Tag Archives: division

Gallery Walks: Engaging Students in Other’s Ideas

One instructional strategy that I love for collaboration and public sharing of student ideas is a gallery walk. In a gallery walk, students create displays of their thinking on chart paper or white boards and then the small groups walk around the room and visit each other’s posters. And even though students create such beautiful displays of their ideas, it is always challenging for me to structure the walk in a way that actively engages them in one another’s ideas. Like any problem of practice, it takes trying out new ideas to see what works, when, and for whom.

The Lesson

Last week, it was the first 3rd grade lesson about division. We decided to launch by mathematizing Dozens of Doughnuts to set the stage for the subsequent activities. If you haven’t read the book before, it is about a bear named LouAnn who keeps baking 12 doughnuts to share with a different number of guests who arrive at her door. We read the book and did a notice and wonder, anticipating we would hear something about LouAnn sharing doughnuts and the number of doughnuts, friends, or plates, which we did.

Student Displays

We then asked small groups to record all they ways that LouAnn shared her doughnuts. We purposefully didn’t specify the representation so they could look for different ways during the gallery walk.

As we walked around it was great to see the various ways students were representing the situations, but some small groups seemed to have settled on only one way. We had planned for them to look for similar and different ways during the gallery walk, but that can be so passive, with no opportunity for them to connect those new ideas to their work. So, instead of waiting for the gallery walk at the end, we decided to engage them mid-activity with each other’s ideas and allow time for them to use those ideas.

Taking a page from Tracy’s book, Becoming the Math Teacher You Wish You Had, we opted for a Walk-Around to cross pollinate ideas. We asked students to walk around and look for ideas they wanted to add to their poster. These could be new ideas or just a different way of representing an idea they already had.

You would have thought we gave them a chance to ‘cheat’ as they walked around with such intention to other’s posters. I wish I had captured the before and afters of all of their posters, but here are just a few where you can see the new addition of ideas.

After they finished adding to their posters, we paused to discuss the ideas they found from others – both new ideas they hadn’t thought about and ideas they had, but were represented in different ways.

Next Activity

Students then independently solved a few problems. It was great to see the variation we saw on the posters in their work. So many great representations to share and connect in future lessons!

More Ideas and Resources

Want to learn more about mathematizing? Check out Allison and Tony’s book, Mathematizing Children’s Literature.

Want to read more mathematizing blog posts? I have written about some of the books I used when coaching K–5.

Want to share your children’s book ideas for math class? Join me on IG!

Supporting Mathematical Habits of Mind

Mathematical Habits of Mind

Students Should Be Pattern Sniffers
Students Should Be Experimenters
Students Should Be Describers
Students Should Be Tinkerers
Students Should Be Inventors
Students Should Be Visualizers
Students Should Be Conjecturers
Students Should Be Guessers

The thing I love most about these habits of mind is the fact that as I read them, I can picture the math content and activity structures that could provide opportunities for students to develop these habits. I also really like the connectedness of them, where I can easily imagine how one habit leads students to engage in another. And because my favorite Math Practice is SMP7, look for and make use of structure, I am particularly drawn to the habit of conjecturing in math class. Excitingly, last week 5th graders were engaging in a topic that provided a perfect opportunity to conjecture.

Fraction Division

This past week, 5th grade students were dividing unit fractions by whole numbers and whole numbers by unit fractions. If you have ever taught this, you probably immediately picture students overgeneralizing these two different situations. In the vein of answer-getting, they often think the quotient will either always be a whole number OR always be a unit fraction – both including the product of the denominator and whole number in some way. And even though students have engaged in a lot of the habits within this work, it was with the two situation types separately.

To address the overgeneralization, we wanted them to engage in mix of the situation types in order to compare them. We launched with the following 2 problems, purposefully choosing the same numbers to elicit the difference in what is happening in the situation and the resulting quotients.

