Powers of Ten and Place Value

My resident and I are pulling our hair out over our Honors Algebra 1 class. They’re happy to work, but they don’t want to think. If they don’t know what comes next, the majority of the class just shuts down. Seriously?!? I’m guessing part of this comes from their backgrounds–they got into the Honors program here in high school, but they’re in Algebra–the top kids passed the Chicago Algebra Exam and are taking Geometry as freshmen instead. That means they have a complex and aren’t usually the very top–just close to it. So they really, really don’t want to be wrong–they’d rather just not try it. I’ll win this one eventually but man am I frustrated right now. Particularly since I think the best way to get what I want is to give a little right now to get them working for me (even if it isn’t really thinking for me) so that they get in the habit of trying things my way for the future.


We wanted to do some early work around estimation, especially in terms of using powers of ten (both for estimating with operations and for place value/number sense understanding). They can all name their place values and identify numbers, but don’t seem to have a strong concept of equivalence–its all procedural.

So we came up with pyramids.

The basic idea is pretty straightforward–find the product of the numbers on the left side and fill in the pyramid (this is the part they found easy). Down the right side, we gave them similar products, but changed the number on the left as well, so for instance 3.2*10000 or 320*10. The back side has only one number each and they had to find all of their own products using different powers of 10.

We ended up also making a matching chain to practice the powers of 10 for in class, which gave us a good idea of where they struggle (anything with a decimal). If anyone is interested, I’d be happy to share it.

(This document was made by my awesome resident Mrs. D after we brainstormed the idea together.)


Cows (Translating Expressions)

Between moving into a new “room”* and three preps, its been pretty crazy/busy. I actually have a team this year (for only one of my three preps, but beats last year when I was an island x 3) and we’ve been trying to get a little ahead on our planning.

My team plans on-level Algebra 1 (I also teach the Honors section), which meets for two periods every day–almost two hours. This past week we moved into writing basic expressions and translating expressions between words and Algebra.

After making a vocabulary chart, we did some basic practice and moved on to the sheet I affectionately (and somewhat nonsensically) call Cows.** It’s the first real group activity of the year, and my scared freshmen still don’t really want to talk to each other (although its nice to figure out who will talk no matter what and who only speaks when spoken to for future group composition).

It starts off easy, with Part I on both sides moving through some basic translation. We did a mix of having students write their answers on the board (selected by a marker left on their desk as we circulated), writing a few things we’d seen on the board and having someone read their answer. Some expressions had several interesting ways of being represented–like “half a number”–and students weren’t always sure which were equivalent. Part II is also pretty straightforward, and then Part III happens, usually on day two of the work (it took me the better part of the double period to get through Parts I & II).

On the Words to Algebra side, the problems get at some basic misconceptions (Larry is four times as old as Bobby) and unit conversion versus equation errors (If three feet equal one yard is the equation 3 f = 1 y correct?). I’m a stickler for defining a variable, and I start in on it here. What is the difference between a variable and a label (or unit)? Is “apples” a variable? Does it represent a quantity? I correct a few “f = Fred” issues and we start to get the idea that f needs to equal a quantity about Fred–his age, the number of plaid shirts he owns, how many fish he caught (apparently I have specific ideas about who Fred is, too.)

And then Algebra to Words happens. 3m + 2b = 60. No big deal, right? “Three of a variable plus….” Nope. Not cutting it here. Define a variable as a quantity with some kind of context, and roll from there. The students really want to turn this in to something like “3 monkeys plus 2 bananas equals 60”. 60 whats, no one knows, including them, and some of them take awhile to catch the issue. I usually do some counting on my fingers; looks like 5 to me.

In class, I did a practice problem first, and modeled how I would think it through to develop a problem, bouncing ideas off of my resident across the room:

Alright, m and b. So I could use monkeys and bananas. What about them? I could do…weights. So 3 monkeys times the weight of a monkey plus 2 bananas….bunches of bananas? Yeah, bunches of bananas times the weight of a bunch of bananas is the weight of the monkeys and bananas…in a van. Going to a new zoo. They’re very small monkeys. Or maybe big bananas. (I realize I could get into specifics around using the same units, like pounds, for both, but that isn’t my focus so much. I had a group catch that on their own in their problem, and handled it then. Since its not my focus, and what we are trying to do is pretty tricky, I try not to overwhelm with details.)

My 1st period class especially worked super hard on this, and came up with some great examples, such as “3 lbs of food for each monkey times the number of monkeys plus 2 pounds of food for each buffalo times the number of buffalos equals 60 pounds of food.” I realize the buffalo are probably starving to death, but the construction is pretty decent.

I know this isn’t perfect, and I’d love to refine it more, so please let me know any feedback you may have. I’ve never done a real revision, and I do like the results I’ve gotten from this version in the past, but I’m positive it could be even more valuable.

What do you think? What would you change? Are there questions you would add? Remove? Tweak?

*Its…basically what you would think teaching in a storage room is like. One of my students told me Friday she had a dream about me. She was telling them to give me my old room back because I’m a good teacher. If only it worked…

**After resistance, my whole team now calls it Cows, even after Conor swore he wouldn’t because it makes no sense. Ha! Also, you can’t even see why I call it Cows on this uploaded one. I’ll try to update with a photo of my beautifully illustrated version.