Hands-on Fraction Division with Play-Doh!

Welcome to my 6th grade math classroom! Today we are learning the algorithm for fraction division with one of my favorite math manipulatives, play-doh!

I’m using the fantastic Illustrative Mathematics curriculum through Open Up Resources. We’ve been doing some work on conceptual understanding of fraction division using diagrams, but today we are building upon the previous lessons and working toward the algorithm. The warm up from Illustrative Mathematics, Unit 4, lesson 10 asks students to work with partners on these questions:

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Students then compare answers for each question to see that dividing a number by a is the same as multiplying by 1/a. This worked great with groups that had answers, but several of my students are coming in with gaps from 5th grade, so we had to review that 12 groups of 1/3 meant that you had 1/3 twelve times. So I had to scaffold here a little bit but most groups were able to see the connection. But what about dividing by 1/a?

From the warm up, I asked students to think about the work that they had done and write on their whiteboards, “What does 12 divided by 3 really mean?” (without using the word division.) I listed their explanations on the board using their names: IMG_0889.JPG

We talked about how these were great ways to explain division, but we were going to focus on the last one as we worked on division today: How many groups of a number are in another number?

Illustrative Mathematics then goes into a nice sequence of dividing by unit fractions. What is 6 divided by 1/2? 1/3? 1/4? any unit fraction? We’ve done quite a bit of work with tape diagrams, and I decided to mix things up by using play-doh! I needed a manipulative that could easily be divided up into different units. After some ground rules (don’t eat the dough, people), I gave a little context also: We have 6 balls of dough (yes, I said “balls” in a 6th grade class, which is always a risk.) We need 1/3 a ball of dough to make a pie. How many pies can we make?

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I will post some more pictures throughout the day, but I saw some fantastic representations. We did this with 1/2 and 1/3. As we worked on 1/4’s, most students no longer needed the dough, and were intuitively able to see that dividing by 1/4 was the same as multiplying by the reciprocal. Yeah for SMP!

But what about non-unit fractions? Come back tomorrow to see part 2 of this lesson!

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Illustrative Math 6.3.14: Percent Strategies

I’m so happy with the work my students are doing on percents!

Today’s number talk warm up was 6 (0.8)(2). I know my kids struggle heavily with decimal computation, so I was expecting them to have a very hard time, and as usual, they blew my away!! “My favorite no” was 0.96. A student knew that 6(2)=12 and 12(8) = 96, but because there was a decimal point before the 8, she put a decimal point before the 96 to make 0.96. I love her thinking! We cleaned it up a little by talking about where the decimal should go, and we will keep working on this with more hits in the warm ups and eventually the decimal operation unit. Lots of kids came up with 4.8 (2) and were able to do that mentally. Nice work! I opted for one in-depth problem rather than rushing through both warm up questions in 10 minutes.

After working on a few problems about % off coupons, we moved on to info gap problems. Info gap problems are the problems that kids love to hate! They are frustrating and challenging and kids are relieved when they finally get it! If you are not using Illustrative Mathematics, I highly suggest looking up info gap problems and trying these out with whatever curriculum you have. I did some scaffolding here, giving most kids problem/data card 2, and giving the higher level kids problem/data card 1, which included managing multiple costs and percents. (The problems were all about purchasing something with a certain amount of money which was a percent of the total.) Again, I opted for one problem per team, with time to share multiple strategies. Here is some of their thinking! (The only writing that is mine here is the purple, showing students that the girl who make the chart was basically using a ratio table, except without the gridlines.) Great work, students!

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Welcome Back! Percent bars, tape diagrams, Illustrative Math, and the end of winter break.

Hello and welcome back to school after winter break! I am bright and cheery this morning on about 4 hours of sleep! Well, not really, but that will be my outward demeanor today. I need to be bright and cheery for the kids who aren’t: for the ones who have had a rough break or a too-much-fun break. In any case, it was a break, and I need to make sure that school and I are pleasant and welcoming as we shift back into our routine. I try to be patient with the kids who ask me where I keep paper and pencils (in the same place I have all year) because I can barely remember my own seating chart and where I put my markers.

Today’s warm up involves talking about winter break and what they are looking forward to as they return to school. In my heart of hearts, I am hoping someone will say “I’m looking forward to math!” but I gracefully accept answers like “seeing my friends,” especially since the first thing I did this morning before I cracked open my math materials was to seek out my own teacher BFFs to say hello to. My main goal today, in conjunction with teaching percents, is to connect with them and help them connect with each other on as we ease back into the routines of school.

Today we worked on percents using tape diagrams, which is Illustrative Mathematics Grade 6, Unit 3, Lesson 12. The warm-up included this picture: Screen Shot 2018-01-02 at 10.40.41 AM.png

We used the notice and wonder routine and it blew my mind that almost every student said, “It’s 60%.” Huh?? I knew they were mixing up the percent and the quantity, but I hadn’t anticipated this misconception, so I got stuck here for a moment. I know they will explore this idea more in the lesson so I do some direct instruction here, reminding them that the total or one whole is 100%, so what percent is shaded in? I know IM doesn’t expect numerical answers here yet, but I felt the need to address this because our break had them mixing up concepts, though they were on the right track. Advice on how to handle this next time?

