For the past four weeks our team has been sharing the Towers Task activity progression in our blog posts. Last week, Darrell shared that, as students become adept at working with more than one unit (tower and cubes), they are able to answer questions that require them to go back and forth between thinking about the towers and cubes as different countable units. In that post, the situation presented was that the sender pretends to add more cubes and tells the bringer how many cubes they added. The bringer then has to decide how many towers were added, how many total towers there are, and how many total cubes there are. So what is the next step after a student has had many experiences with the Towers Task and is able to move back and forth between the two units fluently?
The answer is we give students another experience that extends their thinking about working with more than one unit. The sender sends the bringer to build a certain number of towers; for example, 6 towers of 3 cubes each. When the bringer brings the towers back, the sender hides the towers and pretends to add more cubes. The sender will tell the bringer, “now you have a total of 42 cubes.” Note that the difference here is that the sender reveals the total number of cubes in the current collection, whereas in the previous example the sender only revealed how many cubes were added to their collection. Then the sender asks the following questions:
- How many cubes were added?
- How many towers were added?
- How many total towers are there?
This scenario and set of questions cause students to think about the total number of cubes and then go back and forth within the different units by asking how many cubes were added. They have to think about how many cubes they had initially, find the difference with what they have now, and decide how many towers that would make. They also need to think about how many total towers they would have. For a student to be able to answer these questions they would have to move fluently between thinking about towers as units to thinking about the cubes themselves as units. The ability to coordinate units, the number of towers and the total number of cubes, is an example of what a student needs to do to be successful with multiplication, because in multiplication students are working with “units of units.”
Our students have had experiences with cubes and towers, but what if we change the context of the situation? We know we want our students to be able to take a concept and apply it in different situations. We have tried this and we will share what happened in the next entry of this series.