Fraction Division Models

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Dividing Fractions

One of my classes started a unit on fractions around two weeks ago.  We explored the lowest common multiple and greatest common factor during the first week.  Moving on, the class investigated the different ways to multiply fractions.  Most students had an idea of how to find the products by multiplying the numerators and denominators.  They struggled when using visual models and also when converting fractions to/from a mixed-number form.  We spent a couple days reinforcing how to visualize fraction multiplication and created multiple models to show how to multiply a fraction of a fraction.  We used the folding paper method as well as free-hand drawing of different models.  On Thursday, the class moved to the next topic: fraction division.

I’d say that this is a topic that’s a bit confusing every year.  Many students, and I mean almost a third of my class tend to come into the class with an understanding that when you divide, the quotient will always be less.  This is one of the tricks that expire.  So, as I started to plan out what I was going to do during the introduction, I had to keep in mind that this topic has the potential to be  a misconception minefield.

On Friday, that class started to study the topic of fraction division.  I ended up using a Brian’s amazing resource to put together a Nearpod activity involving fraction division.  Without discussing the topic too much, I asked students this question:

Show 3 ÷ 1/2.  Write on the picture to show your model.

 

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Students worked in pairs with one device to find a solution.  Some groups immediately started splitting up the fractions.  The confidence from these groups seemed to be high.  Other groups were discussing what was meant by the visual model.  Here is one of the initial responses:

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Does this match the number model?

It was interesting as some students wanted to split the entire three sections in half.  This had me wondering if the students understood that each block was one whole. I also had some students that were able to find a solution, but it had to do with using the trick and not the visual model.

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“I used the trick” – visual model wasn’t touched

When pushed to explain their thinking the students weren’t able to move past the process of finding the reciprocal of the second number and multiplying.  The class then moved to the second question.

Show 3 ÷ 1/3.  Write on the picture to show your model.

 

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Students seemed to be more comfortable with the problem.  They were also a bit more careful when splitting up the shapes.  I reinforced with the class that idea of dividing fractions.  The responses showed more detail this time around.

All of the responses used some type of a visual model, which was a positive as this didn’t happen as much the first time.  When asked to explain their reasoning, students were able to tell me with confidence why the model made sense.  Also, some of the students started to find that that they could check their answer with multiplication – 3 ÷ 1/3 = x , then x * 1/3 = 3. There were still students that went to a default of using the trick to find the quotient.  The third problem was designed to see if students could stretch their understanding.

Show 3 ÷ 2/3.  Write on the picture to show your model.

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Students took more time with this model.  Most of the groups started to divide each whole into thirds.  That’s when trouble started to brew.  Students counted up the wholes, but noticed that parts were still missing.  There were a lot of questions here.  I asked students to take a risk and put their mathematical thinking out there so we can all analyze the responses.  The students submitted their ideas and I noticed that the answers were all over the place.

The class noticed that all of the groups split up each whole into thirds.  The groups also shaded in or indicated two thirds of the whole. Students also noticed that some of the third shapes weren’t included in the quotient.  There was a class debate on whether those thirds should be included.  Most of the students agreed and said that they’re part of the whole so they should be included.  This last question took up the most time and I think it was one of the better moments of the lesson.

There was only around ten minutes of class left, so students went back to their seats to work on a few more fraction division application problems. The struggle and perseverance to understand was evident and I’d like to find ways to incorporate this type of instructional routine more often.


 

Overall, I thought this lesson went well.  I recorded the entire lesson for my NBCT video, so it’ll be interesting to see how it turns out.  While watching the videos that I record I tend to remember instances where I could’ve done things differently.  In some of the cases, I could’ve asked better questions or model a bit less and have students make the connections themselves.  It’s a balance though.  I’d say that  watching a video of myself teaching is a humbling experience.  It’s humbling, but the personal reflections that come out of those experiences are worthwhile.  I think I could write an entire blogpost on that reflection process.  Maybe next time.

