Category Archives: Decimals

Attacking The Telephone Game in Math

I think we have all been there (or maybe it is wishful thinking that I am not alone:)…

The class is sharing strategies for solving a problem and all of a sudden, one student explains his/her “shortcut” or algorithm to the class. It’s true, it works, and you know you will get there, but my first thought is always “Do they know why?  while my immediate second thought is, Oh no, now this will look faster to some of my students who will quickly grab onto it to save themselves some time, not caring why it even works.

I deal with this in many ways and it really depends on the situation. Is there time to go deeper into that idea at this point? Is it something that arises later and this could just be a nice “Interesting, that is something to think about”? Is it something that will lose more than half of the class and can be addressed with that student later to gauge understanding? Or is it something that will end up as a version of the Telephone Game?

If you have not had the pleasure of playing the Telephone Game, it goes something like this: One person whispers a sentence to another person, that person whispers the same thing (or as close as they can remember) to the next person in line, so on and so forth until it arrives back to the first person. Typically, when the initial sentence makes it to the last person, it is not the same.

This time, a decimal addition and subtraction strategy has fallen victim to the Telephone Game. During one of our number talks a week or so ago, a student, let’s call her Jane, mentioned that she just “adds and subtracts the numbers as if the decimals are not there and then puts the decimal back into her answer.” She explained it for the problem we were currently working on and we moved on. I made the decision to revisit this with her later in the class period to clarify her thinking and not make it a class discussion at this point. I knew in our upcoming lessons we would get to this “decimal movement” when we started multiplying decimals by powers of 10 and I thought this would be a perfect example to bring back up when we got into that lesson.

Evidently, I waited too long.

Over the course of a couple of days playing Fill Two, Empty Two, and Closest to 1, I started to hear a buzz in many students’ explanations that sounded much like Jane’s. However, it seemed as this idea made its way around my classroom, some very important pieces were missing. Some students were not taking place value into account while many others were losing any concept of sense-making about their answer. For example when adding 3.6 + 2.24, some were adding 36 + 224, arriving at 260 and having no idea where would be a sensible place for the decimal. While Jane was correct, and she understood she needed to put the numbers in the same place value, this part of her reasoning was lost in the Game. Not to mention the “why.”

Hmmm, now, what to do? I didn’t want to explain Jane’s process without going into what is actually happening to the addends when the decimal is “removed” and then “put it back in”, but I questioned whether I would be jumping too far, too quickly for some when we were just getting a handle on adding and subtracting decimals. I hate to ignore ideas or make them feel unimportant when I feel they truly are. I made the decision in my planning yesterday to attack this telephone game head on.

After our number talk today, I typed Jane’s claim on the SMARTBoard and asked the tables to prove if it was always true, and if it was, why can we do that?

IMG_9740Many students started “testing” a lot of problems to see if it worked every time. IMG_9747_2 IMG_9746_2 IMG_9748_2Some tested and tested but struggled with Jane’s “putting the decimal back in.” YEAH, because now we can talk about estimating and reasonableness!

IMG_9741_2 IMG_9744_2Others re-emphasized the point of the same place values combining. This will be a nice discussion of how the base ten system works when combining place values, ie, ten in a place will always make one of the place to its left.

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Two tables did start to talk about the addends multiplying by 10 and/or 100 and then dividing the answer by the same to adjust it. One table jotted down some work, but the others were still in the discussion phase. They did say 2.50 x 100 gives you 250. I asked how they knew that and they said that 2 x 100= 200 and 0.50 x 100 = 50 so it has to be 250.

IMG_9745_2Tomorrow’s conversation will be very interesting! I have so many thoughts about my goals for the convo, but here are my initial thoughts….

1- Reasonableness is SO important, estimate, estimate, estimate!

2 – How adding like place values acts similar across all place values.

3 – What is happening when we “take out the decimal”

4 – How adjusting the addends in the same way affects the sum. Really the bigger generalization for any addition problem.


Subtraction Is So Conceptual

Every year, across all grade levels, I hear (and observe) subtraction being a difficult concept for students. Not just a difficult calculation, but concept. I am not talking about reading a context and knowing if subtraction could be a way to solve it, but instead, what is happening when you subtract and how does a change in the subtrahend and minuend impact the difference? I think students can learn a procedure to “operate” with subtraction (as with any operation), but I always question the conceptual understanding behind their work. I also think that we, as teachers, sometimes make some assumptions about student understanding of subtraction when all of their answers are coming out correctly. It feels really nice to see students read a task and solve it correctly with subtraction, but have they thought about whether the answer makes sense or could they explain what would happen to the answer if I increased or decreased one of the numbers in the problem? This could completely be my own wondering because, I admit,  I tend to question a lot of my students’ understandings until I hear them talking about the idea or working through it in their journals. To get a better understanding of their thinking and attempt to help them move forward in their thinking, I do Number Talks a lot and most recently have really started to listen and think more about what makes subtraction so difficult for them.

