The Pencil Present: Decomposing 5

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SWBAT model the number combinations for 5 in a problem solving lesson.

Big Idea

Students develop an understanding that a set of five objects can be broken into more than one smaller sub-set and still make five.

Attention Grabber/Introduction

5 minutes

My kiddos are funny… we are silly and nice to other kids, but we are tough on pencils! There are some secret little beavers who chew on pencils, and we can make erasers disappear like nobody’s business!

When our university practicum teacher was finishing her placement, she got us a gift of fancy, beautiful, sparkly pencils!!! Never one to just appreciate a gift, I turned it into a math challenge. (Our university student laughed at the idea and gave me her thumbs up!)

I tell the students as we begin math that we got a special present from Ms. Nasr. The students say, “Aww…” I tell them I have a challenge for them, and they perk up even more. We are learning to love challenges, which is a sign of developing confidence. (MP.1)

“Some of the pencils,” I say, “Are red and blue—5, to be exact. I want you to figure out how many of the group of 5 pencils are red, and how many are blue.”

“If you can figure out the secret combination of red & blue pencils, your group can use sparkly pencils at your table!!!” I say.

Guided Practice

10 minutes

I look out to see a bunch of “Huh?” looks on students’ faces. I think 2 kiddos get the challenge and are ready to go, so I decide on the fly to make part of the challenge guided practice.

I tell the kiddos that I made something to help them keep track of the red & blue pencil possibilities, and I show our recording sheet on “the big screen.”

“Of course,” I begin, “There’s a possibility that the 5 pencils I’m talking about are all one color…”

“Who can give me a color that the pencils could be?”

I intentionally ignore the two kids who seem to have the idea of this challenge, which is terrible, I know, but I want to see if someone else can figure out the challenge. After some wait time, I will call on one of my two guys to give me a color, and the kiddo seems relieved to get to at least say, “Red—they can all be red.”

“Yes! That’s right—with red & blue pencils, they could be all red, for sure!” I affirm. “Let’s color in that possibility of all red pencils. We are showing the possible group of red pencils by the row colored red on our recording sheet.” (MP.4)

I ask the other student of the two who seem to have the idea to come up and color in all of the top row of pencils red.

I remind the kids that there are a bunch of other possibilities for our red & blue pencils. “The more combinations or pairs of you figure out, the better your chance is to guess exactly how many pencils are blue, and how many are red,” I remind them (MP.1).

Independent Practice

10 minutes

The helper of the day helps me pass out recording sheets, and I remind the students that we really only need 2 colors of crayons: red & blue.

It has been a while since we worked on decomposing numbers, and maybe it’s because it’s Friday and we are all really ready for the weekend, maybe it’s because this activity was not presented as a sort of “number combination” activity (in my efforts to make it more “real world problem solving”), but I’m seeing some weird color combinations as I walk around. So I ask the kiddos to tell me about their work.

One student tells me she made a pattern of red & blue. I agree that she sure did make a pattern, and I ask how her pattern is going to help her figure out the combination of pencils. She shrugged with a smile, “I dunno!”

Hmm… we have more work to do on this concept.

Some kiddos seem to have the idea, but they’re getting stuck on some of the details. Some kids insist that they have all the possibilities, but they accidentally repeat one or more color combinations, so that while the rows are all full, there may be two or three rows of the same pairing of red and blue.


15 minutes

Our closing actually takes a considerable amount of time, as I realize during the Independent practice that we need a re-teaching on the concept of decomposing numbers.

At this point, the crayons are removed from the tables, because we did have that little incentive to try to guess the actual combination of pencils. We go back to that row of all 5 red pencils, and then I begin thinking out loud to exemplify the process of systematically discovering the possible pairs of red and blue pencils.

“Hmm… they could be all red, like we started, or they could be almost all red… What would that look like?” I ask.

I find a friend who has 4 red and 1 blue pencil, and the student announces the 4 and 1 combination.

Now, I swoop in again. I state that I could put the one blue pencil in the middle, but it would easier for me to keep track of my pencils if I keep the red ones all together and the blue together, so I have a friend help me model to color in the reds together and the one blue pencil at the end of the row.

We do the same thing with the combination of 3 reds and 2 blues, and we even repeat the process for 3 blues and 2 reds. I actually even get out our 2-colored counters to show how the two possible pencil colors are like the counter (chip) combinations we worked on before, explaining that the yellow side of the chips is basically the same as the blue pencils in our pencil combinations. It’s all very systematic, and more than a few kids are saying, “Oh, I remember this!” (MP.5)

Someone says, “I didn’t know that the breaking of the numbers could be a guessing game!”

The students each have a marker to make a star or an asterisk by a row that they correctly recorded. I remind students to check carefully, and that sometimes if we make patterns, what looks different could be a combination that we already have. (MP.6)

Finally, at the very end of our day, our helper of the day reveals the combination of 3 red and 2 blue pencils with a big smile. Every table has at least one student who had this combination, so I announce to the students that every group will get special sparkle pencils. Students cheer.

Even though she is not with us for today's math, we all yell a heartfelt, "Thank you, Ms. Nasr!"