Area and Volume Skills

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My fourth grade students finished up a unit on volume about a month ago.  This past week I gave the same group a cumulative assessment on the first two units.  After grading the assessment I started to notice trends.  Many students had issues with converting square units to cubic units.  Students also mislabeled units related to measurement.

This is my first year using a new version of a district-adopoted math resource.  This year’s scope-and-sequence had students encountering area first and then volume was discussed in a completely separate part of the unit.  I believe that isolation made students think that problems in the different sections were either 1) related to area or 2) related to volume.  The assessments that I graded indicated that students needed some bolstering in applying area and volume.  Combining them would be a bonus.

Early this week I came across Graham’s Tweet about test questions.

I clicked on the article and found some amazing questions.  I definitely geeked out after trying out a few. These types of  questions made me think beyond one math skill or idea and I thought it would move students in that direction too.  I decided to use the area question with my students.

You see, in the past students have been given the length, width, and height, and then asked to use a formula (often given to them) to find the volume.   In this case, students were given the area and had to use that to find the side lengths.  This type of task in Graham’s Tweet was definitely different problem for them.

I gave each student a copy of the sheet and had them work on it individually for about 10 minutes.  Students initially thought of adding all the area sides together, but then they realized that adding them wouldn’t help in the process.  I redirected students to look at what the question is asking.  There were a few minutes of frusturation as students were looking for ways to find the length, width, and height.

Students were then put in groups to work out the problem.  Eventually, students started to think of factors as they started to investigate numbers that  work for the length.  Some went the route of using a factor rainbow, while others used a trial-and-error method.

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More frustration ensued, but students still moved forward.  A few groups were confident that they had a solution.  I briefly looked over all the responses and saw that no one had the correct answer, but I pokerfaced it and had the students work it out at home.  That evening some of the students tackled the problem and came back with a solution.  I was impressed with the perseverance and also how these students applied their understanding of area to find volume.

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Conjectures and Math Arguments with Fourth Graders

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My fourth grade class just started a unit on division and fractions. Students started out the unit by observing the relationship between fractions and division. This was a realization for some students as they perceived fractions as an isolated topic, separate from the division operation.  From there, students moved to adding and subtracting fractions. The class completed an Open Middle problem on Wednesday.

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Last year, students were introduced to this computation skill and many were comfortable transitioning that skill to more complex fractions.  I’d say the majority of my students completed problems using a common denominator strategy.  Very few used a visual model.  For the most part, students were able to find solutions to the computation problems.

The next day, students experienced the idea of using mathematical reasoning to create conjectures.  This was brand new for the students and I recently read about how to use conjectures in class from Tracy Zager’s book.  The class discussed very simple conjectures, claims, and arguments.  After about ten minutes, a student mentioned that someone is being a “math lawyer” when they go through the conjecture, claim, and argument process.  I think the whole class laughed at that response, but agreed.  I then helped model a few different conjectures and clarified the definition.  Students then worked in table groups to role-play how to defend their claims.

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Near the end of class, I was able to have students work on a page related to fractions and conjectures.  Each table group 1) looked at a specific fraction computation problem 2) each student added their own view point of whether they agreed or disagreed 3) the groups came to a consensus and wrote their argument to prove their case 4) students wrote their arguments on the whiteboard.

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I was impressed with how students started to improve the clarity of their mathematical writing during this process.  This is an area of growth for all of my classes.  Writing in math class doesn’t happen as often as I’d like.  I’m hoping to complete more conjecture, claim and argument activities as the year progresses.

Blueberry Cereal and Ratios

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One of my classes is in the midst of a unit on ratios.  Students have been exploring equivalent ratios and using visual models to compare them.  For the most part, it has been a great journey and students are making connections.

One of the common themes of this unit is for students to apply their understanding of ratios in real-world settings.  On Thursday, I introduced an activity about cereal and flavors.   This was my first time using this task so I wasn’t quite sure of what to expect.  Basically, students are given four different cereal recipes.  Each recipe has a certain number of cups of blueberries and cornflakes.

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The ratio is different for each cereal.  For example, cereal A has 1 cup of blueberries and 2 cups of cornflakes.   Students are then asked questions related to doubling, halving, making additional cups, and finding equivalent ratio recipes.   Students seemed to struggle at first and then created models, mostly tape diagrams, to find solutions.  Questions about making a certain cereal taste more like another were also part of this task.  Students had to stretch their understanding of ratios and apply it here. What I found was that some students struggled through this process.  Through the struggle, they moved towards using pictures/models to help themselves.

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This was where I stayed back and let students work.  That was tough. At times, I asked students questions to help move them towards validating their reasoning.  The application piece of taking ratios and using that understanding to see whether a cereal would taste more like blueberries or cornflakes was challenging for students.  They had to perceive the problem in an abstract sense, and then bring justification to why it would taste a certain way.

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One of the more open-ended problems involved a taste test.  Students were asked to create a cereal with a  stronger blueberry taste than all the other cereals.  The kicker was that the new cereal couldn’t have more blueberries than cornflakes.  So, students were required to analyze all of the ratios and create a new ratio where the blueberry taste was more evident.  I observed that students tackled this problem in a couple different ways.  Some students perceived the problem as adding more cups of blueberries than any of the other cereals.  This caused an issue as the ratio wasn’t dependent on the amount of cups, but the actual ratio.

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Compared to what?

This was a red flag to me as students were looking at the value, not necessarily the ratio.  Other students took each ratio from the initial cereals and converted them into fractions.  They then created a fraction that was closer to one compared to the other cereal. This was used in their justification.

This task seemed like a worthwhile activity.  I’m looking forward to what tomorrow brings.

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.

 

 

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.