Here's the link to the Dropbox presentation from today. Please contact me if you have any questions about how you can use Dropbox in your classroom.

# Teaching Systematically

## Tuesday, August 30, 2016

## Friday, July 29, 2016

Was just at #MICTM16 this week and got to see Mr. Dan Meyer speak about creating mathematical headaches before giving students the mathematical aspirin. He even went as far to say that math should be like an infomercial because they do such a great job at selling. (Look up pootrap for a classic infomercial.) Dan pushed us to send him ideas on how we do this in our classroom and so here is an idea that I use in my classroom. Now, I definitely stole this idea a few years back from someone on the #MTBOS but I can't remember who it was. So if anyone knows who had this original idea please let me know so I can give credit to him/her.

I've used this both in Precalculus and in Intro to Statistics. Every time we have great conversations about how organizing, graphing, and understanding data is a big deal right now with big data. I start by asking students to give me a list of countries, and then students look up population numbers using WolframAlpha. I love using WolframAlpha for this because of all the other information that it gives and students really get into some of the other things that come up. The list of countries always varies but it usually contains either China or India, and a smaller country like Sweden or Denmark, or the person I stole this idea from put up Vatican City to really force the issue. What you need is a wide variety of data. Students then graph the data, by hand, and I stress that I need a SUPER accurate graph and the values on the y-axis must be consistent. Students ask if they can use the squiggle mark (what's that thing called?) on the y-axis, or ask for another piece of graph paper so they can fit in China or India. Both of which I refuse to give. Some students just jump head first into graphing, others just sit there and pause because they know they can't do it, but after awhile I grab some of the graphs and throw them up on the doc cam and have some rich discussions. Almost always the students are using a linear scale, but sometimes there's one student who uses an exponential scale. This is great because then we go back to how the y-axis must be consistent and students argue that it's not consistent even though we have usually just talked about exponential functions and even have laid heavy ground work for logarithms. Vsauce has a great video (1, 2, 3, 4, 5, ....., I'm sure you've seen this) that I use to help with this foundation and turning their thinking back to multiplicative that is more natural anyway. After the students come to terms with being able to count multiplicative-ly on the y-axis we then talk about how the numbers are too large to write down and that in math we can use a LOG! (Grumbles here since they all have bad experiences about logs from previous classes.) We then take our data of countries and their populations, and add another column by taking the common log of the populations. Then I let them sit and think about what these numbers mean. It doesn't take too long for students to figure out that the number is based on how many digits which leads to great decimal system talk, but then I pose the question, what does the decimal mean? This takes some thinking and more guiding (depending on the level of the class) to get the students to see that its how close the number is to the next power of ten, in a multiplicative sort of way. This again ties back to the VSauce video. So for instance Sweden has a population of 9 million, which log(9,000,000) = 6.954 and India has a population of 1.29 billion and log(1,290,000,000) = 9.11. So Sweden is closer to 10 mil than India is closer to 10 bil. Obvi. But usually we have the US on the list which is the perfect country for this because the population is 322 million and log(322,000,000) = 8.508. This means that since population growth is exponential the US is halfway to a billion people!! Then all chaos ensues!

I love this activity because it takes out the "mathemagic" when you take a number and hit the "log" button on your calculator. I also do a different activity for "sin" or "cos" on the calculator and getting another weird decimal as an answer. I really think that some students think that these functions on the calculator just randomly select these decimals when they really come from pretty simple ideas. I guess I'll have to write another post about this topic as well, but this is my first blog post in two years. I don't want to pull a muscle.

## Saturday, September 13, 2014

### No Calculators Allowed!

Ok, of course there's a time a place for calculators in math class, but in my classes I try to use them only when appropriate. It is a total shock to my Precalculus students when I tell them that they can't use a calculator on the first day of class. They're preconditioned to walk into a math classroom, grab a calculator from the classroom set, and sit down. The majority of the students say "I can't do this without a calculator" as we start to get into the math. It only takes about a day and half for them to realize I'm serious that you need to convert 325 degrees into radians without a calculator. It's not that I want them to be able to do 325/180 in a blink of an eye, cause I don't, but as I walked around the room this year and had quick conversations with groups of students it becomes clear to them why I'm taking away their "best friend" in my class. Believe it or not, and I know you'll believe me because you probably see the same things in your classes, a lot of students really have to think long and hard before they even realize that 5 is a factor of both of those numbers. That's when I explain to them the reason why I'm taking away the calculators is not so they have to do 180/5 on paper but because it's not automatic for them to know numbers that end in 5s or 0s are divisible by 5. Then we can also have a great conversation about how dividing 180 by 5 might be difficult, but 180/10 is not and my answer is half as large as 180/5.

