Tag Archives: 5th Grade

Rethinking Homework…

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One of the things on my vacation “To-Do” list is to rethink the way I do homework in my classroom. The concept of homework is something I constantly go back and forth with in my head, trying to find the perfect balance of meaningfulness for my students.

Some things I have learned, loathed, and/or questioned about homework…

  • More problems does not equate to more meaningful. If a student can do 2 problems correctly, chances are they can do 20. Conversely, if they can’t do 2, 20 will be extremely painful.
  • Some parents will want more homework, some will want less. This is not necessarily based on any particular demographic, it is varying. I find this more stems from parents reflecting on their own school experience and homework load. What they believe makes students “better math students.”
  • Some students will have help (resources) at home and others will not. Then I end up grading/giving feedback on parent’s work or work that has been corrected by the parents vs the students that did it completely alone. Not equitable in my eyes.
  • Some will never remember to bring their homework home and if they do, will lose it on the way back to school. So, then I feel like I am grading a student on responsibility and not their math understanding.
  • Homework should be meaningful, however what is meaningful varies from student to student. How can I make it meaningful for the 45 individuals in my math classes?
  • Homework can be beneficial to involve parents in what is being learned in class. I do newsletters and parent math nights, but really homework seems more accessible. I want students talking math their parents and questions that may arise from parents. Homework can open those doors, but I don’t want it opened in a negative “my child couldn’t do this and I had to show them how I solve it” kind of way.
  • Am I doing the students a disservice if they hit 6th grade and beyond and the homework load increases dramatically on them? Am I responsible for “preparing” them for what will happen in future grades? (whether it be best practice or not).
  • I feel there is benefits to students being held accountable for completing assignments outside of school. From time management to the organization and responsibility of completing a task are life skills that I feel are extremely important. But, how do grade/give feedback to those skills? If it doesn’t somehow “count,” some students will not really have the motivation to do it.
  • I feel like the time spent grading/giving feedback on homework could be better utilized in planning upcoming lessons.

This list could go on and on, but for the sake of time and actually being able to check something (actually two things bc blogging was also on there) off my to-do list, here is what I have come up with thus far…

I will assign one or two problems for students to solve and explain their reasoning. I will attach those problems to applicable standard(s) (including the Math Practices) so parents who would like, can see why certain things are being done in class. I will have it due over a few day period so students can manage their time and organization. Hence leading to the first part of the page:

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Now the second part is hard because I am trying to break a mindset. I want the students to explain their reasoning to the problem and parents (or any adult) to check off whether the student could explain it clearly or not. NOT FIX IT! Then I left a box for any questions or concerns for me… second part:

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I am hoping this offers as more of a formative assessment for me that connects parents to the learning in our classroom. For the students who do not always have parents available, any adult will do…this way the students could use one of their other teachers in school that they see each day.

I am also hoping that by having students explain their reasoning to someone other than me (because most times I “get where they are coming from mathematically”) it will force them to be accurate and clear in their explanations, putting a big emphasis on the Math Practices.

I am hoping this opens more communication between the parents who I may not have reached the first half of the year.

I am hoping this gives me more time to focus on planning and instruction time than grading or  going over homework assignments.

As always, a work in progress. Any thoughts are appreciated!

Now, time to check two things off my list…for now..

-Kristin

You Never Know What They Know Until You Push Their Thinking….

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Last Friday, at a state math meeting, we had so much fun diving deep into a fraction lesson of a 6th grade teacher. The lesson was on multiplying fractions by fractions and while the conversation started with thoughts about the lesson itself and areas for discussion for the math coach, the lesson really brought to light the fraction progression. I cannot even begin to recap all of the insightful discussion points such as using models and the importance of the representation in mathematics, teacher pedagogy and mathematical understanding, vertical articulation across grade levels….I could go on and on, but I had one brief conversation that leaked its way into my classroom the following Monday.

