Category Archives: Decimals

Decimal Addition Card Sort

When solving problems in Number Talks, the strategies, en route to the solution, are the focus of the discussion. However, not all problems posed during a Number Talk are created equal or solved the same way every time by every student. While I know the majority of students use a particular strategy for one reason or another, whether it be because of the numbers involved or maybe it is the only strategy they are comfortable using, I like to take time and make these choices explicit. I want the students to think about the numbers before just computing and become more metacognitive about their actions.

Last week, the 5th grade teachers and I planned for a card sort to get at just this. The students have been adding decimals and using some great strategies, but we really wanted to hear about the choices they were making. With the help of the Making Number Talks Matter book, we chose problem types for students to think relationally between whole number and decimal operations. While there are no right or wrong answers, these are the cards developed by 5th grade teacher, Eric, and some strategies we thought went along with each. The expectation was not to have the students solving it the way we had listed, but to hear, and have other students hear, the choices being made.

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We gave partners the cards, they sorted, and named the categories whatever they wanted:

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While the card sort conversations were really interesting, the class discussion afterwards was amazing! There were so much questioning of one another about how one strategy is any different than another. For example, some groups used rounding for a problem that another group used compensation and another grouped called it using friendly numbers…so groups had the same problem in three differently-named categories. Again, the category was not important, but more the fact they were actually thinking about the numbers they were given.

The other 5th grade teacher and I are planning to do the same activity with multiplication when they get there! Excited!

 

Fraction & Decimal Number Lines

Recently, our 3rd, 4th, and 5th grade teachers had the opportunity to chat math for 2 hours during a Learning Lab held on a professional development day. It was the first time we had done a vertical lab and it felt like perfect timing as 3rd and 4th grade would soon be starting their fraction unit and 5th would be entering their decimal unit. Prior to the meeting, we read the NCTM article, “Identify Fractions and Decimals on a Number Line” by Meghan Shaughnessy, so we started the meeting discussing ideas in the article. We then jumped into playing around with clothesline number lines and double number lines, discussing what they could look like at each grade level based on where students are in the fractional thinking.

I have co-taught number line lessons in both 5th grade and Kindergarten this year, but both were a bit different in not only number, but organization. In 5th grade we used one clothesline with the whole class, while in Kindergarten we used tape on the floor and students worked in small groups. Leigh, 5th grade teacher, was interested in trying the small group number lines on the floor. As we planned the lesson, the one thing we thought would be difficult about having small groups is getting around to each group to hear their conversations, especially when we were planning cards purposefully to address misconceptions and misunderstandings. However, knowing we would have the two of us circulating, as well as two 3rd grade teachers who wanted to see the lesson (yeah!), we knew we had plenty of eyes and ears around the room to hear the math conversations.

During the lesson, in groups of three, students placed 24 cards on their number line. There were two sets of cards, so after placing all of their cards, each group visited a number line with a different set of cards to discuss. Instead of boring you with all of the number choices we made, here are a few of the choices in cards and the reason(s) we chose them:

1/3 and .3: Students often think these two are equivalent so before the decimal unit we were curious to see how they were thinking around that idea and how they used what they knew about fractions or percents to reason about it.

0.3, 0.33, 0.333, 1/3: The 1/3 and .333 were there to think about equivalency, while the others were there to think about what is the same in each and how much more each decimal has to make it larger. Which you can see caused some confusion here:

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2.01 and 2.08: We were curious about the distance they put between these two cards.

1 6/9 and 1.6: We wanted to see how students compared fractions and decimals when they can’t easily convert 6/9 into a decimal. Then, if they began thinking 6/9 and 6/10, how would they decide on the larger fraction and then how much distance do they put between them?

The group below practically had them on top of one another:

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While this group had a bit of a space between them:

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2.8 and 2.80: Tenths and hundredths equivalency. They all seemed to handle this with ease.

.005 and 1/100: Curious to see the placement in relation to the other numbers. This 1/100 is close to 0 but I wonder about it in relation to the .2. Definitely a conversation worth having!

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2.8 and 2 7/8: To see how they compared the 7/8 to the 8/10.

After they visited other number lines, they had a chance to meet with that group and discuss card placements they agreed with and placements they did not. Groups then made adjustments accordingly…

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Here was a group’s completed number line and my first stab at panoramic on my phone!

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The journal entry we left them with was, “Which cards were the most difficult to place on the number line? Why?”  Many were just as we suspected.

