In this activity, students first concentrate on identifying and informally recording moves used to create equivalent equations. They are introduced to recording the moves by using arrows that connect each equation to the next and labeling the arrows that show the moves. A distinction is made between moves that change how an expression on one side of the equation looks (distributive property, combine like terms) and moves that change each side of the equation (adding the same value to each side, multiplying the same value to each side). In the second problem, students practice making their own sequence of moves, starting from a simple equation, and recording their moves using the labeled arrows.

Students have solved these types of equations in previous grades. In this activity, students are encouraged to be more playful and to write equivalent equations that are not necessarily moving toward a goal of finding a solution.

Students may write “factor” as the reasoning for the last step of the first question. This is okay, but remind students that the distributive property is also a valid description of what has happened and can be used to rewrite as or the other way around.

The goal of this discussion is to begin creating a shared language and way of writing descriptions of moves for writing equivalent equations.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their labels for the arrows in the first question. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond.

Highlight the different ways in which students describe the moves going to the final row. Make sure students understand that when one side of the equation is changing how it looks—such as when using the distributive property or combining like terms—there may not be an equivalent move on the other side of the equation.

Establish a good convention to use with the class. In these materials, we are leaving the arrows blank to indicate that nothing has changed on that side of the equation. This can look unfinished, though. Encourage students to write “copy” or “no change” or use a similar phrase or indicator on the side of the equation opposite one of the moves that changes the look of an expression.

Invite 2–3 groups to share their sequence of equivalent equations for the last question. Display their equations for all to see. Ask other groups to provide the reasoning to label the arrows for each equivalent expression.

In this activity, students select a number and follow a series of instructions to change their number into a surprising result. Students then follow the same series of instructions for a variable to reveal how the result was not surprising at all, but a natural consequence of the instructions. This helps illustrate valid moves that can be done to an equation.

Monitor for students who combine like terms as they go as well as those who do not. 

• “How can you check that each of your equations are equivalent?”
• “How could you use substituting a value or to help?”

Invite selected students to share their equations involving . If all students combine like terms, ask them what each equation would look like if they did not. For example, on the third line, write as well as . Ask students, “What are the benefits and drawbacks of writing it each way?” (The first set of expressions show all of the instructions and how they affect , but there are more terms, so an instruction like “divide by 2” requires more work. The second set of expressions hides the instructions, but is easier to understand how to get the number in the blank.)

Tell students that all of these equations are equivalent because these are valid moves done correctly. Ask students what moves they have seen that create equivalent equations. Direct students’ attention to the reference created using Collect and Display. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. Update the display to create a semi-permanent display of these valid moves to remain available in the classroom until the end of the unit.

• Use the distributive property. ( is equivalent to )
• Combine like terms.  ( is equivalent to )
• Add the same value on each side.
• Subtract the same value on each side.
• Multiply each side by the same non-zero value.
• Divide each side by the same non-zero value.