Peas and Carrots… and Beans

Last year, our first grade team had the opportunity to work with Investigations authors from TERC — the inimitable Karen Economopolous, with Megan Murray in absentia. To prepare for our work on a lesson called Peas and Carrots, Karen asked us to engage in the math as adult learners. Instead of trying to find combinations of 7 — some peas, some carrots — Karen asked us to determine combinations of 7 with some peas, some carrots, and some beans.

While it may seem like one small twist, that extra variable increases the complexity by a lot. But how much?

I had this in the back of my head as we were launching the second Peas and Carrots lesson in first grade recently.

Peas and Carrots

When students are first working on peas and carrots, they might use different tools to help them track combinations. Some students like using unifix cubes (especially color coded ones: green for peas, orange for carrots). Some students jump straight into writing equations. Some students instantly see how they can generate combinations using the commutative property (e.g. 1 pea and 6 carrots, or 6 peas and 1 carrot). Others may work systematically.

If there have to be some peas and some carrots, there are n-1 combinations. For n-1 it looks like this:

If it is all right to have zero peas or carrots, the total number of combinations with two vegetables increases to n+1, to allow for two combinations of 0 and the total number of items.

Some students spend the entire class wrestling with it, while others quickly wrote lists.

Rachel signaled to me as we walked past one another, observing students at work. She pointed to a student who had finished the problem accurately, in less than a minute flat. “Next time should, should we launch the extension problem?”

Peas, Carrots, and Beans

Two days later, we were working again with peas and carrots, but this time with 9 total vegetables. Several students were visibly ready for more.

“Today, we were working with 9 total vegetables. Right now, we’re going to make the number smaller — only six — which might sound like it’s making it easier. But let’s stretch our brains to see what happens when we have 6 items and there are some peas, some carrots… and some beans!

First grader James set off to make a chart, and I did the same, on a sticky note that I concealed with a cupped hand.

How many combinations can I make that include exactly one pea? …exactly two peas? My list contained 10 solutions.

I made up another list to model five items: peas, carrots, beans.

There were 6 solutions. Instantly, my brain started firing: 10, 6… are these triangular numbers? Would four items yield 3 solutions? Yes!

Triangular Numbers

Triangular numbers are numbers of counted objects that could be used to form an equilateral triangle.

from Wikipedia: “Triangular Numbers

We can think about these recursively, with each successful triangular number adding on a new base that matches the stage number, e.g. T_{3} has a base of 3. T_{2} > looks just like T_{1} with an additional base of 2, and T_{3} looks like T_{2} with an additional base of 3, etc.

These can also be expressed as the sums of consecutive numbers, starting with 1.

T_{1} = 1

T_{2} = 1 + 2

T_{3} = 1 + 2 + 3

T_{4} = 1 + 2 + 3 + 4

There are 4 combinations that start with 1 pea. That’s because there are 6 total items, and with exactly one of them known to be a pea, we have 5 unknown items remaining. Assuming that there have to be at least one pea, at least one carrot, and at least one bean, this yields n-1, or 4 combinations.

If two items are known to be peas, that leaves us with 4 unknown items: 4 – 1 or 3 combinations.

If three items are known to be peas, that leaves us with 3 unknown items: 3 – 1 or 2 combinations.

There is only one way to have four items known as peas.

4 + 3 + 2 + 1 = 10

So for six items, and exactly one is a pea with the rest split between carrots and beans, there are 6 - 1 - 1, or 4 possible combinations. When exactly two are peas, and the rest are split between carrots and beans, there are 6 - 2 - 1 combinations, or 3 combinations. This continues until we hit a value of 1 at 6 - 4 - 1, and from there it is impossible to create further combinations using more peas.

So a total of 6 items, with some peas, some carrots, and some beans, results in the same number as T_{4} = 1 + 2 + 3 + 4.

And from there, I surmised that the number of combinations will always be the triangular number for two less than the number of items. If we had 10 items, this would result in:

T_{10-2} = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8

or 36 possible combinations. Add one more item, and there are 45 possible combinations, and. this quickly becomes unwieldy for students to list out.

