Multiplication Fluency in Middle School: Part 4
By Michael Wiernicki, posted
January 4, 2016 –
I’ve spent the last few posts describing how a sixth-grade teacher
and I began building fluency with multiplication facts with a group of sixth-grade
students. So far, the students have investigated—
But they have really done more than that. These students have been
empowered during these two investigations. We, as teachers, showed these
students that their thoughts and ideas matter by asking what they notice and
wonder, rather than telling them what they should notice and wonder. Empowering
these students has not only given them confidence but also engaged them more in
the mathematics and relationships. They seem more interested in learning and
what will be discovered next.
We continued our lesson to build fluency with multiplication facts
by showing students another number line and asking what they noticed in the
diagram that follows.
The students’ comment:
“It’s 4 less now.”
It was what we expected, but not the way we wanted it. So we probed further:
“Why do you think it’s 4 less? Talk with your partner.” Every pair of students
said that it was because there are 2 numbers to the left of 4 and 2 numbers to
the right of 4 and 2 + 2 = 4. The students’ thinking was very surface-level;
there was no depth. Naturally, we told them that they just came up with a
mathematical theory and now they had to prove it. To do this, they first needed
to check and see if this “4 less” idea held up for all kinds of numbers, then,
if it did, they needed to prove why it worked.
We set them on their way to draw some number lines and check if “4
less” worked. They were very persistent and, like before, tried numbers large
and small. After a very short time, they decided that it did work for all
Then we asked if they were wondering about anything we had just investigated.
After a bit of discussion, someone asked, “How come we subtract? We’re going up
and down the number line the same amount, so how come the answers aren’t the
We directed students to draw a number line and number it 0–9. We had
them choose any number to start with (suggesting smaller numbers, in the
interest of time) and had them build an array for that fact using tiles on the
table. When they had all built their arrays, we asked them what they would have
to do to transform their array into one that would be 1 step out in both
directions. They were instructed to—
We had each of them share what they did. Here’s a sample:
4 x 4 = 16 3
x 5 = 15 (“stickie outie” extra gets taken away)
They wanted to try 2 steps out, so we let them! They used their same
strategy, and one of the girls noticed that the 4 tiles they had to take away
were in the shape of a square. We wrote this on the board, as a reminder for the
When the students came back, we built a table of values:
Students helped develop the headings in the table. They were asked
to predict the amounts to subtract for 3 and 4 steps out. Some of their
predictions are noted in red in the table above.
Developing this table while working with other students gave them a
new reason to practice their multiplication facts. They wanted to know if their
predictions were correct.
As with the previous investigations, with 1 step out and 2 steps
out, students tried various numbers to make sure they were on track. They tried
larger numbers and smaller numbers. What we noticed at this point was that it
seemed like the students were already becoming more fluent on the basis of the numbers
they were choosing to compute and their conversations with one another.
This practice was not without discussion. As students worked on
their facts, they shared with their partners the numbers they had tried for
each. They checked their work and determined that some of their predictions
were off a bit.
When we brought the group back together, we asked them what they
noticed. They eagerly shared that the amounts to subtract were 9 for 3 steps
out and 16 for 4 steps out. Not forgetting about the squares from the day
before, students grabbed the tiles, used their same strategy, and noticed that
all the subtractions were square numbers!
To top it all off, all it took, again, was one student saying, “We’re
just taking squares away from squares?” This brought on a few chuckles, but
really he was right.
We ended our sessions with these students with another idea from E. Paul Goldenberg. This provided
students with more purposeful practice that empowered them and encouraged them
to begin working with larger numbers.
We asked students for a number between 10 and 30; they chose 21. We chose
the second number, 19. We challenged students to multiply the 2 numbers
mentally before we could (they were allowed to use calculators). When we told
them the product was 399 before they got their calculators out, their jaws
dropped. The students asked, “How did you do that?”
Instead of telling them, we drew a number line and asked them what
In a very short time, the students figured out that they had
developed this mental math magic trick over the past few days. And they could
perform it for everyone—and they have!
These lessons were eye-opening for all. Students saw connections and
became empowered to do mathematics; my co-teacher saw students eager to learn
and use mathematics; and we all saw everyone’s understandings of mathematics
Wiernicki, email@example.com, is a math specialist in Georgia. He
tweets at @mikewiernicki and blogs about teaching mathematics