Multiplication Cluster Problems
Deborah Schifter, Virginia Bastable, and Susan Jo Russell
We have been working with arrays and with multiplication cluster problems to develop the idea that problems can be broken into smaller parts, which are then solved and recombined to get the answer to the original problem. I have given students clusters of related problems such as the following:
Students solve the problems and tell how they could use one or more of the easier problems to help them solve one of the harder ones. The following response to this cluster is typical:
I just knew the first two. Then 2 X 40 is like 2 X 4, but when you multiply the 4 by 10 to get 40 you have to multiply the answer by 10, too. So 2 X 40 is the same, but you multiplied the 2 by 10 instead of the 4. The answer is the same. For the last one I started with 20 X 4 = 80 and just added on 3 more fours to get 92.
Some of the children talk about "adding a zero" to a number when they multiply by ten. I'm not sure what I think about the language. It disturbs me, but the children seem to know what they mean. Another student used problems from the cluster to create big and small arrays to help solve 23 X 4:
One day students made their own clusters. I presented a problem to be solved, and the students decided what other problems should be in the cluster to help someone solve the original problems. To start, I presented 21 X 4. The class suggested these for the cluster:
20 X 4 =
No problem here, so I upped the ante with 271 X 7.
I heard Donte murmur, "It's going to be higher than 1,400." I repeated his comment and asked why. "Well, 200 X 7 will be 1,400 and there's still more you have to multiply." I asked for other estimates.
Howie said he thought it would be about 1,450 because "the first part would be like Donte's [referring to 200 X 7 = 1,400], and then there would be a 7 X 7 and a 1." Nobody commented at the time that it should be 70 X 7 instead of 7 X 7.
I asked if anyone could give me an estimate that would be close to the answer but more than the actual answer. There were a few attempts, but students couldn't explain where their estimates came from.
We then returned to creating the cluster. Jeannie volunteered the following:
200 X 7 =
We used the bottom three lines to help us solve the top two. Then we added the first three lines to get the product 271 X 7 = 1,897.
After we did another one together, I suggested that those who were ready to try this on their own could create and solve their own clusters. For the starting problem, the first number could have as many digits as they wanted, but the second number had to be a single digit. They were encouraged to discuss their work with others. Most students went off to try out their own problem-solving ability.
I was left with eight students who wanted more help. We worked through two more clusters together, then three of them went off to work independently. Two more had to leave the room for special programs. The remaining three students worked on and off with me and with each other at the board. These students had difficulty keeping in mind the place value for each digit. For example, they did not see the 7 in 72 as representing 70. They knew they were supposed to add some of the subproducts, but they were unclear as to which ones and why. I'm not sure how to help these children.
In the meantime, I started checking in with the other students. They were confident and excited to discover that they could multiply large numbers. Most students used a four-digit number for their first number. Although there were a few multiplication and addition errors, these students understood how to take the problems apart to solve them. Not all were as efficient as Jeannie's presentation on the problem we did together, but they were definitely headed in the right direction.
Marisa and Jiro understood how to take the problem apart but reverted to addition to solve 8,000 X 6 rather than relating it to 8 X 6.
Tynisha challenged herself with 92,512,995 X 5. She made two subproduct errors with order of magnitude. When I pointed out where the errors were made, she was able to correct them and proudly ended up with the right answer.
Amber and a number of others wrote down 5,000 X 5 = 25,000 without needing to write out 5 X 5, 50 X 5, and 500 X 5. Others, like Ricardo, wrote all the steps, either because they thought they were supposed to or because it was helpful to them
Chantel and Tabitha attempted to solve their problems using the algorithm they had been taught the year before. Both had difficulty, but were able to solve the same problem correctly using clusters.
I was struck by how many students understood how to create their own clusters and how to use the clusters to solve hard problems. The students saw this as empowering. There was an air of excitement as they realized that they could tackle increasingly bigger numbers. Awed by Tynisha's accomplishment, a number of students tried really big numbers the next day. (A "really big number" seems to be defined as one that's too big for the calculator!)
I have some worries that the cluster method might become an alternate series of steps that students use without meaning. The small group that stayed with me was attempting to mimic what they had seen their classmates do, but clearly they did not have an understanding of the process. What can I do to build the foundation for these students? Should I provide more of the array activities and cluster problems, or something different? Should I have them build numbers with base ten blocks? Clearly, place value is a problem for them.
Do the successful students who "add zeros" when multiplying by ten or a hundred understand mathematically what they're doing, or only that it gets them the right answer?
I want to extend the estimating we did. I'd like students to be able to come up with a reasonable (and justifiable) range into which the answer will fall. This is a different, but related, way of looking at the magnitude of the numbers. Hopefully, students will use this information to evaluate their actual answers.
Full of enthusiasm, I tried the same "make-your-own-cluster" lesson with my other class the next week. It bombed! I realized I had not done as much groundwork to prepare them. Back to arrays and clusters!
From Developing Mathematical Ideas: Numbers and Operations: Building a System of Tens Part 1 by Deborah Schifter, Virginia Bastable, and Susan Jo Russell © 2000 by Education Development Center, Inc. Published by Dale Seymour Publications, an imprint of Pearson Learning Group, a division of Pearson Education, Inc. Used by Permission.