Tag Archives: Adding 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.

-Kristin

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.

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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!

-Kristin