Day 104: Making Problems Better

Irrelevant, out of date, overly complicated, boring: These are just a few of the ways that I could describe the problems that are in our text book.  I found a great example today that I wanted to share:

The function c = 0.5m + 1 describes the cost c in dollars of a phone call that lasts m minutes made from a room at the Shady Tree Hotel. Graph the function. Use the graph to determine how much a 7-minute call will cost.

**yawn**

Dan Meyer talks about ways that teachers can take bad problems and make them good.

This problem is pretty bad.  Other than the incomprehensible wording, confusing presentation and forced premise, I can't figure out what skills or content this problem is supposed to be practicing.

Is the purpose to get students to see that functions are used in real life?  Is it supposed to imply that phone bills will arrive covered in algebra?  Does it want students to feel that world is a confusing and incomprehensible place?  Does it just want them to be able to graph a function with the basic direction of "Graph the function c = 0.5m + 1" cleverly disguised as a practical, real-world problem?

Does it want them to hate and fear mathematics, making them test averse, driving down scores on the standardized tests designed by those same companies, forcing districts to spend their scarce resources on quick-fix curricula designed by the companies that instigated and perpetuated the "downward spiral" of those scores in the firsts place?

I don't know. I'm just a humble math teacher.

What I DO know is that this problem is terrible.

So I used it as an opportunity for my students to practice the skills that I value.

"What in the world are they asking here?"
 **crickets**
"Does Verizon put a formula like this in their commercials?  Have you ever seen it written that way?"

They had not.  We had a discussion about how we could reword the problem to make it more real and less absurd.  We came up with several ideas.

In order to call out from the hotel phone, the hotel will charge $1 to connect the call and $0.50 for each minute that you talk.  Create a graph that shows how cost and minutes on the phone are related.  What equation would describe that graph?

Not perfect, but much better.  I want my students analyzing situations that they will encounter, using wording they will see.


In geometry, I've been following the guided notes that they use at the high school to ensure that we cover the same content.

Today, the activity was one where the students explored the possibilities of sides that will or won't form a triangle.  The activity in their guided notes had them experimenting with strips of paper that were 2, 3, 4, and 5 inches long, asking them to form a hypothesis about necessary side lengths.  Picking 3 of these sides, there are 4 possible combinations.

I wanted something a bit more than this.  So I made more strips for them of lengths varying from 3 centimeters up to 24 centimeters.  There were 15,600 possible combinations, making the group work a bit more interesting.  It opened up the possibility for one person to get all combinations that made triangles and another to get none.

While they were working on that and discussing it, I build an applet in GeoGebra with sliders that let us input side lengths and see whether they made a triangle.

With the visuals, the conversation got much deeper into WHY the sum of the two smaller sides had to be larger than the third side.  The other upside is that my demonstration, along with a brief video from Pixar about how they use geometry to make their movies, has my kids champing at the bit to play around with GeoGebra.

With it loaded onto the student profiles, I hope I'll be able to get them into the lab in the next few days and let them play around with it.

Day 103: Planning to Fail

When I am asked in interviews about my biggest weakness as a teacher, I always lie.

My biggest weakness is in planning.

I get bursts of inspiration, find cool activities and tasks and engage my students.  But I could do a better job if I were willing to spend more time planning.  One of my biggest strengths is my ability to take teachable moments and run with them.  I can start with a basic activity and turn it into something deep and interesting.

I have come to the conclusion that this is not nearly enough for my Math 8 classes.  I don't need a script for them, but with 90 minutes, I clearly need a more involved plan than I've been using.  My basic plan is usually an outline of the topics that I would like to cover with specific examples that I want to talk about and I let the lesson flow organically from there.

This causes a very negative feedback cycle for me.  I don't plan quite enough, so the lessons don't go as well as I would like, so I just want to put it behind me and think about other things, so I don't plan enough for tomorrow, so the lessons don't go as well as I would like...and so on.

I did the same thing as a student and many of my current students do it as well.

