# 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.

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.

# Visualizing Fractions

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

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.

# Coordinate Grids – Part Two

Last week my students started to plot points on coordinate grids.  They were identifying different quadrants and becoming more confident with drawing shapes on the plane. While reflecting on last week’s activities I noticed a Tweet that was sent our replying to one of my blog posts.

I’m a rookie when it comes to Desmos.  Most of the stories I hear involve middle or high school students. I needed to find something that worked with my elementary kids.  So I started to research and did a little bit of exploring to see how this could be used with my third grade class.  I ended up looking up some of the templates but had a bit of trouble finding an extremely basic rookie-like coordinate plane activity for my students.  I decided to go the route of creating a template and having  students manipulate created points for a project.  Click here for the template.

I quickly found that students had no idea how to use Desmos.  I gave the students 5-10 minutes to orient themselves.  Students were asked to move the points to certain coordinates  on the grid.  As they moved the points students started noticing that the tables on the left side of the screen changed.  Students started connecting how the tables changed and this helped reinforce concepts learned last week.  After this introduction time, students were given a rubric that contained the following:

• Move the points on the grid to create two angles
• The angles need be located in two different quadrants
• The angles need to be acute and obtuse with arcs located in each one
• Indicate the measurement of each angle

Students were then given 15-20 minutes to create their projects.

Students created their angles by moving the points around the grid.  Students then shared their projects with the class.

Students took a screenshot and then added the degree measurements to the angles.  The class reviewed the projects and students explained how they plotted the points.  This project seemed to help students make the connection between points and the x and y-coordinates.  It also reinforced skills related to angle classification and measurements.  I’m looking forward to expanding on this project next week.

# Exploring Coordinate Grids

My third graders started to explore coordinate grids this week. For many, this was the first time that they’ve used them. Some of the students have played Battleship or some other game that involves a grids.  Playing off that background knowledge, I used a road map to show how people can find certain locations by using a coordinate grid. This made sense to some of the students but a few still were unsure of what axis was used first to determine where to plot a point.  This was a reoccurring theme throughout the lesson.

During this process I remembered a strategy that another colleague suggested a few years ago. She borrowed the idea from another teacher and it seemed to work well in her classroom. A colleague of mine used (3,2) as an example of the “go into the building” – first number (right 3) and then “go up or down the elevator” (up 2) method. I decided to use that strategy and a few more students started to grasp the process.  The next activity in the paragraphs below seemed to solidify a better understanding for the rest of the class.

Earlier in the day I created a very short Nearpod lesson involving mostly pictures of coordinate grids. I handed out a iPad to each student. Students logged in and given a picture of a grid and asked to draw and label points.

I then revealed the pictures to the class on the whiteboard. The names of the students were hidden so that we could analyze each response without throwing judgement lightning bolts towards a specific individual. As the class went through each picture they started to notice trends.

• Some were switching up the x and y-axis numbers
• Some were not creating a point
• Some were not creating a letter for the point
• Some were confused by the negative sign in front of the numbers

Students observed these issues from the first question and grid. After a decent discussion on the above trends, the class moved towards the second grid and question. I gave the students that same amount of time and the results seemed to initially improve.

Students started to become better at finding their own mistakes before submitting their creations. I used the same strategy as earlier and displayed the results to the class. There were a few that had some of the same misconceptions, but not as many. In fact, many students vocalized the class improvement since the last question. One of the evident misconceptions revolved around students having trouble plotting negative numbers on the coordinate grid. The class discussed this and completed the third question and grid. The student responses from this question were much better than the prior two. Students were starting to develop some true confidence in being able to correctly plot points on a coordinate grid. I kept a list of the trends that students noticed and will bring it out later in the unit as we’ll be revisiting coordinate grids next week.

After our Nearpod lesson (which was about 15-20 minutes) students played a Kahoot on identifying points on a coordinate grid. I felt like this was helpful as students identified the points and were able to gauge their own understanding compared to the goal.

# Minecraft and Math

Earlier in the school year a group of three teachers at my school wrote a grant expressing the need to incorporate Minecraft in the classroom. The idea actually started last summer when a colleague and I attended a professional development event in Downers Grove. During one of the sessions I met two teachers from nearby school districts that used MinecraftEdu in a school club. What they had to say caught my interest and two other teachers and I decided to start a school club in 2016. We wrote the grant and it was accepted. Last week the licenses were purchased and I’ve explored the potential of using the program in the classroom setting.

