Graphing Polynomials Using Vases 📈🏺

Polynomials is one of the hardest sections to teach, over the past four years I have acquired different handouts, activities, lessons, and tasks for Algebra 2 students and almost no material for the section on polynomials. Adding, subtracting, multiplying, and dividing polynomials always seemed like an algebraic process and not so much visual or hands on.

Now I have one activity!!! Graphing polynomials was always tricky, but teaching quadratics before made it seem like a piece of cake for them. One of my favorites is graphing polynomials using vases, yes vases.

So to preface this I with I spent a week searching all the Goodwills in the Omaha metro area for different vases and this is basically what I found when you take all of the repeats out.

I did replace the big one in the middle and the one on the far right, well you can tell why.

The way I set this up is I provided each group a vis-a-vis marker, ruler, set of measuring cups, and a vase. Students were given the following directions:

1. they needed to mark off every inch on the outside of the glass 
2. make a table for how many mL in every inch.
3. Put the table into Desmos
4. Find the line of best fit on Desmos (I gave them the different equations)
5. Look at the R squared value to find which one is best.
6. Present your vase to the class the following day.

I had students present from their iPads, but having them create a poster would have been better so they could compare and contrast the vase with the graph to identify key attributes.

What is even better the day before the students presented they practiced with a Desmos activity. At the end of the activity students had to create their own vase and graph.

Below are some photos of my students working on their vase.

Transversal Tag 🏃

One of my favorite geometry activities I did this year was Transversal Tag. I set up the gym so it had the pattern of a transversal, like the picture below:

Students were randomly assigned a number 1-8 and then a tagger was randomly chosen. I am sure this game would have been much better to play with 3-4 taggers and play freeze tag, but we played that if you were tagged you became a tagger as well. Also if you went to the wrong area you could be tagged and become a tagger as well.

The taggers had to decide which angle congruency to say to get the most amount of people. For example, a favorite to choose was alternate exterior angles, because half of them had to run to the other side of the gym. The taggers also had to be smart about choosing ones where the runner will be going. 

It was a quick fun game that would have lasted longer if it was freeze tag, but the students had fun, used vocabulary, and had fun running around the gym for 20 minutes instead of being in class.

Head's Up: Review Game 🗣️

Head's Up! is a game of charades where the person puts their iPad or iPhone on their head and the audience performs and nods down for correct and backwards for pass. For students this was one of the best review games we used for vocabulary for geometry.

Students stood up in front of half the class and used my phone to show the students. This was excellent for students coming up with moving actions for such words as vertical angles and supplementary angles. Students were better able to self define vocabulary words better than previous, plus students had more fun reviewing.

It costs .99$ for the app and an additional .99$ to make your own cards. Well worth the money.

Coding Inequalities 💻

Coding in the classroom has always been an interest to me, Hour of Code is a great resource for any teacher especially those just starting out. For the past two years we have been doing Hour of Code during the Hour of Code week and that has been the most coding we do all year. What I wanted in my classroom was more coding, because I think coding could be the future for most of my students. I also believe that coding can be a gateway to other mathematical principles that are taught in the classroom such as: growth mindset, support productive struggle, and promote reasoning and problem solving.

So we have started in the math classroom is using one day a unit to work on these skills by implementing a way for students to apply their mathematical knowledge and coding. Students are given a task write code to make a calculator to solve an inequality. Students had to write code in to write their inequality.

We were going to use Swift by my Seniors do not have the iPads supported to do it, so we decided for everyone to use Trinket. Some students really loved the problem solving aspect of coding, using the blocks to get the numbers to do what they want them to do. Other students were not happy about trial and error process to finding the answer.

I have some other upcoming units to try this out with like Pythagorean Theorem.

How I Teach Direct Variation

I use to teach direct variation by having students take notes, but the past few years I have been using Jon Orr's Water Bottle Flip.

There is an excellent Desmos activity that goes along with it. This year I copied and edited my first Desmos activity which was this one.

I added two slides:

I wanted to emphasize direct variation and ask them deep meanings of graphs. One of the questions I asked was looking at the graph on the bottom, what inferences can you draw?

On September 8, there was the NATM (Nebraska Association of Teachers of Mathematics) Conference. During Lenny VerMass presentation, Smoke and you Croak or Huffing and Puffing to Understand Slope, he had a very interesting task. 

Students had to measure how much air filled their lungs. So we exhaled into a balloon and measured (3) breaths and the circumference of the balloon. Then on a big sheet of paper we had to plot all of our data points for the following graphs and interesting things happened. Try it with your students.

Student Created Kahoot

Kahoot is the first tool that seems universally accepted tech tool in every classroom. I can see why, its fun to play against others. I remember when I was growing up we played a game in middle school called hands down, if you were the first person on the bottom and had the correct answer you scored points, it was my favorite.

But, Kahoot has been placed in a DOK 1 or DOK 2 depth of knowledge when students are playing. It is hard to find Kahoots where students are not only just remembering or applying theorems but creating and evaluating. One of the things I wanted my students to know is how teachers choose Kahoots and for them to not only review but practice and evaluate others Kahoots.

Paper Kahoots

We started in the classroom with paper Kahoots as a lesson. We talked about how long it would take to do the problem, where there answers that were misleading, and what did the student know if they got the question wrong. Here are some examples students made.

You can find a PDF version here:


For students to create their own Kahoots I had to change my username and password, since Google Sign-in wasn't cooperating. Some students took off and were self sufficient other students struggled coming up with questions, because of the content. I had to ask them how to be a teacher and what kind of questions I would ask.

Created Kahoots

Most students took the route of pure vocabulary and no mathematical questions, but I did not specify what type of questions, now I know.

Here were some example questions they came up with:

Artist Sol LeWitt and Points, Lines, Angles

Sol LeWitt was an artist born in Hartford, Connecticut in 1928 he was most known for his conceptual art, however in this overview we are going to focus on his Instructables. Instructables are wall art where the artist has to follow a particular set of instructions. Sol LeWitt came up with a large number of different instructions, some he never did himself.

For example Wall Drawing #65 in colored pencil is of follows:
Lines are not short, not straight, crossing and touching, drawn at random using four colors, uniformly dispersed with maximum density, covering the entire surface of the wall.

This is what Sol LeWitt came up with:

This is bad example, because it does not take in the sheer size of the piece. Since it is a wall piece it is so large that you could not fully see it from one spot.

So how does this relate to math?

Sol LeWitt has hundreds of these instructions were he takes shapes such as squares, circles, and triangles. He also loves lines, some straight some not, and vertical and perpendicular angles. So to introduce and apply the first section of geometry points, lines, and planes. We attempted our own Sol LeWitt.

Our instructions were: On a wall surface, any continuous stretch of wall, using a hard pencil, place fifty points at random. The points should be evenly distributed over the area of the wall. All of the points should be connected by straight lines.

I assigned all students a letter and then had them connect to each other, so we only really had 26 points, but our artwork was just as amazing.

It did take a little bit more time than I was planning, but the picture at the top took 8 days to make.

We talked about lines and line segments and this brought up a good conversation about how we name lines. I would ask a student which one is the longest line, but would not let them get out of their seat. So it was easier for the student to name the line segment than point.

I love using art in the classroom and Sol LeWitt's Instructables are an easy way to get art in the geometry classroom.

Below is a PDF with some Instructions to do you own.