**Who gets to be challenged in class? How does the work we give students impact their status (or how they perceive their status) in the classroom?**

### Meet Calvin

You’ve probably met a student like Calvin before. Calvin wants everyone in his third grade class to know he’s good at math. (They know.)

He is good, too. He looks for patterns, and makes connections. Someone taught him how to add fractions with unlike denominators, too, which looks like a pretty cool party trick to his eight year old classmates.

Calvin wants a challenge in math class — but if something is *too *challenging, he bawks at it: “I don’t like this! Give me a better challenge.” (Sigh.)

Calvin’s third grade class is currently studying fractions. On my first day in the classroom, Calvin finished a task much earlier than his peers. “Ugh,” he lamented in a stage whisper — loud enough for everyone around him to hear. “This is so easy. It’s boring.”

In a pinch, I pulled out my phone and showed him a **Yohaku** puzzle that involved fraction addition/subtraction. They’re a lot of fun, and I’ve used them in classrooms from first grade up to 7th!

To solve a Yohaku puzzle, fill in the empty cells such that they give the sum (or product) shown in each row and column. This is an additive puzzle.

Calvin was thrilled!

### Opening Up The Challenge to Others

The next day, I brought a packet of these problems for Calvin, and offered them to anyone who expressed interest, too. Of course, I forgot: the class was working to understand equivalence. The only students ready to work on fraction addition (with unlike denominators) were ones with experiences outside of school. Students that were not familiar with this content puzzled for a little while, wrote some incorrect numbers, and grew bored.

Although the puzzle was great for Calvin, and pushed his thinking in appropriate ways, it wasn’t a fair challenge for all. Accidentally, I had elevated Calvin’s status — and Calvin didn’t need *any help* elevating his status. He was happy to announce his superiority to others.

I had to rethink things.

**When do we give “challenging” tasks or puzzles to students?**Is it only after they’ve completed ‘other’ work?**What’s the goal of this “challenge work”?****How could I provide a challenge that would push Calvin’s thinking, but also the thinking of students who only had experience with third grade content?**

### The Striped Snake Barbecue

Enter: Math Pickle’s Half Fraction Snake Challenge, aka the Striped Snake Barbecue (@gamesbygord)

This puzzle is about equivalent fractions and halves. There are 3 customers at the Snake BBQ: one that wants a portion of snake that is 1/2 blue, another that wants a portion of snake that is 1/2 red, and one that wants a portion of snake that is 1/2 yellow. Your job is to make two cuts in each snake to partition the snake into 3 portions, one for each customer.

I launched the puzzle with a “notice/wonder” (“what do you notice? What do you wonder?”) about the first snake on the page.

- There are three colors used.
- There are 4 of each color.
- There are 12 squares in all.
- There doesn’t seem to be a pattern. It goes yellow, red, blue, red yellow, red blue?
- That might be a pattern. It’s yellow and blue with red between them. But the numbers don’t make any sense.
- There are always 2 blues in a row.

Then I launched into a story about our Snake BBQ restaurant, and the conceit and constraints of the puzzle. Students were intrigued.

Students worked using “SmartPals” (transparent dry erase pockets). I could see this working with the puzzles tucked into sheet protectors, or even with paper and pencil. The colors make it easier to access, but I imagine it could be done in grayscale, too. There is a black & white version available on MathPickle.com.

Students preferred to work parallel to peers. They mused aloud.

“I cut here, so that the first part is 1/2 blue.”

“But then the rest of the snake is impossible.”

“Maybe this is the impossible snake?” (One of the six snakes on the page *is* “putrid,” and doesn’t work.)

“This snake is 1/2 red AND 1/2 blue. I’ll give it to the red customer.”

“Oh, this part IS 1/2 yellow! There are 3 yellow and 3 not yellow. Half.”

“Is it okay if my 3 yellow aren’t in a row? Is it still a half if they’re broken up?”

“Yeah. I think so.”

Students who completed the first challenge were given 3 “mythical circle snakes,” with no heads or tails. Students could then make 3 cuts, partitioning the circle snake into 3 portions, to appease the customers.

“The circle snake is harder! I don’t know where to start!”

“There are more squares in a circle snake!”