While we were “doing the math” the students would be doing in the lesson video, a colleague and I were talking about where our 5th graders leave off with fractions and how that is built upon in 6th grade. She made the comment that if the students truly understood taking a fraction of another fraction and fraction of a whole number (both 5th grade standards), then they could reason their way through mixed number times mixed number, which is introduced in 6th grade.  She quickly drew out 3 1/3 x 3 1/3 and we talked through the context in which our book uses and how students could reason about that problem.

So, of course, I have to throw it out to my students on Monday because I am curious at this point if they could work their way through the problem and the various ways they would think about it. This is where that “engaging” vs “not engaging” or “real world” vs “not real world” conversation seems void. I used no context, no real world example, I simply said, “I was talking to some middle and high school teachers at my meeting on Friday about your fraction work and they were wondering how you guys would solve this problem. 3 1/3 x 3 1/3.” They went to work and I started walking around to chat with them.

Here are some strategies I saw…

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She started with 3 1/3 x 3 and then added another 3 1/3 and found 1/3 of that to be 1 1/9.

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He used partial products. When I asked him how he figured that out, he wrote the 25 x 25 and explained how he gets his partial products there so he did the same thing with wholes an fractions. Wow. Did not expect this one!

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Same partial products, just a bit neater!

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She used separate bars for each 3 1/3 and then divided the bottom up to find the 1/3 of 3 1/3.

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I was so impressed by the work of these kiddos and they were so proud of themselves! They connected understandings of whole number operations to fractions, applied properties of mathematics, used what they knew conceptually about fractions to model the situation, and most importantly persevered through the problem and constructed arguments about their answer.

Don’t get me wrong, it wasn’t all picture perfect….I did have some who initially gave me 9 1/9 (as I anticipated they multiplied the whole numbers then the fractions and put them together) but that led to a great “reasonableness” conversation. A context in this case helped some students see that if you did 3 laps that were 3 1/3 miles long it was 10 miles, so if you did a 1/3 longer, can your answer be less than 10?

Needless to say, I don’t know how anyone doesn’t just love hearing students talk about math and reason about problems. I find it energizes me, my students, and the climate in my classroom. So, thank you to MSERC (University of Delaware Math & Science Education Center) and the Delaware Math Coalition for all of the hard work that is put into making these professional development opportunities so rewarding for both myself and my students! I think you all are AMAZING!

-Kristin

Math & Minecraft Day 1

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After many days of discovering my HUGE learning curve with Minecraft, I am finally starting to feel relatively comfortable in Creative mode…I can build a house without flooding it, planted a few trees and I no longer have random blocks floating in the sky around my world!  My class has been staying with me during recess to teach me how to play and I am amazed at how fast and detail-oriented they are in their designs, such as putting lava rocks under the water blocks to form a hot tub and putting glass windows in their new greenhouses. I just kept thinking that I would love for them to use this same precision and perseverance in math class.

I must have Minecraft on the brain, because I as I was planning this weekend for the upcoming week (multiplying fractions w/arrays), all of the scenarios were about planting on an acre of land.  For those who may not know, Minecraft is based in cubes that can be planted in the ground to show a square, perfect for our gardens. I came up with this scenario…

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I honestly lost sleep last night anticipating student responses because I knew some students would look at it as fraction of a group of blocks in this scenario, when I wanted it to be fraction of a one whole. Ideally (whatever that really is) I students would build the garden, split the fourths and divide 3 of the fourths in half to result in 3/8 of the garden (being the whole) being melons.  But as they got into groups today, hopped into each others worlds and went to work it was quite a variety of outcomes.

As I expected, many students did it as fraction of a group of however many squares were in their garden. Here is an example of this: http://www.educreations.com/lesson/view/sammy/14346123/?s=sXGl7c&ref=link

This one was interesting because they did a combination of staying with the garden as a whole and then in the end went to the number of blocks were planted with melons: http://www.educreations.com/lesson/view/steve-s-garden/14349675/?ref=link

This one was great because they brought back the fraction bar model we had been previously working with and had it next to their Minecraft garden. (Plus you have to Love their answer): http://www.educreations.com/lesson/view/garden/14361270/?s=tk0bLr&ref=link

Ignore my loud voice in the background on this one, but it is a very great build (and with a key): http://www.educreations.com/lesson/view/dylan/14361243/?s=Qt3Ws8&ref=link

When they completed their garden, I gave them a square and told them that it was one acre and I wanted them to represent the same scenario but on the open square.  I immediately saw confusion in the students who had saw the garden as 16 blocks vs the students who saw it as one whole garden.

