By Marian Small, Posted May 23, 2014 –   

In a lot of school districts in my region, there is an emphasis on building proportional reasoning even before it is formally introduced in the curriculum.

A problem I used recently is the one I’ve proposed here. You would provide students with base-ten blocks, including hundred flats, ten rods, and unit blocks; or students could use virtual base-ten blocks.

Build a number [Note: You might limit the value to numbers under 1000] where all of the following are true:

There are twice as many ten rods as hundred flats. There are one-fourth as many ten rods as one blocks. What could the number be?

A simpler variation that might better suit some students is created by changing the second condition:

• There are three times as many one blocks as ten rods.

Whichever version you use, good things happen. Students think about how to represent three-digit numbers and how to regroup and name them using an equivalent form; for example, they realize that 256 can be represented with 2 flats, 4 rods, and 16 ones—and not just 2 flats, 5 rods, and 6 ones. If the value is limited, for example, to 1000, they start realizing why 8 flats is too many: because 8 flats + 16 rods + 64 ones is too much.

Students will think about what terms like twice as much or one-fourth as much mean. In particular, they will realize that another way to say that one number is one-fourth as much as another is to say that the second number is four times the first. You might lead in to the task with the following questions:

• You model a number with 6 ones and some tens. What could it be?
• You model a number with 15 ones and some tens. What could it be?

I encourage you to try one of the variations with your class and tell us how it went. Encourage students to work through the problem with a partner. As you discuss solutions, include questions like these:

• How many rods did you use before you traded? Why not 3 or 5 or 7?
• How many ones did you use before you traded? Why were these the only possibilities?
• Which did you choose first—the number of flats, rods, or units? Did you have to?
• What was the least number you could have had?
• What did you notice about the possible solutions?
• Were there numbers you tried to get and just couldn’t? What were they?

I’ve presented this activity recently to a third-grade class, and students were fully engaged. Let me know how it goes for you. I look forward to hearing your experiences, and we’ll talk about the Build a Number problem in a couple of weeks.

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.

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