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:

• They’re like multiples.
• The first row counts by ones, the second row counts by twos, and it goes on like that.
• The colors are in a backward L shape.
• There are squares (with an arm bent to show) diagonally.
• The last backward L has mixed colors.
• It’s kind of like a picture of the multiplication chart.

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.”

• The columns count like the rows. The first column counts by ones and the second column counts by twos, just like the rows do.
• The rectangles next to the squares are the same. Like with the orange square, the rectangle on top is 2 × 3 = 6 and the rectangle on the left of it is 3 × 2 = 6.
• And the rectangles two away from the squares are the same, too.
• They all are the same, even on the blue and green square; the rectangles are the same if they’re the same amount away from the square.

Students wondered about the following:

• Why are the colors in that backward L shape instead of in rows?
• How many cubes would be in the next backward L shape?
• Why does it stop at 6? If it’s a multiplication chart, shouldn’t it go to 10?

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:

• Not if we use one 2 × 3 and a 1 × 3 and then use the other 1 × 3 and 2 × 3.
• After figuring out another backward L: Yeah, so we don’t need to go out from the square, we can just find two rectangles in the backward L that add up to the square and set them on top.
• The second backward L has 2 layers and the third backward L has 3 layers.
• So we started with squares and made them into cubes?

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:

• Making tables of data (input/output)
• Connecting products to values represented as arrays
• Investigating patterns of squares and cubes
• Investigating and making sense of volume as the area of a base multiplied by the height (number of layers)

Although this task engaged students in multiplication, we realized that to build fluency, we needed to provide more purposeful practice that empowers students.

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