## Exploring Fractions – Week One My third grade students started a new unit on fractions last week.  They started the unit by learning about part-to-whole fractions and how to identify them.  Student teams explored how fractions are represented in different situations.  One of the first activities asked students to create their own version of a part-to-whole model.  The scissors came out early this week while students cut out fraction area models. They compared the pieces within the area models to create equivalent fractions.  This gave students another way to compare and observe equivalencies.  This was time consuming activity, but so worth the time.  Students made connections and played around with the circles/pieces to compare the models.  The only negative was that some students didn’t cut the fractions exactly on the line so the pieces didn’t always line up.  The next day students compared the fraction pieces using <, >, or = signs. Students were making progress in comparing fraction area models.  The fraction circles were being used consistently and then the class moved to transitioning to identifying and placing fractions on number lines.  This was a challenge.  We started with a 0-1 line and then identified half.  From there students used benchmarks to compare fractions on line.  Students had some trouble when the number line was stretched from 0-2.  The class also explored how the fractions look on a vertical number line.  A different dynamic was at play there.  Students then practiced a bit more with an Open Middle activity. Students were given opportunities to discuss fractions with their peers through a few different fraction math talks earlier in the week.  The time spent today revolved around reviewing different fraction models. On Friday, the class participated in a fraction Desmos Polygraph activity.  Feel free to use the program here.  This was one of the learning highlights of the week.  Students were asked to pick one of the fraction models that they created early in the week, while other students asked questions to help determine the fraction. At first students asked questions related to the color of the fraction.  Then they moved to questions involving less than half and more than half.  Students found that clear questions revealed better answers. I was excited to see that students were using benchmarks and part-to-whole ideas to help uncover the mystery pick.  Students spent around 20-30 minutes exploring the polygraph with a few different partners.   I even snuck in as a participant.  I’ll be keeping this idea in my back pocket for next year’s plans.

Next week, students will start to add and subtraction fractions.  I’m looking forward to seeing how students will use the experiences this week.  There’s plenty more to this unit and we’re just getting started.

## Which One Doesn’t Belong – Fraction Edition My classes have been using a WODB board this year.  The board has been a permanent fixture in my room and it has been up since August.  I came across the idea last year after reading Christopher’s idea and Joel’s example.  I’m finding that it has been a great routine for my 3-5th grade math students.  My goal was to change my WODB bulletin board every  week, but it’s really being changed around 2-3 weeks or so.  My boards started out as mainly shapes, but has moved to numbers and equations recently. Changing it less gives kids time to see other options and add more notes. My third grade students are in the middle of a unit on fractions.  They used number lines to multiply fractions by whole numbers earlier in the week.  The students are becoming better at multiplying fractions using visual models, although some are more wanting to multiply the numerators and denominators.

Today, the students completed an individual Which One Doesn’t Belong task.  I’ve heard of other classes doing something similar, so I thought it might work well with my kids. Students were given a criteria for success page and then asked a bunch of questions. Students were asked to create four different fraction multiplication models.  Students then created two different solutions for the WODB prompt.  After a brief amount of modeling, students started to create their own WODB boards.  Many students had questions about what could count for solutions?  I put it back on the students to figure out if their solutions were appropriate or not. For the most part, students did a fine job finding two different solutions. Students then wrote down their solutions and folded the paper to hide them. The students took pictures and put them in their SeeSaw accounts.  Next week, the kids will look at each others’ responses and see if their solutions match. I’m looking forward to what they observe.

## Fraction Division – Models and Strategies

My fourth grade students have been exploring fractions.  They’ve become familiar with how to add, subtract, and multiply fractions.  They just started to divide fractions earlier in the week.  Whenever I introduce fraction division I tend to have one or two kids that raise their hand quickly.   Their quickly raised hand tends to cause me to slow down and prepare.   They comment that there’s a “fast” way to divide fractions that they learned at Kumon or from someone at home.  Sadly, that trick is infamous number 1 on the NCTM’s Tricks that Expire!  These students can explain what to do, (change the numerator and denominator of the second fraction and multiply) but struggle when pushed to explain why it works.  I feel like at times these particular students inadvertently or purposefly convince others in the class that this method is the quickest.  Some agreed, but introducing this idea at the begging caused unneeded confusion.

