Purposeful Number Choices: Fraction Subtraction

Today, I introduced Number Boxes (blog #1, blog #2, blog #3) to a fifth grade class. (Read blog #1 for the rules to the game.) The teacher was looking to spend a day reviewing fraction subtraction, and this is a great game for practice. It’s engaging. It has elements of choice that encourage strategies and reasoning, but also there’s some luck involved. It’s accessible to many kids.

In my last blog post, I described how some different rounds in another game allowed for great conversations about the need for flexible strategies when adding fractions. When do we want to reason about the size of a piece? When do we want to computer using equivalent fractions? C. Harun Böke wrote the following response:

C. Harun Böke wrote, "Once again, amazing post!.

Apparently, it's quite important to think through about the whole discussion that is intended, hence the choice of fractions as well."

He’s absolutely right that the number choice mattered, and also that it’s important to anticipate different avenues a class conversation might take. I carried this with me as I planned the next learning experience with Number Boxes. I thought about how to structure the game to increase the likelihood of certain mathematical dilemmas, and also how to launch the game using a carefully planned problem string. (You will see me interchangeably refer to it as a ‘number talk’ and a ‘problem string.’ Apologies to people that hate this.)

Planning the Game

If I set up the game like this:

There was a good chance that many students would end up with negative answers. It’s so easy for the second number (the subtrahend) to be larger than the first number (the minuend). (As an aside: I had to look those words up again to make certain I’m using them correctly, and I don’t use them with students.) These fifth graders would have had some light discussion of negative numbers over the years, but nothing formal until 6th grade, so I wanted to stay away from that.

For example, rolling the numbers 4, 1, 3, 5, 6, 1 might result in the following scenarios:

Student 1: $\frac{4}{5} - \frac{1}{3}$ (extras: 6, 1)

Student 2: $\frac{4}{6} - \frac{3}{1}$ (extras: 1, 5)

Student 1 has a positive result, and student 2 has a pretty dramatic negative result.

I decided I’d structure the game like this:

It is still possible for students to end up with a subtraction problem that has a negative difference. For example:

$1\frac{1}{6} - \frac{4}{1}$ (extras: 3, 5)

…but that this would decrease the odds of so many kids getting a negative answer, and it would further decrease depending on the target of the game. I’d map those out after thinking about the kind of student thinking that might surface.

Anticipating What Students Might Need

Students in this fifth grade seemed pretty comfortable using equivalent fractions to add. I anticipated that one sticking point for them might be when regrouping is involved. There are ways to think about it that don’t involve regrouping: for example, students might use a number line to jump backwards to the next whole and then beyond it, or they might think of subtraction as a comparison situation and add up, etc.

I decided I would launch with a number talk that would elicit both thinking of subtraction as moving backwards on a number line, and also using equivalent fractions. That might encourage these approaches during game play.

Launching with a Number Talk

I wanted to start with a problem that I thought all of the students in the room would be successful with, and build from there. That meant dipping into fourth grade content.

$1\frac{1}{6} - \frac{1}{6}$

Students quickly signaled that they had an answer by giving a thumbs up, held close to their chest. I elicited answers and quickly discovered that everyone had arrived an a consensus: 1. Jada shared why: you have one and a sixth and then you take away that sixth and it’s gone, so… just one.

Then I wrote the next problem:

$1\frac{1}{6} - \frac{2}{6}$

I expected that students would see that this was subtracting one more $\frac{1}{6}$ , which would make the difference $\frac{1}{6}$ less. $1 - \frac{1}{6} =$ …

The answer was again unanimous amongst the class: $\frac{5}{6}$ . I modeled Leo’s explanation on a number line.

Because I also wanted to elicit the strategy of using equivalent fractions, pushing us into solidly fifth grade territory, I used the next problem:

$1\frac{1}{6} - \frac{1}{3}$

As students started to signal — thumb next to the chest — a few students blurted out, “wait… but… isn’t that the same?”

A great question! What’s the same about it? Since the values of each fraction are the same as the values for the fractions in the previous problem, yes, the difference is the same.

And lastly:

$1\frac{1}{6} - \frac{2}{3}$

I anticipated that some students would solve this problem by taking away an additional $\frac{1}{3}$ or $\frac{2}{6}$ .

“What do you think the answer is?” I asked, before calling on Thomas.

“It’s $\frac{3}{6}$ ,” he said.