I have a few ideas based on my observations of the students’ conversations and many lie in the fact that we do much relational thinking about addition and subtraction that students assume that the numbers operate in the same manner.

1- Commutativity. When adding, it is so convenient that you could add the tens and ones in either order and still end with the same answer. For example, when adding 34 + 63 I could add (30+60)+(3+4) and still result in the same answer. Even if it changed the context of the problem, it would still result in the correct answer. Whereas, with subtraction if I was subtracting 63-34, I can’t just do (60-30) + (4-3). It now creates a different problem but it is something that students do ALL of the time in order to take a smaller number from a larger one. Which is what I see happening here with the quick subtraction problem I gave students to solve last week before we started looking deeper into decimal subtraction.I just wanted to get a look at what they were thinking, as was not surprised to see this on many papers.


2 – Number Adjustments and the Effect on the Context. This comes out A LOT in our talks. When they are adding, they love to compensate and adjust the addends to make an easier problem. For example, 49 + 33, students would take one from the 33 to give to the 49 to make an easier problem of 50 + 32. Again, it would change the context of the problem they were solving, however not impact the result. Now given 49 – 33, giving 1 to the 49 from the 33 leaves you with 50 – 32 and completely changes the context. Given a removal problem, you are starting with more, but taking away less. Or given a distance problem, you have moved the starting and ending point in opposite directions. There is SO much context in a subtraction problem in just the number adjustments themselves.

3 – Number Adjustments and the Effect on the Outcome. When adding, students understand how adjusting one of the addends affects the solution. If I add one more to this addend it increases the sum by one or if I decrease both addends by 1, the sum will decrease by 2. Again, the context can come into play here, but the students get pretty comfortable with the numbers, stripped of context, in understanding this. Now, subtraction is not so nice in that way. Again, context is SO important. 34 – 12 = 22. If I take one from the 34, making the problem 33 – 12 = 21, it works in the way the students know addition works. However, taking 1 from the 12, making the problem 34 – 11 = 23, it does not. They are so perplexed when they try this and it instead adds to the original difference.

Now, because students do not feel as comfortable with subtraction, I also see less willingness to reach outside of the standard algorithm once they “get it to work”. I appreciate the use of the algorithm, however after this quick formative, I had the feeling that there was some conceptual understanding missing that would really impact our decimal work. Because of this, I decided to start with an Investigations story problem on our grid paper.

“Mercedes had 1.86 grams of gold. She used 0.73 gram of it in a piece of jewelry. How much gold does she have left?

I asked them what this story would look like on grids and I got quite a variety of thoughts but I was very surprised to see students putting all three numbers (the two in the problem and the difference) on three separate grids.

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I did have a quick realization of the difference between “Show this problem on the grids” and “Show how this story looks on the grids?”

These showed the STORY….

This student taped the removed part over top of what she had, to leave the answer in purple:


This student set the whole aside because she knew she didn’t need to touch it and dealt with the hundredths.


These involved some taking away of pieces to leave them with the answer.


This student changed the whole to be the tenth, but represented each number in the equation.

IMG_9667To see if they made a connection between what they had done on their grids to the solution process, I asked them to solve it in their journal the way they would have just given the problem (again, most with the algorithm) and then tell if it was similar to what they did on their grids. Many struggled to see any similarities which surprised me, especially with the way some took away the tenths and hundredths on the grids.



This was so interesting to me especially when I saw so many correct answers in their journals but when asked to explain, it was tough! Subtraction is tough…for students and adults. Not the calculation so much, but the concept of what is happening. It is so conceptual and really hard to break away from methods we know that work for us to truly understand the meaning behind them! I know I still have to think harder about subtraction then I do addition, so I want to make it clearer for my students.

So much to think about and I am sure I have so much to learn about subtraction and connecting representations to their thinking, but this is a stepping stone along the way!


