Category Archives: 5th Grade

Finding Angle Measures

Fraction Flexibility in Number Talks

In my RTI (Response to Intervention) class, I use Marilyn Burns’ Do The Math program, which is wonderful for building conceptual understanding of fractions through the use of fraction strips. Students use the fraction strips to build equivalencies, make comparisons and add/subtract fractions. It does not take long for students to be able to “see” the equivalencies without having the strips in front of them and develop fluency and flexibility with fractions. In addition to this module, I do Number Talks with the group. I do a combination of whole number operation talks and fraction number talks.

This Thursday, I did a Fraction Number Talk in which I wanted students to think about the fractions and make friendly combinations when adding. I never like to pose a problem with one solution path, so each can be solved using another strategy, however my goal was making friendly combinations. Next to each problem I put my thought in brackets so you have an idea of what i was thinking:) This is the string I planned:

2/4 + 2/3 + 6/12 [(2/4 + 6/12) + 2/3]

2/3 + 1/4 + 1/4 + 2/6 [(2/3 + 2/6) + 1/4 +1/4]

1 3/8 + 5/10 + 3/4 [(5/10 + 1/2)+1 3/8 + 1/4]

They did so wonderful with these and some began whining that these were too easy and to give them something really hard. So I gave them my final problem:

2/3 + 1/2 + 3 + 1/4

There were a few groans and “this isn’t hard“s because they went to 12ths and had the answer quickly. I told them if they had the answer, to try to use the strategy they had used in the previous problems to see if they got the same answer. I was thinking they would use a piece of the 3 to make the 1/2 and 1/4 a whole, but of course there is always one who surprises me! He had a beautiful explanation so I asked him to write it down for me so I could remember. He got a little mixed up in his wording, so I will do a translation after you check out his reasoning.

He took 1.5/3 from the 2/3 to add to the 1/2 and make a whole. He then added the 1/2/3 to the 1/4. I, of course, asked him how he added that and his response was so beautiful as he explained it to me. I mean how amazing is it that he knew 1/2/3 is equivalent to 2/12…and this was all mentally!

Let me assure you that this student CAN add these fractions in a much more efficient way, and this was him challenging himself to play around with the fractions. THIS is what I would consider flexibility in operations and also where I want students to see math as fun…playing around with numbers!

– Kristin

Always, Sometimes, Never….Year 2, Part 2

Always, Sometimes, Never….Year 2

5 Replies

Last year, I used this activity with my 5th grade students and blogged about it here. This year, I used the same activity, however tried it a bit earlier in the unit than last. Based on our Talking Point activity before the unit began, I found the students had very good understandings that would emerge naturally in the Always, Sometimes, Never activity. I wanted to see how using the activity before a lot of our classification work would affect the outcome, if it would differ from last year’s.

This year, we had played “Guess My Rule” the day before in which students use attributes to choose two quadrilateral cards that fit their rule and one quadrilateral card that does not, while their partner tries to guess the rule. It is great for thinking about classification by sides or angles and vocabulary building. At the end of class, we did a few rounds together, and chatted about some vocabulary that was helpful and talked about our classification by sides, angles, or both.

Being the day before Thanksgiving, you never know how it was going to go, but they were so engaged in the work. Here is a copy of the cards that I used:

I gave them time to individually read them and underline any words or wording in which they needed clarification. Although I was surprised that some students asked about words we had discussed in prior class periods, I was so much more happy they felt safe enough to ask for clarification. This is a prime example of not making assumptions in what our students know just because we have talked about it before in class.

They cut them out and went to work. As expected, they would have a brief conversation with their group and place them quickly into the appropriate column on their desk. I let them do that on my first round around the classroom and then as I heard some debates starting, I suggested that maybe using their journals to show their proofs may help their argument. For those who were quickly done, I said, “If it falls in the Sometimes category, you should be able to show when it does happen and when it doesn’t happen, right?” I also pointed out the reasoning for Always never being able to be disproved and the opposite for Never. This had them really go back and take a deeper look at the cards and got their conversations going.

