As the students prepare for math class, I put up two multiplication problems, with both the factors and the product, on the board that indicated the commutative property. I call them to the community center and ask them to look at the two problems.
Mathematicians, please look at these two equations. We know that three groups, with seven in each group, is 21 altogether. We say this as 3 x 7 = 21. And we know that seven groups, with three in each group, also is 21 altogether. We say this as 7 x 3 = 21.
We have talked about the commutative property, or the "flip flop" property so we know the products are the same. I am wondering if we can decide on drawings that are not arrays for these equations. What would you like our unit to be? Pumpkins? Perfect. Okay, let's talk about pumpkins in fields. How could we draw a simple sketch for these equations? Please write in your reflection journals and then give me a thumbs up when you have an idea.
I know that my class has a difficult time matching the correct equation to its representation from looking at their work with arrays and writing equations for word problems. Many had the correct product, but listed the commutative property equation. As they share their stories and drawings with me, I will read the equations back to them using the phrase "groups of" when seeing the multiplication symbol. In doing this, they are much more apt to understand if their story/picture is correct.
You may choose to make the decision, as I did, to veer away from the term "times" to using the phrase "groups of" for some time. I found with my students that this phrasing helps them "imagine" the math better and guides them in their representations.
My goal during this section of the lesson is to familiarize the students with using a number line as a tool for modeling mathematical thinking. As I discuss with the students the equations on the board, I begin showing them how I would model the "hops" or "groups of" on the line. We then compare the different paths to the same product.
Following the example and guiding questions about "groups of", I assigned partnerships an activity in which they will roll a die and create equal groups to move along the number line. One of the confusions I expect some students to have is how to organize their equation to properly represent their math model.
I used a worksheet from K-5 Math Teaching Resources for the activity.
Boys and girls, you will now work with your partner to use the number line as a tool to figuring out how to write an expression for the equal groups. You will roll a die to find the number of "hops" for the given groups size. Then you will show your work on the number line and write an expression. if you roll a 1 or a 6, that is a free choice. You may choose 2, 3, 4, or 5 as your multiplier.
As the students are working on their activity, I am carefully listening to their conversations on how they will begin and how they will write their expressions. Debates are the best thing to have in a math class, as they lead to deeper understanding for everyone. I try to find the students having those conversations, listen to their work, and try to guide them to have meaningful math talk, rather than arguments:) It is also a perfect time for me to find out who has it and who still needs help.
Listen to these partners work to understand equal group size.
At this point, I call the students to the community center with their papers. I ask students to share what they found tricky and what they learned that they deem important. Many students will discuss the order in which to write the factors in the equation. I listen to the responses carefully to determine next instructional steps for tomorrow.
Great thinking boys and girls. So many of you took the risk and stayed in that "comfortable struggle" to try and make sense of something tricky for you. I was so happy to hear you talking with your partner about patterns, predictions, strategies, and equal groups. We will keep practicing all of this. I would like to close with letting our friends that were struggling with the even and odd concept to share what they learned. Let's listen and see if you can add to their thinking.