Here are a few example answers:

http://www.educreations.com/lesson/view/kurtis/14360404/?s=xWw8UW&ref=link

http://www.educreations.com/lesson/view/riley/14361043/?s=FbOohc&ref=link

This one has some interesting talking points (a little long). You can forward to minute 3:00 for the blank array: http://www.educreations.com/lesson/view/steve-s-garden-kyzei-and-aiyana/14360241/?s=hRV0CF&ref=link

*We also had some great conversations about deciding about the dimensions of the garden and the denominator being a factor of the dimension since we couldn’t split the blocks. For example, many students built a 5 x 5 and went to break it into fourths. They said, “four is not a factor of five so we can’t.”

Lots of sharing to do tomorrow and discussing strategies, notation and the whole in the problem….stayed tuned for Minecraft Day 2…

-Kristin

Number Talks – Fractions

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Through doing Number Talks with students K-5, I started to realize that one thing I look for students to use in our whole number computation discussions is using known or derived facts to come to a solution. I feel like the problems I have been using are crafted to use the answers from previous problems to reason about the ending problem.

In the younger grades, I would like to see students using the double known fact of 7+7=14 to know 7+8=15. I want them using 23 + 20=43 to get 23 +19 = 42. I don’t want them treating every problem as if they have to “start from scratch” adding all or adding on.

An example in the upper elementary:

18 x 2

18 x 20

18 x 19

This progression leads them to use a known or derived fact (18 x 20) in order to solve 18 x 19. To build efficiency, I don’t want them to the treat the final problem in the progression as a “brand new” problem in order to reason about an answer.

Along these lines of thinking, as I observed students working the other day, I realized that students weren’t using this same use of known/derived facts when working with fractions. For example, a student was adding  3/4 +  7/8. He used 6/8 as an equivalent of 3/4, added that to 7/8 and ended with an answer of 13/8. Don’t get me wrong, I loves his use of equivalency and I am a fan of improper fractions, however I started wondering to myself if it would have been more efficient (or show that he actually thought about the fractions themselves) if he used a fact he may have known such as 3/4 + 3/4=1 1/2 to then add an 1/8 on to get 1 5/8? Or used 3/4 + 1 = 1  3/4 and then took away an 1/8? Is that the flexibility I want them using with fractions like I do with whole numbers?

I thought I would try a Number Talk the following day to see….

1/2 + 1/2

Thumbs went up and they laughed with a lot of “this is too easy”s going around.

1/2 + 1/4

Majority reasoned that 1/2 was the same as 2/4 and added that to 1/4 to get 3/4. Some said they “just knew it because they could picture it in their head” I asked if anyone used what they knew about the first problem to help them with the second problem? Hands went right up and I got an answer that I wish I was recording. It was to the effect of,”I know a 1/4 is half of 1/2 so the answer would be a 1/4 less than 1.”

1/2 + 3/4

Thumbs went up and I got a variety here. Some used 2/4 + 3/4 to get 5/4 while others decomposed the 3/4 to 1/2 + 1/4, added 1/2 + 1/2=1 and added the 1/4 to get 1  1/4.

3/4 + 3/4

Got some grumbles on this one, because it was “too easy” – 6/4…Duh! The class shook their hands in agreement and they were ready to move on to something harder.  I noticed that when the denominators are same, they don’t really “think” about the fractions too much. I waited….finally a student said, “It is just a 1/4 more than the previous problem so it is 1  1/2″ and another said each 3/4 is 1/4 more than a 1/2 so if you know 1/2 + 1/2 = 1 then you add 1/2 because 1/4 + 1/4 = 1/2.” I had to record that reasoning for the class bc it was hard for many to visualize.