Decimal Subtraction

We are doing subtraction work tomorrow in class, so to better prepare myself, I asked the students last week to solve just one decimal subtraction problem to give me an idea of where we were starting. I told the students to solve it as many ways as they could and I got a bit of a range, however most went to the subtraction algorithm. This could be because of my number choice or just their comfort zone. I am posting these here to revisit soon, but after much time reviewing them, I don’t have the time tonight to write about each. I will follow up with a blog post which I am sure will be interesting because subtraction always seems to be.

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What Are They Really Thinking About Decimals?

Understanding student thinking is so hard. I make assumptions. I read into things. I SO want to believe there is understanding behind everything they write on their papers. However, it is so much more difficult than that and my most recent difficulty is addition of decimals.

We have talked about decimals in one frames, shaded grids, and I am confident that every student can compare decimals with understanding of place value and magnitude. They understand decimals independently. Then, enter decimal addition. What is it about computation that sends students right back to not thinking about the numbers themselves and straight back to “lining them up” and adding? I know it is not that they CAN’T think about the numbers, so then my wheels start turning…. is it just ease of use? Great. But is it ease of use with understanding? Or is it ease of use without understanding but just gets them the right answer? This is where teaching is so hard!

We do number talks at least 2-3 times per week and given a problem such as 38 + 47, the majority of the students would say 40 + 45 = 85 using a compensation strategy. Today, given 6.8 + 4.7, I got “I lined them up and added 8 and 7 and got 15, carried the one…” You can hear the rest. Wait, what? Where are the tenths? Where is the place value? Why didn’t I ask them to give me an estimate first (ugh, hindsight)? I ask for any other strategies, nothing emerges. I am left to wonder what they truly understand about addition of decimals. Is it the decimal place value that takes away from thinking about the numbers or is it simply that they see how the decimals operate like whole numbers in a base ten sense. After doing a contextual task the day before, with pencil and paper, I was excited by the outcome, there were numerous strategies. However, if pushed to solve mentally, the students reverted back to an algorithmic feel. I am not saying that it means the students do not understand the place values they are adding, but trying to bring to light how hard it is to interpret their understanding on my part.

I then gave them a problem involving three decimals and asked them to solve it two ways. I was trying to get a better feel of their understanding. The two ways would push those “liner-uppers” to work with the decimals in a different way and also allow me the time to walk around and question students about their work.  I was not shocked to see that the majority went to lining them up as their first strategy, however I was very excited by their second strategy that showed more understanding of place value.

Here are some examples of the students solving 0.98 + 0.05 + 1.06


I loved the number line in the first example and the breaking apart of the 0.05 in the second example. I was starting to see the flexibility and thinking that I want to see in my students.

I gave my second class a different problem involving two decimals, both in the hundredths that were not as “friendly” as the decimals in the previous example. I was happy to see the variety of strategies, including my student who starts assigning letters for each digit. He said he is ready to start doing some algebra 🙂  I love it!

IMG_9607_2IMG_9605I had a few who finished fairly quickly, so I gave them the problem 0.8 + 0.75 + 0.625, and then they started getting creative! This is one answer that was so interesting and will be the way I kick off my class tomorrow. His reference to columns and boxes are the hundredths grids we used for the Fill Two game.

IMG_9606Today was a day that really showed me how hard it is to understand student thinking and how important it is to push students to explain their understanding in more than one way. I could have very easily assumed that every student could add decimals by place value because they lined them up and added to get a correct answer.  However, if not given the opportunity to show another way to think about the problem, how would I truly know? I still have a few students who are getting the correct answer but are not able to articulate their process, so I am going to do a lot more estimating to get them thinking about the numbers before the operation. Going to be a fun day in math tomorrow!


Adding Decimals to the Thousandths

Yesterday, the students played a game called Fill 2 (shown below). Building on that game, today the students worked on their first task involving addition of decimals to the thousandths.

fill2The task involved a jeweler who, after making jewelry each day had pieces of gold leftover. One day she was left with 0.3 g, 1.14 g, 0.085 g and the students were asked how much gold she had left that day. I gave the students some individual time to come up with at least one way to solve this problem before they came together as a group. As always, it is so interesting that even in coming up with the same answer, there were such different approaches to the solution.

This student is interesting because she changed the decimals to thousandths in fraction notation. It definitely is her comfort zone and the conversation with her group when first comparing answers was great for them her to agree that it was equivalent to the decimal notation.