We have also had a lot of practice these first two weeks of school with fractions. Students don't even know where to begin with fractions, and by taking away the calculator they are forced to deal with them, forced to think about how they interact with numbers, and forced to finally learn fractions. One of the first homework assignments is a review of solving equations with one problem containing a fraction of a third and I'm always surprised at how many students change this to 0.33. I usually give an analogy the next day like if you have a dollar and have to split it with two of your friends, everyone gets 33 cents and a penny is left over. No biggie. And if you have $100 everyone gets $33 and only a dollar is left over. No biggie. But when you're rich a successful and you and your two business partners have to split $1 billion.......call me up, I'll take your leftovers. It's a dumb analogy but the students like it and I get a lot of students to stop changing fractions to decimals like that.

I do have a lot of motivation for getting the students to do more mental math since I teach a section of AP Calculus (more than half the AP test at the end of the year is no calculator) and most of my Precalc students will take AP Calculus the next year. It's only my second year at this school but historically our students have done the worst on the no calculator multiple choice part of the AP test. Maybe it's because the middle schools at one point stopped teaching long division, or other basic fundamentals that are going away that probably shouldn't. Anyhow, I'm trying to work on my students number sense and doing so by taking away their calculators.

### High Five Fridays!

It's been awhile since I've posted. A few years in fact. After having going through some big changes in my life, (a move, a kid, a new teaching job) I think I'm starting to get the itch to really get into this blogging thing.

1. It breaks up the monotony for the students and myself and gives us something to look forward to.

2. It shows the students that the world of mathematics doesn't fit into the small box that is controlled by the sequencing of textbooks, ACT College Readiness Standards, or the CCSS.

3. It (hopefully) gets students to appreciate the field of mathematics and understand its vastness.

One of my goals as a math teacher is to get more students interested in STEM related fields. I feel that by only teaching the traditional scope and sequence more and more students will choose not to go into math and science related careers. This is especially true for those students who don't get to the more advanced courses and never get past the idea that math is not just a bunch of rules that don't make sense. I hate that it's ok in our society to say "I'm just not good at math." Hopefully, by showing the students that traditional high school math doesn't define math I'm hoping that students get a better understanding of what math is and create curiosity to want to learn more math.

What I do each Friday isn't that special. Typically it's just a short YouTube video that I found at some point and a short history/background of the topic. Sometimes it's related to the topic we're learning about, and sometimes not. Sometimes it's science related, and others are just a cool, blow your mind video. You know the ones. The Vi Harts, Veritasiums, Minute Physics, etc. I've been teaching for six years now and I can't really remember when I started this, but the past three years I've made it a point to do this every Friday. The students look forward to it, and they also get a celebratory high five as they leave class. This year I decided to create a sequence for each grade/course level I teach with the plans of expanding this to all 9-12 math courses. I mean I've kept a list of the ones I've used in the past but I didn't always keep track of each one and they weren't in any kind of order. The sequences aren't complete, in fact I don't even know what I'm going to do for this coming Friday. I used to just use the same plan for all my classes but I'm thinking that if this is going to be a thing, then they should be different. A lot of the students I have in Precalc I will see again in Calculus. My ultimate goal would be to roll this out so that all the math teachers in my school/district could pull from these lists and use them if they wanted to, which is another reason to have no repeats. Of course I would also love to share this with the MTBoS and would love to get suggestions from you. When I get a decent sized list going I'll post it here so check back soon!

## Thursday, July 10, 2014

### Reflective Teaching

It's summertime which, for me, means a lot of planning a new course (is there ever enough time to do it right the first time?), reflecting, and … teaching summer school. I teach an accelerated Pre-Calculus course for 12 students looking to take AP Calculus their senior year and every time I teach this class, I’m confronted with the concept of reflective teaching. I think reflection is one of the most understated component of not good but great teachers. This summer school curriculum (and most of the materials) was assembled over the years by my mentor teacher, Steve, and is a historical snapshot of what great reflective teaching can be and should be.

In Steve’s 1,826 files for summer school he has titled each document specifically. You might see “Concept of a function”, “Concept of a function v09”, “Concept of a function v10”, “Concept of a function v11”, and finally “Concept of a function v13”. Most of his instructional activities have been rewritten multiple times and then passed onto a group of his mentees. Each iteration has a sometime small but profound change that enhanced his students learning. Sometimes he swapped some problems out for other, more meaningful, ones. Other times he might add a section and other times, take a section out that led to misunderstandings.

Steve had documents in each folder that were titled “0-tn-functions” which means “Teacher Notes”. This document historically outlines these iterations and reflections to then create meaningful changes each subsequent year. He listened and observed student learning then reworked each instructional activity to better bridge his curriculum to his assessments which were vertically aligned to the AP Calculus course (another blog entry). Steve had phenomenal results with the largest group of students taking AP Calculus in Chicago as well as one of the highest pass rates. I would contribute a certain portion of this to his reflective practice.

