Category Archives: 5th Grade

Decimal Division, Running & Why I Love My Tweeps

Yesterday, I posed a decimal division problem to get my students thinking about what division means to them and how that applies to decimals: https://mathmindsblog.wordpress.com/2015/05/05/a-great-day-of-decimal-division/ (It was a really great day)

I was thinking of moving into a context today to see how they would represent the problem and the approach they would take after yesterday’s discussion. So, of course I threw it out on Twitter…

@MathMinds @allisonhintz124 @kassiaowedekind 48 mile trek/hike through mountain range or for fundraiser. want to put a sign every .6 mile

— ekazemi (@ekazemi) May 6, 2015

All evening I was thinking about a context and this one Elham suggested worked great for me! I was still thinking about how to word it to be something that the students may be connected to, then Joe’s tweet came this morning after my run…

@MathMinds So how many .1 miles is that?

— Joe Schwartz (@JSchwartz10a) May 6, 2015

Duh, my runs! Thank goodness Joe was up early too! My students know I run every morning and cannot fathom that anyone actually wakes at 4:30 in the morning, so I knew they would love this.

To start the class, I posed..

“I ran 2 miles on Monday afternoon. Every .4 mile I took a sip from my water bottle.How many sips of water did I take during my run?”

As with most times, I gave them some individual time before consulting with their table mates. It was awesome to see so many of the connections to yesterday’s work and also new representations that did not show up yesterday.

This one was so interesting how he broke up the mile to .4 +.4 +.2 and then combined the .2’s to make 5 four tenths.

This number line was so nice and then I loved how he got to the end and then counted the jumps going back down to zero. Also, at the top he had multiplied up to the 2 miles, nice way to show two ways of thinking about the problem.

There was a lot of skip counting by .4, but this model was especially wonderful. It is an area model combined with a number line. He counted up by .4 in squares that attached until he reached 2. I would expect students to count the number of .4 sections to find the answer, however this one labeled the 1, 2, 3, 4, and 5 at the end of each section.

I then gave them a log of my past five runs. I told them to assume that I still take a sip of water every .4 mile. I wanted to know how many sips I took and then how much further I had to go until my next sip.

I got some awesome partial quotients, number lines and multiplying up.

Now, the conversation of remainders came up. They want to know how to write the answer without the “r.” They wanted to know if they could write that as part of the number that was the answer. For example, could they write “7 sips r .2 as 7 1/2?” Saving that for tomorrow.

And THIS is why I love the #mtbos….my lessons take wonderful twists that make the learning experiences in my classroom so much better for my students! No teacher can do this job alone!

-Kristin

A Great Day of Decimal Division

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Today, I really saw such a beautiful picture of the culture of learning in my class and marveled in the way in which my students had arguments in the best possible sense of the word. They were excited about the math, working so hard on proving their answer, and in the end ok with being wrong because they “saw where they messed up.”

I posed the the question 2 ÷ 1 on the board and asked the students to write what they thought about when solving that question, or any division equation similar to that. We shared and collected our responses…

Going back to their journals, there were some other interesting ones such as contexts and what the symbols meant…

I then asked them to write this equation and answer they thought for 2 ÷ 0.1. I was so excited to have a split class of the answers 20 and 0.2. They worked on proofs with their tables and I got some great thoughts around what the think about division as well as references to visuals they think about in their solution.

I particularly like this answer because of the way it was written…

At this point they begged for chart paper to create a poster to “show the other group why they are right.”

Each group shared their thoughts and there were a lot of “Oh”‘s and “I have a question for…”‘s going around the carpet area. At that moment, they were completely owning the class. They politely waited to ask their questions, politely disagreed with one another, and openly admitted when they “changed their mind.” I loved this moment so much, I just listened.

I had all but one student who still did not agree yet with the 20 so I had him write down why so I can think about how to structure tomorrow to meet his concerns. He is focused on adjusting either the dividend and/or divisor and then adjusting the quotient. I love all of the “how”‘s.

I asked the ones why had 20 to take a stab at a context for the problem and others who changed their mind to tell me what part was the final aha for them.