This is probably the most common approach I saw. This student put all of the decimals into thousandths and added. It was nice to see they combined the correct place values, however this is the reason I have them come together as a group. I cannot tell from this work if the student understands the combining of place values or just has learned a procedure for adding decimals (line up the decimals, put the zeros on, and add). I do, however like the written explanation of putting them into thousandths, which does indicate an understanding beyond “putting the zeros on” to add.

IMG_9510This student did a beautiful job of adding the decimals by place value and writing a description of the process. IMG_9509This one is lovely because of all of the messy work and “notes” to me:) When I walked up to her table, she was thinking about the first two decimals in terms of hundredths (in fraction form), but was struggling with the 0.085. She had written it as 85/1000 but then rounded it to 9 to add with the others, but was getting lost in the meaning of the numbers.  She couldn’t pull the numbers out of place value so well to operate with them and put them back in, but instead was struggling.  She was great in her fractions, but her notation then seemed to bounce between whole numbers and decimals. This felt like the SMP of being able to contextualize and decontextualize. I asked her to talk to me in terms of hundredths and she had no problem saying that it was 30, 14 (she had put the 1 aside) and then 8 1/2. She wasn’t comfortable putting that into fraction form, so she rounded it. After she said 8 1/2/ 100 to me, I asked her to work with that and left her to think. When I popped back into their group, she was sharing her 52 1/2 / 100 with the group and how she translated that into 1.525.

IMG_9508 The groups then came together, agreed upon an answer and then put their strategies on a chart. After each table had finished, I had them go around to each table and jot down any strategies their group had not come up with. Here are a couple of the posters:IMG_9495_2 IMG_9498_2Although the “American Algorithm” takes a lot of my attention here because I find it so cute, the bottom of the page is really an interesting visual of the students’ thinking. The decimal numbers are not in orderly rows which really shows that they were truly thinking about how many tenths, hundredths, and thousandths they had in each number. I think the arrow from the hundredths to the tenths shows nicely how ten hundredths make a tenth. The best part of this was the connection to the algorithm above. It clearly shows why there are two hundredths and 5 tenths.

IMG_9497_2Starting some decimal addition number talks tomorrow, excited!


Fraction to Decimal Division Table Yr 2

After this lesson from last year: a lot of the same patterns emerged from the students. There is, however, one fraction that still drives them crazy…the 11ths.

Here it is showing up on two of the students’ papers…you can tell the 11ths are a thorn in their side!

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The best part of this lesson was the work I found after the lesson was over. They were working on it any free moment they could find in the day! This has become a genuine curiosity for them and I love it! They are still working, but I could not help but laugh at the heading of their work:


I will keep you posted on their findings….


Fractions as Division…Say What?

Last year I learned to appreciate the Investigations lesson in which students explore fractions as division in a Division Table: However, as I was planning this year, I thought I really missed the mark in making it an explicit understanding that fractions represent division before exploring this table. I think I completely just assumed that students understood this from previous years and investigations with sharing situations involving fractional answers. I decided to check it out this year to see what they students knew/understood before beginning the division table work. I thought it could make some really nice connections evident.

I started by putting a few sharing problem on the board: 6 subs shared by 4 people, 9 subs shared by 4 people, 3 subs shared by 5 people, and 6 subs shared by 9 people. I asked how much each person would get if they shared the subs equally.  I gave the students some individual time to work through the problems and, after that, an opportunity to share their answers and strategies with their group.  In the majority of the class, I saw the work I had anticipated based on their third grade brownie sharing work in Investigations. A lot of drawing of subs, people, and “passing out” of the pieces.

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One student thought about the whole being the number of subs, divided it into the number of people sharing and arrived at decimals, however struggled when he got to the 6 shared by 9. (The side written piece is after I asked them to write what they noticed and then he proved it worked with fractional subs to start).

IMG_9375I had a few students that provided the perfect transition between the visual drawings and the fraction being division. They intuitively wrote the problem as a division problem and solved it using what they know about multiplication. After sharing some of the visual representations, I had these students share their equations. They explained to the class that is felt like division because they were dividing it up among people.

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After our sharing, I revisited the original problems, with the students proven answers, and ask them to write what they noticed about the problems. After a few moments, I heard so many “Oh My Gosh”s and “It was really that easy”s echoing about the room. One student exclaimed, “Why did I do all of that work?” pointing to his beautiful sub and people drawings.

Here are some of their noticings (I love that they automatically start proving it to see if will always work without me even asking anymore).

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This one just absolutely cracked my up and proved once again that I cannot make assumptions about student understandings….