“Give a me-too [sign] if you agree with Thomas.” It looked like most students agreed. “Raise your hand if you got a different answer.” And at least half of the class’ hands shot up.

“I think it’s $\frac{1}{2}$ ,” Destiny said. Immediately, students signed their agreement.

“Anything else?”

Yi Chen narrowed his eyes, as if he were weighing the decision whether or not to add to the conversation. Knowing him, I wondered if he was thinking about how there are infinite equivalent responses. The answer could be $\frac{4}{8}$ or $\frac{6}{12}$ or $\frac{512}{1024}$ .

“Yi Chen?”

He hesitated. “Ehh… I think $\frac{1}{2}$ is the best answer.”

“Okay, how did you get that?”

“Well, $1\frac{1}{6}$ is $\frac{7}{6}$ , and $\frac{2}{3}$ is $\frac{4}{6}$ , and $\frac{7}{6} - \frac{4}{6} = \frac{3}{6}$ which is $\frac{1}{2}$ .”

“Hmm… I’m curious about why you changed $1\frac{1}{6}$ to $\frac{7}{6}$ …”

“Oh, well, I want to take away $\frac{4}{6}$ , so I wanted to start with more than 4 sixths,” Yi Chen stated.

“…or it goes into the negatives!” Finn called out. I thought about addressing the precision in this — $1\frac{1}{6}$ is larger than $\frac{4}{6}$ , and so it wouldn’t exactly go into the negatives. It would result in $1-\frac{3}{6}$ , which is $\frac{3}{6}$ . But we had so many other things to work on that day, and I knew that this would be at least a 10 minute detour. I decided to let it go for now. We recorded Yi Chen’s thinking

Number Boxes, Round 1: Close to 1

I wanted to choose a target that would result in an interesting conversation, and would also connect to some of the strategies and ideas in our number talk.

“Okay, so your target is: an answer that is as close as possible to 1.”

“Is it okay if our answer is exactly 1?” Noah called out.

“What do you think, everyone?” I asked.

“I think that if you get exactly one, you definitely win the round!” Ethan offered.

I decided that, to make the numbers a little easier to work with, we would generate the random numbers using a 1-6 die. I used the ones on the smartboard for the sake of novelty, but I generally use either real dice or my phone.

We ended up getting the numbers 5, 6, 4, 5, 1, 4. I played alongside the kids, and here’s what I got:

(Also I like that they campaigned to call the extra boxes, that I usually call ‘throwaways,’ the compost. These numbers aren’t recycled, but I’m happy to let them compost away, to return to and enrich our natural environment.)

I did… okay. I asked students to determine whether my difference would be greater than or less than 1 (greater!), and then we decided that the most accurate way to calculate these differences was to use equivalent fractions. My result was $1\frac{9}{20}$ , which, again, was… fine.

Students did much better. A few students had answers of exactly 1. ( $1\frac{4}{5} - \frac{4}{5}$ — they got lucky!) And then a few students were remarkably close. One student got $1\frac{5}{6} - \frac{4}{5}$ , for an answer of $1\frac{1}{30}$ . Excellent work! I like that this game involves both luck and strategy.

But, most importantly, we were building off the thinking from the number talk. The first problem was all about subtracting a fraction from a mixed number in the form of 1 + that fraction, and arriving back at 1. Students knew that they wanted the fractional parts to be equal in order. Maybe they would be identical, or maybe it would use equivalent fractions.

Round 2: Smallest Difference ≥0

I announced that the next round would follow the same format, but this time our goal would be to get the smallest difference that didn’t push us into negative numbers.

“What if we get exactly zero?” Noah asked, slyly.

I smiled. “That would be amazing!”

Once again, I played along with the kids. Students found it harder to latch onto a strategy for this one. Should the second fraction be larger? What numbers are we hoping for the denominator if the numerator is 4?

Here was my result:

$1\frac{3}{5} - \frac{4}{3}$

Which students helped me rename as:

$1\frac{9}{15} - \frac{20}{15}$

“This is just like in our number talk warm-up,” I mused, “where we discovered that equivalent fractions can be helpful for figuring out fraction addition and subtraction.”

“But,” I continued. “I’m not certain what my next move should be. I want to take away the $\frac{20}{15}$ from the $\frac{9}{15}$ , but that is feeling complicated. What would you do next?” I asked students to think for 10 seconds on their own, and then turn and talk with someone at their table group. I wondered which students might want to rename the first number, $1\frac{9}{15}$ in fraction form, and which might want to rename the second as a mixed number, $1\frac{5}{15}$ .

