Visualizing Fractions

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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
  • Angles

 

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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.

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The Power of Visual Models

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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.

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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.

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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.

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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.

 

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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.

Higher-Level Math Tasks

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.

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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.

 

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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.

Visual Fraction Models

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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.

Visual Fraction Models
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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.

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This student turned a pentagon into 5 triangles. Each triangle represents 18 miles.

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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.

Exploring Fractions

Exploring Fractions

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.

Fraction Blocks

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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 .

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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.

 

Representing Fractions with Thinking Blocks

Many classrooms in my school are in the midst of reviewing fraction concepts. Throughout the school students are finding fractional pieces, converting fractions to decimals, and identifying fractions on number lines.  For the past week students in second grade have been identifying fractional parts.  Earlier in the week students completed the page below during a math station.  Students did well on the first two pages, but struggled a bit when identifying fractions on a number line.

Representing 3/4 in many different ways
Representing 3/4 in different ways

This was a challenge for some students as many are more familiar with identifying fractions within objects (in a circle/rectangle).  Moving from identifying fraction to placing them on a number line can be a stretch.  Many students have already started to decompose numbers and have completed “fraction-of” problems.  These types of activities have helped reinforce the number line and fraction connection.  Next week students will be assessed on the fraction unit and many classrooms move into geometry concepts.  Before focusing in on geometry, I wanted to give student an opportunity to visualize fractions and use them with more complex word problems.

As I was looking for supplemental material I came across a Tweet by Paula (@plnaugle). She referenced Thinking Blocks  as a resource that she uses with an interactive whiteboard. I looked into the site and thought that it might be useful for my grades 2-3 classes since the app allows students the opportunity to solve fraction problems visually.  Specifically, I downloaded the fraction app on the school iPads.   Yesterday a second and third grade class used this app in their classroom as a guided activity.  The app was introduced to the class and I modeled the different steps involved in solving the problems.

Modeling
Modeling

The students were then asked to find a comfy place in the room and complete a minimum of three exercises.  What’s nice is that the problems are picked at random, so students aren’t on the same problem at the same time.  There’s also a feedback box that assists in guiding students towards labeling the correct parts of the fractions.

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Click to enlarge

I helped the students as needed, but many were able to use the virtual manipulatives and generated feedback to stay on track.  Some students completed three problems, while some went beyond and tried out five.  After about 12 minutes the class gathered and we reflected on the perseverance that was needed and celebrated successes. This activity gave students an opportunity to make mistakes and persevere.  I’ll be keeping this app in my repertoire for the future.

Introducing Fractions

Fractions

Today’s second grade math lesson included an introduction to fractions.  In the past I’ve introduced our fraction unit with pie manipulatives.  They work great, but I was looking for a  more hands-on lesson that motivated as well as provided opportunities for enrichment.  While thinking about how I could make my fraction lesson more engaging, I decided to research a few different options.  Specifically, I wanted to find a way to incorporate a lesson with multiple answers.  I thought what’s healthy (district wellness plan), easy to peel, has different sectional pieces, and is relatively easy to clean up?  I ended up deciding on purchasing a bag of clementines.  I also put together this sheet for the activity.

Sheet

The lesson went well and it’s definitely one that I’ll keep in the repertoire.  Students estimated the amount of slices, identified fractional pieces, found the numerator/denominator, turned the fraction into a mixed number and finally ate the clementine.  A few pictures are below.

How do you introduce fractions?