Before the school year started I knew very little about how to use Minecraft. I decided to purchase a copy and explore the Minecraft world over the summer. I quickly learned the controls and watched a number of YouTube videos to become a better rookie. I’m still a rookie. I found the MinecraftEdu community online and started posting questions to the forums. Moderators answered my questions and I started feeling more comfortable using the program on my own. The forum has been especially valuable in giving me ideas to use in the classroom.

I downloaded a few world templates and started brainstorming. I then bounced a few ideas off of colleagues and decided to start using the program for a math scavenger hunt. The goal was to have students get used to using the program in an education setting while reviewing fraction math concepts in the process.  Most students already understood the controls and the game but weren’t used to using it for a different purpose.  I wanted to start simple and I thought a scavenger hunt would be an easy way to start incorporating the program in my math class.

Math scavenger hunt – third grade

Students entered into the fraction world that I created.  Once they entered into the world I froze all of them. I explained the goal of the world and answered questions. The goal was to explore the world and find the signs that were posted. Students were using the MinecraftEdu version where they weren’t able to build or keep inventory of items. Trap doors, caverns and bridges were all part of this simple world. Each sign had a particular math problem on it and students were expected to solve the problem. I then passed out a sheet that went with the scavenger hunt. The sheet had spaces for students’ number models and solutions.

I then unfroze the students and they were off to the races. Students split up and started exploring the area. They soon found that working in teams seemed to be more efficient in finding the signs. All students were finished with the scavenger hunt in 30 minutes. Afterwards the class reviewed the answers.

I created a completely flat Minecraft world for this activity. Students were grouped into teams and given a task related to concepts that we’ve been discussing. The fifth grade class has explored area and perimeter and will eventually be investigating volume in January. Each group was asked to create a building that met a certain criteria. It was stated that each Minecraft “block” was exactly 2 feet on each side. Those measurements were used to meet the criteria.

Students worked together and started building their houses. A few groups had to restart as they found out that the perimeter and area didn’t meet the criteria. After around 30 minutes students are about 50% complete with their houses.  I’m assuming that another 30-40 minutes and the students will be finished with their projects.  At some point after break the class will be presenting their buildings to the class.

In January my school will be offering a Minecraft club to around 25 elementary students. We’re planning on building our actual school from scratch using some type of scale model. The students are already excited to be using this program in school and I’m looking forward to what students create and the process involved in that creation.

# Math Intervention for Enrichment

This school year I’ve been given the opportunity to work with a select group of second grade math students.  Since early October I’ve been seeing two groups of around 20 students for approximately 30 minutes twice a week.  These 40 students were selected based on unit pre-assessment scores and teacher recommendations.  The second grade students that I see tend to be in need of enrichment of the math skills that they’re exploring in class.  This enrichment can take on many forms, but mainly I’ve been looking at have students develop a better understanding of numbers and patterns.  I’ve been asked to expand on the unit being taught in class and report back progress that students have been making.  The groups that I see are designed to be flexible and change depending on a particular math unit.

Here area  few things I’ve observed as the year has unfolded:

1.)  30 minutes twice a week is a short time period.  I’m all for packing in as much instruction as possible, but 30 minutes goes by very quickly.  I’ve had to redesign many of my lessons to overlap the two days in a week.  Retention can also be an issue with this.  I spend each session with a bit of review and that has seemed to help.

2.)  I’ve had to incorporate my own pre/post-assessment to show student growth.  At first I thought this was extremely time consuming as students only have a small amount of time in my class and I want to make sure that the class time is being used appropriately.  This year many of the classes in my school are using the same pre-test as the post-assessment.  I’m using that model right now but it may change as the year progresses.

3.)  I’m not able to meet with the second grade team every week so we decided to use Google Docs as a communication tool.  My students’ pre/post assessment scores are located in the shared doc and can be assessed by any of the second grade teachers.  I also attached a copy of the pre/post assessment to the document so teachers are aware of what topics I’m addressing.

4.)  I’ve been using effect size to show student growth.  I learned about effect size in more detail after attending a Visible Learning conference over the summer.  I feel like this has been a useful tool and has shown some insight into student gains in my class.  This tool has also been important as it brings some finality to the units that I teach and can be used as one data point in transitioning students in/out of my class.

5.)  Student reflection is key.  This year I’ve been giving students a copy of their pre-assessment stapled to their post-assessment.  Students are then able to review their growth and ask questions.  The focus is on student growth and not necessarily on point value or grade.  Thankfully at second grade students aren’t used to traditional grades yet.

I’m looking forward to seeing how this enrichment opportunity develops over time and the positive impact it has on students.

# The Power of Visual Models

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.