3/4 + 5/8

Huge variety on this one and I thoroughly enjoyed it! From 6/8 + 5/8 = 11/8 to decomposing to combine 3/4 and 2/8 to get the whole and then 3 more 1/8s = 1 3/8.  There were many more students who used problems we had previously done.

What I learned (and questions I still have) from this little experiment:

– Students LOVE having the same denominator when combining fractions.

– Do they really “think” about the fractions when the denominators are the same? Can they reason if that answer makes sense if they are just finding equivalents and adding.

– Students can be flexible with fractions if you push them to be.

– Subtraction will be an interesting one to try out next.

– I would much prefer if I remembered to use the word “sum” instead of “answer”…. I tell myself all of the time, but in the moment I always forget.

– Using known or derived fact and compensation are invaluable for students when working with both whole numbers,  fractions and decimals.

– Are there mathematical concepts that present themselves later in Middle School or High School in which known and derived facts would be useful?

Happy Thanksgiving,

Kristin

Modeling Mathematics – Developing the Need

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Today we were talking about things we noticed as we worked with finding a fraction of 1/2. Students are noticing things I expected: that the denominator doubles each time, the numerator is staying the same as the fraction you are dividing the half up into, some are starting to notice that the numerators are multiplying and so are the denominators, and some are just flat out complaining that they have to model it on the fraction the bar.

So the fraction of 1/2 was pulling some great noticings, however I wanted the students to feel the value of being able to model the mathematics, to show what was happening, so I asked them what would it look like if I had 3/4 of a candy bar and wanted to split it with two friends. What fraction of the whole bar would I get?

I was excited that some labeled the 3/4 on a fraction bar with 6/8 and then split that in half and labeled 3/8. They said they “knew half of 6/8 was 3/8” I asked why didn’t they work with 3/4, they explained that splitting the three was not working and 6 was easier because it was even.

Some said they had split the 3/4 in half and it looked about like a 1/3, so it was 1/3. I appreciated the estimations, but looking for them to dig further after they estimated.

I would say that a third of the class had written 3/4 of 1/2= 3/8 with a fraction bar 3/4 shaded and then split in half and labeled 3/8. When asked how the fraction bar modeled their answer, they told me that they didn’t “need” the fraction bar to find the answer, they noticed that you multiply the numerators and denominators. “Can’t we just give you the answer?” “It’s the right answer, right?”

We don’t “need” the fraction bar. Huh.

Then an interesting thought hit me….they see pictures as a tool they don’t need rather than a model of a mathematical situation. It almost seemed as if they viewed the bar as “baby-ish” to use. You know how certain things hit you as WoW?!? That is completely what that comment did. I immediately started to reflect on how I had made the fraction bar sound…did I just make it sound like a way to solve? Did I even use the word model? Am I placing too much emphasis on the modeling piece?

I can see why students view diagrams as a way to solve….when they learn to add, they draw pictures. When they first work with “groups of” they draw pictures. When they first work with arrays, they draw things in row and columns. Once they have learned how to add, the pictures aren’t necessary. When they have learned how to multiply, the arrays and groups of aren’t seen as a necessity.

In that moment, I wanted the students to appreciate how important (and difficult) modeling is in mathematics.  I pushed them to explain how to name that line drawn at half of the 3/4 and we had some great conversations about why this was more difficult than a fraction of 1/2.

In the end, I assured them that I sit with adults all of the time and we struggle (and find MUCH enjoyment) in making models of mathematical situations. They felt ok with knowing it wasn’t just tough for them and I felt ok that they could see a “need” for their fraction bars!

