Lesson 5

In this warm-up, students determine how to rearrange an original equation into 3 different equivalent equations. The purpose is to remind students of moves allowed when rearranging equations. This will be helpful in this lesson when students rearrange other linear equations and in the associated Algebra 1 lesson when students rearrange quadratic equations.

The purpose of the discussion is to recall some of the moves for rearranging equations to create new equations. Select students to share their reasoning for why the equations are equivalent. To continue the discussion, ask students:

• “What are some other equivalent equations we could get by rearranging this one? Explain how you know they are equivalent to the original equation.” (\(7x - 102 = 8\) since it is the first solution with the constant terms combined and that solution is equivalent to the original equation. There are many other reasonable answers.)
• “Two of the equations given have 0 on one side of the equation. Why might it be useful to have an equation in this form?” (In this form, we can set the other side of the equation equal to \(y\), then graph it to find the solutions as the \(x\)-coordinates of the \(x\)-intercepts.)
• “Andre rearranges the equation to \(7x = 110\). Is this equation equivalent to the original? If not, find an equivalent equation that has a similar form. Then, explain why this form might be useful.” (It is equivalent to the original equation. This could be useful for solving the equation. After this, dividing both sides of the equation by 7 would solve for \(x\).)

In this activity, students examine work for common mistakes when solving equations. In the associated Algebra 1 lesson, students solve quadratic equations and must be mindful of possible mistakes in their work. Students construct viable arguments and critique the reasoning of others (MP3) when they identify the errors and explain their reasons to agree or disagree.

The purpose of the discussion is to highlight some common errors students may have while solving equations. Select students to share their solutions and reasoning. Ask students,

• “What other common errors do you have to be careful to avoid when solving equations?” (I sometimes forget to expand something like \((x+1)^2\) and accidentally write \(x^2 + 1\).)

In this activity, students rewrite equations into equivalent versions with a given property. To transform the equations, students will need to be able to distribute binomials and factor quadratics. Students work in pairs and each partner is responsible for answering the questions in either column A or column B. Although each row has two different problems, they share the same answer. Ensure that students work their problems out independently and collaborate with one another when they do not arrive at the same answers. Students construct viable arguments and critique the reasoning of others (MP3) when they resolve errors by critiquing their partner’s work or explaining their reasoning.

Arrange students in groups of 2. In each group, ask students to decide who will work on column A and who will work on column B.

The purpose of the discussion is to recognize different ways to rewrite equations. Ask students what they notice about the starting equations and final, rewritten equations. Students may notice:

• a lot of the final forms end with one side equal to zero.
• distributing can create a form that can be easier to compare to other forms.
• multiplying an equation by a denominator (as long as it is not zero) helps simplify the equation by removing the fractions.