 

 

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Math and Gallery Walks

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My fourth grade students are exploring exponents this week.  Students are learning how to write numbers in exponential form and covert the numbers to standard form.  For the most part, students have had a productive week learning how to write very large and small numbers.  Later in the week, I decided to have students complete a team task enrichment project. Students were asked to compute large numbers.  Here’s the Freight Train Wrap-Up prompt:

Brianna loves freight trains.  She learned that in 2011, there were about 1,283,000 freight cars in the United States.  Brianna wondered whether all those train cars, lined up end to end, would wrap all the way around the Earth.  Help Brianna answer her questions.  Could a freight train with 1,283,000 cards wrap all the way around the Earth?

I reviewed the criteria for success with the students and then placed them in teams.  Students were asked to use an anchor chart to show their mathematical thinking.  They could use markers, and other materials to showcase their solutions.  Students were given around 20 minutes to work in their teams to find an answer.  Some students drew pictures, while others decided on writing out equations.  I heard a number of groups argue about the solution and what to compute.  After about 15 minutes, most groups were close to finishing.

After the time was up, I brought the students to the front of the room.  I briefly communicated all the different solutions that were evident.  Students were asked to participate in a gallery walk.  Gallery walks are used as a standard default activity for district meetings so I decided to try it out with my own class.

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Students were then given three different Post-it notes.  Each note was intended to ask students to indicate whether the chart that they were viewing answered these questions: 1) Did the team show their work? 2) Was there a solution? 3) Was there a visual representation?  The note also indicated a question or an agreement.

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Students efficiently traveled from group to group.  They gave feedback and mostly agreed with what the other groups came up with.  Students weren’t allowed to give feedback on their own anchor charts.

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I then brought the students back to the front and we went through the problem together. We discussed the numbers, operations needed, and possible solutions.  Students then went back to their original charts to read the feedback.  Some students were surprised at the comments while others wanted to debate them immediately.  I was able to touch base with each group and discuss what the constructive criticism might have meant.  Students spend a decent amount of time talking with one another about their chart and process.  Afterwards, students went back to their seats and prepared to leave.

This type of activity went well.  After thinking about it I might consider doing something like this once every month or so.  Also, I might need to think about how to get my hands on more anchor chart paper.

Whiteboards and Math

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Over the summer I changed classrooms.  It was a lengthy process, but great, as I was able to reorganize my classroom.  I moved into a smaller room with less cabinet space.  To boot, the room also didn’t have any carpet.  Starting with a blank slate caused some anxiety at first, but it also gave me time to think of different design ideas.

Over the summer I received many great ideas from my pln about classroom design.  I knew I wanted to add additional group stations and lay out the space so kids could utilize all the different locations within the classroom.  I’m not an HGTV expert by any means, but I thought that some changes in my design might be helpful.  During August, I ran across a few Tweets from TMC about vertical non-permanent surfaces.  It even has it’s own tag – #vnps.  Interested, I researched this a bit and found some great news.  My summer book study and the TMC crowd both confirmed that these seemed to help students.  Thankfully, I ran across a Tweet about getting whiteboard from Home Depot.

I went over to Home Depot a couple days later and bought two 2 x 5 boards.  I wasn’t really sure where I’d place them. Over the next few days I started unboxing my materials and started planning out student learning places.  I put in a work order to hang up the vertical whiteboards and they were installed a couple days early.  Maintenance drilled the boards into the wall and I was a happy camper.

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I labeled the stations the next day.  I explained that the whiteboards were used so students could brainstorm and show their thinking.  Immediately, students were excited to use these new shiny boards.  The quality was decent and they easily erased.  It was interesting how quickly students picked up their Expos markers and got to work.  Some use them solo while other students like to use them in groups.

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My only gripe is that I wish they had a magnetic component.  Some students want to hang up their papers on the board and show their work on the board.  I’m still looking into options to what I can use to attach the work to the board without buying some magnetic paint.  Still checking out alternative ideas.  I’m looking forward to seeing how students use these surfaces throughout the year.

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Cracking the Code

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My third grade students explored different addition and subtraction algorithms this week.  It’s been a challenge.  In the past, students used number lines to add or subtract by the highest place value first and then slowly move towards the lowest.  Moving towards a  standard algorithm has been a process this week.  Students started by using the partial-sums method.  This was similar to what they were used to and mimicked the number line models.  Students seemed fairly comfortable in using this method.  The column addition method was next on the docket.