-Kristin

Fraction of Fraction Day 2

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As I mentioned in my previous post: https://mathmindsblog.wordpress.com/2013/11/15/fractions-of-fractions/ 
I had wondered about fraction multiplication being introduced without a context when the students were coming from lessons in which a fraction of a whole/mixed number had a context. Feeling like the students had a solid grasp on how to find a fraction of a fraction on a fraction bar, I thought I would try having them develop a story context for fraction multiplication problem. They had free reign of the fractions they used and context they chose. Needless to say, it was a learning experience for me. Some showed understanding of what they were doing when finding a fraction of a fraction of something while others unveiled some things I need to go back and revisit.

I have included clips from some of their videos and what I learned from each…. (turn your volume up bc they whispered on these)

http://www.educreations.com/lesson/view/aiyana-fraction-bars/13628710/?s=fv1sHe&ref=link

This one was SO interesting (and a little humorous) because she cut the fraction bar to find 2/3s of 1/2, however when she is explaining her reasoning she used the commutative property. Saying that the answer is 2/6 because that is half of 2/3 was something I had never thought of exploring with students when reasoning about whether the answer made sense. I loved it and definitely added to my lessons for next week!

http://www.educreations.com/lesson/view/riley-s-breadstick-word-problem/13630693/?s=j3i9i5&ref=link

When she introduces the scenario, she says “1/2 of 1/4” so I don’t know if she misspoke or not really understanding the context. I can see she has the process but I don’t know if the understanding is there. I do love how she says “He wanted to find how much of the whole bread stick that was” because she is relating her answer back to the whole. This was difficult for many students. Maybe picky on my end, but I would have liked for her to label the pieces 1/8, 2/8, etc instead of by whole number, even though I know she is counting the pieces.

http://www.educreations.com/lesson/view/ab-word-problem/13627803/?s=Hr3zRv&ref=link

I was impressed how she used a class of students as the whole and did not get confused with the fraction of the class as opposed to the number of students. Many others got caught up in “How many students…” instead of “What fraction of the class.” One thing that just bothered me in watching it was the empty seat in the class! I just wanted to draw a person in for her!

http://www.educreations.com/lesson/view/kyra-s-problem/13628055/?s=aIYKAt&ref=link

This one has such a great context and division of the Hershey Bar that I was so excited, until the end. She seemed good with the context, decontextualized to solve, but then struggled to recontextualize to explain the answer.

I could post and comment all day, but needless to say there is other work to be done and papers to be commented on! It was a great first day with our 1:1 iPads using Educreations! I learned so much that now I must work on readjusting my math plans for next week!

-Kristin

Fractions of Fractions

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This is day 1 of multiplication of fraction by a fraction and I can already see this will dramatically increase my blogging! So much to write about (for reflection, excitement and possibly confusion). With the implementation of CCSS this year, this is new in the Investigations curriculum and I am finding some things I love about it already and some things I am struggling with just a bit.

Before this lesson, students have worked in the context of a bike race of “x” number of miles and found a fraction of the race various bikers have completed.  Looked like this:

frac1This lesson went very smoothly and I found it was more of a struggle to have them model what was happening on the fraction bar since finding the fraction of the whole number was an action they could do mentally.  To some, it seemed like an unnecessary step and to be honest, I wavered between unnecessary and yet completely necessary to make their thinking visual. I knew how important it would be in fraction x fraction, so I made them construct the model of what was happening in the story.

Today we started fraction of a fraction. It incorporates the same visual image of the fraction bar, so I love that continuation from previous lessons. It did lack a context, which at first bothered me but as we continued working, and heard the discussions, I moved past that.  Tomorrow, I am actually going to have them come up with a story to go along with a few problems to see if they can contextualize the math they are doing.  We started with a fraction of a half and then a fraction of a third, writing the expressions (some equations) as we went:

IMG_2426IMG_2411Of course, you always have the students who fly through the work and finish early as I am walking around and having discussions with the students who need some extra help, so I asked those who finished early to think about the denominator each time. Why is the product’s denominator changing from the denominators of the factors? Did you have an idea what the denominator would be before you used the fraction bar?  There thoughts were so interesting:

Absolutely LOVE all of this scratching out, changing her reasoning!

Absolutely LOVE all of this scratching out, changing her reasoning!

