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.
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.
My third grade students started a new unit on fractions this week. They’ve explored fractions before, but more along the lines of identifying different types of fractions and adding/subtracting with common denominators. This new unit involves students finding fractions of sets and a heavy dose of fraction computation. Students need to have a deep understanding of fractions to be able to add them and show a visual model. So on Friday the class practiced skills associated with finding fractions of sets. Students were given this prompt:
Draw four different ways to show 3/4 in the box below.
The student models fell into a few different categories.
A number line
Pie, rectangles, squares
Dots or arrays
The class reviewed the results and we had a discussion about the different ways to represent fractions. Next week the class will be combining these models to add and subtract mixed numbers.
My fifth graders started off the week learning more about fractions. On Monday students used a visual model to multiply mixed numbers. The visual model was a bit challenging for students to grasp. Many of the students knew parts of a multiplication algorithm, but not necessarily how to show the computation visually.
It took a decent amount of modeling and experimenting, but I believe completing the visual models increased students’ understanding of fraction computation. After a decent dose of the visual model, students were introduced to a fraction multiplication algorithm. I tried to make the connections between the algorithm and visual model as apparent as possible. Many students made the connections, but not all.
Around mid-week students started to divide fractions. Again, I started off the discussion around using a visual model to show the division.
This time students were more confident in creating the visual models, although some wanted to jump to the standard algorithm. This stopped once I included the visual model as part of the steps required to solve the problem. Some of the students that had trouble creating the visual model for fraction multiplication started to become more comfortable with the division model. This was good news. The most challenging part for my students was finding a fraction of fractional pieces.
This was solved once students realized that “of” meant to multiply and then they were able to find a solution. The class had many light-bulb moments as students made connections between the visual model and standard algorithm. Also, one of the additional benefits was observing students look at the reasonableness of their answers. This was more apparent when students created a visual model first. The class will be sharing their models next week.
I’m finding that there’s power in using visual models. The opportunity to use trial-and-error with visual models has many benefits. It’s a low-risk opportunity that allows for multiple entry points. Students are making sense of fractions before moving to the standard algorithm. It might not be the most efficient way to compute fractions (as students continue to find out), but I believe students will have a better understanding when they can visualize fractional pieces and then use a process to find the solution. At some point students will be shown a visual model and be asked for the computation. I feel as though students were steadily building their conceptual understanding of fractions this week.
A few days ago I started reading Principles to Actions Ensuring Mathematical Success For All as part of a book study. As I was reading in preparation for our first session I came across a few ideas worth highlighting. Pages 18 and 19 discuss the four levels of cognitive demand in math classes. Along with expectations, these demands are often revealed in tasks or assignments that students are asked to complete.
The book describes lower-level demands as tasks related to memorization and procedures without connections. Memorizing rules/formulas and following procedures is often related to lower-level demands. Students often understand what’s expected when lower-level demands are required. Generally one answer or procedure is evident with this type of task. Worksheets that have students practice rote computation skills without words could fall into the lower-level demand category. Higher-level demands are procedures with connections and often require considerable cognitive effort to achieve. Anxiety is often a part of higher-level demands, although this may be because students don’t see these types of tasks as often.
After reading this page and looking at the different examples I started to reflect on how elementary math classrooms are organized. Math practice is needed, but students should also be given time to explore, discuss and make connections in a low-risk environment. I find more lower-level demands in math classrooms than higher-level, but an ideal ratio is challenging to ascertain.
So after reading pages 1-35 I decided to use an example of a higher-level demand activity with a fifth grade classroom. This particular class is learning about fraction multiplication and division. Students have learned in the past to multiply the numerators and denominators to arrive at a solution. To delve a bit deeper in their understanding I decided to use and adapt one of the tasks in the book. I first grouped the students into teams and gave each team 12 triangular blocks and a whiteboard marker.
