Tag Archives: 5th Grade

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

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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

Attacking The Telephone Game in Math

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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?

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

Others 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.

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.

Tomorrow’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

Commutativity in Fraction Multiplication

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Think about these two expressions…

2/3 x 6 6 x 2/3

Do you think differently about each?

Does your solution approach change?

I had not really given this much thought because we do both in 5th grade, multiply a fraction by a whole number and whole number by a fraction. However, recently, when working with a group of 4th grade teachers and looking more closely at the standards and my curriculum, I am beginning to see a distinct difference. I now look at each expression from a different perspective. Not that both ideas do not arise at multiple grade levels in some form or another, but it is so interesting to me as to which thinking would come before the other.

Let’s first look at the standards…

4th Grade:

5th Grade:

Interesting. For me, taking a fraction of a group feels more “natural” and intuitive than multiplying a whole number by a fraction, however in the learning trajectory of multiplication and building of unit fractions composing a whole, the multiplication of a whole by a fraction feels like the natural next step.

For our upcoming Illustrative Mathematics professional development, I was collecting work samples for the following problem (thanks Jody:)

“Presley is wrapping 6 packages. Each package needs 2/3 of a yard of ribbon. How much ribbon will she use for wrapping the 6 packages?”

As anticipated, I received a wide variety of solutions to arrive at 4 yards of ribbon. Here are just a few examples in what I think is the progression I expect (some of them got finished quickly and opted to show a few ways to solve).

They all finished fairly quickly and as I was walking around I thought it was really interesting to see such a variety in the equations they used to represent the problem. We came together as a whole group and I asked them for the equations they thought best represented the problem. The most common answers were: 2/3 x 6 = 4, 6 x 2/3= 4 and 2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3= 12/3 = 4.

I asked them if there was a difference between the equations and there was a unanimous “No” because they mean the same thing. “They all get 4.” In my head I was very excited that commutativity was something they see when finding a solution, but I was also curious if it worked the same in the opposite direction. I asked if we could narrow it down to two equations and they all agreed that the repeated addition was the same as 6 x 2/3 because it was “six groups of 2/3.” Interesting, so they see that in the numeric representation but not contextually?

I then asked them to write 6 x 2/3 and 2/3 x 6 on the top of their journal page and think about them without the previous context. I posed, “If I gave you these two problems to solve, would you think about them the same way? Do you think about them differently?” I was curious to hear their thoughts on the commutativity.

The conversation after was so great and interesting! There is a difference when going from number to context, however when put in context, I think students use whatever strategy is easiest for them to arrive at the answer. Is this what is truly meant by contextualizing and decontexualizing in the SMPs?

To further intrigue me, I went and pulled a few fourth graders to interview during my planning period. It was so interesting that they saw this as a whole number times a fraction because it was “six 2/3’s.” Their connection to multiplication and “groups of” was evident. I did love how they did 3 of the 2/3s first to get 2 and then doubled that to get 4.

This 4th grader was the most interesting..

She solved it as 2/3 of 6 and arrived at 4. I asked her if she could write an equation for the problem she solved and she wrote 2/3 of 6 = 4. Completely because I am so nosy, I asked her to write 6 x 2/3 under that. I asked how she thought about that problem? Would she solve it the same? She said, “No, that is 6 of the 2/3’s so I have to multiply the 2 and 3 by 6.” She proceeded and ended with 12/18. She saw the numerator and denominator as numbers in and of themselves and used the distributive property to arrive at her answer instead of thinking about the 2/3 as a number. This was something I had never thought of before! I wish I had more time with her because I SO wanted to ask if that makes sense, but since my planning runs into dismissal, she had to get back to class! Argh!

This progression (to me) now seems to be more about building on student’s understanding of multiplication then about what is more intuitive for students to do. That is such a revelation to me. In second and third grade students do so much in “sharing” situations, that I had assumed it was en route to this skill of taking a fraction of a number when in fact it is more about the operations. It builds multiplication and division. Those operations then progress from operations with whole numbers to operations with fractions and from there students start to build deeper understandings of the properties of operations.

This is of course, all my interpretation based on my experiences and perspective of the student work, but how awesome! I cannot wait to share this with the 4th grade teachers along with the video of the kids chatting with me about this, awesome stuff!!

