Second Breakfast: Day 105 Or Thirsty

Some days you teach really hard and at the end of the day you are just thirsty from talking so much and I know that some teachers might say that’s a bad thing and some days I might agree with them but today we did point recovery for the Unit 1 test (as mandated by the school since we cannot do test corrections) so I spent time going over answers that were commonly missed on the exam and I could have done something that was collaborative but I am definitely feeling that pacing guide breathing down my neck and so I could only spend so much time on it and sometimes the facade drops and you don’t even try to hide the fact that 90% of these blog posts are stream of consciousness so you just write in one big, long run-on sentence that just keeps going and then starts being self-referential and self-referential about being self-referential and I’m going to stop the the loop now before it gets away from me and actually this is sometimes how I teach when I get excited about concepts and the words get fasterandclosertogetherandIhavetoslowdownpurposefully and bring it back to a slower pace that is more reasonable. Lately I have been varying my tone and volume more to really emphasize certain points. I have always tried to speak dynamically with my students. (The only thing worse than a math lecture is a mono-tone math lecture.) But I haven’t been able to vary volume very much because I haven’t had enough control of my classroom to really get a quiet group while I am talking. But this semester I have had much more success so I have been able to follow SUDDENLY EMPHASIZED STATEMENTS with soft and slowly spoken follow ups as explanation.

I know lecture isn’t everything but it has been nice to work on improving my skills as a lecturer this semester because I have had more attentive classes. I still need to improve on my facilitation of group activities, but I am making progress there.

Anyway I know I haven’t written much about my actual lessons today, but that’s because they were pretty boring and routine.

Warm Up

Homework Check

Review and Practice from previous day

Introduce New Material

Practice New Material while integrating old material

Skills Check


The bread-and-butter as it were.

Thanks for reading.


Second Breakfast: Day 104

I had a fun time yelling (“yelling”) at my Geometry kids today about Zeno’s paradox and then we did conditional statements to continue the intro into formal logic.

My foundations students responded pretty well to my follow-up lesson introducing the three fundamental rules of algebra for solving one-step equations. On Monday we will revisit them until we’re sick and then move on to two-step equations.

It’s been a full and busy week and I am tired so I think that’s all I am going to write for today.

Thanks for reading.


Second Breakfast: Day 103 Part II Or Geometry Had A Great Day Too

So my CP Geometry classes also went really well today.

I felt really lucky that all of my lessons turned out so well because after giving a test Wednesday I didn’t really think about what I was going to follow up with and so I was scrambling this morning to prepare for all of my classes.

I really love the unit we’re starting in Geometry (Reasoning And Proof) since I get to really get on the soap box about how reasoning and proof is what all of math is. I could tell I was really waxing poetic today in class because my students’ eyes were glazing over as though covered with wax paper.

ANYWAY, I started today with a nice warm-up that I adapted from a John Stevens Would You Rather? tweet. I adapted to how it reads below:

Daily Warm Up

I changed it to read this way because I was concerned about some of my students who can be mean to each other. Although I enforce kind language in my class, I was hoping to avoid the problem and I was afraid that there would be hurt feelings if we started discussing weight.

The downside to this version is that it takes some nuance out of the problem. A very tall and very skinny person might choose differently from me, a stocky guy that’s only slightly above average height.

But as it was it was still good. I was once again amazed at how the same problem can cause such varying reactions among different classes. All three blocks responded differently to it.

After the warm up I went to well that I have often go to before, Jo Boaler’s YouCubed Pascal’s Triangle activity. Asking students to complete the triangle has worked really well in the past when I wanted to show them that math is more than just following an algorithm to find an answer, and I think it’s a great way to start a lesson on inductive reasoning and finding patterns.

This went really well in both blocks. In fact, one student pointed out a pattern in the triangle that I have never seen before and I was blown away. I was very excited.

I let the students explore while I circulated the room. Every time I have done this I have been pleased with the results. Students naturally start sharing ideas and answers with each other. There is something about completing the triangle that hits exactly in the right spot: easy enough that students get started without fear, hard enough that they don’t realize the answer right away with plenty of extensions and side-trips to make even if they realize the basic pattern building the triangle. (What we call a low floor, high ceiling task.) For some things you have to train and help and remind students, but a task like this seems to naturally activate some innate human mathematical ability for pattern recognition. I love it. It’s great.

After a whole class summary and some sharing from students, I moved to a slide deck that started with some GH Hardy quotes about mathematicians being makers of patterns and so on.

I then gave the definition of inductive reasoning from their textbook and followed it with my “simplified steps for inductive reasoning”:

  1. Look for it
  2. State it
  3. Prove it

Where it is the pattern or conjecture. I think this went over pretty well. I really wanted to emphasize with the students that proving a conjecture required showing it was true for all cases or by finding a counter-example. So I introduced the Goldbach Conjecture. I think it’s perfect for this lesson because the conjecture is easy to state and understand and then it’s a bit mind blowing when you see on the Wikipedia article on it that we’ve checked over 400 trillion (400 TRILLION!) cases and we still don’t have a proof. And then I immediately followed this up with patterns from Ben Blum Smith and James Tanton that break very quickly. I think this really drove home the point.