When we joined together, Sarah seemed eager to talk.

“Did you forget that it’s really $1\frac{9}{15}$ ?” Sarah asked.

“Okay! So how will that help me?”

“1 is a whole, so that’s $\frac{15}{15}$ . And then plus the $\frac{9}{15}$ and that’s $\frac{24}{15}$ and you can minus the $\frac{20}{15}$ and it’s only $\frac{4}{15}$ .”

“ $\frac{4}{15}$ seems really close to zero! Did anyone beat me?”

I saw several hands waving, but more confused faces.

Comparing The Differences: Who Won?

“All right, so you’re going to share your answer with your table group,” I told them. “And whatever is the smallest difference greater than or equal to zero will get to represent your table group as we try to figure out the class winner. Think about how you can compare the different answers, and, of course, if anyone beat my dazzling result of $\frac{4}{15}$ .”

Students set to work. I drew a quick map of the classroom on the board, and, after a minute of deliberation, asked each table group to share their smallest difference.

“We got $\frac{1}{12}$ , which was better than you,” Yi Chen said.

Those were fighting words. “Ooooh,” I grinned. The drama. “How do you know?”

“Twelfths and fifteenths are pretty close in size, and you have four and we only had one,” Yi Chen responded.

“Yeah, but ours is better!” Sarah laughed from the second table.

“Really?”

“Yes! Zero!” Sarah trumpeted. I saw Yi Chen spin around in his chair with a slight glare: jealous but impressed.

“How did you get exactly zero!” I asked.

Sarah had me record on the board:

$1\frac{1}{4} - \frac{5}{4}$

“Ohhhh, yes, yes. That $1\frac{1}{4}$ can also be thought of as…” I paused for dramatic flourish.

“ $\frac{5}{4}$ ,” Yi Chen groaned.

“How did you do, table 3?”

They had a smallest difference of $\frac{1}{3}$ , and the last table group matched my $\frac{4}{15}$ .”

“Okay, I think we can all agree that Sarah’s table group is the champion here,” I announced some some tepid applause. The kids at Table 2 smiled. “But who came in second? How can we order these fractions?”

Ethan pointed out that $\frac{1}{3}$ can be renamed as both fifteenths and twelfths, making those comparisons easier.

$\frac{1}{3} = \frac{5}{15}$ , and $\frac{4}{15} < \frac{5}{15}$ . Meanwhile, $\frac{1}{3}=\frac{4}{12}$ , which makes $\frac{1}{12}$ noticeably smaller.

We ordered the four answers on the board, and officially crowned Table 2 the winners of our second round.

As an exit ticket, I asked kids to solve $1\frac{1}{3}-\frac{3}{4}$ . This problem would involve using equivalent fractions and also grouping.

In Reflection: Making Ideas Explicit

That purposefully designed number talk (or problem string) at the launch of the lesson elicited most of the important ideas that we then carried and applied during game play.

It’s easier to subtract when there are common denominators.
We can rename fractions with equivalent fractions in order to create common denominators.
There are several strategies for dealing with ‘regrouping’ with mixed numbers.
- One is to regroup, like we do with whole numbers. The biggest difference between regrouping with fractions and regrouping with whole numbers is that it has less to do with place value. Instead, focus on the value of the unit/whole. For example, $1\frac{9}{15} = \frac{15}{15}+\frac{9}{15} = \frac{24}{15}$ .
- Some people prefer to always rename the mixed number in fraction form. This will always work. Sometimes, the numbers can be a little unwieldy if you prefer to write the answer as a mixed number, but…
- Answers can be in fraction form or written as a mixed number.

It was nice to surface these ideas early in the lesson — during the number talk — so that we could spotlight them during game play. I did this deliberately, and explicitly. I named these things with students. We were using the game to practice, and not as a learning/discovery experience. Using the number talk also allowed me full control over the numbers that I chose, whereas the game involved some (strategic) luck. While we still ended up with some excellent numbers to compare and contrast during game play, it was nice to have control in the initial stages of the lesson, and then again at the close.

Some questions that I’m still considering:

How does anticipating student thinking and strategies impact our plans for a lesson?
We want to be intentional about how and when we surface ideas. Does this change based on the purpose of the lesson (introduction to an idea, practicing, etc.)