Function Notation Same/Different

I have seen other people use the prompt “What is the same?  What is different?” around Twitter (Check out #samediffmath), but I’d never formally used it in my classroom.  I have asked those questions before on the fly, but I’d never created something to put in front of my students where those questions were the main focus.

As I was driving to school Friday and was thinking about what I was teaching that day, I had this thought to create one for function notation.  Function notation is something that some students struggle with, and it sort of surprises me every year because it is so similar to things they’ve already done.  Here’s what I came up with.


I printed these out on half sheets of paper and had my students do a Stand and Talk with them.

Here are some of the responses I got for “What is the same?”

  • Both have an answer of 57
  • The last 5 rows are the same.
  • Both replace the x with 4.
  • Both have 3x^2 + 2x + 1

For “What is different?” we talked about how on the right it has f(x) and asks to find f(4), and on the left instead it says “evaluate…for x = 4”

We talked about how so much of the problems are the same, but if I just gave my students the top row, they would know how to do the left one, but would feel completely lost with the right one.

In my second class, I explained how these questions are asking something very, very similar but the notation is different.  I thought of the example of in elementary school if they were given 5 × 3, they would know exactly what to do.  However, if they were given 5 • 3, they wouldn’t, even though it is asking the same thing.  As I was explaining that, I could see some students making the connection to the two problems we were looking at.

I LOVED using these prompts intentionally in my classroom, and I’m looking forward to finding more ways to incorporate this into my classes.

(Here are some of the other things I’ve done with function notation.)


Good Things: Volume 8

Volume 1 | Volume 2 | Volume 3 | Volume 4| Volume 5 | Volume 6 | Volume 7


Several weeks ago, we got to watch our middle school play.  I love seeing my students excel outside of the classroom.  They did a great job with the play Transyl-Mania!


I still have students asking if they can work on Desmos Mini Golf Marbleslides months after we did it in class.  It makes my heart happy that they not only remember doing this activity, but enjoy it enough to want to work on it in their spare time in class.


I overheard some of my 8th grade girls talking about one of their pens (a Flair pen).  The girl said that I inspired her to get the pens because I have them.  It was cute.  I did nothing to “inspire” her other than use them in class.


I had my students create their own equation puzzles, and they crushed it!  They came up with so many creative puzzles!

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My response to a student’s question was, “We’re all in this together.”  Without missing a beat, a student broke out into High School Musical.


I used Julie Reulbach’s method to introduce equations to my 6th graders.  This is one of my favorite things I do with them.


We were going over a few examples of equations in class, and I had several students with their hands raised before I even asked a question.  I pointed this out, and ended by saying that maybe they do know what I’m going to ask.  They then proceeded to go through the example and ask the guiding questions that I would have through the entire problem.


Not entirely math or school related, but I put my Christmas tree up last weekend.  I know it’s early, but I wasn’t sure what my weekends would look like the next couple weeks, and I LOVE having the tree up and wanted to enjoy it as much as possible.  Christmas ornaments have always been super special in my family.  It started when I was a little girl and every year my aunt would make an ornament each year for everyone and my grandma on the other side would also paint a ceramic ornament each year.  My aunt and my grandma have both passed away, so those ornaments are now extra special.  My mom always gets ornaments to remember special events, whether it’s a trip throughout the year or a big event that happened that year.  I’ve started doing this too.  I smiled when I put up my TMC ornaments


My mom, cousin, and I have carried on the tradition of making an ornament each year.  I was pretty excited that Target came out with holiday building blocks.  I’ve spent a lot of time this past year playing with different mathy toys with my cousin’s daughters, so I thought this would be the perfect ornament for this year.  I’m happy with how they turned out.


Another non-math Good Thing, but I saw this planner, and knew I’d found my planner for next year.



Basketball season has started.  I’m convinced that God made basketball a winter sport because he knew Minnesotans needed it to get through our awful winters.  Basketball is a huge thing in my family since my dad still coaches.  Part of the reason I look forward to it every year is because it means time with family in the gym.  I also love that my middle schoolers ask whenever they have games if I’ll be there doing clock.  I love that it matters to them that I’m there.