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This one brings up the issue of vocabulary….fours instead of fourths, eights instead of eighths. Something I have to bring out in our discussions.

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This one I struggle with because of the words double and triple. I know the number itself is doubling and tripling, but I would like to have them expand that is it happening because there is another half to split or two other thirds to split.

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I love that this makes the fractions factors and products are just like whole number factors and product.

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Again with the “double” word. Is it just me that struggles with this one??

I am thinking this will be one of MANY multiplication and division of fraction posts! I am just amazed at the ease the students work with the fraction bars and I like what Investigations has done thus far with these lessons. One tweak I would like made would be the directions…students are asked to “stripe 1/2 of the shaded portion” and it is becoming a tongue-twister for me 🙂 I keep saying shaded when I mean striped, minor detail but they keep correcting me!

These conversations are so rich and valuable for this understanding that it blows my mind that a teacher could just say “multiply the numerators. multiply the denominators. That is multiplication of fractions.” If I had learned fractions this way, it would have all made SO much more sense!

To be continued…

Reflecting on the Mathematical Practices

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On Thursday, in the spirit of Halloween, I presented the class with a set of vampire teeth and the Pandemic lesson from @Mathalicious: http://www.mathalicious.com/lesson/pandemic. If you haven’t checked it out, you definitely should, great stuff on their site! Also, this post will not make much sense unless you understand the premise of the lesson 🙂

IMG_0196Being a 5th grade class, I knew we wouldn’t get into the exponential representation, but I wanted the students to reason about what was happening each week and look for patterns in the problem. They did not disappoint.

The students were very quick to jump right in…monsters, blood, vampire teeth…they were all in! The majority of the class were fairly quick at recognizing the number of vampires was multiplying by two each week. I found the biggest struggle for them was to keep the total population in mind.  For example, in Week 1 when there were 2 vampires, there had to be 138 humans because there always had to be a population of 140. The following week when there were 4 vampires, the students subtracted 4 from the Week 1 human population, arriving at 134 humans, but the total population would only be 138.  It was hard for them to realize they only needed to subtract the “new vampires” from the human population, not the current vampire population.  It was a struggle and some got frustrated when I would ask them if people left the town? Did you lose dots from your array? They wanted the answer, it drove them crazy and I loved it!

At the end of the lesson, I had two groups who had gotten through the world population piece (they were very surprised that it didn’t take them that long to get to 7 billion)! They predicted it was going to take them forever!

Before leaving, I had everyone reflect on which Math Practice(s) they felt they best reflected their work in math class that day and here are just a couple examples:

IMG_2369 IMG_2370 IMG_2371 IMG_2372Math Practice 1 was by far the unanimous choice because they felt the struggle of working through a math problem. I loved reading their reflections, and it made me realize that I need to really work on asking that question more often and push them to look at the other Practices in their work.

-Kristin

Why We Need Two Teachers in Every Classroom…

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This job takes two brains to handle the thoughts of these students.

In class on Friday, one student made the comment that he didn’t really like adding fractions on the clock because it could only be used for certain fractions.  When I asked him to expand on that, he explained he could only do halves, 3rd, 4ths, 6ths, 12ths and 60ths easily and what if he wanted to do other fractions like 1/8 or 1/24?  He said he couldn’t do that without breaking the minutes up.  I am excited at this comment especially because this student is one whose parents have taken him to Kumon math for years for “extra help” and he is most comfortable memorizing procedures over thinking about the math. He thinks changing to “common denominators by multiplying the numerator and denominator by the same number” is faster and easier than this clock.

Upon reflection, I think it is interesting that he stayed with fractions of the fractions we were working…why not pull out 1/9 or 1/11? But my first train of thought in the moment was changing the whole. I wanted to see if he could put the clock in terms of a whole day, 24 hours, 2 rotations around the clock being the whole instead of one. That way 8ths and 24ths would be more apparent.

