# 99 Bottles of Beer on the Wall

Berkeley Schools have a holiday today in honor of Malcom X’s Birthday and I promised my children we’re go visit dad at work. After decades of work as a computer engineer and nearly as many as a home-brewer, my husband decided to see what commercial brewing is all about.   His work is worthy of a field trip by any pre-algebra or algebra teacher.

We saw a lot of this:

And this:

And a wonderful problem lives here too:

Egan’s favorite activity is watching dad drive the fork lift but we missed that today.  I did ask the kids in the car on the way home how many bottles they thought dad boxed up today.

Egan (age 4): “A Million”

Morgan (age 15): “That’s something a math teacher would ask.”

# Unitizing Your Vitamins

For about two years I have been reading Christopher Danielson’s (@Trianglemancsd) blog Overthinking My Teaching and greatly enjoying the frequency with which he writes about the mathematical conversations he has with his two kids.  While I greatly enjoyed his presentation at NCTM, I was more excited to realize that he’s Tabitha and Griffin’s dad.  Unfortunately I realized this after returning to Berkeley, so I never actually talked to him about this.

Hopefully he won’t mind me utilizing  his style to write from time to time about the developing mathematical thinking of my own son Egan.

This morning after finishing breakfast, Egan, four years old, asked for his vitamins.  He gets 4 of them: two multi, a probiotic and an Omega-3.  He quickly ate two of them.

E: ” I am saving the other two to have with dinner.”

Me: “Oh, so you are saving 1/2 of them?”

E: “No mom.  You can’t break vitamins in half, they are too hard to break.”

Me: “So the only way to get half of something is to break it into two pieces?”

E: ” Of course.”

Me: (holding out 2 hands): “On one of my hands, put out fingers for how many vitamins you just ate and on my other hand put out fingers for the number of vitamins you are saving for dinner.”

He does this with ease.

Me: “Now cut my fingers in half.”

Egan chops his hand between my 4 fingers.

Me: “Hmmm..I thought to take half of something you had to break it into two pieces.  But you didn’t chop up my fingers.”

E: “But I did cut your fingers in half.  I put two on each side.”

Me: “So eating your vitamins today will look something like this.”

E: “That’s cool!”

I was struck by Egan’s ability to refine his understanding of one whole within the course of our conversation.  His initial understanding of 1/2 was when you take something and break it into two pieces.  I am not quite sure he understands the importance of equal pieces, but I know he knows that you get 2 pieces when you break something in half as that idea has come up before with sharing food.  This may have been the first time he was faced with unitizing, the idea that 1 whole can be made up of many pieces.  I think we’ll see where these finger games go at breakfast tomorrow.

# Pre-School Chit-Chatterings

He received many sets of Legos for his birthday and was following the directions.

When he got to this one, he paused.

He said, “Mom, that’s a funny direction.”

I asked why (meanwhile I was dreading finding 50 of these pieces with him).

He excitedly said, “It says the next thing we do is give 5 hugs and kisses!”

His ability to come up with a definition of what it means to have a quantity alongside a ‘word’ was exciting.

I share this as today in the mail I received my copy of Moebius Noodles, a new book by Yelena McManaman and Maria Droujkova on wonderfully creative activities to do with young children to develop their mathematical minds.  I had the pleasure of working with Maria on a web-cast two summers ago and am so excited to share her work.

In the introduction they talk about the four main ways to build the building blocks around the concept of number:

• Subitizing: The ability to instantly recognize quantities without counting.
• Counting: Dealing with objects one by one.
• Unitizing: Using equal groups or units such as in multiplication or division.
• Exponentiating: Self-similar structures such as fractals.

They go on to say that most US curriculum for young children does little or no work with unitizing or exponentiating which leads to difficulty later in elementary school math.  Egan is lucky–most every night at dinner he asks what my students learned that day in school.  I have found ways to show him algebra and geometry concepts in ways he can understand.  That’s what this book is all about.  I have only just begun reading it.  You should too.

# Live crabs in Vending Machines?

### **No, not THOSE crabs**

Of the many projects I have worked on this year as part of my Math For America Teacher-In-Residency, the most exciting and fulfilling has been coaching my middle school’s math department.  I use the word ‘coaching’ loosely as I really see us all as collaborating together, learning from one another.  I just find the grant money so we can be paid.

We have dabbled in all sorts of foci this year, but lately we have been spending most of our time learning from each other as we try out more of the Shell Foundation’s formative assessment lessons (FALS).  Our big focus is always on student’s engagement with the mathematics.  Our more serious conversations focus on our successes (and sometimes utter failures) facilitating small groups work and subsequent whole class discussions.

BUT..the fun is had when we talk about the initial hook–how to overcome the drama of the passing period and the sweat induced lunchtime basketball games.  In my last post I shared a video that the five of us made to introduce the Sharing Gasoline FAL.

Robert MacCarthy found this gem about Chinese vending machines which disperse live crabs.

He used it to introduce a 6th grade lesson on using coordinate graphing to analyze data.  Students work in groups, using this graph

to analyze these cards:

Some students have the ‘Monday’ data which is above.  Others have a different set of ‘Tuesday’ data.  Once groups finish their cards, they exchange cards with a group who had a different day’s data from them and their new task is to construct the graph using the other group’s cards.

This lesson isn’t published yet, but you can try out the draft version if you like.  If you do, let me know your thoughts.

# Revival

There’s a restaurant location in downtown Berkeley which has changed hands several times over the past few years.  It’s newest incarnation is called Revival and I must admit, it’s quite good.  Coining the name “revival” was a bit risky as it reminds us that this spot hasn’t been successful in years past, but their risk paid off.