Today was the day I wanted to have other people in there watching to be as excited as I was. I told the kiddos how proud I was of them and off to lunch they went! I then had to bottle up my excitement until I could get this all out. It was just a really great day to watch my class work, and learn, together.

-Kristin

Conjecture or Claim?

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I have been having wonderful conversations on Twitter recently with Kassia (@kassiaowedekind), Simon (Simon_Gregg), Mike (@MikeFlynn55), Elham (@ekazemi) around the topic of students making claims, more specifically differentiating between claims and conjectures. I have to admit, I have really just formed my own idea of how I differentiate between the two, so it was nice to hear others’ perspectives around this. I consider a conjecture a noticing they think to be true, more on a case by case basis. A claim, to me, becomes more generalized and then followed with a proof. (I have also had great convo with Malke (@mathinyourfeet) around these proofs w/geometry).

The conversation last night started with Kassia…(Look Kassia, I finally learned to embed tweets:)

@MathMinds Do you differentiate between claim and conjecture in your class? If so, how?

— Kassia Wedekind (@kassiaowedekind) May 3, 2015

Mike gave us a nice perspective of claims based on his work with Virginia Bastable….

@kassiaowedekind @MathMinds a conjecture is an expression of a general claim that students have noticed and suspect is true. — Mike Flynn (@MikeFlynn55) May 3, 2015

@kassiaowedekind @MathMinds But we are using the term conjecture when students are exploring this to see if it is true. — Mike Flynn (@MikeFlynn55) May 3, 2015

@kassiaowedekind @MathMinds Credit to Virginia Bastable for this explanation. This is her area of expertise. I’m just lucky to work w/ her. — Mike Flynn (@MikeFlynn55) May 3, 2015

My students have now started to say, “I have a claim to make” when they notice something happening over and over again. In those moments, I don’t really think about “what” they are calling it because I am just so excited to hear them talking about the patterns and regularities they are seeing. But is what they are saying a conjecture or claim? Does it make it to the claim wall to be revisited and proven? This year being my first work in really having students think about making “claims” beyond just noticings, I have made a “Claim Wall.” Students see things happening in certain cases and I ask them if they can write a statement for “any time we…” to see if they can make it more general. I like Simon’s idea to expand on my wall…

@ekazemi @MikeFlynn55 @kassiaowedekind @MathMinds Statements could travel across the wall: ?” -> confident claim -> (triumphantly) PROVEN — Simon Gregg (@Simon_Gregg) May 3, 2015

We all agreed that the proof piece is the difficult piece of going from being a conjecture or unproven claim to a substantiated, generalized claim. I find my students prove over and over again that it “works here and here and here…” but have trouble with the why. It is hard to do, even as adults putting it into words is difficult.

@ekazemi @Simon_Gregg @kassiaowedekind @MathMinds Yes. Very helpful. Here are the guidelines. pic.twitter.com/QQ01IpnmTN — Mike Flynn (@MikeFlynn55) May 3, 2015

What I love most about these conversations is the fact that the next day it continues, but this time with the kids. Simon tweets this morning about a claim that two of his students made while folding paper…

Rod & Samyak’s claim: every even-numbered fold -> square. To prove? @ekazemi @MathMinds @MikeFlynn55 @kassiaowedekind pic.twitter.com/EMyr5oNN3N

— Simon Gregg (@Simon_Gregg) May 4, 2015

Which coincidentally would help my students tremendously to think about when proving their claim from Friday’s number talk…

@Simon_Gregg @ekazemi @MikeFlynn55 @kassiaowedekind One of my Ss work from Friday.. #mtbos pic.twitter.com/I7sf6ez9ar

— Kristin Gray (@MathMinds) May 4, 2015

The coolest part about this claim was that it stemmed from a multiplication of fraction number talk, yet they proof show division. I loved that. Also loved the explanation that accompanied their statement. I did ask them if this was true for taking half of any fraction because they seemed to be just dealing in unit fractions at this point. So is this a conjecture or a claim? I am not sure. How generalized would make it a claim? Could it be “When taking any unit fraction of another fraction…”

Would love any thoughts, conjectures or claims on this…:)

To be continued…

-Kristin

Creating Contexts for Decimal Operations

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Sometimes I have students engaging in math within a context, however at other times, we just explore some beautiful patterns we see as we play around with numbers. I see a value and need for students to experience both. This week was one of those “number weeks” and it was so much fun!

