Multiplication Fluency in Middle School: Part 2
By Michael Wiernicki,
posted December 7, 2015 –
In my last post, students used the image below to notice
patterns using numbers, colors, and shapes (see the list below). Students also
became curious and asked questions about the image (also listed below).
Students noticed the following:
A student, asked
to explain the last comment above, replied, “Well, the first row has a white
cube, that’s 1 × 1, then the second one in that row has 1 × 2, that’s 2. If you
go down a row from there, there’s a square that’s 2 × 2 and it uses 4 cubes. It’s
2 × 2 = 4.”
Students wondered about the following:
Before we had time to guide students to choose a focus question
to investigate, they had already started. This is one of the top reasons I like
to begin a lesson in this way. Students become so engaged in their own curiosity
that they can’t wait to get started. By asking students what they notice and
think, we are essentially telling them that they matter and that their thinking
is important to what they’re studying.
After we discussed their questions, students seemed to be
most curious about the shapes (squares and backward L shapes), so they decided
to work on finding the number of cubes in the next backward L shape.
After discussing the range of estimates that students made
about how many cubes they thought would be in the next backward L shape, they began
working on an accurate solution to the question. Many groups began by observing
the numbers of blocks in the previous backward L shapes and looking for a
pattern in the numbers.
After a short time, some of the sixth-grade students were
struggling to determine the relationship in their tables they had created, so we
directed them back to the cubes. We had set up some cubes in the same arrangement
as in the picture for each group. Most wanted to start with this backward L:
This is where it all began to come together. It only took a
few minutes for one student to say, “Hey, the 1 × 2 and 2 × 1 red pieces will
fit right on top of the 2 × 2 square. Does that work for all the backward Ls?” If
any students were not engaged before, they were now. Some student math talk
began in these ways:
Another math question!
Students worked for a bit longer to determine the number of
cubes in the seventh backward L and were excited to share. They were confident
in their solutions because they were given time to make sense of the
mathematics. Some students did want to build the 7 × 7 × 7 cube, just to see
it, so we left the cubes at the back of the room for them to use to explore.
The goal was to engage students in purposeful multiplication
to build fluency, but the activity ended up being so much more. The students were
engaged in these areas:
Although this task engaged students in multiplication, we
realized that to build fluency, we needed to provide more purposeful practice
that empowers students.
Wiernicki, firstname.lastname@example.org, is a math specialist in Georgia. He
tweets at @mikewiernicki and blogs about teaching mathematics