In this series, we have been discussing a progression of tasks that give students the opportunity to construct meaning for working with two types of units, the towers and the cubes that make up the towers. Last week, Brook discussed an activity in which the students would establish the first set of towers and then their partner would pretend to add in more towers and tell the student the total number of towers after the pretend ones are added. In that activity, the student would have to imagine the additional towers and work with them and the cubes they are constructed from to answer questions.
So what would be the next level of a task? Do you already know the answer?
Remember, the ultimate goal of this task is for students to be able to switch their thinking between the towers (groups) and the cubes that make up the towers (number of items in each group). This is the big switch in thinking for students working with multiplication. With addition and subtraction they are working with the same unit type, but with multiplication they are moving between two different unit types.
The next change in the task promotes the ability to switch between towers and cubes. The sender tells the bringer to bring back a certain number of towers made up of a certain number of blocks (we let them build the towers at this point because it engages the students more) and then the towers are hidden. Next, the sender pretends to add cubes and tells the bringer how many cubes were added. The bringer then reflects on the following questions:
- How many towers were added?
- How many total towers are there?
- How many total cubes are there?
- How did you figure it out?
Notice how the bringer has to now switch back and forth between towers and cubes with the collective experiences of building towers and reflecting on them throughout the series of tasks.
The goal is to provide situations in which the student can construct the understanding of groups and members of a group in order to build toward reasoning multiplicatively. In the next part of this series we will be looking at the next level of complexity in this progression.
As you try these tasks in your own classroom or have questions or thoughts, please share by replying in the comments.