After a little discussion, students started doing a nice job with today’s lesson. Without prompting, one student made his own chart to track the fractions, percents, and quantities: Alexpercent.JPG

Since I haven’t updated my word wall in a while, I think I will use this as inspiration! We are toward the end of unit 3, but need some visuals as we wrap up percents. Stay tuned to see what it looks like! One question I had was whether this format is okay for the work we are doing on tape diagrams. When this student wrote 9,18, etc. I knew he was thinking 9 + 9 more. . now I am up to 18 which is 10% + 10% so that is 20%. Or should it look strictly like  9 9 9 9 9 9 9 9 9 9 with 10% 10% 10% 10% etc. underneath?

Welcome back to work/school to you too! How do you ease into the new year after winter break? I hope you have a wonderful day!

Number Talks and Tape Diagrams: Illustrative Math 2.15

So I am loving number talks more and more!! We have done several number talks this week. I also got asked to model “active engagement strategies” for a newer math teacher. So I came up with a really simple way to do this with number talks:

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I took some of the questions posted in the Illustrative Mathematics teacher guides and put them on a clipboard. Then I write the problems and hand a student the clipboard. These are basically the same questions that I would be asking except now the kids are leading the discussion. They LOVE coming to the board to do this. I barely say a word. Yes, there is some level of randomness to this instead of structuring the responses in a certain order, but sometimes that is even better, as kids tend to pick other kids I might not necessarily have chosen which leads to new strategies emerging. The other day one student responded with, “I just did it in my head,” and the student discussion leader said, “That’s not good enough – tell us HOW you did it!” And the class loved it! If I had said it there may have been moans and groans or “Why is she picking on me?” but they loved seeing a student challenge another student using “friendly controversy.” They are starting to challenge each other; my next step is to introduce more questions and have it not be so scripted, but this is a start. High engagement.

Today we did the tape diagram lesson. While I love the structure of the lesson as introduced by Illustrative Mathematics, I knew as soon as I saw “snap cubes” and “10 minutes” that this would not work with my short periods and large, busy classes. (Plus I just don’t have enough snap cubes. Feel free to donate some to Kodiak Middle School, care of Alex Otto.) So I modified the lesson to suit our needs:

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We used sticky notes to build tape diagrams on whiteboards, then old transparencies (gotta love upcyclying those 1980’s products) to write over the whiteboards. This worked really well. I do admit I used a little more direct instruction than I generally prefer, but with only one day for tape diagrams, I wanted to make sure students really understood how to use this tool. We had lots of success here!

Follow along for more posts about Illustrative Mathematics, or as I would put it, the best curriculum I have ever used. Though I constantly rave about this curriculum, I promise I may not being paid by Illustrative Math! (Though you can feel free to send a Chrismas bonus in large bills,  Ashli, wink, wink.)

 

 

The Power of Number Talks & Ownership

I seriously love that Illustrative Math has incorporated Number Talks as warm ups. I will be totally honest that I have not used all of them, but the ones I have used have been fantastic!

If anyone is following this blog regularly, you can see I recently posted my frustration when maybe one student per class was able to figure out 1/4 * 24. We listed ways to think about this problem and there wasn’t much discussion because so few kids had mastered this in previous years. However, a few days later, I threw in a similar problem. “What is 20/5? What is 1/5 * 20? What is 20 * 1/5?” Unbelievably, almost every student mastered this one, using strategies they had seen from the previous problem they struggled with! I love the grouping of these problems so that students make connections. This was very powerful to watch. I have the Number Talk books by Sherry Parrish on my Amazon wish list now!

The other day, we did a series of problems from Illustrative Math for our number talks: 6*15, 12*15, and 6*45. I took some time with first problem to let students share multiple strategies. I either let students come up and share their thinking or I let them talk and I wrote down what they explained. (In general it’s always better for students to come show their thinking, but I tend to draw out warm ups for too long, so I set a timer and sketched, as long as they did the thinking and talking.) I learned that I have to be VERY intentional in my questioning. At first, I asked students to solve each problem mentally. In my first period class, some kids picked different strategies for the three problems that didn’t necessarily mesh with each other. In 2nd period, I found myself asking, “If you know that 6*15=90, how can you use that fact to help you solve 12*15?” I was much more deliberate about suggesting that they use the first fact to solve the second; otherwise, some students did not see the connection among the problems. By the last problem, students in 5th period were saying, “This is fun; can we do this all period?”