"I got this answer wrong, so I'm going to put my head down and tune out."

This causes exponential problems!  In a race, if you stumble, you can't just lay on the ground.  You HAVE to get back up and keep going.

So I'm making a conscious decision to get back up.

I made a plan for Math 8 today.  It was a solid plan that involved students writing, explaining and working in groups.

It didn't go as I planned.

It went better.

My original plan was to start out with some selected entries from Visual Patterns.  We would talk about the relationships, input, output and how to relate them to functions.  Then, once a level of comfort was developed, students would break into groups, develop their own patterns and have other groups find the functions that describe those patterns.  I originally wanted this to be almost a game or contest with group competing to find the best function.

With this plan in place, I had a goal and a path.  Instead, the discussion moved in a different direction and, while I had them develop their own patterns, it became about identifying slope and y-intercept of the function from the chart.  We talked about how the difference between consecutive values of the output related to the slope of the line.

8th period did just as well, but was a bit more talkative.  I was very pleased with the discussions that we had, even if there wasn't as much depth.  Different groups have different capabilities and limits.

All things considered, I was pleased.

Day 102: Meta Feedback

I have had a toothache for about a week.  This afternoon is the earliest that I was able to schedule an appointment to get it fixed.  As a result, regardless of painkillers, I've been unable to get a good night's sleep.  This means that my resolve to fight my students' disruptive behavior has been minimal.  The results of this have been interesting.

Since my enthusiasm and energy have been low, the students have split into two very distinct groups; those who are taking advantage of my lack of engagement to use my class as social hour thinking I'm not noticing, and those who are now leaning forward in their seats, both figuratively and literally, to hear what I have to say.

What's even more interesting to me is that this latter group is primarily composed of the students with disabilities and their friends.  It occurs to me that with the "faster" students out of the way, I'm able to give the rest of the students the attention they need.  They write down what I ask and try problems on their own, knowing that I have the time and patience for them to move at their own pace.

I wish I could find a way to get those "faster" students to remain engaged.  I have yet to hit on a task or activity that will do so.  Their engagement seems to be random, or at least affected by things I have yet to identify.

The sad reality is that I'm not as good at differentiation as I would like to be.  I don't know how to make the diverse activities that my diverse learners require.  The best I've been able to do is to engage one group at a time, alternating based on days.


I've also been slack lately in having my students write about math.  The activity we did in geometry hit some of those points.

I handed out a sheet with three questions on them.  All three were about students making assertions about triangle congruence based on given information.  My students were instructed to determine if the statements made on the paper were true.  I told them to explain their reasoning however they could to make themselves understood.

I didn't let them write their names on the papers.

I wrote numbers.  Then I collected them.

Then I handed them back out to different students.

Part two of the activity was to have them critique the reasoning of others.  I gave them 7 minutes to read the answers and give feedback.  We talked about what is helpful feedback versus not.  We gave examples of both.  It boiled down to the question: If you received this feedback, would you find it helpful?

In part three, the critics gave the papers back to the original students.  Again, we talked about  what makes feedback valuable, useful and productive.  Then things got REALLY meta!

I asked the students to give feedback on the feedback.

Was the feedback helpful? In what way could it have been more so?  Did it leave you wondering what they liked or didn't like?

We had more of a discussion about the goals of the class and the differences between procedural and conceptual knowledge.

I'm still tired...  I need to get this tooth fixed.

Day 101: Arm Wrestling!

First period was on point today!  I gave them some practice problems to work on around functions and let them go.  The problems were low-floor problems that afforded them easy success.  While they were working, I put up a make-shift number line on the wall.

This group of students has been struggling tremendously with adding and subtracting negative numbers.  When I have the write a number line, they get it without confusion.  Once they have it drawn, they are able to ask themselves two important questions: Where do we start? Where do we go?

Me: "What's -4 + 5?"
S: "9?"
Me: "..."
S: "-9?"
Me: "...  Let's draw a number line."
**draws number line**
S: "We start at -4."
Me: "Then?"
S: "Then we move to the right because we're adding and we move 5 spaces."
Me: "So?"
S: "1."

