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

The first lesson of a new unit always feels like an entire class period of formative assessment to me, which I love! I think finding out what the students know about a topic, especially if it is the first time it is introduced that year, is so interesting.

Since the first lesson of the 4th grade fraction unit starts with fractions of a 4 x 6 array, we wanted to create an introduction lesson that was more reflective of all of the great work they did with fraction strips in 3rd grade to get a better picture of what they know. In 3rd grade they do all of the cutting of the strips, and since we didn’t feel that was necessary to do again, I created a SMARTBoard file so we could build together. [the file is attached at the end of the post if you want to use it].

I posed this slide to introduce the whole:

Then I asked this sequence of questions as we built them on the board:

• If I wanted halves, how many pieces would I have? What is the size of each piece?
• If I wanted thirds, how many pieces would I have? What is the size of each piece?
• etc….until they were all built.

I wanted to reintroduce the language of “size of the piece” from their 3rd grade experiences. Every once in a while I would pause and ask how much I would have if I had more than 1 of those pieces to see if they could name fractions over a unit. For example, What if I had 3 of those fourths? How much would would I have? 

Next, we put up the following questions with the picture of the fraction strips we built:

They recorded them in the journals as a group and then we made a poster to add to as the year progresses. They started with fractions they could show on the fractions strips and an interesting conversation about the fact that we couldn’t list any for 1/8 or 1/12 based on the strips, arose. After talking with their groups, they generated a couple. The conversation about the change in the size of the piece when we make equivalents and how many pieces we would have was really awesome (Yeah, 3rd grade team):)

This was as much as we could fit into one class period, so we asked them to journal about any patterns they noticed or things they were wondering about fractions.

I apologize for the overload of student journals from this point forward, but there were so many great things to think about in planning the unit from here!

These are things that jumped out at me after reading and leaving notes in their journals, I would love to hear any other things that stood out to you:

• A lot of talk about “doubling” and “halving” when naming equivalent fractions. Will want to address what is exactly doubling, what that means in terms of the fraction strips, and how it is affecting the numerator and denominator.
• Interesting noticing and wondering about addition. Some wondering how it works and others thinking they know.
• Love the even and odd talk throughout!
• Some wondering about multiplication and division of fraction!
• The range of fractions – how many we can name, how many unit fractions there are.
• The size of a fraction in different forms – Is the whole the biggest fraction? Is the numerator smaller than the denominator?

In case you want to try it out:

SMARTBoard file for 4th grade

SMARTBoard file for 5th grade

PDF file of the file.

The 4th grade has just started their fraction unit, so I was curious how that may impact their work with the Cuisenaire rods. I started just like I did in Kindergarten and 3rd grade, with a notice and wonder:

There were two of the ideas that really struck me as things I want to have the students explore further later: First was, “2 of the staircases (the staggered rods in order of size) could make a square.” They had them arranged on their desks like the picture below…but I want them to answer:

• Is that a square?
• If not, could we make it a square?

Then I started to wonder, do we call it a square? Should we say square face? Then what about area…would we say, “What is the area of the rectangle?”? That feels wrong because they keep calling the white rod a cube (which it is). But then asking about volume is not 4th grade. BUT, the tiles we use for area in 3rd grade are also 3-dimensional. <–would love thoughts on any of that in the comments!

One student noticed that the orange rod was the height of the staircase and I thought of area again since it was said right after the comment above. This idea would be really helpful for the students above when they are determining if their figure is a square.

I loved that one group noticed that any of the rods could be a whole and another group wondered if orange was the whole. Great lead into what I was thinking I wanted them to explore!

I asked them to find values for the rods based on their relationships. Of course the very first group I call on had 2 as the whole, which blew a lot of students minds, so I want to revisit that a bit later and ask them to explain how that works.

All of the other groups had orange as either 1 or 10, so I asked them to find the other values if the orange was 5 and 100. They played with that for a bit and then I began to hear a lot of aha’s, so I set them off to find more and they could have gone on forever.

I left them with the prompt, “Tell me about the patterns and relationships you notice.” and for those who looked like they were struggling to answer that question, I added, “If you are struggling with that, tell me how you could find the rest of the values if I gave you one of them and which one would you want?”

I loved how this student chose the orange, white and yellow as the easiest end, beginning, and half. I also like the red x 2 is purple, but we need to talk through that notation a bit.

This was the most common response, seeing the numbers get smaller as the rod got shorter.

This student’s noticing could be an interesting number choice question to pose: Why do you think groups chose numbers for orange that were doubles or halves of the other numbers we already had?

This student disagreed with the student who gave the responses in the first column because he is determined the white is 1/10 because the orange is 1. Would be great to pair them up and have them come to an agreement.

This student is seeing the white value adding to each value above it to get the next. I also love how she writes notes about how neat her handwriting is:)

I would love to have them play around with this first pattern in this entry! What other relationships could they find after they explored this one?

So much fun! Cannot wait to get into other grade levels to see if I can begin to find a progression of ideas with these rods!