Middle schoolers’ giggles.  I love the sound of innocent middle school giggles.  I’ve got a group of students who have been coming to my room after lunch to play with the stuff on my play table and just hang out in a quieter space than the lunch room.  This week they made up their own HedBanz game.  I loved the sound of their giggles every day in my room.  On Friday, I had a group of boys first hour get the giggles as they were working.  It was a great way to start the day.


I had an 8th grader come to me before class Friday morning.  She told me she was going to be leaving early and wondered if she could take the test first hour during her study period.  When I brought the test to the teacher, the teacher was unaware that she was leaving and had asked to work on the test early.  That means she took it upon herself to plan ahead and take initiative to get the test taken!!!  I LOVE when I start to see middle schoolers take responsibility.


I had the warm-up on the board Friday, most of my students were working.  I was getting ready to go prompt one student to get started when he said, “I suppose I should probably do something.”  Again, I love when I start to see students monitor their own behavior!


Every Friday we do Hats for Hearts and the proceeds go to one of our secretaries who is battling cancer.  Students pay $1 and get a sticker to put on their hat, which tells us as teachers that they are able to wear their hat for the day.  One student had a sticker on his shirt and wasn’t wearing a hat.  I asked if he just gave money to the cause.  He did.  I love the hearts of these kids!


Our high school play this fall was Clue.  Clue was the game I always played with my grandma when I would go to her house growing up.  We are nearing the 1-year anniversary of her passing away.  Every time I saw the posters for the play up around school, I thought of her.  In my 5 years at my school, the middle schoolers have never gotten to go to the performance of the high school play during the school year.  This year we did, and I don’t think that was a coincidence.  I thought of her throughout the entire performance.

You really do need a play table in your classroom!

When Sara Van Der Werf first wrote about her play table a couple years ago, I loved the idea, but at the same time I was hesitant to do this in my own classroom.  If you haven’t heard of the idea of a play table before, I recommend you go read Sara’s post that I’ve linked above.  In short, Sara has a table in her classroom where she keeps a mathy toy out for students to play with, but her post includes a lot more specifics of how she has this set up.

There were a couple reasons I was hesitant.

1) Time.  Sara mentioned that the main times when students play with the things on the table are during downtime in class, between classes, and before/after school.  We don’t have super long passing times, so I didn’t envision students having time to play between classes.  Because I teach all middle schoolers whose main way of getting to school is the bus, I couldn’t really picture any students playing after school and with breakfast served at school in the morning, I wasn’t sure if mornings would be an option for them either.  I plan things for my students to be doing from the time they walk into my room to the time the bell rings.  I wasn’t sure how this would work into that and couldn’t picture what it would look like with middle schoolers to have them play with this during downtime in class.  This sort of ties into my next hesitation.

2)  Behavior.  With middle schoolers, I wasn’t sure how having a play table would go behavior-wise.  If some students finished their work and started playing during class, would this be a distraction for other?  (Would it be any more of a distraction than some of the other stuff that happens in class?) Would the play table prevent students from getting to work right away at the start of class with whatever task I had on the board for them?  These unknowns were part of the reason it took me a while to set this up in my classroom.

3) Space.  The other reason I didn’t start a play table immediately after reading Sara’s post was space.  I didn’t have a table for this, and I didn’t know where in my room I would put it.

Fast-forward a year.  I now have bigger class sizes than ever before.  I had to get more student desks over the summer to have enough seats for everyone.  Bigger class sizes also means more potential behavior issues, but I added a play table to my classroom this year.

Another teacher had an extra table at the beginning of the year, and I snagged it up.  At the time, I was still on the fence about adding a play table.  Around that time, was Math on a Stick -a math playground at Minnesota’s State Fair headed up by the amazing Christopher Danielson.  I volunteered at Math on a Stick three different times right before school started.  You can read about my experience this summer here.   Watching the awesomeness at Math on a Stick convinced me that I needed a small piece of this in my classroom.

Here are a couple things I’ve noticed/learned so far with the play table.

