If you have been following this blog series you now know more about units coordination, how it relates to fractions, and where it shows up in middle school concepts. So how do we help students progress in their ability to coordinate units?
One thing I have learned from the amazing work of Christopher Danielson is that the best prompts will simply open a dialogue. In his book How Many?, he offers pictures to start a conversation about just that. As the teacher’s edition (which I highly recommend) states, we need to put discussion about unit relationships “front and center for examination and discussion” (40). These conversations lead to insight about what students’ notice, their thinking, and how they see structure. Reading this statement made me realize much of what I already do has the potential for units coordination discussion; I just need to guide the conversations in that direction.
I love using Notice/Wonder as a warm-up. Students talk and share; it sets a tone of collaboration. Choosing visual prompts with items that can be described with multiple units opens students up to this perspective, and they will start to notice it throughout the work done in class. Number Talk Images, #unitchat, and #arraychat are great launching points for these discussions.
Instead of just having students notice, a great question prompt can focus students on thinking about the unit relationships. Here is a favorite of mine:
The picture shows 1/5 of a full row and 1/8 of a full column.
How many sections in a row?
In a column?
How many shoes altogether?
How many pairs?
Photo credit: Erick Lee
Question credit: Sonja Twedt
I would estimate that many of you have used Estimation 180 in your classroom. (If not, here is a great blog series by Jen McAleer you should check out.) From day to day, some of these tasks hold the potential for great discussion of multiple levels of units. Here is a series that shows this:
For us, it seems straightforward that for day 26 we can estimate how many cups we think are in the bag, and we can then multiply by how many candies we know are in the cup. For students who are at the lower stages of units coordination, this is not intuitive. Finding tasks that incorporate discussion of units is the first step. The assessment is a key piece that helps identify where students are in their thinking and how to move them forward. Looking back at the stages of units coordination, we know that it is a progression of being able to coordinate some units “as given” and some “in activity”.
In a whole group discussion, using the phrase “Convince Me” is a great way to build in activity for those students who need it. As students are trying to convince me (and others), they often turn to manipulatives and concrete representations in order to show others their thinking.
In a more structured lesson or focused small group, considering which manipulatives allow students the activity they need to understand the relationships is as important as selecting a task. When it comes to units coordination, I have realized that some manipulatives are better than others. An example of this is the variable cups versus the algebra tiles in my third blog post. Cuisenaire rods, with their many sizes, and linking cubes, with the ability to quickly group and regroup, are two of my favorites. I often represent the relationships on a double number line. This allows students to see how different units are related. It is important to be planned and purposeful about the concrete models students use and about moving from concrete models to representations. As I work to improve in this area, I have written this reflection that might spark some ideas for you.
Units coordination is a crucial skill we need to be working on with students of all levels. I hope this blog series helped to inspire you and provide resources to help you strengthen your units coordination instruction. Please reach out to me in the comments or on Twitter at @a_schindy with success stories, activity ideas, and of course, questions.
Danielson, C. (2018). How Many? : A Counting Book : Teacher's Guide. Portland, ME: Stenhouse.
Norton, A., Boyce, S., Phillips, N., Anwyll, T., Ulrich, C. & Wilkins, J.L.M. (2015). A Written Instrument for
Assessing Students’ Units Coordination Structures. IEJME-Mathematics Education, 10(2), 111-136.