Over the past few weeks, we have been working on decimal multiplication. If you want to see the student experiences prior to this lesson, they are all over my recent blog posts….it is has been decimal overload lately:) After sharing strategies and connecting representations in this lesson, I was curious how students thought about this problem in a context because up to this point, I had not given them one for thinking about a decimal less than one times a decimal less than one.After they wrote their problem, I asked them to tell me what they were thinking about as they were deciding on the context.

I anticipated that many would refer back to what they know about taking a fraction less than 1 of a fraction less than 1, like in this example…

I love how this one said she knew she “had to start with .4” That shows the order of the numbers in the problem create a context for her. It mattered to her, taking .6 of the .4.

This student went with two different contexts and again saying that he started with the .4. This must be something we have chatted about quite a bit about because it showed up multiple times. I loved how this student said he thought about an area model in creation of his problems.

This student was great in listing all of things he was thinking about as he thought about a context..

I had students who attempted to create a “groups of” context. I don’t know if I ever realized how difficult this and how much I, as an adult, need to be able to create a visual in my mind of what is happening in a problem to make sense of it. Here is one example (not the sweetest context but she thought the Mary HAD a little lamb was clever…) She worked a bit yesterday to show what the representation would be, but kept running into problems with cutting into “.6 pieces.”

And then I have these two that had my brain reeling for a bit, for many reasons. First, does the context work with this problem? Secondly, I knew it sounded like it should work, but when I tried to make sense of it, I couldn’t create a visual. Also, as I read them, I thought I knew where it was going and the question I would pose, but it wasn’t the way they saw it ending. I asked them to create an Educreations about their problem so I could check out their thinking around the context.

Yes, Rick Astly. But the question at the end, compared to the total time Never Gonna Give You Up, threw me a bit, not where I was going with it….

His Explanation: https://www.educreations.com/lesson/embed/31398809/?ref=embed

The second one tried it out, and wasn’t so sure of his question after messing with it. The wording “.6 as small” was making me think. I was trying to make sense of that wording, do we ever say six tenths times as small? Then does his question referring back to the .4 make sense?

His Explanation:https://www.educreations.com/lesson/embed/31402039/?ref=embed

Definitely a lot for me to think about this week too! I have some amazing work with them connecting representations to write up later…they are just such great thinkers!

-Kristin

Multiplying Decimals Less Than 1 Whole

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Apologize, not much time to write, but today was so cool I had to share! I am in the midst of using this work to better plan for tomorrow.

Short version: After our predictions yesterday, I posed 0.4 x 0.2 (I changed the problem to 0.6 x 0.4 for my second class) and asked the students to individually jot down what they thought the answer is. I was looking to see how they intuitively thought about the problem. As expected, I saw 0.8 and 0.08 (2.4 and .24) as I walked around. I wrote both answers on the board, asked them to write their reasoning in their journals and then we shared as a class. No telling which was right or wrong, just sharing and listening.

Some great thinking and critiquing of each others’ reasoning ensued and then I sent them off to come to a consensus as a table and create a poster of how they thought about it!

Now, where to go with this work? They could just look at strategies, but I want them to think deeper about the meaning of the problem. After chatting with my colleague Faith tonight, who is coming to observe tomorrow, we are going to have the students walk around to the other group posters and talk about what they saw on the other posters that changed the way they thought about the problem.

From there, I really wanted them to think about a context for this problem and Faith suggested also thinking about what happens are you begin to adjust the numbers and why….really thinking about the reasonableness of answers. What happens when one factor increases? What happens when one factor goes over a whole? What happens if the factors go into the hundredths? Does the product increase or decrease? Why?

So many fun convos to be had tomorrow!

