I shared a bit about the first part of our first unit in 8th grade here. In the second part of this unit, students start solving equations with square roots, x², and absolute value. They are also introduced to the idea that not all equations will have one solution. They learn that equations with absolute value or x² have 2 solutions and then we talk about equations that have no solution or all real numbers as the solution.
One of my goals when I’m teaching them how to solve these new types of equations is to help them understand how it’s similar to solving problems they already know how to solve. I want them to see the similarities in the problems below. A recent conversation with a colleague reminded me that these connections are SO important and that I need to continue to work on helping students see the similarities in these problems.
I sort of want them at first to view the square root, x², and absolute value portion of the problem almost as a variable in and of itself. I want them to understand to use inverse operations to first get that part of the equation by itself. Once they have done that, I want them to understand how to use inverse operations to undo the square root or exponent and then solve the remaining equation. In the case of the absolute value equations, I want them to understand why there are two parts to the answer and how to come up with those two parts once the absolute value is isolated. (This is the one I have the most work to do to improve how I teach it in the future.)
Square Root Equations
We start this part of the unit by reviewing inverse operations, and I tell students that we’re going to focus on squaring and square roots as inverse operations.
Solving equations with square roots typically goes pretty smoothly. Students understand to get the square root by itself, square both sides of the equation. There are two main mistakes I see students make early on when solving these types of problems. In the third example below, some students will want to undo the subtraction before undoing the square root. In the last example, students don’t always recognize that the 2 is being multiplied by the square root, especially if the number being multiplied is negative.
By this point in this unit, I LOVE seeing students applying what they know about solving the types of equations on the left to the equations on the right in the picture below. Most of my students are at least willing to start the top right equation before we do an example together as a class.
Students do usually struggle with the bottom right equation, and when this happens, we discuss how to solve the same equation without the exponents. That is usually enough for students to understand what to do.
For practice on solving these types of equations, I use a worksheet I modified from one of Kate Nowak’s Row Games.
All Real Numbers/No Solution
This year I used the following to introduce these types of equations to students.
When we start solving equations like this, students all of a sudden forget what they have been doing in middle school up until now and want to put “All Real Numbers” or “No Solution” for every answer. I always make sure to include some equations that have one number as an answer when students are doing problems like this, especially equations that have zero as an answer or where there are similar numbers on each side of the equation but the negatives are different. For example -3x + 9 = 3x – 9
We had time for this Open Middle problem in one of my classes. I loved watching students work through this.
Absolute Value Equations
Of the concepts in this portion of the unit, this is the one that I feel I need to improve the most for next year. It always starts out well. Students understand in the first equation below that x can be 5 or -5 and can explain why. The second one goes pretty well too.
We do a few more examples before getting to one like the third example above, but in that problem, students understand to get the absolute value by itself, but then that’s where more of them struggle. After I taught this lesson this year, I thought that this might be a great topic for smudged math, but I haven’t had time to think through how that would go yet.
When I was almost finished with this portion of the unit, I realized that I got super worksheet heavy. It’s even more evident to me know as I put this post together. In the classes that didn’t get to the Open Middle problem, there wasn’t anything other than a worksheet. I now know what I need to work on for next year!
Here is the link to the worksheets I used in this part of the unit.