In last week’s post, David Pearce described a modification of the Towers Task in which the students are asked to build two sets of towers and combine them. For example, the student may be asked to build five towers of four, and then three more towers of four. The student is then asked to find the number of towers, number of cubes in each tower, and the total number of cubes in all towers. It is important to note that students are usually still counting by ones (1,2,3…4,5,6… etc) as they also keep track of how many times they have made three counts. They are still developing a meaningful skip count.
The modification to the task we will discuss this week has a different configuration of towers, as well as a different set of questions for the student. We don’t present the task in this format until students are consistently successful combining two sets of towers of three (or four) cubes each.
Since the students are now proficient at keeping track of how many times they have made three (or four) individual counts, it is time to increase the sophistication of the arrangement of towers. They will still be working on developing a meaningful skip count because this modification will require them to find a total number of cubes, but many students will already be skip counting meaningfully up to a certain point. So here are the steps:
- Student is asked to build a set of towers, and the towers are covered (Example: Please build five towers of four cubes each).
- Student’s partner pretends to add more towers behind the cover, and tells the student the total number of towers that are now hidden (Example: Now that I have added more towers, there are a total of 9 towers hidden).
- The student is asked the following questions:
- How many towers were added?
- How many cubes were added?
- How many cubes altogether?
Here’s what this situation looked like in one of our classroom interactions:
This problem is similar to a missing addend problem, but the addends are towers. In this modification of the task, students work on their ability to switch between units. They have to think about the towers as wholes, and switch their thinking to the cubes, while maintaining a separation in their mind between the towers they built and the towers that were added.
Notice this is the first modification of the task where one of the sets of towers is not built at all. It is a progression towards imagining towers without ever seeing them. Initially, students can struggle to make sense of this problem, and it can be helpful for them to act out the entire task. Their partner can actually build the second set. After a couple of times, students should be ready for the pretend scenario.
Let us know if you try any of these tower tasks and how your students responded.