I shifted the discussion to the meaning of the fraction bar.  One of the students mentioned that the fraction 1/2 is the same as 1 divided by 2.  Another student said that is the same as 0.5.  This conversation was productive and moved the discussion back on course.  Students started to build upon each response and were able to start thinking more about their own understanding of fractions.  I then introduced the idea of fraction as division.  This resonated well with students and I could tell that they were really thinking about how they view fractions.  I then put this problem on the board. Students thought for a little while and then decided to split up each fraction into three pieces.  They then counted up the pieces to find 9. I then introduced students to a common numerator and denominator model.  Students thought about this problem and then started making a few guesses. One thing that seemed to shift this thinking was to look at fraction as division.  In my years of teaching this seems to make quite a few connections   Many students know that a half of a half is a quarter, but are a confused when it comes to dividing a half.  One student mentioned that they both have common denominators and that might be useful when dividing.  Another student said that a fraction is division, so you could divide the numerators and denominators. The class agreed that this will work as long as the denominators are the same. They also concluded that if the denominators aren’t the same, we can find an equivalent fraction to create ones that are. This conversation lasted for about five minutes.  It was productive and not once was there mention of a “fast” method to divide fractions.  I’m hoping that students hold on to visual models and using a variety of strategies when dividing fractions in the future.  Next week, we’ll be investigating how to divide mixed numbers.  That’ll most likely happen after our week long PARCC adventure.

## Fraction Progress My third grade students have been exploring fractions.  For the past month, students have been delving deeper and constructing a better understanding of fractions. Last week, students cut out fraction area circles and matched them to find equivalent fraction pairs. For the most part, students were able to match the fractions to observe equivalency.  Afterwards, students discussed how to find equivalent fractions through different means.  Some students made the connection between doubling the numerator and denominator, while others noticed that they could divide to find an equivalent fraction.

Early this week, students started to place fractions on number lines.  They used the whiteboard and a Nearpod activity to become more accurate when identifying and labeling fractions on a  line. It was interesting to see how students showcased their understanding as the number line increased from 0-1 to 0-2, and beyond.  Giving an option for students to decide which number to use seemed to encourage them to take a risk with showing their understanding.

On Wednesday, students started a fraction task related to computation.  Students were asked to color each fraction bar, cut them out and organize the fraction pieces to complete given number sentences.  Students had to rearrange the fraction pieces and found that there were leftover pieces, which makes this a more challenging task.  You can find more information about this activity here.

This task took around a day to complete.  Students struggled at first and they used a lot of trial-and-error.  Students compared the fractions bars and switch the pieces around quite a bit before taping down the sum.  A few students needed a second attempt to complete this.

On Friday, students used polygon blocks to show their understanding of fractions.  Using polygon blocks, students were asked to take one block and label that as 1/4, 1/2, 1/8, or 1/12.  They then combined at least three different blocks to find a sum of 3 1/2. Students used whiteboards and geometry blocks to combine the fraction pieces.  I observed students using different strategies to combine and then take away blocks to find the sum of 3 1/2.

Next week, students will investigate the relationship between fractions and decimals.

## Fraction Blocks and Strategies – Part 2 Last week my second grade crew explored fraction blocks.  They cut out and used the blocks to compare fractional pieces.  Students enjoyed the trial-and-error component and they started to visualize fractions in a different way.

I decided to use a similar activity with my third graders. Instead of labeling the bars, I decided to leave off the label. This initially confused the students as they expected to see the label. Students moved beyond the confusion when they were given the value of one of the blocks.  They then used that value to compare all the other blocks. Students were asked to cut out the blocks and start comparing them.  I didn’t give them any directions beyond that.  After about 4-5 minutes I placed the sheet below on the overhead projector. We completed the sheet as a class.  I used the document camera and students compared the pieces on their own desk. It took multiple attempts and a number line, but eventually the class was able to finish the sheet.  Students were then off on their own to find the whole or part of certain blocks.  Students used many different strategies since they couldn’t rely on the label. While the students were working I went to the different tables and observed the strategies. Almost all the students compared the shapes to one another to find one whole.  Other students created a number line and placed where they thought each shape would be located.  I had a few students take out a ruler and measure the blocks. I collected the sheets once everyone was finished, marked them up with feedback, and returned them the next day.  I used the NY/M model for this assignment. Every student in the class needed to make some type of correction.  After a brief review, I gave the students back their sheets and they made corrections.  There were few perfect scores after the second attempt, but everyone improved – an #eduwin in my book.

Next week we’ll be learning about equivalent fractions and how to find common denominators.

## Fraction Blocks and Strategies My second and third graders started a unit on fractions last week.  Students are used to identifying typical pie fraction pieces.  Generally, I find students are introduced to fractions using this type of visual representation.  Students then count the amount of pieces and place that number as the numerator.  I find moving towards mixed-numbers has some students changing their strategy as they can’t just count the pieces, but they have to recognize that a certain amount of equal parts are one whole.  Based on their pre-assessment results, it seems as though my second grade and some of my third grade students are at this point.

Using a number line has helped.  Placing the fractions on the line has brought a better understanding of the placement of fractions in relation to a whole number.  Currently, students can identify certain benchmark fractions on a number line.  We’re working on bolstering this skill and connecting it to fraction computation in the near future.  Before that happens I want to ensure that they have a decent understanding of mixed numbers and where they fall on a number line.

