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.

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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.

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I then introduced students to a common numerator and denominator model.  Students thought about this problem and then started making a few guesses.

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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.

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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.

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Understanding Number Relationships

cmparing-01
Grouping Strategies

Today I was able to dig a bit deeper into Kathy Richardson’s book. The first chapter was related to counting and critical phases that are needed as students develop numeracy skills.  The second chapter focuses on number relationships. In order for students to compare numbers they need to be able to distinguish between larger and smaller. Once at this stage students can recognize that numbers are found within numbers.  For example, eight is found within 10. When comparing numbers students generally start to identify differences between the original number and one.

Richardson states that being able to change a number by counting on or adding to a group is a Critical Learning Phase.  Counting on or adding to a group of numbers is a strategy students use when comparing numbers. I believe primary students might use this strategy to find out how many blocks are in both stacks below.

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Assume that each block is the same size.  Now, how do you think primary students would count these two different stacks?  The strategy that they may use to solve this can tell  more about their understanding.  Are they counting each block individually or using the first stack to count on to find the second stack? While comparing numbers younger students often count each block individually.  The model below shows a different strategy.

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In this case the student has taken the five and built upon it to find four more.  The five and four are nine.  This student didn’t count each stack individually.

I find this interesting as it may apply to other areas of mathematics.  After reading this I started to think of how increasing the complexity could apply to fraction concepts.  Specifically, I thought of how theses blocks and fractions are similar:

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If the green stack is one whole what is the second stack’s value?  How would your students solve this?  In the example students may identify that each block is 1/5.  When looking at the parts on the right they might start off at 5/5 and add to that particular block line. Fractions can lead to confusion with a non-linear scale being present.  This is especially the case if students are always seeing 1:1 ratio when counting objects.

I thought a number line might be a better representations for a fraction problem. Richardson notes that number lines are only symbolic relationships.  She also states that when students use number lines they’re most likely not thinking of quantities, but more so using the line to find the solution. They’re using it as a tool to count on to find a solution.  Number lines are used frequently at the early elementary levels so this is something I’m going to keep in mind for the new school year.

Surface Area and Conceptual Understanding

Surface Area

My fourth grade class has been exploring a measurement unit for the past few weeks. We’ve been discussing the difference between area and volume. This has been a bit challenging as many students can apply area and volume formulas but struggle when finding surface area. Students were confusing area and volume and weren’t sure when to use a specific formula.  The idea of area being squared and volume cubed has been emphasized but still not cemented.  It seemed that students knew much more about the formulas and not as much about the conceptual understanding. To strengthen students’ understanding my class started a surface area activity late last week.  Click the image below for the template.

Sheet

Students were asked to pick one box in front of the classroom. I had many different boxes to choose from. Many of them were board games or boxes I borrowed from other teachers. It’s near the end of the school year and some teachers are moving classrooms so there were plenty of boxes. All of the boxes were rectangular prisms. Once students picked a box they took a picture and then found the dimensions. Students then took one piece of butcher paper and created a net based on the dimensions found earlier.

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Students created the net and then wrapped up the box. Students were able to immediately identify whether their measurements were off or on target. It took some groups multiple attempts to find a correct solution. After students wrapped up their box they took a picture. Before and after pictures were sent to me via Showbie. I printed them out and the students placed them on their sheets.

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It would have been great to print these out in color, but at this time of the year our school’s color printer is out of ink.  After the activity students reflected on how their perception of area has changed over the past week.  After listening to a few student reflections I’m deciding to keep this activity for next school year.

Transitioning to the Standard Algorithm

 

When should the standard algorithm be introduced?
When should the standard algorithm be introduced?

 

Many of my second grade classrooms are in the middle of their addition units. The classes often teach place value and addition strategies during the months of September and October. When introducing addition strategies, teachers rarely start using the standard addition algorithm (see # 4). Manipulatives and visual representations are heavily used during the first month of school.  The process below differs per school, but I’m finding that this is often the case in many of the second grade classes that I’ve observed. Keep in mind that I’m missing other approaches, so perceive the following as a few highlighted strategies that are used during the first few months of school.


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Finding the sum of objects

Students are introduced to some from of unifix cubes or counters. Students are asked to compile the groups of counters to find the sum. For the most part students find this task quickly and are ready to move onto the next portion.  This is also a first grade skill that’s reviewed at the beginning of second grade.

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Identifying numbers on the number line

The number line usually follows the counters. Sometimes the number line makes an appearance before the counters, but it’s usually afterwards. The number line is used extensively. Students are asked to find numbers on the number line. This builds number sense and an understanding of the reasonableness of an answer. Eventually students are asked to add numbers with hops showing the addition involved.