So I asked him if he could think of a way we could change the clock to do 1/8 or 1/24 without breaking up minutes? His first reaction was no, so I said “That is interesting because there are 24 hours in a day, so I feel like this should work.” Possibly leading him too much but at that point I could see the glazed look in some of the students eyes and I felt like I was losing the class’ attention. I told him that during math workshop that day he could chat with me about it or he could take that thought and work with some more for Monday.  He said he wanted to think about it over the weekend…I think mainly because he didn’t want to miss the Math Workshop activities, so we will see what he has for me tomorrow.

After school, I am recapping this lesson for Nancy and saying how difficult I thought it would be for them to grasp two rotations of the clock as the whole for the 24 hours that would allow for 8ths and 24ths more easily.  After listening to me ramble for about 5 minutes about this idea, she casually says, “What about military time?” UMMMmmm…DUH. Where was she during that class period??  This job really does take two brains.

So needless to say, I have amended my lesson for tomorrow. I am handing them this military clock and letting them talk about what fractions we can work with easily that are the same as our first clock and which one’s are different. Design addition equations we can solve with this clock that we couldn’t do on the other clock without breaking minutes.

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Don’t get me wrong, I still want to get to changing the whole on our original clock, but I think after working with this clock, it may be more accessible for more of the students. I will post later to update on this lesson to show how it went…but good or bad, the questions and thinking that led to this lesson are so worth it!

-Kristin

A Fraction of our Time in Math Class…

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I absolutely love fraction work with my students because there is always something interesting that leaves me pondering the whys and hows of my practice….

Being a K-5 Math Specialist for a couple years offered me the opportunity to really see the trajectory of our fraction work. Now being back in the classroom, I feel I have a much better grasp as to the work the students have previously done within our math program.  In third grade, they work tremendously with halves, thirds, and sixths using polygons to represent fractions of a hexagon whole for comparison and addition/subtraction. In fourth grade, students use arrays and known equivalencies to compare and add/subtract fractions with unlike denominators by choosing an appropriate array that works for both fractions (common denominator). In addition, at each grade level, students in need of RTI enrichment, work in Marilyn Burns’ Do The Math Program which utilizes fraction strips to compare and add/subtract fractions. All of this work focuses heavily on the students’ understandings of equivalencies.

Knowing all of this still never prepares you for the power of a new model….time! I have to admit, I am a huge fan of fraction strips and array work, however today I felt the power of clocks in developing equivalencies.  I have taught this lesson in previous years and to be completely honest, never really liked it. It felt contrived, like a pizza divided into slices in another form. This year I have realized it was not the context that was lending itself to the “pizza feel,” it was me.

The class began with a discussion of a blank clock face. I asked the class if the minute hand stayed at 12 and the hour hand moved to the 1, what fraction of the clock did it turn? They said 1/12 and we chatted about how we can prove that, divided it up and went from there. Next I asked if the hands were reversed, would that give us a different fraction? Some said no, some said yes and we talked about the equivalency of 5/60.

The student questions that followed took my appreciation of the clock to another level:

“Is this the same as degrees since it is a circle?”

“Could we do the fraction for a whole day (24 hours)?”

“Can we split the minutes in half to do eighths?”

“What fraction does the clock go at the time we go to lunch?”

Holy cow, how many directions could I take this lesson??  I moved forward with having the students work with partners to find all of the fractions they could represent on the clock.  Then I asked them to use that model to add 1/3 and 1/4 on the clock. It was interesting to see the students who know how to “find common denominators” by multiplying the numerator and denominator by the same number were challenged to make a proof of their equivalencies on the clock face, while the students who needed the clock as a tool had it as their disposal to see that 1/4 is 3/12 and 1/3 is 4/12.  That clock face immediately went from something I saw as just one more pizza, to both a tool and model at the same time in my classroom.

The follow up activity is called Roll Around The Clock (http://tinyurl.com/p8sm7wa). It has fantastic variations to the game and I have student work on the positive/negative scoring system that I will post soon, it was the perfect extension for the students who needed it!

So today, in just a fraction of time, I found a new appreciation for the analog clock and hopefully improved my practice by a fraction!

-Kristin