On Thursday and Friday I introduced students to a fraction block activity.  Students were given a sheet with fraction parts.  Each block was split into a certain amount of equal square parts. Each student was given an envelope to put their pieces in once they were finished with the activity.  Students cut out each block and were asked to put them in order from least to greatest value.  Students were able to complete the task. We then had a conversation about quarters, halves and wholes.  I then gave each student the card below. Students placed the A block near the top of their desk and started comparing the different blocks.  The class completed the first question together. I then gave students time to work on the rest of the problems.  Students were then given time to use trial-and-error to find which blocks worked for each problem.  I went around to the different table groups and asked students questions about their strategies. Students ended up matching the squares with other shapes to determine what was a quarter, half, almost a half, and what happens when you combine shapes.  After about 10 minutes the class reviewed the sheet and found that some problems could be answered with multiple solutions.  Students put the sheets in their envelopes since we ran out of time.

The next day students completed some more challenging half-sheets involving their blocks. Students struggled a bit with this as they had to look at A as half instead of one whole.  This changed the value of all of the other blocks.  I allowed students to work in groups for about five minutes and then independently for another five.  This gave them an opportunity to gain another perspective and a different strategy.  Afterwards, I reviewed the possible solutions with the class.

Next week I’m taking this activity one step further and using the blocks without markings.  I’m borrowing this idea from Graham’s post on defacing manipulatives. Students will complete similar half-sheets, but without the evident markings. I’m looking forward to seeing how students’ strategies change and the math conversations that follow next week.   Click here to download the activity that I used. My fourth grade crew has been exploring fractions for the past two weeks.  Students have been making some amazing connections between what they’ve learned before and what they’re currently experiencing.  Last year the same group of students added and subtracted fractions with unlike denominators. The process to find the sum and difference was highlighted and that’s what students prioritized.  That was last year.    Although the process was and still is important, this year’s focus in on application.  How do students apply their fraction computation skills in different situations?  That takes a different skill set.  Being able complete a simple algorithm doesn’t necessarily help students read a problem, identify what’s needed and find the best solution.  More so, I feel like the application and strategy piece trumps the actual algorithm process at this stage.

So, I brought out a fraction recipe problem from last year. Similar to last year, students had to change the recipe based on the amount of muffins needed.  Unlike last year, I didn’t introduce the fraction multiplication or division algorithm.  I had students work in groups and document their strategy to find a solution. Students had to indicate whether the number of muffins increased or decreased, by how much and how to change each ingredient.  The group conversations were fantastic.  Groups had a brief conference with me to discuss their strategy once they arrived at a solution. The conversations that occurred during my 1:1 meetings with student groups were beneficial.  Students took what they wrote as a strategy and elaborated with different examples.  I’m thinking that students will write in their math journals about their experience tomorrow.  I’m assuming that this will also help transition students towards understanding why the fraction algorithms work.

## Estimating as Part of the Process My fourth and fifth grade classes explored fraction models this week.  I enjoy teaching about the concept at both of these levels concurrently.   I can see the linear progression of skills associated with fractions and the different perceptions of fractions.  My fourth grade crew is finding equivalent fractions while my fifth graders are multiplying/dividing fractions.   Both groups are finding success, but I’m also seeing similar struggles.  Students are fairly consistent with being able to convert mixed numbers to fractions and combine fractions. Issues still exist in being able to estimate fraction computation problems and determining which operation to use while completing word problems

This year I’ve been focusing in on making sure students are using estimation strategies.  This is especially important when dealing with fractions and eventually decimals.  Unfortunately, I tend to find that time spent on the process (algorithm) trumps the reasonableness (estimate) from time to time.  Part of this is due to past math experiences and time management.  After the last assessment on fractions, I started to look for additional ways to incorporate estimation within my fraction unit.  I came across Open Middle last year and I’m finding their fraction resources to be a great addition.  Both, my fourth and fifth graders completed a few different Open Middle fraction problems this week.

I’m finding that students are estimating a lot more when they are involved in these types of activities.  The tasks I use from OpenMiddle emphasize the need to estimate first and calculate second.  These types of puzzles are interesting for students.  They are low-risk, but yet have a high ceiling.  I also found this to be evident with an activity that I found out of this book. I can’t say enough good things about the ideas and resources found within that resource. Students had to find the missing numerator, denominator or variable.  In both, the Open Middle and Make it True activity, student worked in groups of 2-3.   I gave them about 10-15 minutes to collaborate.  The sheet below was adapted from the book above.