Many students come into my classroom already knowing how to use this method.  It’s often referred to as the traditional/standard addition algorithm.  Students complete the steps and out comes an answer.  Does it make mathematical sense to my third graders?  In some cases the answer is no. So, in an effort to bring a bit more meaning to why it works I decided to use a code activity to reinforce the idea of base-ten and place value.

Cracking the muffin code is an activity found in the Everyday Day math curriculum. A quick Google search will also bring up many different threads related to this activity.  I’ll be paraphrasing the lesson throughout the post.  Basically, students are given a scenario where they’re in charge of a muffin market.  At the market the muffins are packed into boxes.  The boxes only hold a certain amount of muffins.  When someone asks for muffins, an employee fills out an order form.  That order form contains a code.  The largest box needs to be filled first and the employee needs to send boxes that are full. Here’s an example that I paraphrased from class:

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I had students work in groups to figure out the code.  I gave them around 10 minutes and at the end of time a few groups were fairly confident with their answer.  We discussed the code and students started to notice a pattern.  They used trial-and-error to figure out which column matched the box size.

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There was a lot of excitement in the air as students solved the puzzle. Afterwards, students connected this to the idea to place value for the next problem.  This puzzle was designed differently. Now, students were asked to pack boxes of granola bars. The packages hold 100, 10, or 1 bars.  The employee uses a coding system.  Here’s another example:

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Almost immediately, students were able to see that the first column was designed to package one bar.  The second was for ten, and the third, one hundred.  I gave time for students to look at the similarities and differences between the granola and muffin codes.  Students were then asked if the base-ten system was similar to the granola code.  Students nodded their heads and I even had a student say that when we regroup numbersit’s like adding another package.  Another student stated that sometimes not all the numbers fit in a package so we have to find another place for them.  Students were making connections to how the base-ten number system works and why regrouping is sometimes necessary.

This was an eye-opening experience for some students as they started to look at the place value positions as bins or containers.  This lesson had students talking about how place value can be perceived as “containers” or “boxes” for numbers.  Each box needs to be filled to it’s capacity until a new one can be used.   I’ll be referring to this activity throughout the year as it seemed to help students make connections when exploring the base-ten system.

Afterwards, students used the column addition algorithm with a bit more confidence.  Next week we’ll be discussing multidigit subtraction.

 

Volume and Missing Blocks

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On Monday my fourth grade crew worked on tasks related to perimeter and area.  Today they discussed volume. The volume discussion began when I brought out a large container filled with water.  I asked the students what they thought volume was.  Was it the amount of substance contained in an object or the space inside of the object?  It was interesting to hear their perspective.  I wrote down their thinking on the whiteboard. Some students were positive that volume measured how many cm cubes that could fit into the cylinder.  Another student stated that there wasn’t anymore volume left because the water took up the space.  I then took the same cylinder and dumped the water out.  Another student mentioned that you didn’t need to use cubes to fill up the cylinder.  You could’ve used sand instead to measure the volume.  Throughout this class conversation I thought students were testing their understanding of volume and not just regulating it to filling up objects with cubes. The class then made a math journal entry and created a t-chart of examples and non-examples of volume.

Afterwards, students went back to their table groups to discuss volume and I used Steve’s image from the tweet below.

I asked students to think about the shape and how many cubes might need to be added to create another layer.  Students were confused at first, but then gradually came around to thinking about how to add another layer.

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Some students wanted to add a layer on top, but then realized that making that top layer would mess up the stair sequence.  Eventually, after some major perseverance, I asked students to create a model.

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That model proved helpful as students could see and start thinking about how many blocks could be added to the bottom.  I noticed that students started to think of arrays and how helpful they were in creating another layer.

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At that point class ended.  We’ll be discussing this problem again tomorrow.  I’m looking forward to seeing what the students discover, their solutions, and what strategy they end up using.

Triad of Responsibility

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My students finished their seventh day on Friday.  The students and I are in the midst of a timely three-day weekend.  For the most part, I’d say that students and teachers are starting to get into their school routines.  There have been a few bumps in the road (there always are), but the school, teachers and students are making progress and we’re off to a great start.

