Are You Looking for Piece-wise Functions

My Struggles With Completing the Square

This year I have been struggling with this quadratics unit. I feel as if there are no cohesive elements tying one concept from one to the other. The outline from this unit has been as follows:

11/63-11Graphing Quadratic Functions
11/73-11Graphing Quadratic Functions
11/83-21Solving Quadratic Equations by Graphing
11/93-41Solving Quadratic Equations by Factoring
11/143-31Complex Numbers
11/151Complex Numbers
11/161Complex Numbers
11/193-51Solving Quadratic Equations by Completing the Square
11/201Solving Quadratic Equations by Completing the Square
11/263-61The Quadratic Formula and the Discriminant
11/273-61The Quadratic Formula and the Discriminant

With 35 minute periods I have been trying to include different strategies, but overall I want them to know that by completing the square we are finding vertex form of a standard form of a parabola. 

I had students use a visual model of completing the square with algebra tiles from Bob Lochel's site:

They came away from this with the idea that they are actually trying to fill out a square. The next day I borrowed Julie Morgan's ideas and used this tweet.

I felt like it tied it more directly to properties of a parabola, but I feel like I need a better way to assess what they really know about parabolas.

I think I am going to use Nat Banting's post about writing everything they know about a specific parabola and print out each one for groups or partners to discuss.

I need to find a better way to teach this where I am developing conceptual and procedural understanding evenly. If you have any thoughts let me know.

A Flavorful Application of Mean, Median, Mode

I was looking for a different way for students to apply their knowledge of mean, median, and mode in Algebra 1. I wanted some application where they can use mean, median, and more in a different context. 

I found this article a couple of months ago and found it really interesting.

I had students read this article and groups and come up with what they noticed/wondered about the article. I asked the students if I gave them a pack of M&M's what they would be to extrapolate from the pack. They said they needed to know the total M&M's in the package to determine the location where they were made and the color breakdown inside each one.

Each group of students were given a package of M&M's and had to count how many of each color they had.

Then we put all of our data on the board. Students had to come up with the mean, median, and mode for each color and had to decide which of the data sets to use if they had an outlier or not.

They had to look back at their data and examine which factory their set of M&M's came from. We also talked about what might be different from their graphs to ours and how you might be able to tell the difference between each plant.

This was a great activity for students, they were excited because they got to share and eat their M&M's when they were done.

3 Act Task: Bacteria Growth

Act 1:

How many bacteria are there in 300 minutes?

Act 2:

  • Pause the video at 0:00 time frame. There is 1 bacteria at 0 minutes.

  • Pause the video again at 0:01 time frame, when there is 2 bacteria at 38 minutes.

  • Pause the video again at 0:04 time frame, when there is 4 bacteria at 88 minutes.

  • Lastly, pause or show a screen shot at 0:05 time frame, when there is 8 bacteria at 114 minutes.

Act 3:

There are a total of 471 bacteria in 300 minutes.


  • When does the number of minutes and bacteria the same?
  • How many bacteria would there be in 500 minutes?
  • When will there be 1,000,000 bacteria?
  • How else could we represent this information?

My #1TMCThing

I've been going back and forth on what my takeaways from the conference were, because every session I went to felt like I took something away. I have been going to quite a few different conferences mostly around the state of Nebraska and went to ISTE last year in San Antonio and felt like I haven't taken away anything in a long time. Since I have to narrow it down I will talk about two big takeaways I felt permeated through my TMC experience.

1. Using equity and social justice as a way of teaching mathematics. Marian Dingle and Wendy Menard took us on a path of self discovery and identity to teach students better. Using social justice standards to help frame lessons and discussions to help build classroom activities.  The last day we split into small groups and the group I was apart of focused on ideas for classroom activities that promoted social justice, some of the notes taken from that are here.

This is a conversation that I have wanted to have with others in my building for four long years. Are we doing what is best for all students? What practices and approaches can we take to include all students? This is a discussion I want to keep having, especially with others in my state.

Also I want to thank Dr. Robert Berry for being there and representing NCTM, I haven't been a member for three years now, but seeing him there shows the trajectory that I think NCTM will go and I think it is a place that all teachers should evaluate. I will definitely be renewing my membership to NCTM.

2. My second thing that I took away from TMC was how incredibly nice everyone is. When I showed up that first day early to register I was given my badge and a button to wear that said, "FIRST!" and since I showed up early I thought to myself that this might be a badge of shame, that this would have been my first time there. However, wearing that badge the first day other people came up to me and made connections with me. For one of the first times in the #mtbos community I felt like I belonged, all because of a simple badge.

I would love to go to #TMC19 in Berkeley, but others deserve a chance to go and I will wait till it makes its way back to the Midwest to go again.

Reflection: Representations of Relations

The one thing I am really nervous about when starting this new school year is the bell schedule, 35 minutes for one period. What can I get done in 35 minutes? Can I get a Desmos activity done in 35 minutes? Could we do a class project in 35 minutes? I feel like I will be cutting out important discussions or big "ah-ha" moments with less time.

So I went back to thinking about when I start the second day of class with mathematics, I want there to be context in what the students see and do in math. In my last post about Representations of Relations I received this comment:

They changed the focus of that first lesson to make the rest of the year one cohesive goal.

Image result for gif that's what i want

As part of the curriculum group for Advanced Algebra, we set the pacing guide and decided that representations of relations along with the distributive property should be taught first. One thing I want to get across to students that first week is everyone having their voices heard and problem solving.

So each wall of the classroom will have the same layout from the previous post,

  • one wall will have ordered pair along with a piece of butcher paper with notice/wonder at the top. 
  • second wall will have a mapping with a different piece of butcher paper with notice/wonder.
  • third wall will have a table with butcher paper labeled the same way.
  • fourth wall will have the New York Times graph and butcher paper.
I want to have students stand (and gather by the board) and take 1 minute to look and 2 minutes to discuss with a partner what they see. I will ask what students notice first, then wonder. At the end I want them to discuss what was similar or different with the four different relations.

I still have to cover distributive property at the end, but as an exit ticket I want them to reflect on the experience and answer the following question:
Why did the New York Times select a graph to represent this relation?  
I want a connection more to relations, what question should I ask that encompasses what they learned and that representing functions is useful?

Representations of Relations

This is going to be my fifth year teaching Algebra 2, this year I am changing schools so it will be called Advanced Algebra. I was doing some curriculum writing with my new team of teachers and the first section we are going to cover as a class is Representing Relations my first reaction was...

Image result for why gif

At this point with just coming back to school students may not remember what relations are, what functions are, or domain and range.

I thought back to Dan Meyer's talk of headaches and aspirin and why do students need to know there are different ways to represent relations: ordered pairs, tables, graphs, and mapping.

I took the second graph from New York Times: What's Going on in this Graph?  and re-organized the information differently.

Give the students the following information:
This data is organized from by: country (guns per 100 people , mass shooters per 100 million people).
United States (85, 28)
Canada (26, 9)
Afghanistan (2, 20)
Iraq (37, 4)
France (35, 15)
Yemen (55, 40)

Ask the students what they notice? what do they wonder? As the teacher write down everything they say. One question I have is what is the data saying? Is there a different way to represent the data?

Give the students the following information:

What do they notice and wonder now? What has changed? You can show them mapping as well, but eventually you will need to introduce other things, but the last one is the graph from The New York Times.

How do the three representations differ, do they all tell the same story? Do some tell the story better? I'm not sure if this is the way to start the year out, but nothing is perfect. I want students to feel that there is some context to mathematics other than its day 1 therefore we do lesson 1.

"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.