I will begin by stating the essential question: How can you use inverse operations to solve an equation? I will tell my students that we will be solving the equations using a model (MP5). Most of my students solved one-step equations in the 6th grade, but they only solved equations with whole numbers. Now we will be solving equations with rational numbers - today the focus is on integers.
The model uses familiar integer counters. I will quickly model a few sums for students as a warmup.
x - 3 = -4
I will model this as : X - - - = - - - -
Students may have the algebra tiles but will need to be able to draw the model as well. Students may think of this as x + -3 = -4, but of course that is okay.
I'll explain that when solving an equation we want to isolate the variable on one side of the equation "= sign". I'll ask: What needs to be removed so that the variable is isolated? Answer: -3. How can we do that? Answer: Add +3. I'll point out that what we do to one side of an equation we do to another side. (If necessary I'll explain this with a simple example: 1 + 2 = 3. If we add 5 to each side we get 1 + 2 + 5 = 3 + 5; both sides remain equal.)
The point is for students to see that to solve this subtraction equation we use addition.
I will solve this in a similar manner. We'll conclude that to solve addition equations we use subtraction.
Common error: I notice that students often use the wrong value in the inverse operations. For example on example 2 students may try to subtract -5 from both sides. While that creates an equivalent equation, it does not isolate the variable.
Also it is worth pointing out that -5 = n + 2 is the same as n + 2 = -5.
Students are asked to solve 8 problems with their partners. While students may still use the algebra tiles, I want them to model the equations by drawing plus (+) and minus (-) symbols to represent quantities being added or subtracted and also negative numbers.
For each problem the conversation should be centered around what is being done to isolate the variable. Using GP1 as an example, students will likely say they added -10 to both sides. Some may say they subtracted 10 from both sides.
The final four problems ask students to write an equation represented by algebra tiles, then solve.
When going over these problems with the class, I recommend writing the equation using notation and showing how the inverse operations are applied, so that students start to see the connection between the model and how it looks using the more abstract symbols of algebra.
For independent practice, we will remove the tiles and have students draw the models only. Students will work on these 8 problems independently. I'll circulate to make sure students are drawing the models correctly. When we are done, I will do a quick cold call round. I'll call on a student and ask the student what operation and what number was used to isolate the variable. Example: on problem 1, the student should say they added 6. I will then call on another student to answer. This should go swiftly.
Before we begin the exit ticket, I will ask a student to summarize what we learned about solving equations involving addition and subtraction. Because I am asked to enter exit ticket results in SchoolRunner I will assign points to each problem. The first two problems will each be worth 2 points and the third problem will be worth 1 point. I will explain this to my students. My thinking is that the point of the lesson is to model equations (problem 1) and identify how to solve addition and subtraction equations (problem 2). The solution, while important, is less of the cognitive load for this lesson. So if a student can answer problems 1 & 2, but somehow misses 3, they can still earn 4 out of 5 points or 80%.