 

The conversation as I walked back over to the other building with the 3rd grade teachers was, what does this look like in 3rd grade? Could we use array images to place on the line instead of the fractions? Could the pictures include over 1 whole? What whole numbers would we use? Do we play with equivalent pictures with different partitioning? Being mindful of the students’ second grade fraction exposure, below, we are planning on trying out something very soon! I am thinking the cards like these on Illustrative, with the pictures but no fraction names at this point.

CCSS.MATH.CONTENT.2.G.A.2
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
CCSS.MATH.CONTENT.2.G.A.3
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
…and then could the journal could be, “Can you name any of the fractions on your number line? How do you know?”…or something like that!

 

Remainders: Division & The School Year

Looking ahead in planning for the remainder of the school year, I am currently finishing up my decimal unit and excited to end the year with Growth Patterns. I was planning on finishing decimals this week, however, I have one more thing that I feel is missing from the unit that I am curious to see how students are thinking around it. In the unit, within decimal division, the students are very comfortable estimating quotients and thinking about a variety of strategies in finding how many of the divisor are in the dividend. However, one thing that is not addressed is remainders, and how we notate them. I had not really thought much about it because in the context of the problems we were doing, the remainder made sense. However, during a division number talk, not within a context, the “r” was still there. It bothered me a bit. When I asked how they could write the quotient as a number, I got blank stares. I know fourth grade really spends a lot of time on interpreting remainders, but do we spend equal time on various notations of the quotient?

I have decided to extend my decimal unit just a bit longer because I feel this is something my students can definitely reason about and I am curious the connections they can make between whole numbers, fractions, and decimals. I decided to start with whole number divisors and dividends and move to decimals from there. Today, I gave them the problem 256 ÷ 20. They estimated somewhere just over ten and then I asked them to solve it. If they finished early, I asked them to write a context to match the problem.

The majority of the class’ work looked like these and contexts involved a sharing situation…

IMG_0590_2IMG_0594_2When pushed to write their quotient as a number without the “r,” most said this…(I do love the way this student divided:)

IMG_0585I did get a few 12.8 and 12 16/20, which interestingly fell more in money contexts…

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All of these, I had anticipated, but then I got some really great unexpected answers that allowed students to think about the connections between notations…

12.5 r 6         12 16/256         12.75 r 1

I wrote these responses on the board and asked the students to see if any of the answers meant the same as 12.8 or 12 16/20 or 12 r 16, that we had established were the same. They did also mention, which I loved, that certain situations my use different notations.

I had some amazing proofs that we are kicking off the day with tomorrow before moving into decimal divisors. While I was hoping for students to look for equivalencies in the quotients themselves, most groups went back to trying out division in a different way to prove the answers. This group went back and solved the problem using the same method every time, just changing the breakdown of the quotient.

IMG_0591_2This group used multiplying up to see that 12.5 r 6 worked as a correct answer. IMG_0588_2

After asking them if they saw any relationship between the quotients, I got this…(much more what I was hoping to see in their reasoning)

IMG_0593_2This group had a nice, simple explanation at the bottom of this page…

IMG_0584This student is still sticking with 12 16/256 and quite honestly I don’t know how to approach this one. It is a different way of writing the remainder and I cannot decide if there is a time when this would be an appropriate notation?

IMG_0587_2The most perplexing quotient for most of the students was the 12.75 r 1 so I asked the student to write out his thought process because he was having trouble explaining it.

IMG_0581_2Now, while the entire class period seemed to focus on the remainder in a division problem, this explanation represents the remainder of the school year! I asked the above student to go in the hallway and record his thinking through the problem because he had such a beautiful way of starting to explain how he decided how much to add based on the distance from the dividend…but then I got this….heehee: https://www.educreations.com/lesson/view/kewl-aid/31841872/

And here’s to the remainder of the school year….

-Kristin

Creating Contexts for Decimal Operations

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…

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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.

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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.

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

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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.”

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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….

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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?

IMG_0345His 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

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.

Cliff’s note 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.

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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!

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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

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…

IMG_0263 IMG_0259I 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.

IMG_0262We 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.

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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

Investigating Patterns

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…

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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…

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

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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…

IMG_0205Other groups used a variety of patterns, noticing that some would work nicely for certain decimals and not others…

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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…IMG_0210He 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…

IMG_0196She 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…

IMG_0195It 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