To close the problem, we discussed strategies for proving that the fractions are equivalent. This is precisely where we are in our fraction unit! How perfect!

### When and how do we “challenge” students?

I *love* Yohaku puzzles. (I might write about them another time.) But the fraction addition Yohaku weren’t the right choice for this particular third grade class.

Meanwhile, the Striped Snake Barbecue was accessible to just about everyone in the class. Some students already had a good sense for equivalent fractions, and were able to reason that 4/8, 5/10, etc. would be worth 1/2. Other students needed some extra supports, like red, yellow, and blue cubes that he could organize into equal piles. If the number of cubes for the color you want (e.g. yellow) is equal to the number of the other cubes (e.g. red and blue), then your snake is 1/2 yellow. This allowed them to draw on 1st and 2nd grade understandings of halving.

There are still days when I offer something to students that I know the other students in class probably cannot access. It’s a decision that I know comes has the potential to yield negative consequences for our classroom community. So, increasingly so, I try to find challenges that are more open — like the Inaba Place Value Puzzles, which I have used in 2nd graders, 3rd grade intervention, and with 6th grade accelerated students! It’s all in the framing (the launch AND the close), the math you draw out for students, the way students interact with the puzzles (including available tools).

### What’s in a word?

Maybe thinking of these as “*challenges*” is not even the right word or framing for us, as educators. It implies that there’s “regular work” and then there’s “extras.” Students only get to work on a ‘challenge’ after they complete some other required work. Some days this happens, but some days, like with the Striped Snake Barbecue, it just becomes the task for everyone. It’s not extra or for enrichment. *It’s just how we learn together.*

**Resources**

- Yohaku Puzzles
- Half-Fraction Snake (Striped Snake Barbecue) from MathPickle.com
- Posts about Inaba Place Value Puzzles (Feb 2017) (Nov 2018)

I think this sidesteps the problem a little bit. Let’s assume that the class is always doing the most engaging problems like the snake barbecue. There still may (probably will) be kids who finish early or who perhaps already know what’s being focused on.

Thus despite the tasks you always reach a point where you have to decide whether you are going to extend or not for a student and how to do it. And even if you don’t the problem of kids groaning and claiming this is easy still also remains.

What resonated most for me is I have a girl this year who thinks she is misplaced in Math7 (“I know everything already”) She actually is a great problem solver and often demonstrates interesting thinking. So in my limited role I’ve been trying to give the advice to find the interesting part in the math you’re working on and at Math Club things are free ranging enough that I think she encounters lots of problems that challenge her thinking. But I imagine given I’ve heard this that it probably comes out a bit class too. And while I know time will make things right its hard.

Thanks for the post

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I agree that there will never, ever be a class where students all ‘finish’ at the same time! One thing about this particular puzzle (and a number of others I found) is that they have some built in ways to extend them. There are some harder versions of the puzzle, or students could create their own, etc. I think it’s important to plan for the sorts of questions to ask and ways to extend.

I will often extend a student’s thinking with a question that other students in the class may not be able to access, but I feel like that’s a very different decision in the context of status. Asking a question to extend thinking privately should have minimal impact on others. I’ll often sneak to the side and flip a 2nd grade place value question to be able decimals for students that I know are ready for that. I try not to give out an explicit assignment that is about decimals for 2nd graders.

And I don’t think children who are working on ‘above grade level’ content should have to hide that! The other day, first graders in a number talk wanted to go deep about negative numbers. (“10 – 3 is not the same as 3 – 7. 10-3=7, but 3-10=-7. They’re like opposites.”) We had a good portion of this conversation in front of the class. It’s such a difficult balance to strike. I don’t want students who are ready for those conversations to feel burdened by censorship — whether self-imposed or from the teacher — while I also want students who are ‘struggling’ to feel comfortable and empowered. Part of feeling comfortable in our classrooms is going to have to be feeling comfortable that some students know things you don’t know. Part of feeling comfortable is understanding that just because other students understand something that I haven’t been exposed to (or that I struggle with) doesn’t mean that I’m not capable of achieving things, too.

The other day, I worked with a group of fourth graders that wanted to explore negative fractions on a Yohaku puzzle. Clearly the rest of the class wasn’t ready for that… so… my thinking about this topic is still very ‘rough draft!’

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