IMG_9372From this point, we tested out a bunch, talked about why it will always work and then starting looking at representing our “benchmark” answers as decimals. Tomorrow, I feel great knowing we will start looking into the division table with a deeper understanding of fractions as division. The word “explicit” sometimes makes me cringe in the way of “telling” students things, however I feel in this case the understanding of fractions as division was made explicit to the students through their own work group sharing and noticing today. I think that may be the piece I have missed before… I assumed they knew and could arrive at an answer, however never made the idea explicit as a whole group.

Today was a great day in math…Say What?


Comparing and Ordering Decimals

It is always so interesting to me what students take away in terms of strategies for doing various tasks in math class. In this particular case, ordering and comparing decimals. We all did the same shading activities, played the same comparing games, however the way this shading is applied to student thinking is so different among the students in the class.  In our assessment today I saw quite a variety in thinking that I just love.

These three are a sample of the most common strategy I saw in the work today. The students first thought about it in terms of how many tenths each decimal had. We talked about this a lot while shading in terms of full tenths, partial columns for hundredths and then parts of hundredths for thousandths so it makes sense that they would think about which decimal had the most tenths shaded first and move on from there.

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This student has a comfort in fractions and changed each to a fraction in the thousandths. It is interesting that certain students like to stay in fractions, where she could have just as easily made them decimals to the thousandths.


This student explains using the hundredths grid in words. I love the use of the word blocks and 1/2 blocks. I just want to check back in on this one to see if there are connections to that 1/2 block representing 5 thousandths.


“I thought about it like whole numbers.” This is something I would be extremely worried about if she had ordered them like: .6, 0.8, .55, .125, .875 because then the decimal would have been irrelevant in their reasoning in terms of “whole numbers”. She really multiplied each by 1000, which is something I would like to revisit tomorrow with her.


I also had quite a few that compare using percents. This is a nice connection back to our fraction/percent work on the 100s grid earlier in the year.


This was a nice way to ease back in after an extended weekend of snow and ice!


The “One Frame”

I love this introduction of decimal addition so much from last year, that I had to relive it again:  It was just as amazing this time!

I opened with the same discussion about the ten frame, why we call it a ten frame, and then changed it to a one frame. We discussed the value of each box and were on our way. This year, I really pushed the students more into the equations that matched the frame on the board. We did .9 as a group in a number talk setting with a lot of revoicing and restating to be sure the students could explain how their equations matched the one frame image. I then put up a frame showing 0.7 ( four tenths on the top row and 3 tenths on the bottom row) and sent them to their journals to write some equations by themselves before sharing out. Here are some examples… (Some went crazy:) I think it is so interesting that without any formal work with decimal multiplication, students intuitively can see that any number of groups of some tenths can be written as multiplication.

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The one below was so interesting when he said, “.35 x 2” I asked him how that matched the picture and he said, “..since I like symmetry, I took the fourth dot on the top row, split it in half, and put the other half on the bottom row.” I asked the class how that gave him .35 and another student explained that because half of a tenth was 5 hundredth, it became .35 on each row. YES!

IMG_9275 - Version 2I think put up two frames, one with .9 and the second with .3 and asked students to write down how much was represented in the picture. Like last year, it was a mix of 1.2, 12/20 and .12. I asked students to prove the one they got as their answer and then explain where they think someone got confused with one of the answers they do not agree with. They did a beautiful job with this.

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It was so nice to kick off our adding decimals with students identifying what the whole it, looking at decomposing numbers, being aware of place value and reasoning about what makes sense. I am SO looking forward to the rest of this work!


Which One Doesn’t Belong?

After all of the interesting conversation around Christopher’s (@trianglemancsd) Shapes Book and a conversation with Faith (@Foizym), I thought it would be fun to take this thought about “Which One Doesn’t Belong” into my students’ decimal work. With these decimals, I wanted to draw out reasonings about closeness to benchmarks, equivalents, and properties of numbers in relation to decimals. It did all of that and more! I wrote the following four decimals on the board and had students talk about which one they thought didn’t belong:

woIn brainstorming these decimals beforehand, I knew that 0.49 would be the most obvious because it is the only one that went into the hundredths, so I go that out of the way as the sample response and asked them to see if they could find another reason for 0.49 or any of the other three decimals. They brought out some pretty great stuff and definitely gave me insight into how they think about multiplication of decimals! It was also so nice to hear, as I walked around during their talk, the freedom students felt expressing their ideas when they knew there was no right or wrong answer!