Fraction Blocks and Strategies – Part 2

photo-jan-24-10-41-47-am

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.

screen-shot-2017-01-21-at-8-12-34-am

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.  screen-shot-2017-01-28-at-7-53-24-am

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.

photo-jan-24-10-49-14-am

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.

Download the file for this activity here.

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

Fraction Blocks and Strategies

cover.jpg

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.

Screen Shot 2017-01-21 at 8.10.45 AM.png

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.

photo-jan-17-9-26-40-am

We then had a conversation about quarters, halves and wholes.  I then gave each student the card below.

Screen Shot 2017-01-21 at 8.12.34 AM.png

Students placed the A block near the top of their desk and started comparing the different blocks.  The class completed the first question together.

LetterA.jpg

 

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.

screen-shot-2017-01-21-at-8-22-58-am

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.

Screen Shot 2017-01-21 at 8.29.05 AM.png

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.

 

What’s your strategy?

screen-shot-2016-12-01-at-6-33-06-pm

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.

screen-shot-2016-02-26-at-5-26-13-pm 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.

screen-shot-2016-12-01-at-6-15-18-pm

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.

2016-11-29-15-29-39

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.

screen-shot-2016-12-01-at-6-32-08-pm
Can you tell that I like my new stamps?

Estimating as Part of the Process

 

screen-shot-2016-11-18-at-9-35-44-pm

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.

photo-nov-14-6-02-35-am

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.

screen-shot-2016-11-18-at-9-39-39-pm
Fifth graders worked on this for 10-15 minutes.  Class discussion followed

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

Screen Shot 2016-02-26 at 5.47.33 PM.png

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.

Screen Shot 2016-02-26 at 5.26.13 PM

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.

Screen Shot 2016-02-26 at 5.37.12 PM
Recipe modification

 

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.

Screen Shot 2016-02-26 at 5.35.39 PM
Showing number models

Screen Shot 2016-02-26 at 5.45.30 PM

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.

Photo Jan 28, 9 17 09 AM

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.

Screen Shot 2016-01-28 at 9.01.22 PM

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.

Photo Jan 28, 9 17 29 AM

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.

Visualizing Fractions

Screen Shot 2016-01-23 at 8.42.19 AM.png


 

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

 

Screen Shot 2016-01-23 at 8.23.07 AMScreen Shot 2016-01-23 at 8.23.25 AMScreen Shot 2016-01-23 at 8.23.36 AMScreen Shot 2016-01-23 at 8.24.09 AM

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.