Tag Archives: Math

To PD or Not PD..That Is the Question

The past two years as Math Specialist, I was in a position in which I was continually planning and attending Professional Development on a regular basis. I am a learner, so I frequently got frustrated and a bit upset when teachers complained about attending the PD. I would hear such things as, “I need time to grade my papers” or “Sub plans are such a pain to write.” How could they not love these learning experiences as much as me?

Fast forward to this year, I am back into the classroom, and I completely feel their frustrations. I have papers that need to be graded, I despise sub plans, and most importantly, l have lesson plans that I need time to think about & dig deeper into. Time, as always, is a high commodity. So, as I was in my classroom Thursday evening, writing sub plans (or more accurately procrastinating by finding anything in my classroom that needed to be done BESIDES writing the plans) I found myself thinking that it would be so much easier to not attend the PD (it was by choice I was going) and just stay in the classroom on Friday. No sub plans, and I would have my planning time to get the paper grading and lesson planning done.

This was it, this is the point where teachers (me included) need to step out of their immediate surroundings, look at the bigger picture, and ask themselves the following questions….

1. How can I continue to improve student learning in my classroom if I don’t dig deeper into my content area(s)?

2. How can I grow as an educator alone?

3. How can reflecting on my own teaching with others improve my classroom experiences?

4. How can what I know about teaching help others in my network?

5. There is always SO much more to learn. Not a question, I know, but it is my driving force as an educator.

And….How great is it to have breakfast and lunch made for me and I can use the bathroom anytime I want ūüôā

Needless to say, I always try to attend professional development when offered the opportunity and after leaving my PD on Friday I just found myself smiling. I love talking to others with the same passion for mathematics and teaching as myself. I learn so much and just flat out have fun while talking about impactful issues in education. We all want what is best for our students and staff and work together to make great things happen.

Don’t get me wrong, I am picky when choosing my PD. It must be relevant. I have sat in a mandatory PD or two (hundred) that have not been what I needed, but I try to find at least one thing I can take away. Even through the bad experiences, I grow. If a presenter is not engaging, I think about what I can do when I facilitate to be engaging. If the content is confusing, I think about how I can clarify things when I facilitate a professional development. I don’t let one bad experience kill all professional development opportunities for me. They are independent variables, like a die. One roll does not impact the next. One bad professional development does not impact the next one.

In the end, I owe it to my students to go. If I am learning more, they will be learning more.

Happy Saturday,

Kristin

Math & Minecraft Day 1

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…

mc

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

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.

РUsing 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

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

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

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!

IMG_2413

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.

IMG_2414

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.

IMG_2415

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

IMG_2416

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 ūüôā 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

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 ūüôā

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