Of course, we did not get to come back together as a group and come to consensus but here were the table card arrangements as the class ended:

I had them do an individual reflection on which card they are still really struggling with and these responses are going to help in framing how I proceed from here on Monday after vacation:

It is really interesting that for some students the orientation of the shape makes the biggest difference, others strugglw with the vocabulary, and, like last year, that rhombus one is blowing their minds. It is so interesting to me that a student can apply shape attributes to make a conclusion that a rhombus is a rhombus, but then to take that reasoning and apply it to another shape, is extremely difficult. This led to a very interesting conversation between a colleague (who was in observing) and myself about students knowing definitions versus descriptions….still wrapping my own head around that one…will probably be a blog post coming soon:)

With all of this information from the students, on Monday, I plan on putting them in groups based on the related cards they were left grappling with. I think rearranging the groups will lead to interesting conversations and more detailed proofs. Each table will get three cards to create an argument for the placement of that card. They will present their argument to the class and we will try to come to consensus as a class. Last year we did this share as a whole class, and I didn’t feel like it “wrapped up” and things were left hanging out there that needed to be a bit more solidified in future classification work, so hopefully this will be change that.

Happy Thanksgiving all!

-Kristin

Articulating Claims in Math

8 Replies

This summer I was fortunate to hear Virginia Bastable keynote about the work in her book Connecting Arithmetic to Algebra. If you have not read this book, it is a must! It explores the process by which we have students notice regularities, articulate claims, create arguments and representations, and make generalizations.

It is something, as elementary school teachers, we need to really be thinking about more in our math classes. Are we creating environments that encourage students to think about the math behind the strategies and make generalizations based on the properties of operations? I have taken this recent reading and made it a priority in my classroom.

I always have students notice and discuss patterns and regularities but I don’t often have them create generalizations for us to revisit as we move through the year. For example if their claim works for whole numbers, shouldn’t I revisit that as we work with fractions? Does your claim still hold true?

As a class routine, I posted this on the board and asked students to fill in the blanks to make it true:

12 x 4 = ___ x ____

Quickly, students wrote down answers, had their hands up, and one student blurted, “This is easy, you don’t even have to solve it!” Typically blurting out answers before others are done thinking drives me a bit crazy, but this time I was thinking…Yes! I asked who else thought the same thing. I had at least one hand up at each table so I asked them to discuss with their table how that is possible. We came back together and each table said they could double/half to fill in the blanks. I took answers on the board and got the expected: 6 x 8, 24 x 2, 48 x 1, 3 x 16 and then I even had a 96 x 1/2 and 36 x 1 1/3! I asked about the 48 x 1…did you get that by double/halving from the original problem? What is happening there? They noticed that it was x 4, ÷ 4, and then the same with 3 x 16. I asked them to take some individual time to see if they thought their strategy would always work and could prove it with a representation. They then talked at their tables and I asked each table to write a claim, something they think is true about this work.

I got some who kept solving problems to prove it works:

I had a couple try out the representation (exclaiming how hard it was to draw what is happening:)

Here are some of their claims:

This is the class list from my second period class. I especially liked that one of them said it only worked with multiplication. How fun to revisit!

Instead of losing these, I started a Claim Wall to post and have students add to and revisit throughout the year. I am trying to think through how to have students comment on them, possibly agree/disagree post its?

If you would like more information about Virginia’s work, there are courses available here: http://mathleadership.org/programs/online-courses/ Check it out, great stuff!

-Kristin

Talking Points – 2D Geometry

9 Replies

We are about to start out work with Polygons, so I decided to kick it off with Talking Points. If you have never read about them before you can check out my post or Elizabeth’s post to learn more.

Here were the points my students were discussing:

I had gotten these points by looking back at their fourth grade geometry unit work and thinking about what misconceptions or partial understandings students have each year when we start this unit.

This time, I made a few changes from past experience. In each group I had a facilitator to be sure that everyone got a chance to speak without interruption during Round 1, and a recorder to keep the tally for the group. Also, after the first talking point, I had advise from a math coach in the room filming with me to add individual think time after the reading of the point. LOVED IT! During think time, they were jotting in the journal and getting their thoughts together. I got things like this from just the think time:

It was nice to see them take ownership with their journal without being told to write anything down. They were working on proofs before they started. After the six talking points, I posed three questions on the board for them to reflect upon individually:

1 – What talking point are you sure you were right in your answer? Explain your reasoning.

