At first when I read this standard and knew I was expected to teach students to be able to list factor pairs to 100, I couldn't imagine how that would be accomplished! They often come to me weak in understanding math multiplication facts having just been drilling their 2's through their 12's. How would they be able to conceive factor pairs? And to 100? When they were just drilling...they weren't thinking; not really.
Knowing "what makes" a given product really is the way we think of multiplication. We figure out groups to make a product when we are figuring seating arrangements, packing boxes or a myriad of other life grouping problems. I do it every time I figure out how to group students! I take the total and figure out the best size group! CCSS 4.OA.B.4 simply states they must "find" factor pairs. But, through finding them, comes a level of fluency that is really useful.
Full Class Lesson: I taught this lesson whole class because the concept of factor pairs is new to them and even my highest students have not been exposed yet. Therefore, it warrants whole class instruction in order that everyone masters the standard.
I wrote the number 12 on the board. I posed this question to the full class: How many ways can we make twelve by multiplying? List all the ways to make twelve in your notebook using multiplication. I was interested in seeing if they noticed the commutative pairs. ( 4x3, 3x4)
I gave the students a few minutes and soon the hands shot up offering up answers like 12 x1, 4x3 and 6 x 2. I asked if they thought there were anymore. I listed each of their answers on the board.
One boy said: Well, there is 1 x12 and 2x6. I listed them next to the factor pairs with an equals sign. I wrote Commutative Property on the board. I also wrote Commutative Pairs on the board and explained that each factor pair has a commutative pair. I told them that I call the commutative pair a "turn around" to help me remember, because the factors just turn around the other direction. It is an easy way to remember where to stop when listing factor pairs. But, I wanted them to remember that it is an example of the Commutative Property and that they need to remember that. The end list looked like this:
2x6 = 6x2
3x4 = 4x3
They seemed to understand that because they chimed in as I wrote. We said the word Commutative three times to help us remember the sound and pronunciation.
I asked these questions: How do we know twelve has 3x4 as a factor pair? They talked about memorizing math facts and that is how they knew. One student said that it means there is three groups of four...or four groups of three and that always makes twelve. While the standard expects mastery of listing factor pairs to 100, fact fluency is dependent on the success of mastery of this standard too. The factor pairs will also affect the fact fluency because they have to use number sense and not just rote memory. CC develops this beautifully!
I moved on: What is the inverse operation of multiplication? They easily answered: division. Good. They should know that. I asked this because they need to understand the relationship between the two operations right from the beginning in order to fluently solve factor pairs.
I drew a math mountain for each of the factors pairs for 12. I labeled the operations on the math mountains and asked them what the top of this math mountain represented. They answered: Product. Good. I now knew that they know how basic facts work. Factor pairs should be a snap.
Through the math mountains students see the direct connections between division and multiplication and how they are related and so I always like to bring it up. I never take for granted that this concept is completely understood in this transition to CC. Eventually it will be!.
Guiding students to understand the strategy: I wanted my students to undersand that finding a factor pair could be challenging if you didn't have strategies so I decided to start with the number 72. 72 has 6 factor pairs and factors that students aren't used to working with. It's challenging! This adds rigor to the lesson and makes them a little uncomfortable, but joins those high end students easily into the whole group lesson. Sometimes differentiation looks like whole group, but I can infuse it in and get results. This is an example of that.
I put the number 72 on the whiteboard and asked them to find the factor pairs for 72. One student jumped at 72x1 and the next said 9x8. Then is stopped until someone said there had to be factor pairs with 2. One of my high- end students piped up and shared: 36. I asked if there were anymore factor pairs. They didn't know. I told them that all had to be found!
I asked this question: What if we had a slick way of figuring out all of the factor pairs?
Mrs. K's Slick Systematic Method of Listing Factor Pairs: I listed 1x72, 2x36 and told them if we kept listing factor pairs starting with 1 and testing each number numerically as a factor, we could find out how many factor pairs 72 had.
Is there 3x something for 72? They weren't sure so I led them by telling them I had a slick trick for finding out if there was a 3x something for 72. It was time to teach divisibility rules. I know that I cannot proceed without these rules being presented. I have chosen to proceed by introducing divisibility rules as another tool for becoming more fluent in finding factor pairs. It reduces a little of the possible stress of not knowing if 3 is a factor. An array will help us find what would be the missing factor.
Using Factorize, a web site that uses a virtual graph and a factor pair calculator, I introduced the concept of finding missing factor pairs or missing factors and the concept of using the array to do so as an extension from Introduction: Discovering an Array. This little interactive game is really slick because it graphs the factor pairs exactly using rows and columns! Yesterday we learned that we can graph factors using an array and graph paper. Today, I started with this activity to introduce a virtual tool that would help them find a missing factor of a product up to 50. Together, we would discover how to do that when we got stuck on factors we didn't know.