  • One good thing about waiting a year to start my own play table was it gave me time to build up a collection of mathy toys.  After seeing Sara’s, I started picking up different things when I saw them on sale on Amazon, at garage sales, or at Goodwill.  It’s nice now that I have several, that I’m not in a panic to try to find something new for my students when I’m ready to switch up the toy.
  • I’ve followed Sara’s advice and only have one type of “toy” out at a time.
  • On average, I’m switching what I put out every two weeks.  I’ve noticed that the first week a new toy is out, I have the “regular” students play with it -the ones that always come check it out, regardless of what’s out.  The second week, I see students who may not play with this stuff as much at the play table.  I also see more creativity the second week.
  • Before school and right after lunch are when 95% of the “play” happens, which means only about two classes are using the play table, but I am ok with that.
  • I haves some students who come to my room early on Monday just to see the new toy for the week, and I now have a group of students who will come to my room 15+ minutes early after lunch to play.

Here are the things I’ve put out so far this year.

Hexagons:  Unfortunately, these hexagons are no longer available at Target.  I keep waiting for Target to bring them back.  I was fortunate enough to have a friend give me some of theirs.  This was the very first thing I put out, and I think it took almost a week before a student *finally* became curious enough to start playing with them.  Once one student did, it didn’t take long for several others to start playing too.

As Sara mentioned, one downside to these is that they take more time to break apart.  Like Sara suggests, at least once a day I take apart what students have made so that other students can create.

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Pattern Blocks:  I think I only had these out for a week, and I wish I had kept them out there for two.  They’ll show up again later this year.




Brain Flakes:  This was one of the toys that got many students hooked on the Play Table.  After putting this out, I started seeing a lot more students playing.





Hashtag Blocks:  Run! These are currently in my Target’s dollar section for the holidays.  They originally had them for back to school, and I checked Target a lot for them during that time.  There are also Plus Plus Blocks, which are similar, but a lot more expensive.

One downside to these is that they do take a bit more time to break apart than some of the other things I’ve had out.





Magformers:  These have definitely been the favorite so far this year.  They are also the most expensive, but worth it.  Every once in a while Amazon has them as their Daily Deal.  It was a fun day when students realized these stick to the whiteboards!




Mosaic Mysteries and Magic Mosaic:  I think the Mosaic Mysteries is a Discovery Toy.  I’m not sure what Magic Mosaic is.  I’ve picked up one of each of these at thrift stores.  At first I wasn’t sure how this would go over because I only have two of the hexagon boards, but I found that students will play with the trapezoid pieces without the board itself.

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I am SO glad I decided to add this to my classroom this year.  It has become one of the highlights of my day to see what my students create.

It’s fun to hear the math that comes up as students are playing.  I often hear students talk about making patterns.  When the hashtag blocks were out I heard one student ask another, “Oh, you’re making a giant pineapple?”  The student replied,  “No, I’m making it to scale with this carrot.”  (We haven’t talked about scale at all this year.)

If you are like I was and have seen things about a play table on Twitter or in the MTBoS community, I can’t encourage you enough to start one of your own!

If you have a play table and have other ideas of things I could put out, please share!

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.

Equation Scales

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.


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.

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.




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


One Good Thing: Volume 7

My students are starting to get into the play table more, and it’s like having a little piece of Math on a Stick in my own classroom!  Right away as they discovered these, they started doing things I never would have thought to do!  I love it!


We’ve been working on area in 6th grade, and I’ve shown students several GIFs to help them understand the formulas.  One student commented, “All of these GIFs are just so perfect for this!  Where do you find them?!”  I love when they appreciate things I’m sharing with them.


A pair of students called me over and asked, “Can you help us?  We’re arguing over here.”  Me:  “Arguing?  Over math?!  I LOVE it!”


I’ve done a worksheet like this in a couple different classes this week, and I LOVE how persistent my students have been.  I know that these types of worksheets can be frustrating because students are forced to get a correct answer rather than just move on, but for the most part, they’ve embraced the challenge.


We played the game linked in this post this week and a students’ comment as she left class, “PLEASE tell me we’re going to play that again.  THAT was fun!”



