# 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…

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

# 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…

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

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

# Subtraction Is So Conceptual

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.

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.

To 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!

-Kristin

# 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.

-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.

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.

Starting some decimal addition number talks tomorrow, excited!

-Kristin

# Fraction to Decimal Division Table Yr 2

After this lesson from last year: https://mathmindsblog.wordpress.com/2014/03/18/fraction-to-decimal-division-table-noticings/ 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!

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….

-Kristin

# 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: https://mathmindsblog.wordpress.com/2014/03/18/fraction-to-decimal-division-table-noticings/. 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.

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).

I 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.

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).

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

From 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?

-Kristin

# 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.

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!

-Kristin

# Unanticipated Student Work…Always a Fun Reflection!

As I was planning for a summer PD, “Decimal Fluency Built on Conceptual Understanding”, I was going through pictures of my students’ work. I focused on the very first multiplication problem I had presented to them in which both numbers were less than a whole. I presented them with 0.2 x 0.4 and asked them to do a “Notice/Wonder” and think about the product. I had anticipated some may reason using fraction equivalents, some may know that .4 is close to half and take half of .2, and some may try fraction bars or arrays to solve. Here are samples of their initial work….

As I circulated the room, the two products that showed up were 0.8 and 0.08, as I anticipated. I put them on the board and had the students work through it as a group and try to prove the product they thought was correct and disprove the one they thought was incorrect (I did not tell them at this point, that was their job!:)

During the share out, this is the one response I did not anticipate at all and now, going back, I wish I had spent more time with…grrr….darn hindsight!

For all of the nerdy math peeps, like me, who like to “figure things out” I am going to leave out her explanation here! I will gladly recap it for anyone who would like to hear it in the comments or via twitter!

Needless to say it left many students a little baffled, and we did revisit it the next day for her to re-explain her reasoning. I just wish I had extended this by asking students if this model would work for any two decimals less than one whole? Why does it work with .2 of .4?

I highly recommend snapping pictures of your students’ work all year long because reflecting back on this work over the summer has taught me a lot about anticipating student responses and how to handle those responses you just don’t expect! It also just makes me smile at the way my students reasoned about the math we were doing!

-Kristin

Here are a few pictures of the follow up group work and Gallery Walk they did with 0.5 x 0.3….