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.

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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.

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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.

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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.

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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.

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Puzzles, Puzzles, and More Puzzles

One of my projects last summer was going through all of the different places I had resource stored (emails to myself, Twitter likes, Google drive documents, etc.) and compile them into one place.  I created a separate Google Doc for each of my preps and sorted the resources by unit.  It took quite a bit of time to go through all of those things, but I am reaping the benefits already.  As I start a new unit, or a new concept within a unit, I am able to check this document for resources to add to or replace things that I have done in the past.  I’m no longer spending time searching for these things in 4 different places or looking for new resources when I’ve already found things in the past.

Already in our second unit in 6th grade, I tried several new things this year that I’ve loved! We’ve been talking about exponents, prime and composite numbers, factors and multiples.  I found several puzzles that my students have been enjoying and have been SO persistent with.  I had one student at the start of the week tell me he hates puzzles.  I think probably because he’s a student who picks up on things quickly and doesn’t always like that he can’t figure out a puzzle right away.  By the end of the week he was asking for more puzzles.  🙂 Success.

It’s been fun to listen to their conversations as they’ve been working and to see them excited to share the progress they’ve been making on them.


The first task I added to this unit is the following problem from Open Middle.

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Look at the amount of work by one student!! Wow! So proud of him!


Then I used this Open Middle problem from Bryan.  I LOVED this one so much!  It was so challenging, but it really got my students thinking about exponents.

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I know there is at least one mistake in this one, but again, even despite that, look at all the great thinking that took place here!


Then we started getting into prime and composite numbers and multiples and factors.  I came across this puzzle on Twitter.  I love that it incorporates so much vocab into one puzzle.  After using this in one class, I realized that dry erase pockets would work well for this one so students could more easily change the numbers.

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Such great group work on this puzzle!


Lastly, I found this puzzle here and here.  Again, I love how it incorporates so much vocab into one puzzle, and I love the extra challenge of placing the headers versus having them already placed on the puzzle.  I did type up my own version of the puzzle.  You can download it here.  I used this on a Friday in one of my classes and they were so disappointed when it was time to clean up.  I haven’t had any students solve it yet, but several have come so close.

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Again, not everything is perfect in those, but what great thinking by my students!