Student Thinking

As anticipated, we saw wonderful diagrams that generally matched each situation, but we could tell by the shading and erased work on Situation B that students were thinking that because they were working with fractions, their answer had to be a fraction.

We focused our discussion on the questions, “Where is 1 cake in your diagram?”, “Where are the people in your diagram?”, “Where are the servings in your diagram?”, and “Where is your answer in the diagram?”. Through those questions we saw a lot of labeling revisions to their work to make it clearer.

Mathematicians Talk Small and Think Big

“The simplest problems and situations often turn into applications for deep mathematical theories; conversely, elaborate branches of mathematics often develop in attempts to solve problems that are quite simple to state.”
Cuoco, Al & Goldenberg, Paul & Mark, June. (2010)

While the discussion was productive and we saw a ton of sense-making, visualizing, describing, and revision, I was left wondering how this moment transfers to the next time a student engages in one of these division situations.

I love this idea of tinkering around with smaller ideas to conjecture about larger ideas as a great way for students to deeply understand a concept and be able to transfer their understanding to the next time they engage in that concept.

So, for the tables done their discussions early, I asked them to write things they think are true about the division and lingering questions they might have. Here are a couple examples:

Next Steps

The question I am always left with after students have such amazing insights and questions is, ‘How do I keep this math conversation alive?’ With the pacing of curriculum, it can be challenging to dig into each of these moments for an extended period, so we need ways to let this thinking extend across the year.

One thing we could do is ask students if we can launch the next class period with their ideas. For example, I might ask the first student if I could post, “The order matters in division.’ at the start of class the next day and have the class discuss if they think that will always be true and why. This would be a great way to elicit the difference in quotients when we divide a whole number by a fraction and vice versa.

Another option that I used in my classroom, was posting the ideas on what I called a Class Claim wall. When students make a claim or conjecture, we posted them on the wall and then anyone could revisit them at any point and time.

I think both of these options are a wonderful way for students to continually think small and big about concepts while allowing us the opportunity to communicate to them that just because a curriculum unit of study wraps up, the learning about that concept continues.

-Kristin

If you want to read a bit more about claims and conjectures, I was kind of obsessed with it when I was teaching and blogged a lot:

Developing Claims – Rectangles With Equal Perimeters

Geometry Is Worth The Extra Time…

The Meaning of Subtraction

Remainders: Division & The School Year

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Looking ahead in planning for the remainder of the school year, I am currently finishing up my decimal unit and excited to end the year with Growth Patterns. I was planning on finishing decimals this week, however, I have one more thing that I feel is missing from the unit that I am curious to see how students are thinking around it. In the unit, within decimal division, the students are very comfortable estimating quotients and thinking about a variety of strategies in finding how many of the divisor are in the dividend. However, one thing that is not addressed is remainders, and how we notate them. I had not really thought much about it because in the context of the problems we were doing, the remainder made sense. However, during a division number talk, not within a context, the “r” was still there. It bothered me a bit. When I asked how they could write the quotient as a number, I got blank stares. I know fourth grade really spends a lot of time on interpreting remainders, but do we spend equal time on various notations of the quotient?

I have decided to extend my decimal unit just a bit longer because I feel this is something my students can definitely reason about and I am curious the connections they can make between whole numbers, fractions, and decimals. I decided to start with whole number divisors and dividends and move to decimals from there. Today, I gave them the problem 256 ÷ 20. They estimated somewhere just over ten and then I asked them to solve it. If they finished early, I asked them to write a context to match the problem.

The majority of the class’ work looked like these and contexts involved a sharing situation…

When pushed to write their quotient as a number without the “r,” most said this…(I do love the way this student divided:)

I did get a few 12.8 and 12 16/20, which interestingly fell more in money contexts…

All of these, I had anticipated, but then I got some really great unexpected answers that allowed students to think about the connections between notations…

12.5 r 6 12 16/256 12.75 r 1

I wrote these responses on the board and asked the students to see if any of the answers meant the same as 12.8 or 12 16/20 or 12 r 16, that we had established were the same. They did also mention, which I loved, that certain situations my use different notations.