-Kristin

Making Decimal Predictions

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Over the past weeks, I have done a lot of blogging about our work with decimal multiplication. All of this work has been focused around contexts that involve multiplication of a whole number by decimals both greater than and less than one. The students have very flexibly moved into using whole number strategies in order to multiply decimals during our number talks. Today I asked them to think about how whole numbers multiplication is similar or different from multiplication involving decimals. I was hoping to hear the relationship between the factors and the product and they did not disappoint. These are the findings from my two math classes…

I asked them to prove that a decimal greater than 1 times a whole number will have a product that is greater than both factors OR if a whole number, less than one, times a whole number will have a product that is less than one factor but greater than the other.

We shared out and ended the class predicting what they think would happen when we multiply two numbers that are less than one. This is where I saw an interesting difference in the way students thought about the problem. Some focused on the numbers and what it means in an “of” sense, while others connected to what happens with the multiplication process.

This makes for such an interesting conversation tomorrow! Excited to see the fractions come out and for students to revisit their predictions! This is the work tomorrow from last year’s experience: https://mathmindsblog.wordpress.com/2014/07/25/unanticipated-student-work-always-a-fun-reflection/

-Kristin

When My Students Uncover Something I Never Learned….

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As teachers, we don’t typically like to admit when we don’t know something in front of our peers and especially in front of our students. Luckily for us, if we can stall long enough to get to our phone, Google has made it quite handy in making those moments extremely short-lived. The unique opportunity of being a teacher however, is using those moments to reflect on how or why you never learned that particular idea, and in this instance, what the answer really is!

After working through this choral count: https://mathmindsblog.wordpress.com/2015/04/20/choral-counting-decimals/ and https://mathmindsblog.wordpress.com/2015/04/22/investigating-patterns/ my students have come to some really interesting noticings and looked deeply into some proofs of why those patterns are happening each time. Most of this has been focused on properties of multiplication and division and thinking a lot about relationships between factors and multiples. One group of students, however have begun to really play around with the “switching” of the digits in the multiples of .3 (and 3’s since they noticed their similarities) and will rest until they understand why.

I see them working so hard because they WANT to understand. I will completely admit, the closest thing I could come up with is the divisibility rule I “was taught” for 3’s. I wasn’t going to tell them this “rule” because I realized, in that moment, they uncovered something I could not explain to them at all because I never truly learned it. So instead, I sat with them, and we thought through it together. We played around with partial quotients and noticed we could always make dividends that were divisible by 3 any way that we moved the digits around. But, why? I had one student finally just ask…

“Mrs. Gray, do some things just work in math because they just do?“

I quickly said no, but that was exactly what my problem was, I never truly learned why numbers were divisible by 3. I thought it worked because it just did, why would my teacher tell me otherwise? I completely remember copying down all of the divisibility rules, memorizing them and acing the test I took on it. It seemed like a really cool trick that just worked because it did. Today, I know I could easily Google it, find a video with an explanation, but I want to think about it more. I want to play around with the numbers and understand why this works with 3’s, so I can really learn it this time around. I want to be like my students…struggle, persevere and learn.

It is moments like this that make me feel so amazing about the thinking and learning that happens in my classroom and the classrooms of so many of the wonderful colleagues I have in person and on Twitter. We want our students to truly understand the math, not simply just be able to do the math. This is especially true for me in this moment. I could easily have told the class that they can switch the order because the sum of the digits will still be divisible by 3 and that is the rule for determining a multiple of 3, it just works. But I don’t want my students ever thinking math is a series of things that “just work because they do” or something we learn in school and never revisit to think deeper about it. I want them to see us all as learners, which is why I continue to play around with this 3 thing…I will get it:)

-Kristin

Investigating Patterns

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Due to ELA testing, I luck out with an extra 45 minutes of math time twice this week, and today was one!! I wanted my students to revisit the choral count we did on Monday and look deeper into the patterns they noticed. To extend that thinking, I wanted them to make some predictions about decimals that may or may not show up if we continued counting by 0.3 (Thanks so much Elham for the suggestion:)!

