We’ll start this lesson having students explain what is being done to the variable in different equations. This will help tie into the idea that when solving equations, we need to isolate the variable in the opposite order that things are being done to it. We’ll review inverse operations before dividing into different examples of two-step equations.
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Students will be able to solve two-step equations.
Hand out the warm up to students and give them a few minutes to work on it independently. You can have students work with a partner if they’re stuck or want to check their answers. If the class needs additional help, try going through the first example together to show the format you’re looking for. An answer key is provided below and in the warm-up document.
You can go over the answers as a class, having students explain how they wrote out the order of operations. The key is for students to recognize that multiplication and division always came before addition and subtraction. Students should already know this from simplifying expressions using the order of operations, but it’s helpful for them to apply that to expressions with variables.
The idea behind the warm-up is getting students to think about the order of operations, so that we can talk about how we “undo” operations being done to the variable in the opposite order. Some teachers like to refer to this as “reverse order of operations”.
Students should already be familiar with inverse operations from solving one-step equations, but give students the opportunity to review with slides `1-4`. First ask students, “what is the inverse operation of addition?”.
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Remind students that if subtraction is the inverse of addition, addition is the inverse of subtraction. It goes both ways! Then ask the inverse of multiplication and allow students to recall that multiplication and division are inverse operations.
Display the first example and ask students to think about the left side of the equation and what is happening to `p`. Explain that we need to “undo” those operations in the opposite order. This will help them to understand that we need to undo the `6` first by subtracting `6` on both sides. Once you simplify, students will quickly recognize this has now become a one-step equation. Divide both sides by `-3` to get `p = -11`.
Using the same strategy in the last example, allow students to determine what needs to be “undone” first. Students should recognize that we need to add `4` on both sides first. Then once we have `a/2 = -4`, we can multiply by `2` on both sides to get `a = -8`. Division examples tend to sometimes give students trouble, so remind students to think about how since `a` is being divided by `2`, we would need to multiply by `2` on both sides as the last step to isolate `a`.
For the next example, the coefficient is a variable. There was a similar expression in the warm up, so students should be able to use the same reasoning to determine that we still need to subtract `18` as the first step. Once we have the equation as `2/3y = 8`, students may get stuck. Ask students if they can remember the inverse of a fraction. Students might recognize that the inverse of a fraction is called its reciprocal. You may need to prompt students by asking, “what can we multiply `2/3` by to cancel it out to `1`?” Once students recall how to find the reciprocal, you can show how we multiply by `3/2` on both sides. That gives us `y = 12`.
This last example may not immediately seem tricky for students, but it is the type that I have seen students make the most mistakes with!
Mistake #1: Students will try to add `8`, since they see subtraction.
Once students decide they need to get rid of the `8` first, be sure to carefully ask how we can cancel it out. If students say we should add `8`, ask, “does `8` plus `8` cancel out to `0`?” This should help students recognize that they need to subtract `8` to cancel.
Mistake #2: Students will not bring down the subtraction sign in front of `3m` after subtracting `8`.
Remind students that this subtraction sign is the same as if we wrote plus negative `3m`. The key is for students to recognize that the coefficient of `m` is `-3`, not `3`.
Once you’ve talked through both of those possible mistakes, students should arrive at the answer of `m = -2`.
Now it’s time for some independent practice! You can assign a ByteLearn online practice to your class using the link below. Students will get immediate feedback and step-by-step help if they need it. Set a due date and allow students to finish the assignment for homework. Once complete, you’ll see detailed reports of students who may need additional support, students who are ready for a challenge, and other interesting insights!
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