-Kristin

Fraction Number Talks

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Two days a week we have a Math RTI period built into our school schedule. It is 50 minutes in which students receive additional math support through Marilyn Burns’ Do The Math Program, as well as the use of Number Talks. The groups are smaller than the regular core classes, allowing for more individual time with each student. In 5th grade, we focus heavily on the fraction module and building reasoning within the structure of our number system. When we implemented this structure about four years ago, the majority of the students in the more intensive groups had an extreme aversion to fractions and really just a lack of confidence in their ability to do math. They were just looking for a “way to solve” the problem to get it over with, rather than reasoning and working through a problem.

The fraction module, through the use of fraction strips, encourages the students to think about the size of fractional pieces, creates a visual for fraction equivalence and looks at the relationships between fractions. Students use these understandings to compare, add and subtract fractions and most importantly build their confidence in their ability to do math. The Number Talks I do with fractions really focus on getting the students to THINK about the fractions before just operating left to right and looking for a common denominator each time. This week I was doing a number talk on adding fractions with my group and put up this problem: 3/4 + 5/10 + 1/8 + 2/16. My thought in choosing the problem was there was some great decomposition and equivalence that could happen.

We usually do these problems mentally, so I don’t typically give them white boards but since I really wanted to see their thinking, I did this time (and I am so glad). Seven students came up with six different answers. It was awesome. I had them lay their boards down and look at them all before they started to explain their strategy. It was all of the great decomposition, equivalence, and addition I was hoping it would be. I especially love 3/4 + 5/10 = 1 1/4 and the bottom left where the student rewrote 5/10 as 4/8 + 1/8 to add to the 6/8.

I started to hear a lot of “Oh”‘s and “They are the same”‘s but the student who got 24/16 thought she was wrong because hers “looked different.” They all agreed the others were equivalent but I asked them to explain to their strategy and discuss the 24/16.

It was such a great discussion and as I was listening to them, I wondered how in the world any teacher could ever want to teach a group of students how to solve problems in only one way when there is such rich conversation in their individual thinking. They loved matching their answer to the others and proving how it was the same. Not to mention the confidence, independence and reassurance in their own math ability when they arrived at the correct answer.

-Kristin

Decimal Subtraction

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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

Flexibility, Efficiency or Starting From Scratch?

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I ask myself this question numerous times during the course of school week. During number talks and in class conversations, the students show such amazing thinking and strategies in solving various computation problems. But, just when I think they are constantly thinking about the numbers, their values and sense-making, they seem to start a new problem from scratch without connecting to any of their prior reasonings. Is it flexibility in their thinking, efficiency or seeing each problem as a new one? I was SO glad to see I am not alone when I read Tracy’s tweets yesterday….

The conversation was an interesting one that then seemed to moved into number choice and thinking about what the students were thinking and what we do as teachers from here. We all definitely had a lot more questions than answers, which is always fun to explore!

So, of course I had to test out some of our questions into my number talk today. I had the students do the number talk from their seats so they had their journals readily available. I gave them 36 x 7, asked them to solve mentally and really think about the strategy they were using. I took answers, they all got 252, and I asked them to jot down how they solved it. We shared out and the majority had solved it just as Tracy had mentioned in her tweet, (30×7) + (6 x 7). Then I gave them 36 x 25 to see if, when given a 2-digt x 2-digit, they changed their thinking. I was also interested in the influence of the number choice of 25.

I don’t think it was the two-digit times 2-digit number that changed their approaches, but more so the influence of the 25. A lot went to double/halving because they could get to 50 and 100 and others used the 100 made of four 25s. One student multiplied 40x 25 and subtracted 100 while a few others used the associative property that Tracy had mentioned (4×25) x 9.

The final problem was 39 x 25. Unlike a typical number talk in which I push students to connect to previous responses in route to an answer, I instead asked them to not solve it, but just think about how they would solve the problem. After they had their thumbs up with a strategy, I asked them to complete one of the following prompts: “I used the same strategy I had used before because….” or “I used a different strategy in this problem because…” Here are some of their responses…

My conclusion is: the more students talk about their strategies, reasonings, and choices, the more they think about the numbers and what “makes sense” in the solution pathway. I think some students definitely get into a comfort zone with a strategy that works for them, and that is ok with me, but I definitely want to expose them to other ideas and things to think about. I loved that 25 and 39 influenced their thinking about the way to approach the problems.