Find the sum of 25 and 15.
Find the sum of 25 and 20.

Base-ten blocks are then introduced to emphasize place value. Students are asked to combine base-ten blocks to find the sum. They are asked to find the sum of the base ten blocks and place them on the number line.

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What is the sum?

The four processes above aren’t necessarily mandates, but it’s found in the sequence of the textbook.  I should mention that the same process is used with subtraction later in the year. What I’m finding is that there’s rarely a mention of using the standard addition algorithm to find the sum. I don’t necessarily think that’s a problem, but it raises the question of when should the algorithm be introduced? In what cases should the algorithm be introduced and is it only used in certain circumstances.

Think of 200 + 198. Would your students use the standard algorithm for this?  Some might, other might prefer to use another method. Regardless of when the standard algorithm is introduced, there will still be students that would prefer to use the algorithm.  Is that the most efficient method?

The topic of this post was tackled during last Thursday’s #ElemMathChat. Most of the questions revolved around when the standard algorithm should be introduced and mistakes that occur when students focus on the steps.  There were many useful answers, but I’m not positive if one right answer climbed its way to the top.  There are many factors at play here. Some teachers feel pressured to move through the curriculum at a high pace because of testing.  They might teach the algorithm sooner, while others might not mention it and scroll through the prescribed lesson sequence.  Most teachers would like students to have a conceptual understanding of numbers and systems before moving towards a standard algorithm. How much time is spent developing that truly depends on the teacher and student. I’m not judging any teacher is these situations as we are all in this together, but I believe having this discussion is important.  I think this also plays a role in how other algorithms are introduced.

 

Exploring Rules and Patterns

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This past week my upper elementary classes started their equations, patterns, and rules units.  The units are composed of patterns, special cases, student-created rules, and solving equations.  To be honest this is one of my favorite units and involves a good amount of pattern exploration.  Through exploration, students construct their own understanding of how mathematical rules can be developed by analyzing patterns.   Many of these activities involve manipulatives or visual representations of various patterns.  I’m going to highlight three specific activities that seemed to work well this past week.

Analyzing the Perimeter

What's the Rule?
What’s the Rule?

Students were given a handful of square geometry blocks.  They were asked to find the perimeter of one block.  This was quick as students just needed to count the sides of the block.  Four!  Students then put together two blocks and found that the perimeter didn’t double, instead it was six. Students continued the patterns and discussed with their group what the rule could possible be.  Some groups used the whiteboards to write possible solutions.  Throughout this activity students struggled at first and then came to an understanding that the rule just didn’t include one operation. After the rule was discovered the students found the perimeter of 100, 200, and even 1,000 squares put together in a horizontal row.  I believe this activity also helped establish the reason for having mathematical rules.

Rule Tables

Students used four dice, a whiteboard, iPad, and dry erase marker to complete this activity. Two of the dice were operation and they had + and – on the sides.  The other two were typical six-sided 1-6 dice.   Students rolled all four dice and created a rule.  For example, if a student rolled a 6, 2, +, and – then he/she could say the rule is + 6 – 2.  Students wrote the rule on top of the whiteboard and used one of the die to roll five numbers that would be included in the in column.  Afterwards, students were asked to find the out column using the rule that was created.  A few examples are below.

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The students then took a picture of their product and sent it to Showbie.  Later on that day the class discussed how to combine rules.  So instead of + 6 – 1 this rule could be + 5.  The students were then combining all of their rules.  This activity led to some productive discussions on how to simplify or expand rules.

Visual Patterns

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I came across Fawn’s Visualpatterns site a couple years ago.  This is a fantastic resource that I introduced this past week.  I printed out some of the patterns and placed them in manilla file folders.  The picture of that is located near the top of this post.  The six folders were placed around the classroom.  Student groups visited each folder and determined the rule. While in the group students worked together and filled out the sheet below.

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Modified from this site.

Students took whiteboards and started to build possible rules for the pattern. Once they accomplished this they filled out the table and graphed the relationship.   I appreciate that students are asked to graph their findings.  This could lead into so many other math topics. Students only rotated through two folder stations so we’ll continue this activity next week.  By the way, the students were stoked when I showed them the visual patterns site and not because it has the answers.  A few students even said they were going to check out the other patterns on the site.  I’m looking forward to utilizing this resource a bit more next week.


How do you introduce patterns, rules, and equations?

 

Bridging Procedural and Conceptual Understanding

Yesterday I was putting together a few math projects when a Tweet caught my eye. The Tweet below started a short conversation that I thought was interesting.