Students were asked to show a visual model of 1/6 of 1/2. Some students knew the answer already but seemed unsure of how to show the answer visually. Many of the groups weren’t quite sure on how to approach the construction of the fractions. They understood the abstract and procedural but had a challenging time visualizing the fractions.
After seeing the students struggle a bit I’m glad that I decided to have them work in pairs. Students started to build models of 1/2 using the 12 triangles. Some of the groups came to a conclusion that two different sets of six triangles shows half. Then from there students started to think of what’s 1/6 of the 1/2. Students took out 1/6 but then debated on the value. Some groups said that the answer was 1/6 while others were confident that it was 1/12. Eventually the students decided that 1/12 was the correct solution.
I went around the classroom and took some pictures of the different creations. Not everyone created the same type of model. This was a great opportunity to highlight some of the different models that arrived at the same solution.
Afterwards, I thought that offering exactly 12 triangles helped but limited the choices for a visual model. The student models were somewhat similar as a result of the level of scaffolding. As students reflected on their actions in this activity I heard some interesting conversations. Students were aware of the procedure to multiply fractions less than one, but started to visualize the model through this activity. I thought this might be one way to introduce fraction multiplication at the fourth grade level. I also thought that this activity was well worth the time and I’m looking at incorporating additional high-level cognitive demand activities in the future.
Today students explored fractions using number and visual models. Students have been practicing how to add and subtract numbers for the past few weeks. Most of the students have an understanding of how to find common denominators and add or subtract problems. Yesterday students answered word problems involving fraction computation.
What I’m noticing is that students are understanding and compiling their number models but aren’t as comfortable with visual representations. Being able to model fractions is important and a key ingredient in understanding fractional parts. As the class progressed I felt like there was a disconnect between fraction representation and computation. Eventually, the lack of conceptual math understanding impacts a student. I’ve found this to be especially clear with fractions. So today’s class focused on showing both, the number model and visual representation. Students worked in groups on the page below.
Students worked together on the two problems. There was a lot of struggle, especially with the visual model portion of problem two. I was tempted to lean in and help the students but I wanted them to use strategies and their partner to find a solution. I let the students work and even debate strategies with each other. Near the end of the class students presented their final number models and visual representations.
I gave them feedback and asked questions in return. Two of the better examples are above. Tomorrow the class will be exploring visual fraction models via Thinking Blocks. Overall, I felt the productive struggle was worth their time and I hope that another layer of conceptual understanding is starting to cement.
Fourth grade students explored fraction computation last week. Since the beginning of the year they’ve been periodically reviewing how to add and subtract simple fractions. About a month ago this same group of students used fraction pieces of a pie to show a visual model of adding/subtracting different fraction less than one. Last week students identified and compared fractions and mixed numbers. They started to convert mixed numbers into fractions and vice versa. I’m finding that as the students became more comfortable with converting fractions they’re becoming better at fraction computation. Not all the students are at this level, but many are ready to add/subtract mixed numbers.
Over the past few years I’ve used a fraction computation activity that I often refer back to throughout the year. Every year I tweak it a bit more to fit better with my students. This year I felt my students were ready for the challenge. The students cut out the fraction pieces below. Students are then given time to explore how different fraction pieces are equivalent.
I asked the students to model different types of fractions with their pieces. The class came to a few different conclusions on how fraction sums were calculated. I didn’t really hear students talk about finding common denominators; instead I heard students saying the words “equivalent” “matches” “is the same as” throughout the conversation.
Students were then asked to combine their fraction pieces to find certain sums. For example, students were asked to show 1 1/2 using 7 pieces. Students wrote the number model below their visual representation. I was encouraged to see that some of the students showed fraction multiplication in their number model eg. (5 x 1/6) + 1/3 + 1/3 = 1 1/2 .
Through trial and error students started gaining traction in finding the sums. Students had to place all the questions out on their desk and match the fraction pieces to find the sum. After all the fractions were found students taped/glued them to their paper. The class then discussed how this activity could be completed in a variety of ways. Next week students will reflect on this activity in their math journals. The activity described in this post can be found here.