I’ve been sick this week, and one day my eyes get really watery several times throughout the day.  One student was in the middle of a sentence when she looked at me and with genuine concern asked, “Are you ok?”  These kids.  I just love them so much.


One of my lessons this week did not go how I hoped.  We were working on simplifying expressions.  Every year I’ve taught this I’ve forgotten where students are when they come to me.  When planning this lesson, I always think of where the prior year’s group ended -not remembering how much that group had learned over the course of the year.  Anyway, the lesson combined too many new things together too quickly.  As I was teaching, I knew it wasn’t going that well, but I didn’t know how to improve the lesson on the fly.  I made a slight change from the first class to the next class that helped a bit, but I ended the day frustrated with myself for forgetting how this lesson had gone the previous year.

We came back the next day and did an Open Middle (more on that later), reviewed briefly before going to the whiteboards for some Vertical Non-Permanent Surfaces.  They ROCKED it that day!  They were so focused and worked so hard!  I was SO proud of them!


I introduced students to Open Middle.  Because it was the first time students were introduced to this type of problem, I wanted them to feel success without it being too easy.  I think this problem worked well with my goals.  At first students were confused, and I sensed some were hesitant to even get started because they were unsure of themselves.  However, once they grasped what was being asked and dove in, they were able to solve the problem, and most often come up with more than one solution.  Cheers of excitement could be heard throughout my room as students successfully solved the problem.

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We followed that one up with another one the next day and they dove right in.

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The 8th grade football players were practicing as I was leaving school one day this week. A few of my 8th grade boys noticed me leaving and started waving at me -it wasn’t just your typical wave.  It was the practically jumping up and down because they’re waving so big kind of wave.  What a great way to end the day!


A student as he left class, “Thank you for teaching me today.”


Friday High Fives have turned into a Friday hug with one of my 6th graders.  6th graders are just the sweetest.


I wore my Desmos shirt for the first time this year.  I got by far more comments on that shirt than anything else I’ve worn this year -both from current and former students.  Some students thought it was cool.  Some asked, “WHY do you have a Desmos shirt?” and now think I’m even weirder than I was before.  🙂  One girl asked, “Does Desmos sponsor you or something?”  When I told her no, she replied, “Well they should.”  Haha!


A couple of groups finished what they were working on earlier than the rest of the class. Without prompting, they started helping other groups.  It was like having 3 or 4 teachers in the room as these students walked about the room asking groups if they had questions.  These kids.  I tell you.  How did I get so lucky that I get to spend my days with them?

Math on a Stick

This past summer was FULL of wonderful math things!  I finished up my last two math grad classes (Yay!), went to TMC, and got to learn from Fawn Nguyen!  And if all of that wasn’t enough, I got to end my summer with Math on a Stick!

Math on a Stick started thanks to Christopher Danielson.  It’s essentially a math playground that is open every day of our State Fair.  You can read more about it here and here.


Tiling Turtles

You can also read a reflection from Laura’s daughter here.  I so enjoyed meeting her and the conversations we had.  Here she is running the spiral machine.


This was my second year volunteering, and it is always a great time, rain or shine!  If its rainy, that usually means more time to reconnect with the other volunteers, but if the weather’s great, Math on a Stick fills up quickly!

Above:  The Chris, Christopher, Annie, and Phil having tetrahedron building contests when it was raining.

Below:  It was so fun to see how many people came to play!

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Every day there is a visiting mathematician with some activity for people to do.


The math bus that Christopher and his crew put together when he was the visiting mathematician.


The visiting mathematician this day was from the Fractal Foundation.

It’s so fun to see all the different things people create throughout the day.

Casey flew in from California for Math on a Stick!  When we weren’t volunteering, we went to a few “must sees” at the State Fair.


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I was so fortunate that I was able to see so many people from the MTBoS community  that flew in for math on a stick as well as reconnect with many MN friends.

It is always so fun to meet Twitter friends in real life!


I’m already looking forward to next year!  The last weekend of the fair is Labor Day weekend.  Come visit!!