Unit 4, Lesson 1
Writing Equivalent Equations
Grade 8
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In this activity, students create equivalent expressions to given expressions. Students are encouraged to be creative in their answers, but should ensure that the new expressions are equivalent to the original. This work prepares students for writing equivalent equations in later activities.
Arrange students in groups of 2.
Display for all to see:
For the task, encourage students to be creative, but check with their partner that their expressions are still equivalent to the original. Tell students to try to give examples that they think nobody else in the class has written.
Write as many equivalent expressions for each as you have time.
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Invite 3–5 students to share examples that they have written for each expression. After each example, ask if other students can explain how it is equivalent to the original.
MLR2: Collect and Display
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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.
Arrange students in groups of 2.
Display this equation for all to see:
Tell students that equivalent equations are equations that have the same solution. For example,
Ask students, “What is another way you could write an equation equivalent to
Ask students to think of another equation that is equivalent to the new equation. After a brief quiet think time, select a student to share their answer and description of the move. Record the response for all to see with arrows connecting the equations describing the move.
Tell students that it is important to explain their reasoning for writing equivalent equations and that we will begin by using the arrows.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to describe the moves. Display words and phrases such as “to each side,” “distributive property,” and “do the same thing on both sides.”
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.
MLR2: Collect and Display
to see where instructional routines fit in your journey.
None
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.
Begin the activity by telling students not to communicate with one another until instructed to do so. This includes talking or showing each other what is on their papers.
Tell students to select a number and write it on the first line. Then follow the instructions to write the correct number on the next line. Continue filling in the blanks until all have been filled.
After all students have had a chance to complete all of the instructions, tell students to compare the number on the last blank with one another.
Ask students, “What do you notice about the final answers?” (Everyone got 3, regardless of their starting number.)
Demonstrate how to begin the next part of the activity. Write "
To see why this works, tell students to write "
Select students who combine like terms as they follow the instructions, as well as those who do not. Ask them to share later.
Use Collect and Display to direct attention to words collected and displayed from an earlier activity. Ask students to suggest ways to update the display: “Are there any new words or phrases that you would like to add?” “Is there any language you would like to revise or remove?” Encourage students to use the display as a reference.
Invite selected students to share their equations involving
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.
Display the partially completed set of equivalent equations.
Ask students,
Then display this pair of equations.
Ask students,
Equations are equivalent if values for the variables that make one equation true also make the other equation true. One way to create equivalent equations is to correctly use valid moves.
Valid moves include:
For example, all of these equations are equivalent:
For these equations, the valid moves are used correctly, so all of the equations are equivalent. The last equation shows that 3 is the value for
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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
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