The other day, Jamie tweeted about this example problem from a 4th grade program:

After looking at the Standards, Learning Progressions, and discussing the 3rd grade fraction work, all of us on the thread agreed it was an appropriate question for 4th grade but then started to question whether we would give this problem as is, or adapt it. I really appreciate these conversations because they move us beyond ‘this problem sucks, don’t do it’ or ‘do this problem instead, it’s more fun’ to thinking about a realistic thought process teachers can use when working with a text that may not be aligned to the standards or lessons that may not best meet the needs of their students.

While all of the adaptations we discussed sounded similar, I couldn’t help but wonder how each impacted so many classroom things in different ways. What may seem like a small change can easily impact the amount of time it takes in class, students’ approaches to the problem, what we learn about student thinking, and the follow-up question we could potentially ask.

There is no right or wrong adaptation here, but I wanted to sketch out how each change to the problem impacts teaching and learning.

**Change 1: Remove the context and only give the expression 1 – ^{2}⁄_{6. }**

This would be the quickest way to do this problem in class. With this change, I would be curious to see how students think about 1 in the expression.

Do they record it as ^{6}⁄_{6? }Do they draw a diagram? If so, what does it look like? How is it partitioned? Are the pieces removed in the diagram or is there work off to the side? A potential follow-up could be to ask students to write a context to match the expression. I think in their contexts I would have the opportunity to see how they thought about 1 in a different way.

**Change 2: Remove the expression and only give the context. **

I find anytime there is a context it takes a bit longer because of the time to read and reread the problem so this change would take a bit more class time than the first. This change would give students more access to the problem and I could possibly learn how they make sense of a context, but I wonder what I would learn about their fractional thinking. Since the context pushes students to think about 1 as the whole pizza and also tells them that there are 6 equal pieces, the diagram, partitioning, and denominator are practically done for them.

Because of this, I may not learn if they know 1 is equivalent to ^{6}⁄_{6 }and may not find out how they represent fractions in a diagram because I imagine most would draw a circle. Since they could do the removal on the circle, I also wonder if I would learn much about how they saw this problem as an expression so I would add that as my follow-up question.

**Change 3: Remove the expression and the numbers from the context.**

This one would definitely take an entire class period as a numberless word problem and probably the longest to plan. Because it takes the longest to plan and implement I really have to think about what I learn above and beyond the two changes previously mentioned if I were to do it this way.

I imagine the scenario could sound like this:

*“Sam ordered a pizza cut into equal pieces. He ate some of the pizza and put the rest away for later.” *or *“Sam ordered a pizza. He ate some of the pizza and put the rest away for later.” *

When I do a numberless problem, my goals are to give students access to the problem and see how they make sense of a context without the numbers prompting them to feel like they have to do something. I have to plan for how I craftily find the appropriate time to let them notice and wonder and plan questions that elicit the subtraction from 1 that I hope to see. I also like to give students a chance to choose their own numbers for problems like these in order to see how they think about the reasonableness of numbers, which adds more time. The hardest part here is getting to the fraction work because I think students could stay in whole numbers as they talk about number of pieces. I can hear them wondering how many pieces it is cut into and how many he ate – neither of which guarantees fractions. So, while this has the potential to get at everything Change 1 and 2 do, a teacher must weigh how much the making sense of context portion meets the needs of his or her students.

All of this for one problem and I haven’t even discussed the two most important questions we need to ask ourselves before even making these changes – what understandings are students building on? and what understandings are students building towards?

There is much to think about in planning that is often hard to think about all of the implications of one tiny change to a task, however, thinking about how each of these changes impacts teaching and learning is the fun and exciting part of the work!

After the post, Brian had another change that he posted on Twitter! I wanted to capture it here so it is not lost in the crazy Twitter feed:

I wonder if another change could be tweaking the problem to where you don’t tell them it was cut into six slices: Sam ordered a small pizza. He ate 2/6 and put away the rest for later. How much of the pizza did he put away for later?

— Brian Bushart (@bstockus) February 1, 2018

Oh, and still get rid of the expression, in case that wasn’t obvious. Perhaps after Ss share solutions, if no one uses 1 – 2/6, you could share it and ask how it also represents the situatuon, assuming that fits with your overarching goals.

— Brian Bushart (@bstockus) February 1, 2018

justmsuKristin, Insightful post, much of food for thought. I appreciate your consideration of the impact our choices make on time, as well as how the different changes still need to address the same learning outcome. Not so simple, really, but I assume easier with practice.

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mathmindsblogPost authorThank you so much – I agree, not so simple at all and definitely gets easier to anticipate with practice.

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Becky HoldenI have been thinking about something you also mentioned in this post. What choices do I make to best meet the needs of the students. I can choose a task, offer that same task in four different classrooms, and the learning intention for each classroom becomes different. .

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mathmindsblogPost authorI agree, each classroom and change in task can impact the learning that happens so much. I really enjoy thinking about all of the different ways we could introduce, teach, question, and facilitate whole-group discussions in relation to what students leave the classroom understanding that day. So much fun!

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