By Marian Small, Posted June 9, 2014 – 

I hope you have had a chance to try the Build a Number problem with your students. I had lots of fun with it when I tried it with some third- and fourth-grade students.

Recall that students were going to use base-ten blocks to build a number that has—

•  twice as many ten rods as hundred flats, and
• one-fourth as many ten rods as unit blocks.

We heard from a couple of you who saw the potential of this task for differentiation. It was a great idea to suggest using the blocks to represent decimals as well.

Someone asked about modifications for kindergarten or grade 1. One possibility, more likely for grade 1, is to propose only one condition; for example, there are twice as many ones as tens.

Most students start with the hundred flats. Once they put out 1 hundred flat, they realize that they need 2 ten rods and then 8 units to go with it, i.e., 128. You might ask students if the problem could be solved by starting with the rods or units first. Of course, it can, but it may take more experimenting. For example, if students start with 1 unit block, that won’t work. They might start with 4 unit blocks and 1 rod. But then they realize that, oops, there wouldn’t be a flat; they have to go back to the beginning to start with 8 unit blocks. It might be interesting for students to realize that sometimes the order in which you solve a problem matters, but not always.

After students get to 128, many just stop there, but they can easily be encouraged to look for more numbers. You might ask them to use 2 flats, and they soon see they would need 2 flats, 4 rods, and 16 units. Some students will wonder whether it is “legal” to have 16 units. Of course, it is, but the number will need to be written as 256, not 2 4 16.

At this point, most students just keep adding 1 flat, 2 rods, and 8 units, realizing that each time, they get a new correct answer. Once a few of these numbers are created, there is usually some excitement when students notice that the numbers are 128 apart; if you keep adding 128, you get more and more answers, namely 128, 256, 384, 512, 640, 768, 896, . . . . A class of older students might notice that these numbers are, in fact, multiples of 128.

It would be worth exploring why no other numbers are possible. In effect, if there have to be 2 rods and 8 units for every flat, any increase in a number has to come as a package of 128.

In one classroom, a student told me that there was actually a number lower than 128—namely 0. He said 0 flats, 0 rods, and 0 units does the trick, and so it does. That was quite insightful. It would be equally interesting to ask if there is a greatest number. There is not, because 128 could continue to be added.

An alternative where there were three times as many blocks as ten rods was also offered in the earlier post. Here the values are all multiples of 126. The problem isn’t really simpler; it’s just the language of saying “three times as many” instead of “one-fourth as many.”

What did you like about this task? What would you have changed?

Marian Small is the former dean of education at the University of New Brunswick, where she taught mathematics and math education courses to elementary and secondary school teachers. She has been involved as an NCTM writer on the Navigations series, has served on the editorial panel of a recent NCTM yearbook, and has served as the NCTM representative on the MathCounts writing team. She has written many professional resources including Good Questions: Great Ways to Differentiate Mathematics Instruction (2012), Eyes on Math (2012), and Uncomplicating Fractions to Meet Common Core Standards in Math, K–7 (2013), all co-published by NCTM.