I started the lesson with students seated in front of the SB so they could easily get up and participate in forming the array with their hands. One by one we worked on "factorizing" different products as I guided them to start with listing 1x the product, first and then moving onward trying 2x, 3x, etc. until we had all of the factor pairs.
When we got stuck on a factor pair, we could reverse the process and draw out the rows and columns as we watched the product change until we got the one we wanted. The students can then count the rows and columns to reveal the missing factor pair or factor.
For example: My one student doesn't know the factor pairs for 48 very well. She tried 1 x 48, 2x___? and got stuck. She created two rows and then moved her finger over until the graph revealed 48, showing her 24 columns.
It takes the frustration out of learning these and quickly wraps their minds around finding factor pairs which can seem so daunting at first. Here's a peek at us Factorizing! Factorize!
*Just be sure you don't put spaces between your factor pairs...type it in: 2x24 and not 2 x 24 because it will " boing" and make you think you are wrong. It won't sketch the array if you do that.
* It's a great game play but will not run on iPads because it needs Flashplayer.
* It's a good game for centers and for lower students who are struggling.
To add to their skills, I wanted to introduce divisibility rules to students because they understand that the inverse of multiplication is division. While the graph paper array method works well, I knew later on when we list factor pairs to 100, graph paper or array models would be cumbersome. In talking with them, they already knew rules for 2,5, & 10. I decided to teach rules for 3 and see if they could practice knowing just these 4 divisibility rules. While divisibility rules are not part of the standard, using them along with an array model guides them along a little bit to find missing factors. Later on, the Sieve of Eratosthenes will be a discovery lesson that reinforces their understanding of why.
Divisibility of 3
I told my students to add the digits of 72...7+2. 9, one student said. I asked if 9 was divisible by 3?I got a resounding yes. So, we now know that 3x something is 72. I told them that since we don't know how to do long division yet, there are other ways to figure out how to group. I walked over to the Smart Board where I had left the Factorize ap up. The Factorize ap will only factor up to 50, but we can use the Smart Board crayon to draw the array. I showed them how to list it by 3 rows and continued to find the other factor. I showed them how to count the top number and complete the factor pair. We found it to be 24.
We continued on with 4 and then discovered that it worked too! 4 x 18. By using the Factorize graph paper, students had a visual they would transfer into their homework today to help them master listing factor pairs to satisfy the standard.
We continued to 5. They knew the divisibility rules was that the last number had to be 5 or 0, just like they knew that 72 was an even number and that all even numbers are divisible by 2. I was pleased with this. So, the only divisibility rule they needed was 3's.
We kept going and tried 6. I decided not to teach them divisibility rules for 6 today since just mastering the 3's would be tough enough. So we used the graph again and found it to be 6x12. They had forgotten that math fact! I smiled and told them that sometimes we get caught up in the process and forget what we already know!
We worked our way to 8x9 knowing that 7 wouldn't make sense because they knew that 7x10 was 70 and 7x11 was 77...so it wouldn't be logical. I was pleased that they understood that. So I guided the conversation in the direction of understanding that we also use what we already know to solve!
I asked them how they knew we didn't need to go on any farther. One boy said that it was the "turn around"...9x8. He remembered from the first example of 12.
This method of solving to find factor pairs is appealing because it is like a puzzle.
Independent Practice: Gradual Release
I had them write the number 21 in their notebooks and asked them to follow the rules we had noted and try to list them on their own. I roved and checked notebooks. All but one student understood.
We tried 16. Again, students listed the factor pairs. This time I stopped them when most were done and asked if they found 16 to have doubles. They knew to stop at the doubles.
I questioned: What makes sense to you about stopping at the doubles? One answered: it wouldn't make sense to say 4x4 twice. 4 groups of 4 is four groups of 4. Nothing changes, so that means it automatically is the turn around. It is the commutative pair.
I noticed they were mostly independent at this point and could manage homework practice.
I asked students to come and sit up by the Smart Board once more as we played the Factorize game using the number 12 again. I noted the arrays that were drawn, the rows and columns, and told them that I would give them graph paper and they would create the arrays on it listing the factor pairs in order first in their notebook and on the graph paper, using arrays to solve any factors they got stuck on. I wrote 18, 20, 24,28,32, and 38 on the board and assigned these products for them to solve and create arrays.
I asked them to factorize 18 first, before going on, and I would look at it to make sure they understood and give them support. I was happy to see that every student was able to list all the factors and get through creating arrays well enough that I felt comfortable sending them home with it. This particular student had moved on to 24 and I stopped to help her because she had gotten stuck. Finding all the factor pairs using our array shows us what was going on.
Closing: I asked students if they could see how listing factor pairs in order helped them feel that they had control. I asked them to think about figuring out factor pairs for a number like 86 and not understanding where to start after 2xn?
This activity and lesson made my class feel secure in listing and drawing factor pairs. They could see how if they didn't know the missing factor, the array becomes a tool in answering that problem.