# Equations and Inequalities with No Solution or Infinite Solutions

I was looking for something a little bit different than what I had done in the past to introduce equations that have no solution or infinite solutions.  I came across this post from Sarah who blogs at Everybody is a Genius, and it was exactly what I was looking for.  I also liked this because when I had these students as 6th graders, I used scales to introduce them to solving equations, so this wasn’t a new idea for them.

I gave this sheet to students and told them to fill in the boxes to keep the scales balanced, and that for each scale, the number in the box must be the same.  Students have done a few different Open Middle problems this year, so some students struggled with the idea that they could no reuse numbers since they are used to not being able to reuse them for those problems, but they eventually understood what to do. As I was walking around, exactly what I hoped would happen, happened.  Students got two number 3 and I heard, “What?  This doesn’t make sense.”  “This is impossible.”

As we went over what students came up with, we discussed how in #1 and #4, we could pick any number we wanted, in #2 and #5 only one number works, and in #3, and #6 no numbers work.  Then we took some notes on this.  In the notes sheet I handed out to students, I included a picture of the scale and we wrote out the equation and showed what was happening to the scale as we did the algebra.

I liked that introducing this topic this way to students gave students a visual to help them understand these types of equations.

The next day we did some practice at the whiteboards.  I always include some problems that have one solution (especially ones where x = 0) because some students want to start saying every single equation either has no solution or infinite solutions, even though I stress that this only happens when the variables are eliminated.

Sarah Carter has created a nice Open Middle style problem to go with this topic.  Here students can use the numbers -4 to 4. Last week we worked on solving inequalities with infinite solutions and no solution.  I really liked what I did last year for this, so I did something similar this year.  I started the day by having students solve an equation that had no solution.  Then, I asked students which inequalities would make that true and which would make it false. We briefly discussed which would make it true and which would make it false, and that was pretty much the only instruction I gave students that day.  They had little to no trouble transferring the idea of equations with no solutions or infinite solutions to inequalities.

I shared at the end of this post a Desmos card sort I use as well as another Open Middle style problem on this topic.  Overall, I’m really happy with how students are doing with these types of problems.  I think that introducing this idea using the scales really helped my students to see what was going on.

# 6th Grade Unit 2: Intro to Algebra (Part 3 -Equations)

You can read more about our Intro to Algebra unit here (part 1) and here (part 2).  In this portion of the unit we get into solving equations.

There are several things I do prior to actually solving equations to get students thinking algebraically.

I absolutely LOVE puzzles like this.  One of my students caught something in this puzzle that I had missed the first time I solved it. In one of my classes we had some extra time, so I had my students create their own.  It was super fun to watch them get excited over this and to see their creativity in what they used in their puzzles.

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Then we used one of my online puzzles –Solve Me Mobiles, and one of my favorite movement activities, Balance Points.  I blogged about both of those activities here.  When we play Balance Points I put an equation up on the board and with a partner, whatever the answer is students need to have that many body parts touching the ground.

When we get into solving equations, I stole Julie’s idea found here.  If you teach middle school and haven’t read that blog post yet, you need to stop and do that right now.  I’m not even going to say anything more about it to force you to go read it.  Most of my students will only work on one-step equations.  In some classes they’re ready for multi-step equations or some students in the class are.  Here is the link to download a couple worksheets I use and an add-em-up activity.

Open Middle also has some great one-step equation problems.  # 8th Grade Unit 1: Equations (Part 2)

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.

# 8th Grade Unit 1: Solving Equations (Part 1)

The first unit we do in 8th grade is on equations.  I start by reviewing order of operations, evaluating expressions, and simplifying expressions.  Then we get into solving more basic equations.  Here is a semi-brief overview of the first part of this unit.