I am not sure this answers any questions in our Twitter conversations, but I am always SO incredibly curious to see what the students actually do after anticipating their thoughts. The even better part is, they love sharing what they were thinking without the worry of being wrong. I even had one student who said she changed her strategy for the last problem because she got the one before it wrong after solving it twice. In her words, “It definitely was not working.” 🙂

Hope this gives you something to think about Tracy, Christopher, Sadie, Simon and Kassia!

-Kristin

#PiDay2015…Circle Fun

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Some of my students this year were excited to “celebrate” pi day and were very disappointed that it fell on a Saturday, so we decided to have some pi fun on Pi Day Eve. I am not one for “gimmicky” holiday lessons and wanted whatever I decided to do, to not just be definitions of circles and their properties or a formula for how to use pi to find measurements, but instead an activity that allowed students to discover all of the cool things about circles and patterns that arise from that work.

After brainstorming with a colleague, she suggested I just have the students try to create a prefect circle. Loved it. I put out tape, scissors, rulers, paper, string and told them if they thought of other tools they wanted to use, they had to pass my approval first (I wanted to keep the protractors and compasses out of the equation for right now). Off they went! It was soooo interesting to see all of the great approaches and all of the cool ideas that emerged from their work.

I found it so interesting that quite a few first drew a square and tried to find the center. They said they knew that the circle could be made inside of it because a circle is 360 degrees and each angle of the crossed lines was 90. The problem became figuring out how to get the “rounded edges to be the same.”

Quite a few groups had seen a compass before (but didn’t know what it was called) and tried to recreate one with the available tools. Some started from finding a center and going from there, while others created the center by just placing the scissors on the paper and going around from there. After many attempts, they were starting to realize how important keeping that constant distance in the scissor opening really was.

To solve the constant distance problem, one group used tape to keep it the same while another group used string (and chopsticks she just happened to have in her lunchbox that day:).

This group solved the constant distance with two pencils attached with string. The funniest part of this one were the trials as the string kept wrapping around the center pencil as they went around and never meeting exactly back at the start. They eventually figured it out after blaming the “center holder” numerous time for “moving the pencil.” Another group kept a constant distance by taping their string to the center of their paper and putting a pencil on the other end.

This group created a center from overlapping rulers and attempted to put string around the the ruler corners to make an arc, but couldn’t agree with how to get them all the same. While another group tried to use the ruler ends as the center but ran into the same problem with the rounded edges.

This idea was interesting to watch evolve. She had seen the group on the floor (in the pic above) and said she realized that any rectangle rotated would make a circle. She then grabbed a ruler, taped two cap erasers to each end and thought the caps would leave eraser marks she could go back and trace after rotating the ruler. That didn’t work, no marks. She then cut her pencil to get some lead and taped that to one end.

The final product….. After sharing their circles and approaches, I had the students jot down some things that were important when constructing their circles.

From these, I realized (and was surprised) the students have some circle vocabulary in their toolbox. I decided to get that out so we could be sure everyone in the class had exposure to all of this great stuff. I asked them to share their findings and what measurements they used or could find in their circle.

One group had finished their circle early, so I asked them to find some of these measurements. They found the diameter and circumference with the ruler and string they had used in the construction. It was so interesting to see the intuition students have around finding the diameter. They knew it had to go through the center and that no matter where they measured from, it would be the same. It makes me wonder why we, as teachers, sometimes think that we need to give students definitions for things before they get to demonstrate their intuition around these very ideas. I could have told them “diameter is distance across the circle through the center” before the lesson started, but they already knew that, love it.

After testing a few circles, this group started to see pi emerge…

For the the last circle in this list, they measured the diameter of their large circle they created and I asked them to estimate the circumference. After seeing that each circumference was “about 3 times as much,” they estimated 46 x 3 to be circumference. They haven’t had a chance to test it on the actual circle yet because we ran out of time, but that will be some fun on Monday!

Happy Pi Day 2015!

-Kristin

Adding Decimals to the Thousandths

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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.