We are working on composed units and place value in my math course for future elementary and special ed teachers. We were discussing examples of composed units that are composed of things exactly like one another. A pack of gum is an example of this, while a car is not. A car is composed of a whole bunch of parts, but those parts are not all alike.
A student asked whether quarters being composed to make a dollar count. I said yes and then took a poll. I now share that poll with you, and invite discussion.
The poll
Both of the following are correct. However, one of them probably matches more closely how you think about this. So vote for the description that feels closest to how you think:
- A dollar is the original unit, and it is partitioned into smaller units called cents. In this case a cent is a partitioned unit.
- A cent is the original unit, and one hundred of them are composed into a larger unit called a dollar. In this case, a dollar is a composed unit.
The Results
I won’t tell you the results of my poll yet. I will tell that I had an instinct about it, that my students favored one of these outcomes by a ratio of 
So how about it? What are your thoughts? And what are the implications for using money to teach decimals?
Overthinking my teaching 
I am going to guess that your students picked option 1. However my choice is option 2. Implications for using money?? Needs to happen AFTER students are very comfortable with place value because money brings another aspect of value. How does ONE nickel represent FIVE cents? It is quite perplexing for students who aren’t ready for it. I have lots more to add here so I will be back later with more comments 🙂 unit chat is my favorite!
Is there a difference between a 2:1 ratio and a 2/3 : 1/3 ratio?
Will I EVER type the word “ratio” without reflexively typing “ration” and having to delete the ‘n’ afterwards?
I don’t have a definite intuition as to which option would dominate. And from my perspective as a math thinker, it makes little if any difference how I conceptualize it. My sense is that it’s a fluid relationship and being able to move between those conceptualizations easily and comfortably is a distinct advantage over insisting upon only one being “right” or dominant.
I would guess because “cent” means 100, they started with the original unit dollar and then chose to break it up into 100 pieces which they called “cents.”
I suspect if they started with the copper piece, then when they got to 100, they would have lent the “cent”ish name to the thing that represented 100 of the copper pieces. So, it would have perhaps been 100 coppers = 1 centurion.
It also seems to support the notion that breaking that original unit into four pieces would naturally yield “quarters,” a name that makes very little sense if we are focusing on the 25-ness of it instead of the 1/4th-ness of it.
I think using money is a fine way to get kids to think about place value, it grabs their attention – especially if you bring a bag of change and have them handle real coins….
I think as long as you prioritize decimal currency (penny, dime, dollar, $10 bill), you’re in pretty easy waters.
I think nickels, quarters half-dollars, and Fivers would make for good in-depth conversation, but I wonder if that’s best done in the latter part of the lesson, after they’ve had time to play a little and more robust instruction… once you think they understand the basics. A nice way to differentiate for your early finishers?
Full Disclosure:
As one of Christopher’s previous students, I submitted that very example of pennies/dollar units during my semester… It was my position that the Dollar was the unit. I argued similar positions, I belive (a “quarter” refered to the 1/4th ness) and the name wouldn’t be “dollar” if the base unit was the penny.
No surprises/spoilers here: Christopher had us explore the conversation, but didn’t impose a position on us.
and Michael, for what it’s worth – typing “ration” is universal it seems – thanks for coming out first.
AND look how we write it. $1.00
I’m no fun because I cannot pick one over the other. Fine, I’ll go with choice 1 that the dollar is the original unit. I mean as a math teacher I want to be receptive to lots of wiggle room in kids’ thinking. It’s more about how one defends her argument.
I would have to agree with Andrew (because of his name and) because of the names of the currency we use, and with Fawn about encouraging a good mathematical defense.
I wonder if the cent-as-a-unit mindset would really endure for a student much past 1 dollar though. Is someone who sees the cent as the unit really inclined to think of 100 dollars as 10,000 pennies, or are they unit switching? Is that a bad thing?
I prefer the dollar being the original unit and the cent being a partitioned unit. I do an entire “cubie” lab with a gallon ziploc of linking cubes, 10 in 10 columns. The bag is the whole, the cubes are 1/100, 1%, 0.01, etc. The students and I have great fun ‘fair sharing’ the one bag among a given number of friends. The fun begins with the cubes must be divided (8ths and 16ths). We go to the “cubie” bank and exchange a big cube for 8th’s and continue to fair shard. Helps students understand thousandths place.
I agree with Fawn: the dollar is the unit.
Mostly because…
A) I view a handful of change as not quite one whole unit. And
B) That’s HEAVILY the basis of how I teach place value to remedial CAHSEE students, and if that falls apart, I may have to live under the overpass in a cardboard box.
*CAHSEE is the California High School Exit Exam, one of the checkboxes to a high school diploma. It’s 7th and 8th grade math. I had my middle schoolers take it last year and 75% passed.
This post reminded me of a scene in Julie Brennan’s chapter, Tying It All Together, in the soon-to-be-published book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. I’ve excerpted that scene on my Playing With Math Facebook page. You might enjoy it.
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