I also decided to slow down a little this week to allow for some processing of information and reflection. One of the lessons in Illustrative Math involved having students make a visual display of their understanding of equivalent ratios for 15 minutes. I decided to have students do this as a draft, and then gave them another period to make a “final copy” using color, neat work, etc. This really allowed me to come work with groups who still were struggling with the concept of equivalent ratios.  It also gave kids ownership. They had to explain to each other what equivalent ratios were and give each other examples. By the end I felt like there was a much stronger understanding. I will post some pictures of their work here later on.

 

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I am loving these Illustrative Math ratio lessons!

Quick recap: After having been out with the flu for several days, we are a bit behind. There’s no way I would have left these lessons with a sub (any advice on that, by the way??) so kids did some prep for parent teacher conferences and some skills practice on our skills practice program, Stride.

When I got back, we worked on the set of problems for equivalent ratios. I brought in Tang for the first hands-on lesson.

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I followed the lesson plan pretty closely. One minor difference is that instead of just showing them the picture of the diagram, I had them draw it on white boards. I feel like that gave them a little more ownership. Plus it’s a quick formative assessment to see who can make ratio diagrams and I can chat with the kids who drew a picture of me standing with a pitcher about what a “diagram” means.IMG_6279

I “dramatically” mixed A & B together after making two equivalent mixtures. I was BLOWN AWAY at how many kids said, “Whoah!! That is going to be SO sweet!” Even though kids worked on ratios and ratio tables in their 5th grade Bridges curriculum last year, this statement was SO telling. They still hadn’t grasped that mixture C would NOT be sweeter. We had a REALLY good discussion about it. We connected it to the coffee shop next door. What if we got a small slush puppy vs a large slush puppy? The kids realized the large one might have more syrup in it but it also has more ice. This lesson seemed so basic to me but I think it was truly eye opening for our students!

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Mixing green paint came next. Many kids quickly realized that the double recipe would be the same “because this is just like the Tang!” (Some of them even drank it to see if it tasted the same. Oh my. Gotta love middle school!) One issue we had was that when we doubled and tripled the recipe, the green DID look slightly darker! One kid finally said it’s like the ocean or a swimming pool; as it gets deeper it looks darker but it’s really the same. Ideas on how to deal with this for next year?? We finally ended up splitting the recipes into smaller cups to see if they looked the same.

For the tuna casserole recipe (thank you for NOT making this hands on), kids did a nice job but I definitely stepped in and drew diagrams on the board to make explicit the multiplicative nature of ratios. Lots of kids tried to say “you can multiply or divide” and I tried to steer them toward multiplying by the reciprocal as per the warm ups we’ve been doing, but I’m wondering if I’m doing them a disservice because for some of them it’s still easier to see that 2:6 is equivalent to 1:3 because you can divide each part of the ratio by 2.

What better way to introduce a double number line than with clothesline math today? Stay tuned for more!

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Illustrative Math 2.2: Ratio Diagrams

Word wall has been updated!

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Kids did GREAT on Friday when we introduced ratios in 2.1 of Illustrative Math, 6th grade! They brought in all kind of fun things. My rule was they could bring food as long as they waited to eat it until we were done.

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The warm up today for lesson 2.2 was a struggle. Kids for the most part had NO trouble with 24/4=6, and even had some creative ways to prove it when I asked them to. But 1/4 * 24?? Maybe 1 kid got it in each class. Is this typical that almost no one mastered this in 5th grade? I had that kid share what he or she was thinking and then I definitely jumped in and helped clarify that child’s thinking to model how he or she explained it. (Each one struggled to explain it.) I decided to do 24 * 1/4 but skip the last question in the problem set. It was just taking way too long and I wasn’t sure that the payoff was there. We did talk about how dividing by 4 was the same as multiplying by 1/4, but again, this was much more teacher-led. Suggestions?

The first part of the lesson went fine. I called up some of my best artists and had them make (quick) elaborate sketches and we talked about how they were fantastic, but how we wanted to make something simple for ratio diagrams. I may just model this myself for my later classes for the sake of time. Part 2.3 of the lesson went fine, though I did have them work alone for the sake of time. (We were already behind due to the warm up struggles.)

The ratio card sort exposed many errors. Lots of kids picked matches based on the number, but did not look at the order of the items. Tomorrow we are doing a self-reflection to prepare for parent-teacher conferences, so we will have some time to review the card sort activity.

For next year, I would do the following:

*Maybe update the cards to make recipes for slime! The kids are STILL obsessed with it! Of course, by that time, they will probably move on to being obsessed with something else. 🙂

*It would be really fun to actually make a recipe in the kitchen! I did this one year. We made “Improper fraction brownies” with ingredients like 5/4 cup flour, etc. It was really easy to see with the finished product which groups miscalculated! I wish I had more time for math instruction. Maybe I can do this with my homeroom.

Who else is using Illustrative Math for 6th grade? How are the ratio lessons going? Are you seeing any of the same things I am seeing? Let’s keep in touch! @alexandraotto