##### Order of Operations

We start off with order of operations.  I use the following Notice/Wonder to lead into our discussion/review of order of operations. We also review absolute value as well as square roots as part of our order of operations practice.  These are great problems for vertical nonpermanent surfaces (#VNPS) This Desmos activity from Cathy Yenca is also a great review of squares and square roots. After a couple days of absolute value problems and square root problems, students work on a worksheet similar to the one below.  You can download it here.  I’ve thought about changing up this worksheet since it doesn’t include square roots or absolute value, but it is a good challenge for students, since students are only allowed to use the numbers 0 through 9 once, and I like that about it. Students also see their first Find the Flub warm-up in this unit. ##### Evaluate Expressions

Then we spend a little bit of time on evaluating expressions.  I use the worksheet below as practice for students.  I blogged about this type of worksheet here.  You can find the link to download it in that post. ##### Simplifying Expressions

Both years I’ve taught this, I forget that students aren’t as comfortable simplifying expressions as I expect them to be.  I start by having students simplify expressions that don’t involve the distributive property, and then I add that in a day or so later.  I found a Desmos activity in the Desmos Bank that I modified and uses on one of the first days on this topic.  Here is the link to the activity I modified. Then we do a couple days of simplifying expressions with the distributive property.  Again, I use a “One Incorrect” Worksheet.  You can download it in this post. The Notice/Wonder I used below was GREAT to discussion some common mistakes I was seeing students make when simplifying expressions.  For example, I had students who would say that 5x² was 25x.  We had a really good discussion about the differences in the expressions below and how that changed things. ##### Solving Equations

Then we get start solving equations.  A few years ago, I had a group of students that struggled to plot points on a number line, so when we got to solving equations, I saw that as an opportunity for them to get more practice with that by having them graph the solution to the equation.  They also struggled with order of operations/evaluating expressions, so again,  I decided to have them practice this by checking their answers to the equations.  I’ve never looked back, and now I have students graph and check their answers to nearly every problem they do for me. If you’re interested in the worksheet I use, you can download it here.  Below are a couple of warm-ups we use when we’re talking about solving equations.  # My Favorites: 2 Equation Activities

I decided to give the #MTBoS #MtbosBlogsplosion blogging initiative a try, at least for this week.  This week’s topic is “My Favorite” and in typical me fashion, I couldn’t pick just one, so here are a couple of my current favorites related to solving equations.  Ask me in a couple weeks what “my favorites” are, and I’d likely give you a different answer depending on what topic I’m covering then.

### My Favorite App:  SolveMe Mobiles

I’m probably super late in discovering SolveMe Mobiles, but I love it!  I’ve used it with 6th, 7th, and 8th graders and all students have been enjoyed it.  One of my 6th graders came to me this week, “Can you please help me with this puzzle?  I was working on it last night with my brother and we couldn’t figure it out.  We stayed up until 10:00 trying to figure it out.”  You know you’ve found a winner when a students voluntarily does math at home, gets his brother to play along, and they stay up late trying to figure it out!

If you haven’t checked this app out yet, I strongly encourage you to do so.  (There is also a website if your students don’t have iPads.)  I love that the app is simple and easy to use, but it also offers many different features.  Students can draw on the screen, zoom in and out, and you can drag down from the mobile and it creates an equation with the shapes.  I have only used the “play” mode with students, but they are also able to build their own.

If students type in an incorrect answer the mobiles tilt to reflect that one side is heavier than the other. I’ve also had some great conversations with kids when they do something like I’ve got pictured below and they tell me that it’s balanced and wonder why it doesn’t work. ### My Favorite Movement Activity:  Balance Points

My first year teaching I went to Minnesota’s annual math conference and attended Sara Van Der Werf’s session on movement.  The entire session was a game-changer for me.  Not only was it the first time I was able to experience first-hand Sara’s awesomeness, but I also took away a ton of stuff I instantly applied to my classroom.  Right away, I saw the impact movement had on my students and my lessons, AND realized that movement activities don’t have to take away from the time we’re doing math.

To play Balance Points, students are paired up, and I put an equation on the board such as x + 5 = 9.  While staying connected in some way, such as holding hands, students must show the correct answer by having that many body parts touching the ground.  I encourage them to be creative and won’t accept “boring” answers. I wish I had pictures to share of my students playing this.  They LOVE it.  There isn’t a person in the room who isn’t smiling and laughing, myself included!   There’s usually a lot of  “Come quick!  Check our answer!” because students are about ready to fall over from the crazy positions they’ve come up with.  So fun!!  I haven’t done this, but you could use this for order of operations as well.

What other ways could you use this with students?  Let me know what you try!