Fraction Addition: Playing a Game With Many Shades of Fluency

Today in grade 5, we played Addition Compare with Fractions (Investigations, Grade 5, Unit 3). Students draw fraction cards, calculate the sum, and compare sums. My description there is deceptively simple — this game is ripe with mathematical possibility.

Some rounds ended almost instantaneously. “I can tell that Jada won,” Destiny sighed within a few seconds of reviewing the cards. Other rounds seemed to involve laborious calculations, with children hunched over desks like 1950s accountants in tax season.

The Addition Compare with Fractions game promotes a robust version of fluency: accuracy, efficiency, and flexibility. (Russell 2000) Sometimes, students used reasoning to determine a winner. Sometimes, they needed to crunch through the calculations. Fluency is knowing when to employ each method.

Round One: Using Reasoning about Size

Jada invited me to join her and Destiny in a round. “We can figure out how to play with three people!” She offered cheerfully.

Destiny dealt each of us six cards, two cards for each of the three anticipated rounds. “Pick up only the first two,” she cautioned. “Then add them.”

Destiny revealed her two cards with an amused smile: “They’re kind of the same.”

“What do you mean?”

“Like 8, 8… 3, 3…” She trailed off, and then gleefully added, “oh, but they’re actually the same!”

“Can you figure out your sum?”

“It’s 1 whole plus 1 whole, so… two wholes!”

I revealed my cards: and .

“Hmm… mine aren’t as nice to add together,” I mused.

“Yours aren’t wholes,” Jada added.

I wrinkled my nose. “Oh, yeah, these numbers are hideous!” There’s something about degrading the numbers: it makes students laugh, but also seems to increase buy-in. They’re validated but suspicious. Like, “oh, good, we can all agree that we want nothing to do with this problem. So why are we still going… this lady has a lot of tricks up her sleeves.” Just kidding. Fifth graders in 2024 do not use phrases anything like “tricks up her sleeves.” Have you even met a child? All the same, the way that they narrow their eyes speaks volumes.

“But I think we can figure out which sum is bigger without calculating anything.”

Challenge accepted. Jada traced her finger cautiously along the edge of the table, and then, “Wait. Okay. Wait.”

Destiny looked up.

“Okay, so… is less than a whole. And is less than a whole. There’s no way we can get two wholes like Destiny did. You’re going to end up with one whole and something.”

“And sometimes,” I told her. “We will care about what that something is. But right now, I think we can declare Destiny the winner!”

“But what about my cards?” Jada asked. She flipped over the top two on her stack, and showed us the result: .

“Ten eighths and another eighth is eleven eighths,” Jada said. “I didn’t make it to two.”

“How many eighths would you need to get to make two?”

The girls weren’t certain, so I drew a quick diagram of a whole as , and another whole as an additional . They quickly agreed on .

“It seems like it’s much easier for you to add fractions when the denominators are the same.” I was helping the girls consolidate. “Like you were talking about adding an eighth to some other eighths, and you have some number of eighths in all.”

“Yeah, it’s way easier,” Destiny agreed.

“Okay, let’s see what we get next time!”

Round Two: Using Reasoning about Equivalence

I got two garbage cards: and

“Did you even make it to a whole?” Destiny asked. I think I caught her smirking.

“Anyyyyway,” I continued. “Let’s focus on your cards.”

Destiny got and . “I got one that’s bigger than a whole, and one that’s less than a whole.”

“Both of mine are bigger than a whole, so I probably won,” Jada stated. She had and

“But my is way bigger than your ,” Destiny argued. It’s like 1 and a bunch of extra. extra!” She paused. “Oh, that’s like one and a half! And then is a half! So I have one and a half and another half so I have two again. In this game, is the answer always two?”

“No, because I don’t think my answer is two,” Jada stated crisply.

“So last time, you had two numbers that were both eighths, and you combined them really easily. Ten eighths and one more eighth is eleven eighths. Can you do that here, since you have two elevens?”

Jada paused. “No…” she started. “Because… no. Oh, because this is 11 tens and then 11 eighths, and I don’t know what that is. But they’re both bigger than one, so my answer is definitely going to be bigger than two.”

“Drats!” Destiny said. Sometimes fifth graders in 2024 still say some hilarious, old-timey things.

Round Three: Yes, We Need to Calculate

We had made it two rounds without needing to calculate the sum using a rigorous process of finding equivalent fractions with a common denominator, and scaling.

But now it was time.

We each revealed our cards.

• I had .
• Jada had
• Destiny had

“These are alllll hideous!” I announced, to some polite chuckles. They were onto me.

“All of us have a car that is less than a whole and a card that is bigger than a whole,” said Jada.

“So how can we do this without figuring it out?” Destiny asked.

“Well,” I leaned in. “Sometimes we just have to figure it out. Let’s take a field trip to the smartboard to calculate!” Gone were the polite chuckles, and in their place were some groans of mathematical despair.

We started with my figures. . Jada wrote out multiples of 8: 8, 16, 24, 32… “Oh, three also goes into 24,” she said, stopping herself shorter than she had anticipated. “So I can make 8ths into 24th and 3rds into 24ths.”

“Eleven twenty-fourths is really close to a half,” Destiny observed.

“Yes, so my sum is really close to . Can you beat that?”

We calculate Jada’s sum next. With denominators of 6 and 10, we considered using a denominator of 60. I asked the girls to consider what we’d do if we renamed as , and they agreed that they’d rather work with thirtieths than sixtieths.

“Our fractions have the same numerator,” said Jada. She pointed to the pair of elevens. “So you have one and and I have one and . Your number is smaller so I think the pieces are bigger, right? So your fraction is bigger.”

We had just gone through the trouble of computing an exact answer, and even then we were able to use some reasoning to take us through the very last step. This game was working shockingly well!

Destiny’s was last. Her final sum was

“ is a whole, and is another whole…. oh, my answer is bigger than two! I win!”

Reflecting with the Class

When it came time to consolidate the game-playing experience, I called on Jada and Destiny’s experiences.

“I noticed that some of you sometimes finished a round so fast! You didn’t have to compute anything. You were able to figure out which sum was larger using reasoning and thinking about the size of pieces.” I saw Emma near the front of the room vigorously offering a ‘me too’ non-verbal sign, with pinkie and thumb extended out. “I see Emma agrees. Did that happen to other groups, too?”

I heard a few students mumble yes.

“But sometimes, you got stuck! You couldn’t reason your way out, and you had to actually do the computation.” Once again, Emma was emphatic: ME. TOO. “Did that happen to more of you? So: what sorts of numbers were you able to use reasoning on? When did you need to compute?”

Students agreed that they used reasoning when they could compare a fraction (or the sum) to a benchmark, like knowing that they were both greater than one. Students were also more likely to use reasoning when their fraction card was equivalent to a landmark number, like how .

And when did students need to know how to compute?

When the numbers were hideous, of course. And none of these fraction cards are truly hideous. (All numbers are beautiful!) What made them less appealing was their incompatible denominators, that required us to find common denominators. Often, these common denominators are not as easy to hold in our heads for mental math. Students need to know how to do this! I want every single one of those fifth graders able to add fractions with unlike denominators consistently, accurately, and efficiently. I don’t want them to shy away from those hideous denominators.

…and I think it’s also important that they can identity opportunities to not use that consistent, accurate, and efficient strategy. There are times when using reasoning is actually much more efficient. (Efficiency is also a relative and personal term, that reflects on our own prior experiences as well as some generalized ideas around effectiveness and organization.) It was nice to play a game that allowed students to practice both.