We revisited the count and the noticings…

I then wrote some decimals on the board, shown inside the rectangles (kinda) in the first picture above. I asked them to try and use the patterns they discovered to decide if the decimals would show up if we kept counting by 0.3. I was sure to choose a range of options so everyone had an entry into the investigation and focused on the patterns we had discussed. I loved the way they explored their patterns and it completely intrigued me the manner in which they do so.

Some explored by multiples of 3 by looking at wholes and then tenths…

Some used the patterns that involved just one place value but did not look at the decimal as a number…

This group looked at the decimal as a number and chose one pattern they know would work for any number. They broke each decimal into partial quotients to see if each part was divisible by 3…

Other groups used a variety of patterns, noticing that some would work nicely for certain decimals and not others…

The next two especially caught my attention because I had not anticipated the connections being made (I ADORE the way they think:)..

Let’s look at the first one…He saw the “switching the digits around and the other decimal always shows up” pattern working every time and decided to examine the why. His approach was so interesting. He decided to look at the missing addend between the number and its “switch” each time. He noticed the missing addend was always a multiple of 0.9. He then started to look at the relationship between the original numbers and their missing addend. For example (and I so wish you could hear his thinking on this) the missing addend from 1.2 and its switch was 0.9 and the missing addend from 5.7 and its switch was 1.8, so what is the relationship between 1.2 and 5.7 that explains why the missing addend doubles? My curiosity is..what makes that be the next step for some students while others just notice it the missing addend is a multiple of 0.9 and are content. Loved this moment today because I got such insight into how students look at different pieces of a “puzzle” and choose to explore different relationships.

This one was so funn…

She noticed that any two numbers in her list (table), added together, had a sum that also appeared in the table or would appear, if extended. I asked her how she knew that and she showed me a few examples. “Ok, but why?” She thought for a while and then said, “Okay, it is kind of like the even plus and odd number will always give you an even number.” I could tell she was starting to make sense of the structure of numbers but having such a struggle in explaining it. To her, it seemed to just make sense and I think (hard not to make assumptions) that she was thinking about that 0.3 being a factor of both so duh, it just is.

She came back up, an hour later (she kept working on it when she left me:), and said she had it…”it is like DNA.” Ok, now I am intrigued. She explained it to me and I asked her if she could write that down for me because I thought it was so cool…

It seems like a stretch and I am still thinking about the connections, but I am stuck on the piece in which she says, ” …may look different but act similar…or act different but look similar….”

How many connections to factors and products, addends and sum and such ring true in this statement?? I love when they leave me with something to think about!!!

Another great day in math!

-Kristin

Connecting Whole Number Operations to Decimal Operations

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I planned today’s number talk to draw out a variety of strategies for students to reflect on as they worked through their decimal work today. I used this series of problems:

4 x 18, 12 x 18, 39 x 18

After a variety of strategies such as partial products, area model, double/half, triple/third, friendly numbers, and adjusting a factor and product I paused when I posed the final problem and asked them to estimate. Thumbs went up right away and I go predominately two answers, 800 (from 40 x 20) and 720 (from 4 x 18 x 10). We discussed if it was going to be more or less than the actual answer and then we finished with a student subtracting 18 from 720 and arriving at 702.

While they were still on the carpet, I told them to be thinking of all of these strategies as they were going through their work today because we would be reflecting on them later. I posed the following problem and asked them to think about an equation and answer, “Bob is running 7 miles each day for two days, how many miles did he run.” I got 7 x 2 = 14 and then 2 x 7 = 14 because “it is two groups of 7 miles.” It was nice when a student said the commutative property makes that not matter for the answer. So I followed with, “What would it look like if he ran 0.7 mile each day? More or less than 14?” They said less because it is much smaller and we wrote 2 x 0.7 = 1.4.