I've been meaning to put up a number line for a while, but the layout of my room and the various objects hanging on the wall make placement a bit awkward.  I really wanted it to extend across the entire room, but I couldn't, so I made it work.

Before the end of the period, we had used it at least 4 times.  They worked while I put it up and finished just before I did.  So they started an arm wrestling contest.  The girls.

After a particularly tense match, I jumped in.  I toyed with the one girl a bit, letting her think I was struggling, then I CRUSHED HER!!!

The second girl, however, gave me a run for my money and I honestly wasn't sure I was going to win.  She had a solid grip and was VERY strong.  I beat her too, but only barely.

That's right. I beat two 13-year-old girls at arm wrestling.

I'm amazing!


I bring this up not to brag (although I am amazing and every should know) but because as soon as the contest was over, all of the kids went back to their seats and we went over the problems.  I didn't have to wrangle them or beg them to get back on task.  I didn't even have to ask.  I picked up my copy and they got back on task.

This was quite unusual and I chalk it up to the fact that I've been working solidly on building relationships with the kids.  Arm wrestling with those students showed them that I value what they value.  It would have been so easy to yell at them to get back in their seats and to stop goofing around.  There are times when I do that.

But not today.


Two or three times in the past week, my own children have wanted to do "experiments."  This involved them mixing various liquids they found in the fridge.  The idea of this makes me cringe, just thinking about the mess and the clean-up.  But this past week, I not only let them, but I set out a tray and helped them pick out some instruments for the experiments.

I try not to think "Oh god! This is going to be terrible! It destroys the order of my kitchen and makes things sticky."

Instead, I think "Why do I need to fight this fight? They aren't really hurting anything and it makes them happy."

I had that same thought today in class.  They had completed their work and I wasn't quite ready to move on yet.  They weren't screaming or disturbing the kids who were still working.  I was able to let them be kids.

And I could be a kid with them.



In geometry, we started a supposedly simple question that I had thrown up on Twitter.

Incenter: center of inscribed circle Circumcenter: center of circumscribed circle Centroid: center of mass Orthocenter: ...? #mtbos
— Justin (@JustinAion) February 6, 2015

After a prolonged conversation and internet search, I got it!  Here's the tidbit that blew my students away:

The orthocenter is the center of the circle that is inscribed in the anticomplementary triangle!

This is the GeoGebraTube sheet that I made for the kids.  We played around with it for an hour, looking at various other points on concurrency.


Exploration is fun!

Day 99: PLN Love and Pizza!

I LOVE my Professional Learning Network!

The people with whom I interact with over Twitter, Voxer and this blog are some of the most amazing teachers I know.  It is no exaggeration to say that they have helped me to change my life and outlook on teaching.  They provide me with resources and push my thinking in directions I would not have otherwise considered.

Today's activity in geometry was a prime example of a PLN provided resource.

On my post yesterday about Conceptual Knowledge, one of my colleagues from California, Jed Butler, linked me to an activity on Illuminations.  It's an extension of the "finding an apartment" activity that they enjoyed with a slight twist.

Today, I posed the following situation:

You own a chain of pizza places in Squareville.  Orders are taken by a central switchboard and sent out to the franchise which is closest to the order.  Your task is divide the town into regions so that you will quickly know which franchise will handle which order.

They start out with 2 franchises in the town, then 3, then 4 and finally 5.

This exercise pushed the outside limits of what they were willing to do.  Admittedly, it may have been more productive as a culminating activity when they had more confidence with the compass.  Regardless, it was interesting.

I still struggle with timing.  In this class, but also in my other ones, I want to give time for students to struggle, but not so much that the kids who are fast get bored.  With such a diverse group of learners, it's a difficult line to walk.

In this activity, one group was on to the exercise with 5 franchises while another was still struggling with 3.  If they get too far apart, the slower students get frustrated and give up, which I don't want.

