Lesson 6

The purpose of this activity is for students to begin building linear equations and solving them.

Arrange students in groups of 2. Give students 2 minutes quiet think time, then 2 minutes to discuss their solutions with a partner. Instruct groups to explain to each other how they came up with expressions and an equation to represent the situation.

Ask groups to share their strategies for solving the question. Consider asking some of the following questions:

• “What expression represents the perimeter of the triangle? The perimeter of the square?” (The expression for perimeter of the triangle is \(5x-8\), and the perimeter of the square is \(4(x+2)\).)
• “What was your strategy in making an equation?” (If both perimeters are the same, we can say their expressions are equal.)
• “What does \(x\) mean in the situation?” (It means an unknown value. None of the sides or perimeter is represented by \(x\), so we cannot say it represents a specific thing on the figures.)
• “Looking at the figures, are there any values that \(x\) could not be? Explain your reasoning.” (Since the triangles have sides that are \(2x\), \(x\) cannot be 0 or a negative value. Triangles cannot have sides with 0 or negative side lengths. Since the third side is \(x-8\), we can use this same reasoning to realize that \(x\) must actually be greater than 8.)
• “How does this information help when solving?” (If I make a mistake in my solution and get a value of \(x\) that is less than or equal to 8, then I know immediately that my answer is not reasonable and I can try to find my error.)

The purpose of this activity is to shift the focus from solving an equation to thinking about what it means for a number to be a solution of an equation. Students inspect each equation, looking at the structure, the signs, and the operations in it to decide if the solution is positive, negative, or zero. Some questions are paired with another question (for example, the last two questions) so students can take advantage of their thinking from one to the next.

Arrange students in groups of 2.

Display the equation \(5x=6x\) for all to see.

Ask students, "How might we know whether \(x\) is a positive number, negative number, or zero, without solving the equation?" (The variables can be combined into one term, but there are no constant terms. That means eventually the variable term has to equal 0, so \(x\) must be 0.)

Display the equation \(5x=\text-16.5\) for all to see and ask the same question. (Without solving, we can see that a positive number of \(x\)s has to equal a negative value, so \(x\) must be a negative number.)

Instruct students to inspect each equation carefully and use reasoning to answer the questions in the activity rather than trying to solve each equation for a specific value. Give 5 minutes of quiet think time, and then ask students to compare their work with their partner. For any questions they disagree on, students should work to reach an agreement.

For each equation, invite groups to share how they decided if the solution was positive, negative, or zero. After each group shares, ask if any other group reasoned about the problem in a different way and invite them to share their reasoning.

The purpose of this discussion is for students to practice talking about equations, the operations within them, and use logical thinking. There is no need to try and formally generalize student thinking for all cases at this time. In later grades, students will continue the work started here looking for structure in equations.

The purpose of this activity is for students to think about what they see as “least difficult” and “most difficult” when looking at equations and to practice solving equations. Students also discuss strategies for dealing with “difficult” parts of equations.

Keep students in the same groups of 2. Give students 3–5 minutes quiet think time to get started and then 5–8 minutes to discuss and work with their partner. Leave ample time for a whole-class discussion.

The purpose of this discussion is for students to discuss strategies for solving different types of equations. Some students may have thought that an equation was in the “least difficult” category, while others thought that the same equation was in the “most difficult” category. Remind students that once you feel confident about the strategies for solving an equation, it may move into the “least difficult” category, and recognizing good strategies takes practice and time.

Poll the class for each question as to whether they placed it in the “most difficult” category, “least difficult” category, or if it was somewhere in the middle. Record and display the results of the poll for all to see.

For questions with a split vote, have a group share something that was difficult about it and something that made it seem easy. If there are any questions that everyone thought would be more difficult or everyone thought would be less difficult, ask students why it seemed that way. Ask students, “Were there any equations that were more difficult to solve than you expected? Were there any that were less difficult to solve than you expected?”

Consider asking some of the following questions to further the discussion:

• “For equation A, what could we do to eliminate the fraction?” (Multiply each side by the common denominator of 6. Then the terms will all have integer coefficients.)
• “Which other equations could we use this strategy for?” (Any equations that had fractions, such as B, C, E, and G.)
• “What steps do you need to do to solve equation D? Which other equations are like this one?” (There is a lot of distributing and collecting like terms. F and H also have to distribute several times.)
• “What other strategies or steps did you use in solving the equations?”