They went back to their tables and I asked them to think about how we could represent these two equations on a number line. My thought was that it would give them a visual of the size (magnitude) of the jump and help in determining reasonableness. Eh, mistake on my part…I forced that number line on them and, while it was fairly easy for most, some really struggled. The upside was, it was a great formative assessment for me to see how students approach number lines (many like putting the 1/2 in the middle, yet had 14 on the end and were confused). We used number lines a lot in our fraction unit and definitely went past a whole on them, however I guess I did not really make that connection to fraction multiplication on a number line. Mental note for future work:)

After we had our number lines up on board and talked about determining reasonableness based on the factors. I posed this, “Let’s say Bob ran 2.8 miles a day for 8 days. What would be a reasonable estimate for his total miles.” They talked at tables, came back with 24, 20-24, and 17. We talked about the actual being more or less than each estimate.
Fabulous, now I want you to show how you could find the actual answer as many ways as you can.”

They went around to the different tables and talked about which strategies they had that were the same, one they maybe had not thought about and then which one they connected most with. After they finished walking around, I had them reflect on any of those questions in their journals…

Aww….”a bad number you can round to a happy number.” 🙂

I love this student picked this strategy up from another group!

“The distribution property”:)

Can you tell they did not take to that number line at all? Not one number line. I also anticipated some fraction work, but they were really working with the decimal in connecting with whole number multiplication. It was a really fun day of math!

-Kristin

Choral Counting – Decimals

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The first day back to school after NCSM/NCTM is definitely an exciting one! I was excited to see my students, hear and see them doing math again, and incorporate the amazing things I learned at the conference with them. It is always great when I can go to a session, regardless of the grade level focus, and be curious how my students would engage in the activity. For example, I went to an amazing session on counting by Kassia (kassiaowedekind), Elham (@ekazemi) and Allison (@allisonhintz124). While the session focused on whole numbers, I began thinking about how I could take this same practice of Choral Counting and use it in my classroom. I have to admit, while my first thought was what my students would think about during this activity, I also had my own curiosities in the teacher organization of the work. Does writing them horizontally vs vertically bring out different noticings or patterns? or How does how many I put in each row or column affect their thinking about it?

Luckily, we are currently working on multiplication of decimals and I thought this would fit in just perfectly. I did some brainstorming and decided for my first class we were going to choral count by 0.3, record horizontally and vertically and have 1/2 of the class focus on the horizontal while the other 1/2 focused on the vertical. I was curious to see if they saw different patterns emerge. I started at 1.5 because I wanted a number that would hit a whole a couple times in our round but not make the “10” of them makes a whole number so obvious.

I did find that many of the same patterns emerged, however it definitely looked more intuitive for the students to look for patterns in the direction they had recorded.

Here are a few students who used the vertical recording…

You can see this student first noticed the 1.5 going up and down each column. She then noticed a diagonal pattern and could place the 9 where it would go had we continued.

This student started with thinking about them as whole number by multiplying them by 10. I love the last noticing because it makes such a beautiful connection to his first statement. When I asked him to clarify his thinking he did stipulate that you had to start at 3 for that to be true.

There is a beautiful statement in here that says she knew 0.3 is 10% of 3 because between each whole number there are ten 0.3’s. Lovely.

Here are some examples of horizontal, again, many of the same patterns…

This one was not so much focused on the patterns of numbers increasing or decreasing, but instead found that if you switched the whole number and tenths, the number would also be hit by a multiple of 0.3. Interesting to figure out why that works and when that doesn’t work. They left REALLY excited to keep working on this one. How much do I love the “I thought of this!” next to it!

I asked one student, who seemed content with his noticings before they shared as a table if he could think of any equations that matched the number patterns he saw while he waited.

I asked him where he saw the last one in the numbers and had him record it in Educreations: https://www.educreations.com/lesson/view/multiplication-decimals/31049585/

When the second class came in, I decided to switch up the number in each row to five (thanks Elham for that suggestion) to see if differences came out. Here was our board:

i definitely like the 5 in each row better than the 6, a lot more patterns emerged, quickly. It pretty much screamed patterns! We shared them all and I asked each table to pick one they wanted to explore deeper and figure out why it was happening.

This student said, “If you pick any number, go up and then over two the tenths digit will be one more than the starting number. It also works if you go down and then over two.” He explored that one here:

It was a wonderful first day back! My students and I really enjoyed the choral count (although they all spelled it coral:)! It was a very safe feeling knowing they were all saying it together, a bit different than the counting around the class.

-Kristin