# 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.

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.

Students in third grade are exploring measurement this week. As students progress through the unit I feel as though they are becoming more efficient in converting Metric units. Near the end of the class today students started debating the differences between US Customary and Metric. The class than started completing an activity where they had to measure different insect lengths.  Students worked in groups to accomplish this task.

During this time I traveled to each group and intentionally eavesdropped on the conversations. Students asked me questions and I listened and asked questions back.  I then moved on to the next group. I wanted the students to work together and persevere. Some students started to talk about the measurement of different objects around the room.  I especially paid close attention to the questions that students were asking each other. This was a great opportunity to check-in on some of the misconceptions that were flying around the room.  I jotted down some of the conversations as the students came back to the large group.

We had around five minutes left in class to review the questions that I noted. I wrote the questions that I heard on the whiteboard.  I was able to clarify some of the responses and answer other questions. This time was definitely worthwhile. The students seemed to appreciate the time as well. During our next group activity I’d like to do something similar, but not completely rely on my less-than-stellar eavesdropping skills. Instead, I’m thinking of having the students periodically use Post-it notes to ask questions. This could turn into a “wonder wall” type of activity. The students could then place the questions in a bin and we can review them throughout the unit. I think this type activity is one way to proactively address misconceptions and answer questions as students grow in their mathematical understanding.

# Using Multiple Strategies in Math Class

Last week a few upper elementary classes started to explore different methods to divide multi-digit numbers.  Many were familiar with repeated subtraction for numbers more than one and some even had experience with the long division algorithm.  I asked students to explain their reasoning for the steps needed to divide numbers using repeated subtraction and long division.  All of the students were able to explain the reasons for using repeated subtraction. Students gave quality answers and were able to communicate why each step was performed. Some students even related repeated subtraction to a number line.  I then asked the students the reasoning for using the long division algorithm.  I heard responses like “it’s quicker” or “that’s what I was told to use” or “you just do this and this.”  I could tell that there was a disconnect between the shortcut and having a conceptual understanding. Students understood the steps but couldn’t provide solid reasoning to why you would bring down the next number.

The class then had a conversation about the importance of being able to clearly explain their mathematical thinking. The students that knew how to use the long division algorithm were getting correct answers, but couldn’t tell me why.  Blindly following procedures can lead to holes in understanding.  Explaining the reasoning behind completing a problem is important. Honestly, I don’t mind if the students use algorithms like the above picture if they already have a descent understanding.  The problem I have is that sometimes this is the only way students are taught how to divide large numbers.  The problem becomes steps –> answer without understanding.

After the class had a discussion about long division we explored the partial quotients algorithm. I explained to the students that this was another method to divide larger numbers.  As we  progressed through and explained the steps, students became more aware of how partial quotients seemed to make sense to them.  For many, this was the first time exploring the partial quotients algorithm.  The students were able to explain why each step was taken in the process.

I believe some of the students were also relieved that they didn’t have to get the partial quotient “correct” the first time.  By breaking apart the problem students were starting to see the correlation between repeated subtraction and the partial quotients algorithm.  More importantly, students were able to explain their reasoning for completing the problem with this method.

As we move forward, I feel as though students are becoming better at explaining their mathematical thinking.  It doesn’t matter to me what method the student decides to use in the future as long as they are able to justify their reasoning.  This thinking could also lend itself to just about any type of computation skill.  Last week reminded me of the need to expose students to multiple strategies to complete problems.  Providing these strategies can assist students in becoming better at explaining their mathematical thinking.

# Math Genius Hour – Research and Presentations

This is my first year using a math genius hour model. My third, fourth and fifth grade classes all started their genius hours at the beginning of September. The beginning of our journey can be found here. Students narrowed down their question and have conducted research over the past month. The research process has been an eye-opening experience. Before beginning, I was able to set aside some time to have a conversation with students about finding appropriate resources for their project. Even though the classes were during the math block I thought discussing this was important, especially if we’re having more than one genius hour per year. I thought that having the conversation would pay a few dividends later in the year.

The majority of the research will be conducted online. The class discussed the importance of reviewing the ending of website addresses. We reviewed the url ending (.gov .edu .org .com .net) and how to conduct research in an effective and meaningful way.

We analyzed different red herring websites (1, 2) and I believe students are getting better at identifying sites that seem legitimate. This took a large amount of time and many questions were asked.  I feel like an entire course could be dedicated to this topic. After a while, the class and I created a sheet that the students would fill out to organize their sources.

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Although my district provides a facts database for students (www.facts4me.com), the majority of the research that needed to be conducted was beyond the site. So students began to explore research outside of the box. I soon found out that many students were under the impression that they could Google their question and use the first link that appeared. Students also found that Yahoo Answers wasn’t necessarily the best source either. Through a good amount of exploration, students found sites that were adequate and provided legitimate information that they could use in their project. The students became much more independent once they understood the research parameters.

At this point students are starting to explore how they will present their project. Last week the classes took time to review different presentation tools. Many of the students used a variety of presentation tools last year so they were fairly comfortable in picking a tool. Eventually the class decided to use the sheet below to help make an organized decision.

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After students pick a tool they will start creating their presentation. Some students are at this point, others are not. My fifth grade classes helped create a rubric that students could follow. I wanted the rubric to be flexible in allowing students to present in a way that they wanted yet a minimum criterion was established. I wanted to also make sure that students’ creativity and voice were part of the presentation. A self-reflection piece is also incorporated into the rubric.

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I’d like to thank Denise Krebs and the genius hour Wiki contributors for the self-reflection sheet. Addition resources on genius hour can be found in Joy’s genius hour Livebinder site.

Overall, the math genius hour is a work in progress and I’m assuming the students will present at some point in December. I continue to look forward to how this project progresses throughout the year.

# Exploring Rates – Part Two

One of the primary goals this year is to find opportunities to use rates in a more practical setting. This past week my classes started to explore rates in more detail.  Students studied visual patterns and rules last week and this has led up to using/converting rates with formulas. While planning I dug out a rate activity that I used last year.

So on Monday I took masking tape and made a simple racetrack around the classroom. The track measured approximately 62 feet. Students took turns timing each other and documented how long it took them to quickly speed-walk around the track twice. We used an online stopwatch to time each student as they sped around the circuit.

Students documented their time and started to fill out the sheet below. The sheet is an upgrade from last year and I feel like it addresses more skills.

Click for Word file

Students were asked to convert their time into feet per second and then how many feet would be traveled in one second. After the feet per second conversion, students converted the seconds to minutes. I gave students an opportunity to find this conversion by exploring and then checking their work.

Similar to last year, this section was challenging for many. Understanding that 1.8 minutes isn’t 1:08 or 1:80 was addressed. I was proud to see students use perseverance to work through this section and use formulas to find solutions. The last section on the front asked students to find how long it would take for students to walk one mile at the 124 foot pace.

The most challenging part of this was converting the minutes and seconds to actual time.  Once students understood the formula they became pros, or at least closer in understanding rates. Some even found how long it would take them to walk 5 and 10 miles.  We shared the data as a class and found that our times per mile ranged from 9:45 – 16:00 per mile.  Then students graphed the information on the backside. I actually thought of using a graph after reading through Fawn’s visual patterns template sheets.

Afterwards, the class had a conversation about all the different math skills that were utilized while completing this activity. We made a list that included conversions, formulas, graphing and many others. I’m hoping to reference this activity and concepts experienced throughout the year.

# Transitioning to the Standard Algorithm

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.

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.

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 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.

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

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?

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.

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

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.

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?