In Chapter 3, one of the big ideas that was discussed was the idea of unitizing. “The whole is thus seen as a number of groups of a number of objects – for example, four groups of six, or 4 X 6. The parts together become the new whole, and the parts (the groups) and the whole can be considered simultaneously.”

Last week, I was co-planning and co-teaching with a 3rd grade teacher. She wanted to introduce the idea of multiplication and create a situation where students would begin to think about repeated addition in terms of “groups of”. We crafted a scenario where a farmer had cows and chickens. His barnyard creatures had 22 legs in total. How many chickens and how many cows might he have?

Students eagerly set about figuring out how many groups of 2 legsand how many groups of 4 legs it could take to total 22 legs. Some students used stylized drawings and counted the legs, many students used repeated addition and a few students used the words “groups of”.

During the congress, one student shared that they figured out that it could be 9 chickens and 1 cow. After everyone agreed that this combination worked another student chimed in that it could therefore also be 1 chicken and 9 cows. They went on to explain that this was true because 9 X 1 and 1 X 9 was the same thing (something that they probably learned at home). Several students disagreed and said that 1 chicken and 9 cows wouldn’t work.

The teacher and I were very excited about this unexpected learning opportunity. We asked students to work with a partner that was of a similar opinion to prove why they agreed or disagreed. Many ah-has occurred as students grappled with the number of groups and the number in each group.

It was so exciting to listen to the students discuss their proofs to support their point of view. Some students who had previously used number lines to model addition situations, used a number line to show the number of jumps of 2 and the number of jumps of 4 and how 9 chickens and 1 cow landed on a different number than 9 cows and 1 chicken. It lead to a lot of discussion about the number of groups and the number in each group.

In the end, many students had moved ahead in their development of the big idea of unitizing. Others continued to struggle with simultaneously thinking about the number of groups, the number in each group and the whole. For the classroom teacher, the challenge is to think about what provocation to use next to move all of the students forward in their learning about multiplication.

*Featured image by Annie Spratt on Unsplash*

Listen as Melanie explains this work further:

Hi Melanie, thank you for sharing this blog post with us. After reading chapters 1-3, I started working with 2 boys in grade 3 (friends kids who were willing to let me borrow them a couple of times a week for my own learning lol). I have been leaning on Fosnot’s Minilessons for Early Multiplication and Division as I only work with them 3-4 times a week for 15 min. I am actually going to blog this week about what I am learning from working with them (an amazing learning journey for me) but what resonates this evening is that as we have been working through some of the Fosnot problems, I have been writing down the multiplication notation (as per Cathy’s directions) and now….when I go to put the X both of the boys are now saying the words “groups of”. For e.g when working on the Boxes of Fruit on p.11 ……2 x 6=6+6=12…..they both are now saying that means 2 groups of 6 or in a subsequent problem… skip counting 5-10-15-20……then announcing that there are 2 groups of 20 … and that 20 and 20 is 40 tomatoes. I knew this was a significant moment when they started this but your blog post has helped me to really connect back to the landscape and the importance of noticing, naming and thinking about the unitizing as part of their development. I am going to try the problem you shared this week and will share next week back on your post…..such a great task you have developed for the students. Can’t wait to try it. Thanks again for sharing and please…..keep sharing if you can find the time….as I believe it helps many other educators think deeply about what they are reading and trying in their classrooms, or in my case with my neighbours kids! This week we are partially obscuring the drawings….stay tuned! Asima

LikeLike