Yesterday we spent a lot of time discovering the relationship between multiplication and division. I want students to have a strong foundation of creating and breaking apart equal groups before we get further into division. With the common core, students must develop a deep understanding of the relationship between multiplication and division and develop fluency in solving problems, so I want to ensure that my students had time to construct examples of this relationship. I put a multiplication problem on the board with no product asking students to quickly solve the problem.
Now that you’ve solved this problem, who can tell me what 3 numbers might be in this family? Great. Please draw a model to match this problem. I expect students to use repeated addition, to create equal groups or to draw an array.
Now what does the commutative property of multiplication tell me? So I already know what I need to create the other multiplication problem in this family. Please record your other multiplication fact and draw a model to match.
I stop here for a moment to highlight the similarities and differences between these 2 problems, for example, that there are the same number total objects, that the groups are broken apart differently but are still equal groups, or that they are repeated adding a number.
What did we learn yesterday about division?
Here I will expect students to respond about dividing things into equal groups, making groups and finding how many will be in each etc. I ask students to record the 2 division problems that share a relationship with these 3 numbers and to draw models to match.
Look at this! After only 1 day you’re on your way to becoming experts! Please trade papers with the person next to you and take 1 minute to look over their work. Look for any similarities and differences between your paper and theirs, and have some thoughts to share with the rest of us when the time is up. Go!
After a quick minute, I ask students what they noticed. They should notice that the pictures are all of equal groups, that the commutative property of multiplication shows the same number of total objects, and that with division they still have the same number of total objects but that they were describing their models in a different way.
I always plan for a lot of practice on day 2 of a new concept, but I vary the way students will show what they know. I want students to have time to think about the information and relationships and make sense of them with numbers and models.
So what I’m hearing you say is that we all had the same total number of things in our pictures, and we all created models with equal groups that are essentially the same? Well you’re exactly right! This relationship in a fact family tell us that the problems will all involve the same total numbers of things, and our groups and the amount of each can be moved and broken apart into groups. Now I want to see you challenge yourself by writing down 3 different multiplication problems on a piece of paper. Make sure they are correct, because they are what your team mates are going to use to show a fact family relationship!
I dismiss students back to their seats to write down 3 different multiplication problems. Once they finish, I have a student collect them and re-pass them out to different students. I want students to practice identifying the relationship with numbers they are not generating, as they often think of a problem they know well to display work.
First, check your teammate's work and make sure their problems are correct! If they are not, please write the correct product down. You’re going to need 3 pieces of paper, 1 for each problem. We’re going to divide each piece into a 4 square and you’re going to have decide what the other problems are in each fact family. Be sure your model matches your work!
As I walk around the room I look for students who are labeling their groups, or adding additional understanding from our previous work in multiplication. When I see these things I stop and highlight the different things I’m seeing to help other students see those relationships.
I really like to let students share their work. It gives them a sense of pride, and they produce better work when they know they might get to share it with their team. I call on a few students to pick one problem to share and I allow other students to ask them questions. This helps students develop the ability to construct arguments and question others (MP3) and model the problem (MP4). It is also important for students to be able to justify their thinking when questioned by their peers.
Well I’m impressed with your work! I saw great examples of the relationship between multiplication and division and this will be useful when we really dive into division!