A Snapshot of the Quadratics Folder |

I find that we have so many things on our plates that we don’t take time to reflect. I encourage everyone this summer to start somewhere - reflect on each unit in a course: what worked? what didn’t? Also, set yourself up for success - add a “Reflections” box to your lesson plans or unit plans or put a note in Google calendar to reflect once a month. I’m working toward these goals but am no where near where I need to be. I’m open for suggestions on what

**you**do to make this a part of our daily practice.## Sunday, March 23, 2014

### Why we can't find Malaysian Flight 370

*This entry is with full respect to those individuals and their families who have been missing since the flight's disappearance.*

Our textbooks like to give “Real
World” problems. Too many times these are at the end of the
section or under a section called “Modeling” and they are so
canned, they are unrealistic - thus defeating their purpose. This
pedagogical paradigm that our textbooks frame assumes that we should
teach skills before content but this is like teaching vocabulary
without context and in the world of vocabulary learning, context is
everything (think spelling bee... Can you use this in a sentence?) We
need to be always teaching context for math just like any other
language always uses context and the Malaysian Flight 370
disappearance is a perfect example. Many people are wondering –
why haven't we found the floating stuff that we see with the
satellite?

What a perfect related rates problem.

If I wasn't on spring break right now,
I'd ask my calculus class this exact question. Here's the
information:

-7 days ago the Australian government
spotted a big object floating in a location 1,400 miles south of
Perth

-The current in the South Indian Ocean
flows at 1m/s.

-We can imagine that the search area
becomes a sector area (assuming its still floating) – we can make
up some reasonable angle such as it can move 15 degrees in either
direction from the center line of the current.

Now ask the question again given this
information and see what happens in a calculus class. A follow up
question might be: what is the rate at which the search area is
changing every day/hour/minute/second. I would love to see students
problem solving (making diagrams, asking questions, etc) and seeing
why its so hard to find debris in the ocean.

I found this information on CNN.com so
why not integrate literacy into the lesson by having students read
for information (your English/Social Studies/Science teachers will
love you because they always assume that students are learning math
the way they did 25 years ago). As of yesterday, CNN also states “
the current search area is 2.97

*million*(Thanks to Dan Meyer for checking me on the details) square miles” - that's roughly the size of the continental US. Can you imagine 15 planes and 30 ships trying to find something 100 feet long somewhere in the continental US?
A great follow-up
discussion would be how to maximize the efficiency of searching,
hoping that they find the objects soon would be even more remarkable.

(Want to make
this an Trig/Precalc lesson? Ask students about how the sector area
and how height changes the distance which a person can see on the
ocean – how many ships are necessary and of what height to cover
the search area?)

For more problems like this check out Stu Swartz's "Ripped From the Headlines" at mastermathmentor.com

## Sunday, March 9, 2014

### Horizontal Asymptotes & End Behavior

It's been a while since we've updated this blog, but we're back!

This is a lesson that we did way back in October (better late than never!) in my honors precalculus class. In this short activity students investigate what happens to rational functions toward infinity and negative infinity. The activity is fairly simple, but it helps kids improve number sense a whole lot more than memorizing "

The "Quick Solve" was fairly straightforward for the students. It was easy for them to see that as x gets extremely large the "plus 4x" starts to matter less and less. As x approaches infinity the "minus 3" barely even matters at all. Once they move on to the rational functions they can start thinking about how the numerator and denominator grow in relation to one another, with just looking at the terms that has the most impact on the fraction.

I liked that this activity got the students away from trying to memorize rules to find horizontal asymptotes of rational functions. Once the students completed the table on page 1 we discussed as a class what these functions look like in the coordinate plane as you approach positive and negative infinity, ignoring what's happening between the extremes. This got us into the whole discussion of end behavior (they have already been working with limit notation) and the kids started to think about the "ends" of these functions when there was no horizontal asymptote.

All in all, I think this activity did its job - there's no real practical applications embedded here, but it makes the students just

This is a lesson that we did way back in October (better late than never!) in my honors precalculus class. In this short activity students investigate what happens to rational functions toward infinity and negative infinity. The activity is fairly simple, but it helps kids improve number sense a whole lot more than memorizing "

*if the degree of the numerator=the degree of the denominato*r"....The "Quick Solve" was fairly straightforward for the students. It was easy for them to see that as x gets extremely large the "plus 4x" starts to matter less and less. As x approaches infinity the "minus 3" barely even matters at all. Once they move on to the rational functions they can start thinking about how the numerator and denominator grow in relation to one another, with just looking at the terms that has the most impact on the fraction.

I liked that this activity got the students away from trying to memorize rules to find horizontal asymptotes of rational functions. Once the students completed the table on page 1 we discussed as a class what these functions look like in the coordinate plane as you approach positive and negative infinity, ignoring what's happening between the extremes. This got us into the whole discussion of end behavior (they have already been working with limit notation) and the kids started to think about the "ends" of these functions when there was no horizontal asymptote.

All in all, I think this activity did its job - there's no real practical applications embedded here, but it makes the students just

*think about numbers in general*; something I've been really trying to push with my students this year.
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