma 100. 5 comma 125. 6 comma 150. 7 comma 175. 8 comma 200. 9 comma 225. 10 comma 250.

In this section, students focus on writing expressions and equivalent expressions. This is also where the distributive property is formally introduced.

First, students write expressions that represent situations, using tape diagrams for reasoning as needed. Then equivalent expressions are introduced. Students use tape diagrams to identify when expressions are equal. For example, this diagram shows that is not equal to when . Changing the length of the rectangle to 1, however, would create two rectangles of the same length, so the expressions are equal when .

Two tape diagrams on a grid.  Top diagram 2 parts, x, 2. The part labeled x is composed of four unit squares.  The part labeled 2 is composed of two unit squares.  Bottom diagram 3 parts x,x,x, each part is composed of four unit squares.

Students learn that equivalent expressions with variables are always equal when the same value is used for the variable in each expression. They use properties of operations to identify and write equivalent expressions. For example, is equivalent to because of the commutative property of addition. These two expressions are still equal for every value of their variable despite the addends switching order.

In prior grades, the names of the properties were not emphasized. In this grade, the names of the properties are used, and students can use them to justify equivalent expressions. However, students may continue to use more informal explanations that demonstrate an understanding of the properties.

In the latter half of the section, students explore the distributive property. They use rectangular diagrams to first represent equivalent numerical expressions like and , and then to reason about equivalent expressions with variables. For instance, the following diagram shows that is equivalent to because both expressions represent the area of the same shaded region.

The final optional lesson offers additional reinforcement in using the distributive property to identify and write equivalent expressions.

Area diagram of two attached rectangles, one with a shaded area. The height of the rectangle is 6 and has a total width of m. Smaller attached rectangle shares the height of 6 and has a width of 2.

In this section, students extend their understanding of exponential expressions. They learn to evaluate expressions that include both variables and exponents.

Students first explore perfect squares as areas of squares and perfect cubes as volumes of cubes with whole-number edge lengths. Next, they learn that the exponents ² and ³ can be used to express the multiplication of edge lengths of these figures. For example, the area of a square with a side length of 5 units is square units and the volume of a cube with an edge length of 5 units is cubic units.

The exponents ² and ³ can also be used to express square units and cubic units. The area of a square with 5-inch sides is inch² and the volume of a cube with 5-cm edges is cm³. Students practice using and interpreting this new notation in the context of squares and cubes.

​​​Then students work with whole-number exponents beyond 2 and 3. First, students use exponent notation to represent repeated multiplication in patterns and situations. Next, they use prior knowledge about operations to practice evaluating exponential expressions with various bases, including whole numbers, fractions, and decimals.

Next, students evaluate expressions that include exponents and another operation. A real-world example involving surface area of a cube helps students see a rationale for following the order of operations.

Then students practice following the conventional order for evaluating expressions with and without grouping symbols. Finally, students encounter exponential expressions with variables and evaluate such expressions at specific values of the variables.