David’s Tweet had many responses.  Most responses revealed that educators tend to side with solving one problem ten different ways rather than having students solve ten similar problems.  I started to reflect on how teachers give assignments that ask students to complete repetitive problems that often reinforce procedural mathematical thinking.  I also started to think how in an effort to provide practice, teachers may focus on procedural aspects first and then move towards practical application.  I find this happens frequently with math concepts at the elementary level.  What I don’t find often is the viewpoint that practicing procedural aspects can be embedded in solving specific problems multiple ways.  This type of thinking reminds me of number collection boxes.

Regardless of the assignment I want to be able to give specific feedback.  A larger problem that involves multiple steps can provide opportunities for teachers to pinpoint where misconceptions are and give direct feedback.  This isn’t always possible with ten similar shorter problems.  Below is an example of a few problems that you may find in a fifth grade classroom.  I don’t condone using these types of problems as they are definitely utlized, but I think we need to ask what’s being assessed when students complete this type of problem?  Students are simply asked to find the volume and show a number model.  I appreciate how the problems ask students to show their number model, but these types of problems seem to measure procedural understanding.  Do students know the formula?  Yes, well then they can answer many of these problems, even 10 in a row.

Procedural

 

I think the above problems have a place in the classroom, but shouldn’t necessarily be the norm.  Usually these types of problems are found on homework sheets.  The problem below which was adapted from a recent fifth grade test is more challenging, but gives students opportunities to showcase their own mathematical understanding and persevere.  Some would say that these two problems are completely different.  I would agree, but similar concepts are being assessed.  They do look different and the second requires more skills to complete.  Students need to be able to use their procedural understanding and apply it to the situation.  Also, one key element that’s missing from the first problem is the student explanation.  Students are required to show their mathematical thinking in the second problem.  This is big shift and can reveal student misconceptions more clearly than the first problem.  I struggled with the decision, but eventually had students work in groups to complete the problem below.  Students were allowed to use any of the tools in the classroom to find a solution.

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At first, all groups struggled with this problem.  Near the end of class all the groups presented their findings.  What’s interesting is that all the groups had different answers and ways in which they came to their conclusions.  I was able to offer opportunities for students to see and ask questions about different math strategies.  During the next class I was able to pull each group and give feedback.  This activity took a good amount of time to complete, but I feel like it was worth the commitment.

Through this experience and others I’m continuing to find that it takes a “bridge” to connect the procedural and application pieces.  At times I feel like there’s an assumption that if students are able to answer 10 similar procedural problems that they will be able to simply apply that knowledge in a multi-step problem.  This isn’t always the case and sometimes the bridge doesn’t fully form immediately.  Performance tasks, similar to the problem above can be one way in which teachers can help the transition from procedural understanding to practical application.  Being able to apply that knowledge to a math performance task can be a challenge for some students.  When teachers focus so much on the procedural, that’s the only context that students see and practice.  A blend between procedural and application needs to be established within the classroom.  I feel like activities like this help bridge this gap.


How do you bridge mechanical and conceptual understanding?

Understanding Volume

Constructing Conceptual Understanding
Constructing Conceptual Understanding

This past week second and third grade students at my school are learning about measurement. The students are making progress. The classes have become more fluent with understanding perimeter and area, and are now starting to explore the concept of volume. Throughout the process students have used various manipulatives, such as prisms and nets to deepen mathematical understanding.   Even with all the activities  some students that are still struggling with the concept of volume. In about three weeks or so students will be assessed on this particular topic. Providing extra sessions for students to develop a conceptual understanding of volume is important. I wanted to find or create a math task that gave students intentional time to review geometry and measurement terms, while at the same time allow opportunities for students to create different products. After reviewing different options I decided on having students use the project detailed below.

Students were given a full sheet of colored centimeter graph paper.  They were then asked to read through the directions.

Directions: Create a net for a rectangular prism using the graph paper provided. The rectangular prism you build should have a volume of 20 cm3. Cut out your net and build a rectangular prism using glue or tape. Write the dimensions of the prism you built in the charts below.

Looking back, it seems like there were more than enough questions about what was expected.   Students always seem to have questions when there are multiple solutions/products.  After I answered their questions I took about 10 minutes to model the activity with the students.  This was important as it cleared up expectations for the activity.  I then passed out the assignment.

Click for sheet
Click for sheet

Students then used the centimeter grid paper to create a rectangular prism net.  They then filled out the top portion of the sheet.

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Some students had to use multiple attempts to create a net with a volume of 20 cubic centimeters. This was great opportunity for students to show perseverance and find a solution that worked.  I went around the classroom and asked students questions to help them think of a solution.  The students then cut out the nets and constructed their prisms.  The students then presented their rectangular prisms to the class.

 

 

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How do you construct meaning in geometry?