**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—

- a visual representation of the multiplication table, looking for and describing patterns; and
- multiplication of facts on a number line, looking for and describing relationships.

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 numbers.

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 same?”

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—

- not touch the tiles or the array they built until they had discussed a plan with their partner; and
- not build a different array, but transform the array they already had.

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 next day.

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 they noticed.

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 grow!

Mike Wiernicki, mjwiernicki@gmail.com, is a math specialist in Georgia. He tweets at @mikewiernicki and blogs about teaching mathematics at http://mikewiernicki.com.

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