Here’s a string I designed with a team of fifth grade teachers who were looking for creative ways to encourage a multiplicative understanding of place value. When students are reasoning additively about number, they might see 321 as equivalent to (3 x 100) + (2 x 10) + (1 x 1). And this is very common in classrooms, and is often confused as thinking multiplicatively because it does involve some multiplication. But in essence you are splitting the number into place value parts and adding those together. When students are able to reason multiplicatively about number it looks more like this: 321 = 32.1 x 10 or 3210 ÷ 10.
So, we wanted a context in which to start to explore this shift. First we showed the stack of bills above.
What do you notice? How much money do you think this might be? What makes you say that?
What if I told you this was 97 one dollar bills? How much money in the whole stack?
$1 x 97 =
Easy one. Okay, so what about if I replaced the all of the one dollar bills with ten dollar bills? How much money now?
$10 x 97 =
or
Kids might say: It’s ten times greater than the last stack of bills.
($1 x 97) x 10 =
And how about if each of those was replaced with one hundred dollar bills? How much money now?
$100 x 97 =
or
Kids might say: It’s ten times greater than the last stack of bills.
($10 x 97) x 10 =
Or kids might say: It’s hundred times greater than the first stack of bills.
($1 x 97) x 100 =
Let’s suppose that I make a tower of coins instead. What if I took 97 dimes and made a really tall tower. Crazy, right? How much money would that be? What are you picturing in your mind? How are you thinking about this one?
Kids might say: I knew every 10 dimes was a dollar, so I made groups of ten dimes. 90 dimes would be….. nine dollars, plus seven leftover dimes which would be 70 cents, so $9.70.
I’m going to write what you said in two ways. Turn and talk to your partner about whether these are related or not:
($0.10 x 90) + ($0.10 x 7) = $9.70
Another student might say: Yeah, I know a dime is one tenth of a dollar, which was the first one, so I think you could do 1/10 of 97, though I’m not totally sure what that is.
Are you saying that 1/10 of $97 could also be $9.70? Turn and talk to your partner about what you think of this. Is it true? Could you convince us?
1/10 of $97 =
Well, it turns out I don’t have 97 dimes but I sure have a lot of pennies at home. If I made a crazy tower of 97 pennies, how much would that be? What are you picturing now? How could I represent that?
Based on what kids say you might write:
.01 x 97 =
1/10 of 1/10 of 97 =
1/100 of $1 x 97 =
($1 x 97) ÷ 100 =
Someone might notice that we decreased the value by 1/10 when we replaced dimes with pennies:
1/10 of $9.70 =
Kids might also say: I know it’s a little less than a dollar, because 100 pennies is one dollar.
Great, last one. Let’s say I had nickels instead of pennies. Still a big ol’ tower of 97 of them. Think about how you could figure out how much money the tower is worth now.
[After some substantial think time] Turn and talk with your partner to hear their thinking and share your own.
Kids might say: So nickels are just half of dimes, so I tried to use the one with dimes to figure it out. And so if 10 dimes makes one dollar, ten nickels makes 50 cents. So I made jumps of 10 nickels until I got to 97 jumps.
So you might return to the model for the dimes, and build the model of the nickels, based on this thinking.
Once you get to $4.50 (90 nickels), you might ask how to think about the remaining 7 nickels. If kids need another model below, you could build the one below:
Or someone might say: It’s just half of the dimes, which was $9.70. So I split everything in half. Half of $9.00 is $4.50 and half of 70 cents is 35 cents. So $4.50 plus $0.35 is $4.85.
Number Strings 