They shared ideas, estimated and came to a consensus on what the solution should be. I had the student groups write their answers on the board and the class discussed all the different solutions afterwards.  The class conversation incorporated a decent amount of review and also gave an opportunity for students to ask for clarification.  I’m looking forward to having more classes like this. The class conversation component that occurs after a collaborative effort is starting to become an even more valuable piece of my math instruction.

## Fraction Multiplication and Division My fourth graders are deep into a unit on fractions.  They’ve been multiplying common fractions and tiptoeing into fraction division.  That was until Wednesday of this week.  On Wednesday students explored different ways to divide fractions.  Students used visual models to divide, but that didn’t seem to help students understanding them better.  They encountered abstract problems and used the “flip” method to find the quotient.  This still didn’t help improve much in the conceptual understanding department.  Students wanted me to show the exact process of what to do to solve fraction division problems.  I wasn’t thrilled.  It was evident that students needed more exposure and practice with fractions. So I took a step back and reviewed fraction multiplication.

The class reviewed fraction multiplication and scenarios that are needed to find products.  Students were aware of many different situations where they might need to multiply fractions.   They were able to show visual models and computation strategies to find solutions involving multiplication.  I had a few students also indicate that it’s important to simplify the product.  So the class was rolling in a  positive direction and I decided to bring the lesson back to division.  The break through moment occured when the class connected fact families to the current lesson.  Similar to addition and subtraction, multiplication and division fact families can also contain fractions.  This helped students make connections.  Students wrote out different fact families using unit fractions (1/2,1/4,1/3…).  Students then changed the fact families related to only multiplication and division.  The class was starting to wrap their heads around fraction division with a bit more ease.  I felt as though students were ready for the next activity which was related to food.

Students were placed in teams of three and given a blueberry muffin recipe. Students reviewed the sheet and wondered where this was going.  Each group then received a sheet related to the original recipe.  Each half-sheet asked students to modify the recipe based on the serving size.

Some students were asked to make 12 muffins, while others said 36, 24, 60, 96 or 72.  I felt as though some students were relieved when they were asked to half or double the recipe.  Other groups tackled the problem with some major perseverance.  Students were asked to show their number model and explain why their answers were reasonable.  Some students wrote number models that multiplied fractions by the recipe amount. Groups also used fraction division to show a number model.  The majority of groups connected how multiplication and division of fractions can be part of a fact family.  This was especially apparent when students started to see that 1/4 * 4/1 = 1.  I feel like this is laying groundwork for next year’s class when we start pre-algebra equations. Having a solid understanding of how to “undo” operations is a great tool to have in the math toolbox. Once students found the fractional reduction or addition they changed each ingredient accordingly.  After showing their work, students took a picture of the whiteboard and recorded their voice.  Student groups explained how they found each answer and why it was a reasonable answer.  Some student groups were amazing when communicating their reasoning.  They actually explained that the ingredients needed to be increased by a factor of 4.  Other groups were very general with their reasoning in saying that the recipe increased because they were asked to make more muffins.  I can tell this is an area that’ll need strengthening throughout the year.

Overall, this activity seemed to help reinforce skills taught earlier in the year.  The most complicated part was where to start.  Students had trouble knowing what do do with the problem at first.  Students seemed comfortable with the number model and computation components.  Explaining their reasoning needed some tweaking, but that might also be an expectation that needs to be set more in the future.

## Representing Fractions

During the past few weeks my students have been studying fractions. I feel like the class is making a decent amount of progress.  The class has moved from identifying fraction parts to adding the pieces to find sums. Pattern blocks have been especially helpful with adding fractions. I feel like students are becoming more confident with the computation and we haven’t used the word common denominator yet.  I don’t want students to by relying too much on just the algorithm.  Throughout this process I’m noticing that students are struggling with fraction word problems. Students are having trouble identifying what the fractions represent in the problems.

Yesterday we had a class meeting to discuss this topic. This fit in well with a book that I’ve been reading.  Chapter 8 emphasizes how to teach fraction concepts and computation.  The chapter begins with misconceptions and the different meanings associated with fractions. The class reviewed all the different ways that they view fractions. We documented the class ideas on an anchor chart. Do you notice any trends? The class looked at the list and had no complaints. This is how they visualize fractions. When asked how they use fractions they came back to this list and didn’t have anything to add.  Keep in mind that this is from a group of third graders.  The next step in the class conversation was to discuss different ways that fractions are represented in problems. I started with part-to-whole representations. Most kids were familiar with this type of model. After all, students have been using this model for the past week and most of last year. I then moved onto how fractions can be used to measure objects.  Students nodded their heads in agreement and asked questions as I went through the other representations. Connections were made through this process.  Students created examples of each representation in their math journals. Students are planning to revisit the word problems that I discussed earlier in this post.  They’ll be reading the question and match the context to the representation.  I’m looking forward to having students use this strategy moving forward.