2 – Which talking point are you unsure about your answer? Why?

3 – Which talking point did your group agree upon easily? Why do you think it was easy for your group to agree on that one?

Here are their reflections:

Never anticipated so much “left angle” talk in my life 🙂 I learned SO MUCH about their understandings and wish I had time right now to add my comments to each journal, but I will very soon!

So, I have a moment here at lunch to reflect on what I learned from these talking points:

TP1 – As I anticipated, this one is always a source of confusion. Every year it seems as if the students know the sentence goes one way or the other but can’t remember it because there is little understanding of the WHY piece. Later on in the unit after we have done more classifications, I do more of these statements with Always, Sometimes, Never, so this is a something I wanted to see how students were thinking about it. Most tables said something to the effect of “I remember last year we said a square is a rectangle or a rectangle is a square, but I can’t remember which one.” Another conversation I heard was that a rectangle has to have two short sides and two long sides.

TP2: I loved this question and was really pleasantly surprised to see some trying to draw it and ending up with unconnected sides. One thing I was so surprised about was the “left angle.” They were not thinking the degrees changed so much from the left to the right angle, but more the orientation of the angle (left side, right side). Interesting.

TP3: I got a great sense that most students knew what area and perimeter were and the best part was that if they didn’t remember, someone at their table did and gave an example. Regardless if they knew they could be the same, I was excited to see a great understanding from most here.

TP4: This one was great. I saw some students drawing a square on their paper, showing the group, rotating the paper and saying, “See, now it is a rhombus.” They all seemed to be in the mindset that a rhombus is a diamond shape, but really not reasoning about the attributes that make it a rhombus.

TP5: They did a very nice job with this one. A lot drew examples of combining two shapes, while I heard others asking their group if the “inside connected side counted” when they were trying to name it. Also realized that the term polygon was not familiar to most students. I am wondering what they called them in earlier grades? Pattern Blocks? Shapes?

TP6: Interesting one here and it is where we start our 5th grade work with polygons, classifying triangles. Again, the left angle reappeared:) I did hear a few struggling with the name of the angles, obtuse, acute, right but then I had some that said there are other 3-sided shapes that aren’t called triangles. Hmmm, can’t wait to find out what they are! Of course, you always have your comedians who say agree because it could be Bob or Fred.

Can’t wait to start planning this weekend!

-Kristin

What Happens When You Divide by Zero?

Volume with Fractional Dimensions

2 Replies

Before I began our volume work this year, I blogged about my planning process here: https://mathmindsblog.wordpress.com/2014/10/20/unit-planning/. As anticipated, I had many students who quickly developed (or already had) strategies for finding volume and could articulate a conceptual understanding of what was happening in the prism. In my previous post, I was throwing around the idea of giving those students dimensions with fractional length sides, so the other day I thought I would try it out. I did this Illustrative Task as a formative assessment of student understanding. Many students were done in a couple of minutes, with responses for part b that looked like this:

As I walked around the room and saw they were finished quickly, I asked them to revisit part b and think about a tank with fractional dimensions. Because of the great work they had done here I thought they would have some interesting thoughts. These are a few of the responses I got:

So, what did I learn from this work? I saw they had some great understandings about taking a fraction of one factor to make a number that they knew they needed to multiply by a third whole number factor to get 240. In the first two pictures, there is a great pattern happening that I want to explore further with the whole class. I also loved seeing that a student took the question “fractional length sides” to include decimals in his work. In my question, however, I had wanted them to consider more than one side in fractional lengths, however not being more explicit, they took it and ran with one side being fractional. In the next lesson, I thought I would push them a bit with this.

