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.
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
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
- 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
- What did you notice about the possible
- 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.