If all kids were robots, it would be so much easier!

Maybe that's why the educational system tries to make them into robots...

Day 98: Conceptual Knowledge

I grow weary writing about Math 8 all the time, so I'm going to devote today's post to Geometry.

Last night, Michael Pershan (@mpershan), Geoff Krall (@emergentmath) and I had a discussion about the difference between procedural knowledge and conceptual knowledge.  It was a very lengthy conversation with increasing frustration for me because I was having trouble making my view clear in 140 characters.

My stance is that conceptual knowledge is what is used to set up a problem and procedural knowledge is used to solve it.  Even this explanation doesn't adequately state what I want, so instead, I'll use the geometry class to illustrate.

Over the last three days, we have been doing geometric constructions.  They have been using compass and straightedge to find the incenter, circumcenter, orthcenter and centroid of triangles.  They had to define those terms and tell me what lines or segments were used to find them.

"Find the incenter of the triangle" is a task that requires procedural knowledge.

Today, I handed them a map of Los Angeles.

Me: "You and two of your friends have graduated from college and been offered jobs in Los Angeles.  It's an expensive city so you've decided to get a place together.  What kinds of things do you need to think about?"
S: "Expenses, rent, groceries."
Me: "What else?"
S: "How far we are from our jobs."
Me: "Why would that matter?"
S: "It wouldn't be fair if one person was next door to their job and the others have to go across the city."

I gave them the addresses of their jobs and told them to find a place to live.

I didn't give them any other criteria or directions.

S: "Will we need a compass for this?"
Me: "If you choose a method that requires a compass,  I suppose you will. If not, no."
S: "What method are we supposed to use?"
Me: "I don't care. I care about results and reasons.  Be able to justify your choice."

What I was hoping for was "Our apartment will be here because this point is of equal distance from all of the jobs.  We found the circumcenter of the triangle formed by the addresses because the circumcenter is equidistant from the vertices."

I happily accepted several different answers and reasons as long as they were justified.

This is the kind of problem that I consider as one that requires conceptual knowledge.  Yes, they would need procedural knowledge to find the circumcenter, but to know what they were looking for is my goal.

I don't want my students to ask "when am I ever going to use this."  I worry that we focus heavily on procedure to the exclusion of the conceptual knowledge.  It's great to know how to find the circumcenter, but it's better to know when you would need to and why.


In my opinion, procedural knowledge is the how, where conceptual knowledge is the when and why.

Conceptual knowledge and understanding is where the fun and beauty of mathematics lives.

Day 97: Race

Class started today with a question:

"Mr. Aion, what are we doing for Black History Month?"

As the students started arguing about how this is math class and they should be doing something in history, I suggested that we do a research project about black mathematicians.  Instead, we had a lengthy discussion about racial inequality in the public education system.

We talked about white privilege and why, with 350 faculty members in the district, we could only come up with the names of 5 teachers of color (not including administration).  We talked about what it means for them and their families to see primarily white faces as educators.

We talked about what it means for them to leave their neighborhoods to come to a building where white people tell them what to do, how to behave and how to speak.

We talked about how, even though I have their best long-term interests in mind and do my best to act accordingly, I recognize that they don't always believe me.  I expressed my understanding for their reluctance and how I hoped that over the past few months, I have proven that.

We talked about decision making and trying to do what's best for themselves in the long-run versus social pressure now.

It was an important discussion and I was happy to take the class time to do it.

When we got back to content during the second half of class, things were status quo.


Change can't happen overnight, or during a single period.  Each of us is on our own journey and I need to be aware that my students won't always move at the speed that I would like.

That doesn't mean they aren't moving.



In amazing news for me, my speaker proposals have been accepted to the regional conferences of the National Council of Teachers of Mathematics that will be held in Atlantic City in October and Nashville and Minneapolis in November!  I'm pretty pumped!  I'm not sure I can afford to do all three in terms of money or time off of work, but I'm going to try to swing it!