In the following lesson, students were finding the volume of an unmarked prism in cubic centimeters. They had rulers, cm cubes, and cm grid paper available to them, and went to work. Every year this happens, the Investigations grid paper works with the box to be whole number dimensions, however the cm are a bit “off” when using a ruler or cm cubes. I knew this, however, I do love the discussions that evolve from students who used different tools. I also thought this is the perfect opportunity for my students who were beginning to think about fractional sides. What transpired in the whole class lesson is a blog post in and of itself, however this is what came about from the fractional sides work…

Sooo much great stuff here! I had a group who was using the cubes, coming out with halves, but not wanting to round because it was “right in the middle” of the cube. I let them go and came back to see they were multiplying whole numbers, multiplying the fractions, and then adding them together to get their product. I asked them to think about another multiplication strategy to see if they got the same product, then came the array. Another student in the same group solved mentally to get the products. Unfortunately, the class had to leave me to go to their next class, also leaving me with so many things to think about. From here, I want to be sure students start to think about reasonableness of their solutions, compare their fraction multiplication strategy to whole number multiplication strategies, and think about how we multiply three numbers (Associative property). So much to do, I need full day math classes!

-Kristin

Things I Have Noticed and Wondered

1 Reply

Every day, I notice and wonder things about both the students thinking and the mathematics in my classroom. Over the past couple of weeks, however, there are a couple of things I have noticed that still have me wondering….

We do Number Puzzles in our first Investigations unit, serving as a review of properties of numbers that students have worked with over the years. For those who have not done number puzzles, this is an example:

At the end of this activity (14 puzzles), I have the students design their own number puzzle, trade with another pair of students and talk about the solutions. It is a great formative assessment of their learning and it is fun to see some students creating puzzles that are impossible or have more than one possible answer. The conversations are amazing. However, this year, a student completely stumped me. I wish I had saved his puzzle, but it was two of his four clues that got me wondering. First clue: My number is odd. Second clue: My number becomes even when you make it a decimal. Hmmmm. Thankfully I do not have to recap the conversation here because Christopher Storified our conversation here: https://storify.com/Trianglemancsd/it-s-even-when-i-make-it-a-decimal

My first thought, after I got done reeling over his thinking, was, when was the last time the students have revisited even/odd numbers? I know in second grade, students investigate odd/even numbers in terms of being able to break a number (positive integers) into two equal groups or share between two people. The exact math focus points are…

So, they establish that a number is even if it can be broken into two equal groups and if there is a leftover, it is odd. But, at this point, they have only dealt in whole numbers. Do we ever revisit that when they begin working with rational numbers or negative integers? Do we just assume that students keep the understanding with positive integers and don’t try to apply it to other numbers? What an interesting thing I have never thought about before! It blows my mind that after 19 years of teaching (12 of those years in 5th grade) that I have never had this conversation before with a student. This whole conversation has me digging back into the math introduced in earlier years to see if there are other things that we never revisit in our work as we move into decimals, fractions, negatives…etc. Just for some added fun, Christopher said he thinks he could master “Billy’s” even/odd quiz. Billy made a quiz here: pic.twitter.com/p24gMG4Uwd and you can check your answers here: https://www.dropbox.com/s/2chzj2n3534stjo/billy.parity.answer.key.pdf?dl=0

My second wondering stems from my students’ work yesterday with Volume. Up to this point, we did a lot of building with unit cubes and in Minecraft of rectangular prisms in route to formalizing our strategies for finding volume as l x w x h or area of b x h. Students were very comfortable with finding the volume of rectangular prisms, with or without the cubes drawn in the diagrams. They could talk about the arrays in each layer they saw, and how many layers they saw “stacked” in finding the volume. I was feeling so confident that these understandings would follow us right into our work with two rectangular prisms combined to form a solid figure. It was looking great as we worked with the gridded diagrams. Here is an example of one student’s work:

Then hit the unmarked models…

Even anticipating there would be some confusion, I was honestly blown away by the number of students struggling to break apart the picture and think about the lengths, widths, and heights they needed to reason about layers of cubes. I found that some of the students felt the need to use every number in the picture in their solution while others were making no connection to their reasoning about arrays and layers. They could describe how they broke the shape apart into two pieces, however when I asked how many cubes were in the bottom layer of the prism to the right, they could not see the 3 x 2 as 6 cubes in that layer, and if they did (and had broken it horizontally) would do 9 layers instead of the 5.

