By Michael Wiernicki, posted December 21, 2015 –

So far, the sixth-grade students we’ve been working with had an opportunity to experience multiplication and the multiplication table as a series of arrays. Students also recognized and defined square and cube numbers and their roots.

Students were reminded of the concept of an array or introduced to the ideas of working with arrays, so they needed to be given an opportunity to use arrays to make sense of something that they could wonder about and investigate to engage in purposeful practice to build fluency. The next portion of the lesson was adapted from some ideas from E. Paul Goldenberg and colleagues’ recently published Making Sense of Algebra: Developing Students’ Mathematical Habits of Mind.

We started by drawing an empty number line on the white board. After discussing what number should begin our number line, we got started with our lesson. We began by drawing on the number line as shown below and asked students what they noticed. They were quick to say that 35 should be placed above the 36 because 5 × 7 = 35. We asked them how they knew to multiply, and one student raised her hand and said that to get 36 with 6, you had to multiply two of them.

Everyone  that the number between the arrows should be 35, so we wrote it in there. We asked the students again what they noticed. The following is a close approximation of what happened next.

“It’s one less.”

“What’s one less?”

“The number on top.”

“Did everyone hear that?”

“Yes.”

“Talk to someone near you about that, and see if you can say it another way, using the numbers on the number line.”

This slowed them down a bit. The students in this class were not used to being the ones talking in math class. We gave them some time and prompted them as we walked around the room. They then shared what they noticed using much more mathematical language: 

“ 6 × 6 = 36 and 5 × 7 = 35, and they’re one less.”

“The number smaller than 6 and the number bigger than 6 are multiplied and make a number 1 less than 36.”

We asked them to choose another number. Students were given small white boards, markers, and erasers. They drew their numbers lines and tried it with another number that they chose. They found out that their numbers produced the same result. They were surprised, so we asked them to test some other numbers, whatever numbers they wanted. Their goal was to find some numbers where this wouldn’t work.

Below are samples of work from students who struggled with multiplication:

The students in this class had a lot of multiplication practice and while some stuck to smaller numbers, others really branched out to check larger numbers. They were only told to try some other numbers. No one asked how many they “had to try.” Most students tried several numbers, and many ventured into double digits multiple times.

We asked students to share the numbers they tried and tell what happened. Every student’s attempts produced similar results. One student did ask if it worked with really large numbers, so we asked him to choose a large number to try. When he chose 1,378, we asked what he would do with that number and how he would do it. He said he would multiply 1,378 by itself on a calculator. We told him, “So would we,” and handed him one. He multiplied and told us the product was 1,898,884. Before he could continue, we asked the class what the next step would be and what the product would be. They proved to themselves (some by drawing number lines) that he should multiply 1,377 by 1,379 and that the product should be 1,898,883 because it would be one less. The student read the product and the class shouted out, “Yes!” and “I knew it!”

This is what purposeful math practice looks like. The practice is embedded within a task that students find engaging. The practice is more like a puzzle, the students’ ideas matter, and students   challenge themselves to try larger numbers because they want to be the ones to find one that doesn’t work. Although   the students engaged in some meaningful practice, they need more. Find out the conclusion in my final post.   

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