I hope that pouring life back into my blog will also be sustainable.  This is the second time in my blog’s short lifespan that I have taken a long time away from it. Reconnecting with PCMI mathy friends (which made me just 1 degree of separation from a whole slew of other cool mathy folks) at NCTM in Denver, most all of whom blog and tweet regularly, made me promise to recommit myself to writing with some degree of regularity.

I have a few deadlines coming up where audience participation would be helpful:

Deadline #1: May 1 for submission for proposal to present at NCTM next year.  I would like to share some of the work that my department is doing developing lessons based on the FALS (formative assessment lessons) written in collaboration with UC Berkeley and the Shell Foundation.  Do you know of them?  Do you use them?  If so, what kind of presentation would help those unfamiliar with them learn to use this rich resource?  I would love to brainstorm this with someone.

If you’re unfamiliar with them, they’re found under the ‘lessons’ tab here.

I have piloted these lessons for the past three years and currently most of my department is piloting them as well.  While at NCTM this year I found that many folks have written ‘tasks’ which correlate to Common Core standards, but I found few which were as rich as the lessons produced by the Shell Foundation.  In my department we are using these lessons to explicitly teach the Mathematical Practices and are working on a series of videos to engage students around this work.  By the end of the year we hope to have videos of our students, grades 6-8, as they are exemplifying one or more of the practices.  We will show these to our new students next September, to help teach them what each practice looks like.

In the meantime, our first production is a video to engage students around one of the FALs entitled Sharing Gasoline.  Teachers in grades 6-8 taught this same FAL and together we made this video to hook them.

Here’s the lesson: Sharing Gasoline FAL

If you have feedback about what would be most interesting in terms of sharing formative assessment lessons from the Shell Foundation, please let me know.

Deadline #2 is a Math For America forum where I need to present on my teacher-in-residency year at UC.  More on that soon.

# Prove Me!

Yesterday in geometry students were analyzing proofs of the Pythagorean Theorem using this lesson by The Shell Foundation.  It ties in nicely with my favorite proof which they discuss while watching this video (there’s something about doing proofs to classical music which delights me).

At the end of class, a student came up to me and asked if there was a theorem which says the following:

I responded that I had no idea if it was a theorem, but if he could prove it, then it would become a theorem and be named after him.

It sure SEEMS like it’s true.

But then again, there’s this:

And the above diagram actually satisfies his initial question which was if you have a square, and you draw 4 congruent segments inside the square which intersect the sides of the square, will the resulting figure also be a square?

Thoughts?

So…I wouldn’t exactly say this went viral, but last year The Teaching Channel came and filmed the work I did with my students around integer addition.  Since then, on and off, I have gotten emails asking to hear more about how I teach integer operations.  What I have decided over my 14 years of teaching, is that with integers, I prefer to use just one model: a number line.

The 6th and 7th grade teachers at my school use multiple models, like plus and minus tiles and colored discs.  But as an 8th grade teacher, who cannot afford to spend a whole lot of time reteaching integer operations (but nor can I ignore the fact that you really can’t learn algebra without it), I find that sticking with a single model works best for me and my students.

This year, for the first time, I used only generic number lines.  By that I mean a number line where zero is the only number written on it and it extends in both directions.  We started with temperature and used a vertical number line, looking at temperatures rising and falling from am to pm and finding the change (absolute value).  We looked at elevation gain and loss and also at sea birds and dolphins which dive under and jump out of the water.  Oh year, and we talked about my alma mater, Dartmouth, where the stacks in the library extended many floors underground and how I had an office down there, but the Peruvian literature that I needed for my research was way up on the 6th floor. So my students got their integer fingers wet through real world scenarios.

BUT, from day one of these scenarios, we only used generic number lines. I really discouraged them from making tick marks for each integer (remember…8th grade here).

I got a brilliant idea from a blog (which I suspect was from Everybody is a Genius, but looking back, I’m not totally sure it originated there), of using plastic sleeves as white boards.   You make a sheet (in this case, a generic number line), slip it into the sleeve, hand out white board markers, and shazam, you have a personalized white board for the lesson you’re teaching that day.  So, my students all used these to think about questions such as if the temperature was -10 degrees in the am and the afternoon temperature was 25 degrees in the pm, how much did it rise.

So, we actually didn’t start with questions like -3 + 5.  We started with generic number lines and counting by 5s and 10s on much larger numbers.

Which brings me to my revisions of the “Hey Baby, What’s your Sign” lesson featured in my video.  This year, I decided to use the stickers to mark the ‘jumps’ which happened when you add two integers.  It looked like this:

Same idea as last year…make a jump from zero for the first addend.  Look at the second addend and decide which way you’re going to move.  Then ask yourself (in your sexiest possible voice), “Hey Baby, What’s your Sign?” to determine where your second jump will land.  THEN, use the orange sticker to note the sign of the sum, and only then, add the sticker for the second addend.

After students finished this worksheet they had a choice of 2 differentiated ones.  One of which had more practice of single and small double digit addends.  Or, they could choose the extra challenge one which had all double and triple digit numbers.

A few kids were really stubborn and didn’t want to use the stickers.  I quickly learned that 100% of the complainers were already quite proficient in integer addition of big numbers. Of the kids who needed the time to think through these problems, none of them complained about the stickers slowing the pace of their work.

So, really, there’s just one major change from last year: at no time did we use number lines where integers from -10 through +10 (or something similar) were written on the number line, nor did I encourage kids who were confused to do that.  I tried to be patient, and really help them think about each addend as one single jump and work towards ‘seeing’ when you’d add the 2 absolute values together and what you’d find the difference.  By no means do we have 100% mastery after just a single block period.  But I don’t plan to introduce any new models, nor encourage those who are confused to draw each tick mark. I want to continue to be patient with this generic number line model.