I was struggling with how to help them go back to their amazing work with cubes. Why was this visualization disappearing? In hopes of having them have a better visualization, I asked them to build the above picture in Minecraft. Done. Every single struggling student automatically laid down a 2 x 3 array, made 5 stacks of them before moving into four 12 x 2 arrays on top of that. What?!? This has me completely perplexed. How can they see it to build it without a second thought, but cannot communicate about it otherwise? More importantly, where do I go from here? They obviously cannot build every figure they need simply because when the numbers are larger, it is completely time consuming and I want them to connect their concrete building to a more abstract reasoning. I am thinking from here, I will have some continue to build until they don’t need it, pushing them to abstract with questions such as, “If you were to build this right now, describe it to me.” Pausing occasionally to ask how many cubes that would be.

These two wonderings, although so obvious at times, have seemed so complex to me over the past weeks. I love every moment of looking into my students’ reasonings’ because it challenges me to be a better teacher and look deeper into my own understandings about how children learn. Great stuff.

– Kristin

Good Questioning in Math is Tough!

10 Replies

Today, I had one of those opening activities that was planned as a 10-minute math talk, turn into something much more than that. I am always excited when the students drive the conversation that way, however today, I felt at a bit of a loss for the “right” question in our discussion.

We started with an Estimation and Number Sense activity in Investigations in which I flip four digit cards that form a 2-digit by 2-digit multiplication problem. Students estimate a product, we discuss strategies and whether our actual answer will be more or less than our estimate and why. It always sounds so simple in the planning part.

I flipped cards for the problem 81 x 82. Thumbs went up right away (our symbol for having an answer and strategy in our Number Talks) and we got the initial estimate of 6,400 because 80 x 80 was 6,400. A few students explained and revoiced that our actual answer would be more than that because we rounded both numbers down. One student said she had the actual answer of 6,642 and proceeded to take us through the partial products she used. So far, so good. Then a student says he used the estimate to get an actual answer, but came up with 6,644. He said since he only used 80 x 80, he needed one more group of 82 and two more groups of 81, which gave him 82 + 162 = 244. He added that to the 6400 and arrived at 6644. This is our board:

Students are nodding their head in agreement but then start to wonder why it is not matching the answer when we found partial products. It is off by 2, so some students think it is a calculation mistake somewhere but soon realize it is all correct. I send them back to their journals because they want to know which is right and I am not telling them. We come back together in a couple minutes and everyone has 6,642, even the student who gave me the 6,644. So, I ask them, “Then why isn’t “Billy’s” strategy working?” because that is really the fun part :).

This is where I was at a loss for a good question. Everyone could prove to me why the answer they got was right, but I didn’t know what to ask them in that moment without completely giving them the solution path. It is SO hard to question students. I say it all of the time and today, no matter how prepared I was, I was at a loss for a good question. Don’t get me wrong we shared some great work…

I got a example of why it doesn’t work when you have another problem…

I got the student who gave me 6,644, trying to compare what he did to what he knows the answer should be…

But even after we went through our volume lesson and they left me for the day, I still am thinking about what I could have asked them to push their thinking. I was feeling, in that moment, that certain questions would have pushed quite a few students to disengage while at the same time, I was not wanting to let it go. They left not knowing why that was happening, which I am completely fine with, but I don’t want it to be a missed opportunity either. I am thinking that tomorrow I will try to connect it to an area model another student had used to solve it and see how that compares to the problems that “Billy” used to get 6644.

I am up for any and all suggestions as to how to pull this conversation together. What would you have asked in that moment? What would you follow it up with tomorrow?

Thanks!

-Kristin

MathMinds

by Kristin Gray.

Category Archives: 5th Grade

Finding Angle Measures

Fraction Flexibility in Number Talks

Always, Sometimes, Never….Year 2, Part 2

Always, Sometimes, Never….Year 2

Articulating Claims in Math

Talking Points – 2D Geometry

What Happens When You Divide by Zero?

Volume with Fractional Dimensions

Things I Have Noticed and Wondered

Good Questioning in Math is Tough!