"The Grasshopper and the Ant" teach Exponents

What do you notice, what do you wonder when you see this image?

It looks cold, how do you know which one is the ant? Which one is the grasshopper? Which one looks warmer? How do you think the discussion is going?

Next have the students read the fable that is associated with it:

A Grasshopper gay Sang the summer away,
And found herself poor By the winter's first roar.
Of meat or of bread, Not a morsel she had!
So a begging she went, To her neighbour the ant,
For the loan of some wheat, Which would serve her to eat,
Till the season came round. "I will pay you," she saith,
"On an animal's faith, Double weight in the pound
Ere the harvest be bound." The ant is a friend
(And here she might mend) Little given to lend.
"How spent you the summer?" Quoth she, looking shame
At the borrowing dame. "Night and day to each comer
I sang, if you please." "You sang! I'm at ease;
For 'tis plain at a glance, Now, ma'am, you must dance."

What do you notice? What do you wonder now?

Where math comes in to play is the idea of where the grasshopper says, "I will pay you, she saith,
On an animal's faith, Double weight in the ." When you can't afford something say, the full price of a car, how do you afford it?  Most students will talk about saving money or a loan. If the grasshopper wants to survive, she wants a cut of the ants food savings and next season the grasshopper will give double back. Ask your students is this fair? Is this how a bank works?

There is a way to calculate it mathematically, but right now I want students getting use to the idea of exponents. The equation I would have them use is the compound interest equation.


Where A is the amount
P is the principle
r is the interest rate
n is the number of times it is compounded per year
t is the time in years.

I would have students think about what each of them means and how the rate effects how much the grasshopper would way in the long run.

Ruth and the Taco Cart

I love Dan Meyer's Taco Cart if you have never heard of it, here is the task it is great it is wonderful.


We were recently doing this activity in Algebra 2, because I love how it uses the Pythagorean Theorem and one of my students asks, "what if you split the angle in half and walked that, because that is what I would do." I thought that would be a wonderful geometry question.

Ruth and the Taco Cart

I posed this question to my geometry students, we weren't really learning about angle bisectors, but we had already learned it as well as trigonometry. It took them about 15 minutes to use pythagorean theorem and find the time it would take. However, Ruth was much more difficult.

After students completed the time section I asked what they needed to find the angle and the distance Ruth would have to go.

There were lots of good questions. I gave the rest of it as homework and only a few were able to do it completely.

For students that did the work, I had them work in a group and work on a couple of extension problems:

  • If Ruth didn't want to talk alone, how long would she have to wait for the other to catch up?
  • What if she took a different angle?
  • Does changing the angle effect the time it takes for her to get to her destination?
The other students continued to work on the problem and were given some of these extension problems as they got the answer.

I loved the use of geometry, I will definitely use this when we are learning trigonometry.

3 ACT Task: Beat The Freeze: Circumference

The Situation: 
During Atlanta Braves games, one fan has a chance to race "The Freeze." Who is the first to reach the end?

Act 1: First 10 Second of the Video:

The video shows the contestant running the warning track in the outfield. It shows the lead the contestant has over "The Freeze."

Have the students discuss who they think will win.

How can we prove who will win or lose? What would we have to know in order to solve this problem? Are there properties of a baseball field that we need to know before beginning?

Act 2:
At the beginning of the video it shows the distance to the left field wall. The distance to the right field wall is 325. You can use 335 or round to 330. 

The Freeze runs at 22.5 ft/sec
The contestant runs at 19.1 ft/sec

The contestant is given an 70 foot head start.

Act 3: 
Show the full video. 

Extension Questions:
How long can The Freeze wait and still win?
What if the rates changed at the midway point?
What is the biggest lead you could give the fan and win?
If you were the contestant what strategy might you use to win?

Peer Teaching with ELL Students

I have been teaching one student who is an English language learner who came to me at semester how to add and subtract proper fractions for the past three days. Everyday they come in I feel like I am starting from square one each day. I tried teaching by one example at a time, didn't work. I tried teaching using visuals like fraction circles and bars, that failed. The student was getting more and more frustrated, because they weren't moving forward.

I tried a different way. The other students in my class are to graphing linear inequalities. A majority of students in this class speak limited English and/or struggle with mathematics. One of the students finished early and I asked them to help this student.

This was their discussion back in forth in Spanish. It was a great way for both students to move forward mathematically and feel confident going forward.

Link to conversation in Spanish: https://chirb.it/ntn5mD . The sound byte is a minute and a half of the whole conversation which took about 5 minutes. 

The girl in the audio does an excellent job of breaking down the problem and used fraction bars to represent the fractions in the problem. You can hear her counting out the fraction bar in the first part of the audio, eventually she moves towards release of instruction where they did a problem together, then she watched as the student did one guiding through the entire process.

I need to find ways of incorporating more peer teaching for my other students, I wonder how I can help guide them through the steps of asking questions and dialogue between each other better?

Orthographic Projection with Merge Cube

Merge Cube has been a hit with stores like Walmart offering the simple flexible cubes for a dollar a piece. Merge cube is a simple way to get students using augmented reality in a QR code way. Students scan the Merge Cube with an app and a magical world appears.

One of my favorite apps using the Merge Cube is Dig!

Using the app changes a simple cube with a bunch of symbols looking like hieroglyphics into another world. You can build and deconstruct the cube that looks almost exactly like Minecraft. The reason I like this app the most is that students can build using the app.

My students at the beginning of the year struggle with this concept of orthographic projection and being able to correctly sketch the block layout. Having students use the Merge Cube students can grasp that conceptual understanding that they don't get from a sketch.  Last year I borrowed some of the wooden blocks our construction teacher uses and it was a great way for some of the students to see the finished product. The app allows students to see around each object looking at it from the sides and from the top.

What I would like to see is have students create their own and have stations at each group where students correctly draw the orthographic projection of their groups creation. 


TIP: If you can't get a Merge Cube for each student, there is a shortcut. I printed a picture of one side of the Merge Cube and you can't rotate it like a Merge Cube showing each side, but students can use that one side to create especially using the Dig! app.

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. 

Especially closer to the end of the game the taggers had to come together to talk about which one would move the most amount of people and get a specific person out.

Students used the following:
Alternate Exterior
Alternate Interior
Consecutive Interior
Linear Pair

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.