We finished up the last 30 minutes or so with a problem set from the text and then a skills check over some unit 1 ideas.

All three classes today went well but now I am sitting here at home writing my blog at 1030 PM and once again I am going to have to get up in the morning and come up with something. It’s hard to get ahead, but I really love this job on days like today.

Thanks for reading.


Second Breakfast: Day 103 Or The World’s Easiest Math Worksheet

What a day.

Having wrapped up the first unit for Foundations In Algebra (Real Numbers, Rational Exponents, and Simplifying Expressions) it was time to start the next unit, solving equations.

This will be the third time that I have taught Foundations because of our semester/block system. I have finally started to figure out how to start the unit the way I want. Ever since I discovered Glenn Waddell’s Three Fundamental Rules of Algebra I have been in love with them. There are posters of them in the front of my classroom. Little papers with them printed are on the other three walls. I preach it every time I solve an equation. I think they are an elegant and wonderful way to help students understand the why without sacrificing the how.

But the problem I have run into is that my Foundations students are not strong enough in their understand of the roles and properties of 0 and 1 as identities. I have tried to begin the unit with lessons on 0 and 1 in the past, but with mixed success. Today I tried something new. Introducing… (fanfare)

The World’s Easiest Math Worksheet

As you can see, it’s just a bunch of x+0 or 0+x problems on the first side and then a bunch of 1*x or x*1 problems on the second side.

I played the title up in class with same hammy acting and then some reassuring. “No, really, gang. I’m not joking. It’s the world’s easiest math worksheet. Look at it.” Almost all of my students whipped through this in less than 2 minutes after they realized I wasn’t joking. After I gave them some time to complete it I called out for answers and got chorus responses. They started to laugh by the second page. I don’t know why it was funny, but it was. I started laughing too as they said the answers together, giggling.

So far, so good.

Then, I introduced… (more fanfare)

A Worksheet Slightly Harder Than The World’s Easiest Math Worksheet

As you can see, I use some very questionable terms at the bottom of side one, but in my defense, it was early and the inspiration for this idea came to me in the shower the morning that I did this.

The pre-loading done by the world’s easiest worksheet did what I hoped. The students were able to answer the first four questions. I did have some confusion from one of my quicker students about the last two questions because he was confusing identities with commutative properties, but that produced some nice conversation between him, myself, and a few other students. It wasn’t a whole class debate like would be ideal, but still, I was pleased with the direction we went because of his confusion and willingness to voice it.

And this led us to the back page of the slightly harder worksheet. The students were able to quickly produce the numbers that would make a zero or make a one. In the past, I really pushed the “additive inverse” and “multiplicative inverse” vocabulary and I think insistence on precision of language was a hindrance rather than a help. With this worksheet I am hoping to set up a reference so that if I say “How do you make a zero?” or “How do you make a one?” the prompt “Remember the world’s easiest worksheet?” will be effective.

Tomorrow we will start with some more “make a zero” and  “make a one” practice problems and then I will introduce solving one-step equations with the three fundamental rules of algebra.

I am excited.

Thanks for reading.


Second Breakfast: Day 101

In Foundations I wanted to help the students model combining like terms. Mechanically speaking, my more vocal students seem to be pretty solid on the idea, but I wanted to make sure the class understood what they were actually doing. I tried this yesterday in groups (I don’t remember if I wrote about it in yesterday’s post, and I’m too busy / lazy to check) but today I had them work individually. I placed tiles on each student’s desk and had them model the tiles with an expression. Then I added or subtracted tiles each time I walked by. Students had to model the action that I did AND THEN combine like terms. Finally they had to check to make sure the expression that they wrote actually matched the reality that was on their desk. This seemed to go pretty well.

In Geometry I gave a practice exam and then we played a review game to prepare for tomorrow’s exam. I made the mistake in 3/4 block of not asking students to turn in their work from the game. Luckily that block is very on-task so I didn’t have problems behaviorally, but I think a few students may have checked out. My 6/7 is still very good, but not as motivated so I corrected that error and the game went very well.

I think I am still on the fence about review games. I like the engagement they can build and it’s a nice break from endless practice problems in whole class, group, or individual, but I am not sure how learning effective they are other than to give that brain break.

In any case, Semester 2 is still smooth sailing so far.

Thanks for reading.



Second Breakfast: Day 100

Today was a tutoring day and it was busy day for once. I had a table with 3 students. Two of them were in Algebra 3 and required me to remember Cramer’s Rule and determinants of 3×3 matrices, of which one of them spoke English as a second language, and then the third from a Geometry class who spoke no English at all. So my time was spent helping Algebra 3 in English, then helping Algebra 3 in English and Spanish, and then enlisting Algebra 3 to help me help Geometry in Spanish.

Anyway, all of that is to say that I didn’t have time to write my post this afternoon and now that I am home I need to do taxes.

Thanks for reading.