What did students do?

I was surprised to see that students who had written somewhat haphazard lists of combinations before were all of a sudden creating tables to organize their thinking. Many of them were generating each combination one at a time.

“Okay, so 3 + 1 + 2 works,” Sergey said, recording 3 peas, 1 carrot, and 2 beans.

“You can also flip it around!” Bella clapped. “1 + 3 + 2!”

James paused, then said, “And even 1 + 2 + 3.”

“That reminds me of the work you did when it was just peas and carrots, how we could use 4 peas and 3 carrots, or 3 peas and 4 carrots,” I told the kids. “Do you think this flipping around strategy will always work?”

But they had moved past me. “How about 2 + 2 + 2,” Sergey suggested. James and Bella frantically wrote it down. Meanwhile, other students were starting to gather.

“What are you working on?” Jeremiah asked.

“Now there are beans!”

“What? Beans?”

“You can have peas, or carrots, or BEANS! There are so many answers!”

The teacher, Rachel, signaled to get the classes attention, and let them know that they could stretch their thinking with the group at the back. “Challenge your brain to figure out what happens when we add in beans.”

Red and Blue (and Green) Balloons

The next day, students solved a how many of each problem about balloons:

I have 5 balloons.
Some are red and some are blue.
There are 4 possible combinations.
Can you find them all?

Classroom teacher Rachel suggested that everyone start with red and blue, but then stretch yourself: what if there are some red and some blue and some green?

Here are three student work samples. Use the left and right arrows to navigate.

I was fascinated to see that many students wrote a list of equations when asked to have two parts, but all moved to a table to show three parts. Our classroom discussion the previous day had focused on the commutative property (e.g. if 4 and 3 make 7, 3 and 4 must also make 7. So 4 + 3 = 3 + 4), so it was unsurprising to see this creep into their work with three variables or parts.

Today’s Number

A few days later, I introduced the routine “Today’s Number” to students. I was pleasantly surprised to see one student organize his work like this.

I noticed that the column of two number sums to 10 seemed systematic: 1 + 8, then 2 + 7, then 3 + 6. And while the column for three numbers at first seemed disorganized, I noticed that the first addend was 3, and then 4, and then 5…

Interestingly, all of these expressions sum to 9, and not 10 — which had been the target number. It makes me wonder if he was building successive expressions from the previous one, like a string. The sums in the column for 4 numbers also sum to 9, and many seem to connect to the ones in the column with three numbers. 1 + 6 + 1 is next to 6 + 1 + 1 + 1.

Extending Student Thinking in the Math Classroom

How do we stretch student thinking in the math classroom? Fawn Nguyen (@fawnpnguyen) recently wrote on twitter about pushing depth over breadth, and accelerating students ahead.

Meanwhile, @cheesemonkeysf wrote about the need to acknowledge the limits of (and challenges with) heterogeneity, and that some students are better served by being allowed to accelerate.

I do not feel fully grounded in my opinions about this, but I think that I agree with both. I would rather that teachers enriched the classroom experience with gentle and motivating twists like “peas, carrots,… and beans” rather than a cursory introduction to something that will happen in a future grade. But also there are times when I tell kids that we’re deliberately going to do a “sneak preview of next year” with a connection to our current content.

There are also cases where I feel like acceleration just makes sense. In fact, I used to teach an accelerated cohort within my school of elementary school students doing middle school math in a serious and comprehensive way. There are, of course, major issues around equity and access with acceleration, but that doesn’t mean that it’s never appropriate.

Which is all to say: my ideas aren’t fully formed about that, but I do feel pretty formed that things like “peas, carrots,… and beans” is a way to stretch thinking for both students and teachers that feels natural, and very mathematical. It’s taking things to the next level, even if it’s not the next grade level.

These kind of stretches are elegant, and also not always easy to come up with on the spot. So when we have some ideas, we share them.

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