At two of my school’s recent PLC’s they have been focusing on building content knowledge in the area of multiplicative thinking. The one school has also been going pretty in depth with using the Five Practices to plan their math lessons and focus their consolidations. I wanted to share some of the work the PLC group did recently looking at practicing the last three parts of the Five Practices, selecting, sequencing and connecting. The group had asked for this to be part of their last learning session. Since we have also been working on building multiplicative content knowledge the task used was a multiplying task. The class that did the task was a grade 1/2 class. They did this task in groups but even though grade ones don’t have to learn multiplication the teacher has many students that are exceeding where they should be and are starting to pick it up. I think this clearly shows when students are challenged they don’t just meet the bar but can raise it!
The learning goal for the task was this: We will understand that “groups of” can be counted as units to tell us how many in total.
Why: This will help us understand how multiplication works.
The “Why” part is something that we are starting to add in. feedback from our district support visit told us that our students could say what they were learning but often could not say why they were learning it.
Here is picture of all the finished solutions from the task that was chosen:
Close up of the task:
Selecting and Sequencing
As a group we discussed the learning goal and how we wanted to select samples that would really help us bring out the mathematics that we wanted the students to learn. Since the focus was on unitizing (seeing groups of) and then counting the groups we thought that we would start with these solutions:
We agreed that these examples all showed groups that had really identified “groups of” and used them to find the total. Two of the groups used groups of 4 and one group used groups of 2 to find the total squares. The first example on the left is what Alex Lawson describes as a students using the composites units inside of a composite unit to make it more manageable for them to count the equal groups. They used the 2 groups of 2 in each group of 4 to get to their total. We chatted about how this is ok to do especially at this stage in grade 2 but later we may challenge a student that is doing that to move to more efficient way of counting the groups.
We also selected this solution:
We selected this one because this student listed both ways you could solve this by using the commutative property. This little guy is whole other blog post, he is in grade 1 but is picking up multiplication very fast and has already reached auto-recall for some of his facts. He totally understands the situation and can model it. We chose this one because we were thinking that after we used the first three solutions and asked our prompting questions that we would follow with this one. Our idea was to use his number sentence of 4 groups of 5 and ask this prompting question: How could Loic’s number sentence of 4 groups of 5 help us solve this problem more efficiently? Our thoughts were that most of the solutions used 5 groups of 4 and either skipped counted by 4’s or 2’s. No one but Loic saw the commutativity of the array and thought to count by groups of 5.
We now had done our selecting and we had decided on a sequence. Now came the chat about how we are going to connect the consolidation to our learning goal. The plan was to use this prompting question once the first three solutions went up:
What do you notice about how these groups figured out how many pieces it took to make the quilt?
Then follow with these questions:
How many groups of 4 did it take to solve the problem?
How many groups of 2 did it take to solve the problem?
Which one do you think is most efficient? Why?
Then the last solution would be put up and the question I mentioned above would be asked:
How could Loic’s number sentence of 4 groups of 5 help us solve this problem more efficiently?
Then follow with these questions:
What do you notice about the number sentences 5 groups of 4 and 4 groups of 5?
Connection to past tasks: How does the array model help you solve this problem?
The class has been working with arrays in past lessons so this question can help link previous lessons to today’s. That is the plan we came up with, will see how it goes when the teacher does the consolidation. One thing the teachers shared with me is how much more focused their consolidations have been since they started working with the Five Practices. Still they feel they need more practice doing the selecting, sequencing and connecting part of the process to gain more confidence. Using your colleagues to bounce questions off of and to ask for help to look at student work also came up as being super important in this process. Hope you enjoyed a little peek into our journey with the Five Practices. It is work in progress but we are getting their and it is all based in student work. Any comments or thoughts would be welcome!