This past week started off with discussions about expectations and routines for all stakeholders. Staff emails and student assemblies reinforced these expectations for teachers and students.  Early in the week I had the opportunity to have a class conversation about responsibility.  This stemmed from Caitlyn’s blog post and her experience with the NYC Math Lab.  Students were placed in groups and given a marker and anchor chart paper.  Each group was expected to create a list of at least five statements related to what ______ sounds/looks like.

  • What is  your responsibility to your class?
  • What is your responsibility to your partner?
  • What is your responsibility to yourself?

I gave each group around 10 minutes to discuss and write down their thoughts.  It took a while for the groups to decide on what to write, but they eventually came to somewhat of a consensus and documented their answers.

After the ten minutes, I brought the class back together and hung up the anchor charts around the room.  Students were given two stickers and asked to visit an anchor chart that wasn’t their own and place their stickers next to two statement that they thought were the most important.  Students were then given an additional two stickers to place on the remaining anchor chart.  Basically, students weren’t allowed to vote for their own anchor chart.  Afterwards, the class met as a group and analyzed the two most important (as surveyed by the students) statements.  Those statements were used to create the responsibility expectations for the classroom.

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I used this activity with three of my classes and compiled the results. I thought it was a decent activity and it had students thinking about their responsibility.

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Looking back, I probably could have took a math angle to this activity and ask students to think of how each responsibility applies to them as a a mathematician. Maybe next year.  : ) I’ll be referring to the “triad of responsibility” as the year progresses.

Making Plans and Reality

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Three days are in the books. Students arrived on Wednesday and it’s been organized chaos for the last few days. Maybe not organized chaos, but it surely has been hectic and fast-paced. With all the scheduling, drills, trainings and professional development it can be challenging to find time to sit down and reflect on what has happened.

A few weeks ago I made my first day plans. The plans indicated a number of different activities and beginning of the year strategies.  Here’s what actually happened:

Day 1

I planned to have the kids for an hour, but that didn’t work out. Instead, I had around 30-40 minutes with my third and fourth grade classes. My fifth grade class was involved in expectations training so I missed their class on Wednesday. I ended up having students come into the classroom, find a seat, and use the “get to know you” and “get to know your teacher” activity.

Both of these activities were great. Students caught up with their peers and learned a lot of new information about their teacher. Even though I’ve had some of the students for five years, many were left guessing.  Major Kudos to Sarah for sharing this idea. I’ll be using it again next school year. Even more, the students were straight-up giddy about creating a quiz for their teacher.

I ended up having a very short amount of time to review the arrival/dismissal flow charts. By then, the time was gone and students had to move to their next class.

Day 2-3

During the second day I had students follow the arrival routine and sit in the same seats that they occupied yesterday. I randomly grouped students for their assigned seat for the first unit. Students graded their quiz and I was lucky to get a couple answers right on each page. I learned a ton about each student though.

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The class moved into the Number Trouble game. Their new seats already grouped the students into table teams. I distributed the 100 paper to the teams and explained the instructions. Each group had two minutes to find all 100.

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At the end of time every group found around 15-30 numbers. I gave the groups time to look for different strategies/patterns and create a plan. They were given an additional two minutes. Most groups had between 50-90 the second time around. One group made it to 100. I later found out that one of the members of the group used this activity last year.  Afterwards, I brought the classes to the carpet and we discussed what went well and what didn’t. The class created a list of positive behavior interactions and then that discussion spilled over into what the expectations should be for the school year.

Students signed-off on the expectations and then went back to their seats.

I then introduced the name tents and feedback forms. Students wrote their names on the front and I explained what students should do on feedback side. Students weren’t quite sure what to write, but eventually they jotted down what came to their mind. Time was up after this activity and the second day was finished. Later that evening I wrote back to each student.

Students also started their puzzle piece. This is a bit different than in past years. Most students haven’t finished and that’s something that we’ll work on next week. We’re making progress, but I’m hoping when it’s all said and done, the door will be outlined with the student pieces.

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This was my puzzle piece trial.  Eventually, each student will receive one puzzle piece and it will should make a border around the door.  If any remain, I’ll most likely start filling in the border.

I’m looking forward to planning a few lessons and recharging over the weekend.