Exponential Problem

It has been a couple of weeks since I taught exponents and logarithms. We do all the normal things we do with them in Algebra 2. We changed from one to the other, we graphed, we looked at their properties, we did half-life and doubling problems. When my students took the test I was happy to see good scores this late in the school year when motivation can run low.

Today I found a great problem on a site called FiveThirtyEight it has a weekly puzzle people can submit to and it was relevant because Avengers EndGame had just come out. The problem looks like this:


I was going to use this as a bell ringer. I thought it would take them about 2-3 minutes to do. I gave them no further instruction than answer the question. They could use their notes from the last chapter, anything that could help them. After 5 minutes, some students had started to give up. Some students were having rich-mathematical conversations with each other. Talking about halving, did the number of Thanoses also halve, does the original Thanos die when he snaps his fingers. This bell ringer that was supposed to be taking 5 minutes at the maximum is now going on 10 minutes. I felt like there were some misconceptions that students about this problem and that more time would help solidify exponential equations.

I let students struggle for the next 25 minutes, working with partners or in groups, using their notes. Some of the students and groups persevered through the problem, other groups were harder to get moving through the question. Here are some examples of the work my students did.

 



I taught this class three times, after the first time I went to some of the teachers that teach the same thing and asked them if this was beyond their thinking, after a while they decided that the students should be able to answer the problems from what we went over in class, but they added that the student’s wouldn’t be able to get an answer. After two days of sitting on this I went back to them, I wanted to know why they thought the students couldn’t answer the problem and what we could do next year to remedy this.

Here were some of the teachers responses to why the students weren’t able to answer the problem:
  • One teacher said we did exponentials and logarithms too long ago for students to remember. Also students only remember or do what we teach them.
  • Another teacher said students don’t know how an exponential equation works.
  • Another teacher said students can’t persevere or formulate questions.

Here were some of the teachers responses to what can we do differently for next year so we can solve this problem:
  • One teacher said we need to be more connected across Algebra 2, we should start with a question and answer it over the course of the unit or chapter, it needs to be an expectation that students can problem solve, students need to get use to out of the box problems from the start of the year.
  • Another teacher said we need more modeling and that students need more practice at it because the capability is there.
  • Another teacher said we should start off the chapter and year with questions like these, we need to change the way vocabulary is taught especially when it comes to a problem like this, we need to teach students how to use a simpler problem to solve.

I agreed with all of the responses that we should be doing next year. When you value problem solving over speed and simplicity you help your students become better mathematicians. This was a good problem for my students to work on for a day, this will help me become a better teacher for them in the future.

Creating a Polynomial Function

One thing that I have been looking forward to trying in Advanced Algebra is more tasks. One of my goals for this year is doing more mathematical talking and orchestrating better discussions in the classroom. One task I created recently that still needs work is this creating a polynomial function from zeros task. Previously students have only factored and solved polynomials with P's and Q's. I was thinking they could work their way backwards from the previous days material.





Here is a link to a better looking version.

I gave them the task to work on for 15 minutes and I went around the classroom getting students started and answering questions.  If I had more time I would have done a better job of calling students up and presenting their work, but I was running short on time. I was hoping their thinking would help them think backwards especially with day before lesson.

Some places they might have gotten stuck at that I had posed responses for:

  • Students will have a hard time getting started (it feels like a high entry task)
    • I will ask them to go through an example problem we have done before, such as x^2-4x+3
  • Multiplying the "i's" together, it has been about a full quarter since they have seen imaginary numbers multiplied together.
    • I will ask them what i squared is? What do they remember about imaginary numbers?
  • Distributing correctly.
    • Ask them how they would distribute (x+2)(x+2)?
If you are really digging this lesson, you can view 28 minutes of the lesson here

I still feel like the entry to the task is too high, I would love comments on how this could be a better lesson.

Trig Lasers

One of the things I don't get to do often, but love to do is observe other teachers. I get to see how they run the classroom, how their routines are set-up, and sometimes see really cool lessons. In Basic Geometry Mr. Fick and Ms. Groth had a lesson they were talking about that seemed really cool to observe. They had a bunch of different targets printed, lasers, and protractors.

To review the trigonometry they have been practicing in class. Ms. Groth had one student come up and set a target on the front wall of the classroom. She had already set a piece of tape on the ground, she wanted to know the angle that would make it so a laser would shine and hit the target. She asked the class what angle they thought it would be and students guessed. She then asked what other information is needed to know so they could accurately calculate the angle. A couple of students said the height and the distance. Those two students came up and measured both of those. Students were given a handout to write down and then calculate the given angle.

Ms. Groth then did another example where she already gave them the angle and height and they had to calculate the distance away.

I think a great extension to what I saw would to give each pair of students a target, laser, and compass. Then have students calculate their own. Or possibly have some sort of putt-putt kind of station moving from one target to the next.











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
11/121Review
11/131Quiz
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: https://mathcoachblog.com/2017/06/05/area-models-and-completing-the-square/

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.

https://qz.com/918008/the-color-distribution-of-mms-as-determined-by-a-phd-in-statistics/

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.

Extensions:

  • 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.