## 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:

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:

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?

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!

## 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!

## 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:

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:

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:

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:

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.

## Illustrative Math, 2.3-2.6

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.

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.

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!

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!

## Illustrative Math 2.2: Ratio Diagrams

Word wall has been updated!

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.

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

## Illustrative Math: Surface Area of Polyhedra (1.14)

I will keep this blog short. Kids did great with matching nets and polyhedra. I’m hearing lots of great vocabulary in class, and most kids have an understanding of how to find surface area.

BUT THEY COULD NOT CUT OUT A NET AND ASSEMBLE IT!

This part of the lesson was supposed to take 30 minutes. It probably took 30 minutes for some kids to cut their nets out. They cut off parts they needed, had to restart, couldn’t figure out what the flaps were for, assembled them inside out, etc. etc.

I genuinely love the CCSS. But a consequence of increased academic rigor is that some kids haven’t yet mastered basic hands-on life skills. One of the only lessons I remember vividly from kindergarten was the bunny-ear show tying lesson. My own daughter could do the standard algorithm for multiplication before she could tie her own shoes (which she finally learned from a YouTube tutorial.) How do we handle this? My students did great yesterday lining up pre-cut pieces to make nets. They struggled with scissors, glue, and folding. How essential is it for students to master the cutting and assembling? Can they get pre-assembled shapes or is something lost by not having the experience of making it themselves? I’m hoping some of you at IM are reading this and will comment.

We probably got to figuring out the surface area of 1 shape today but I don’t think they are ready to move on. What I noticed in my 43 minutes (on average) class periods is that if we don’t spend enough time working with the concept on a grid, they don’t own it when I remove the grid. So I’m going to call today practice with hands-on skills and revisit the mathematics of this lesson on Monday.

## Illustrative Math: Polyhedra

I’m on a roll with blogging! 2 out of 2 days. Today’s lesson had a LOT. OF. CONTENT.

I love the sorting activities that Illustrative Math uses! Students looked at polyhedra vs nonpolyhedra, then immediately sorted 3D figures into those two categories in groups. Because I had limited supplies, I had students stand up and huddle around their figures at their tables to share. This ended up working out well, as they had to be active and involved instead of passively sitting there while someone else does the work. Justifying your answer seemed implicit in the task directions; if kids disagreed, they immediately spoke up, as in, “No! This one can’t go there because it’s round!” Some table groups built a polyhedra tower.

I wish I had more “weird” polyhedra, but the blackline master didn’t print properly.  I did a quick check by holding up some figures and having students stand up if they thought the figure was a polyhedra and sit down if they thought it was not. We quickly listed some attributes as a class, referring to vocabulary a lot, and switched gears into types of polyhedra.

I had students write “prism” and “pyramid” on their whiteboards. (Whenever IM says to give students “quick think time,” I hand them a board and marker. That way I know they are not quietly thinking about lunch, or their crush, or PE class.) Students did a really good job defining each term. I had an “assistant teacher” (a student) write what they came up with on the board. This mixes things up quite a bit; while I try to hold everyone accountable by making them show their thinking on whiteboards, I wonder sometimes if I tend to call on the same students frequently. When a student has to make those decisions, they tend to call on different people. Here’s what they came up with:

Nice job on the vocab! I had to clarify a few things: AT LEAST 5 faces and 6 vertices, and by “parallelogram” I think they meant that the top and bottom faces were parallel polygons. (I hope I described this correctly; please comment if there is a better way to state this!)

We then went into nets, and kids had no trouble composing nets in different ways from the polygons. Again, we did this standing up and huddled around the pieces. To justify their thinking, I had them show me how the nets would fold up, which automatically required group participation because one person just doesn’t have enough hands to do this alone.

For next year (note that I’m really keeping my fingers crossed that I can use this curriculum next year), I would change the following:

*GET MY WORD WALL PREPPED BEFOREHAND!! When I saw how many vocab words there were, I couldn’t pull it off last night. I wish I had these up for today, but better late than never.

*Talk less (a good general rule in math). The less I talked about the polyhedra, the more the kids did. The more I talked, the less they thought they needed to. Again, the word wall will help them have things to say.

Let’s see how they put this all together tomorrow when they calculate surface area of polyhedra!

## Illustrative Math: Surface Area

I originally intended to blog daily about my journey with Illustrative Math, but quickly fell behind. Like that kid with a month’s worth of missing work, I doggedly avoided the task because I was struggling to remember the nuances of my lessons from weeks ago. So my new resolution is to blog when I can and if I get behind, just pick up with the current day.

Today’s lesson was on surface area. I LOVE Andrew Stadel’s Filing Cabinet lesson. Last year I went ahead and covered my own cart with stickies. This year I stuck with the videos. Since this is middle school, I had the normal share of the abnormal today at random points in the lesson: a case of head lice, a mechanical pencil incident, and two girls who decided to paint their nails to celebrate their friendiversary (we decided tomorrow at lunch would be a better time.) But overall it went great!

Here’s what the kids came up with in period 7.

“Surface Area” was listed as the learning target on the board, so someone wondered if the video had to do with this (why, yes!) I loved that they noticed the brand name of the marker. It gave us a chance to talk about the fact that there is a lot to notice, and then we picked through and figured out what might be mathematically interesting.

Kids then did estimates for the entire cabinet, ranging from about 150 to 4,000. Most were in the 500-1000 range. A handful of kids “estimated” the correct answer. . hmmm. . eventually revealing that a friend had told them. This always happens when I use Estimation 180. At first I was frustrated. Then I quickly realized that: 1) they spent their free time actually talking about math, and 2) some kids without a lot of number sense now can see what 935 of something actually looks like. In these cases, I just casually nod and ask them to justify their answer.

During my first few classes, many students still calculated a completely wrong answer, even after the discussion. One kid finally spoke up and said, “I forgot how to multiply 2-digit numbers over the summer.” Yikes! While this is important, it wasn’t the point of the lesson, so I finally chunked it more. During the discussion, I selected a student intentionally who said, “I think we need the dimensions of the front.” I only gave them those dimensions. Once we figured out the area of the front, we then watched the video revealing the answer – THEN they asked for the dimensions of the side. I let them use calculators if they needed to. What I would tell new teachers is that students are going to want to show you their answers – don’t go over the answers. You will know right away if students got it because you will hear cheering when the answer is revealed in the video.

I loved that IM immediately moved the students to application of this concept by working with blocks. Most students got the idea immediately with blocks. I first asked them to build and then calculate the surface area of the 2x2x3 block structure from the lesson. Many kids correctly got 32 square units, but a few immediately reverted to LWH and got 12. So I quickly made some mini sticky notes and told them to think of this structure as a mini version of the filing cabinet:

That seemed to help.

Here’s what I would do differently for next year:

*Get Surface Area posted on my word wall in advance, maybe with a picture. It’s not going to give anything away if they see the vocabulary. It will just preview the content. Maybe not the definition yet, but with a picture.

*Try not to rush (hard to do when I have 43 minutes to teach, and after required 6th grade team stuff like filling out our daily agenda, it’s more like 40.) The more we talked through what kids noticed and wondered, the more estimates we listed (with kids’ names), the more interesting our discussion and the more kids were engaged.