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.
The 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.
This student did a beautiful job of adding the decimals by place value and writing a description of the process. This 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.
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: Although 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.