I had some amazing proofs that we are kicking off the day with tomorrow before moving into decimal divisors. While I was hoping for students to look for equivalencies in the quotients themselves, most groups went back to trying out division in a different way to prove the answers. This group went back and solved the problem using the same method every time, just changing the breakdown of the quotient.

This group used multiplying up to see that 12.5 r 6 worked as a correct answer. IMG_0588_2

After asking them if they saw any relationship between the quotients, I got this…(much more what I was hoping to see in their reasoning)

This group had a nice, simple explanation at the bottom of this page…

This student is still sticking with 12 16/256 and quite honestly I don’t know how to approach this one. It is a different way of writing the remainder and I cannot decide if there is a time when this would be an appropriate notation?

The most perplexing quotient for most of the students was the 12.75 r 1 so I asked the student to write out his thought process because he was having trouble explaining it.

Now, while the entire class period seemed to focus on the remainder in a division problem, this explanation represents the remainder of the school year! I asked the above student to go in the hallway and record his thinking through the problem because he had such a beautiful way of starting to explain how he decided how much to add based on the distance from the dividend…but then I got this 🙂 https://www.educreations.com/lesson/view/kewl-aid/31841872/

And here’s to the remainder of the school year….

-Kristin

Decimal Division, Running & Why I Love My Tweeps

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Yesterday, I posed a decimal division problem to get my students thinking about what division means to them and how that applies to decimals: https://mathmindsblog.wordpress.com/2015/05/05/a-great-day-of-decimal-division/ (It was a really great day)

I was thinking of moving into a context today to see how they would represent the problem and the approach they would take after yesterday’s discussion. So, of course I threw it out on Twitter…

@MathMinds @allisonhintz124 @kassiaowedekind 48 mile trek/hike through mountain range or for fundraiser. want to put a sign every .6 mile

— ekazemi (@ekazemi) May 6, 2015

All evening I was thinking about a context and this one Elham suggested worked great for me! I was still thinking about how to word it to be something that the students may be connected to, then Joe’s tweet came this morning after my run…

@MathMinds So how many .1 miles is that?

— Joe Schwartz (@JSchwartz10a) May 6, 2015

Duh, my runs! Thank goodness Joe was up early too! My students know I run every morning and cannot fathom that anyone actually wakes at 4:30 in the morning, so I knew they would love this.

To start the class, I posed..

“I ran 2 miles on Monday afternoon. Every .4 mile I took a sip from my water bottle.How many sips of water did I take during my run?”

As with most times, I gave them some individual time before consulting with their table mates. It was awesome to see so many of the connections to yesterday’s work and also new representations that did not show up yesterday.

This one was so interesting how he broke up the mile to .4 +.4 +.2 and then combined the .2’s to make 5 four tenths.

This number line was so nice and then I loved how he got to the end and then counted the jumps going back down to zero. Also, at the top he had multiplied up to the 2 miles, nice way to show two ways of thinking about the problem.

There was a lot of skip counting by .4, but this model was especially wonderful. It is an area model combined with a number line. He counted up by .4 in squares that attached until he reached 2. I would expect students to count the number of .4 sections to find the answer, however this one labeled the 1, 2, 3, 4, and 5 at the end of each section.

I then gave them a log of my past five runs. I told them to assume that I still take a sip of water every .4 mile. I wanted to know how many sips I took and then how much further I had to go until my next sip.

I got some awesome partial quotients, number lines and multiplying up.

Now, the conversation of remainders came up. They want to know how to write the answer without the “r.” They wanted to know if they could write that as part of the number that was the answer. For example, could they write “7 sips r .2 as 7 1/2?” Saving that for tomorrow.

And THIS is why I love the #mtbos….my lessons take wonderful twists that make the learning experiences in my classroom so much better for my students! No teacher can do this job alone!

-Kristin