The common core places heavy emphasis on students' ability to interpret products of whole numbers and whole number quotients, so I wanted to ensure that my students had a visual example of the groups and number of objects in each one that they could compare side by side to relate to division within the same fact family. We have used a lot of modeling of groups to create a deeper understanding of multiplication so I also included a model with the activity to introduce this concept.
One of the really cool things about multiplication is that it teaches us about division. So much of what you have done in multiplication has a relationship, I like to call it a family, with division. See, in division, we take things and put them into equal groups. We divide things out, like placing them into equal piles to see how much we will have in each one.
Here we pause to talk about how multiplication has equal groups, repeated a specific number of times, and that will give us a total number of objects (MP8). I expect students to be able to articulate this clearly, and to use their understanding of the relationship between repeated addition and groups in multiplication (MP1). I let a few students share and then bring them back together.
Can anyone tell me a multiplication fact they know? Great, I’ll use that (whatever they tell me I use, in this case they told me 6 x 3 = 18). I’m going to write 6 x 3= 18 on the white board, and I’m wondering if anyone can create a model for me with these counters? While the student creates a model I ask other students to explain what the student is making.
I use questions such as, “Can you tell me why you think they’re arranging them in that way? What do those piles represent? How many total counters do you think he/she will have in that model?"
Then I ask another student, "Using the commutative property of multiplication, what is the other fact in that family?" I repeat the process of writing it on the white board and asking another student to build a model. It is important that students can explain themselves and use tools strategically, which is what I’m looking for here (MP5).
Ok, great. You’ve got a great start already to learning about these families when we add division in!
I start by pointing to the first multiplication problem and ask, Now, how many total things do I have in this model? Ok, I’ll write that here. Now how many groups have I created? So I’ve put those 18 things into 3 groups? Ok I’ll add that here. And how many counters are in each group? Great, I’ll put that number here to show that there are 6 in each group. Wow, wait a second. You just told me a division problem just from looking at what you know about the multiplication problem. Does anyone think they can make a model to match this division problem? As the student is making the model I call on other students to help explain what they are doing (MP2, MP4).
I repeat the same process for the 2nd multiplication problem in the fact family. Once the 4 facts and their models are laid out I ask students to turn and talk to a partner about what they notice about the problems and their models. With the common core students must apply properties of operations and here we are illustrating the commutative property of multiplication and looking for a relationship with division. I spend a lot of time introducing this concept and letting students work through making sample problems. We repeat the same process for 2 additional examples and I let students write on the white boards and other students make their models (MP6).
Great! Now it’s your turn to create some brilliant examples of what you know about fact families for your classmates to see. I’m going to give you three numbers, four pieces of white paper and manipulatives to work with.
Here I send students back to their tables. On their tables, I have three numbers, four pieces of white paper and a box of manipulatives (counters, domino’s, linking chains, etc). Students must write out all four facts in the family of numbers, each on one piece of paper just as I did on the white boards, and then create a model to match each one (MP6). I walk around the room and ask students to describe what they are doing, and why they are doing it.
As we are wrapping up, we do a museum walk and observe each other’s work, models and the relationships between the numbers at each table.
I saw some fantastic works of art! We were able to take 3 whole numbers, use what we know about the commutative property of multiplication to write problems with our numbers and create models to match. We even took it a step further, by finding their family members in division and creating models to match the problems!
I feel it is important to close this lesson because we are introducing a new relationship with division. I reemphasize how this connects to their previous understandings and I want to tie the information together again after they have had an opportunity to view other group work.