Coordinate Grids and Houses

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My fourth graders just ended a unit on decimals and coordinate grids.  The unit lasted about a month and a test is scheduled for next week.  This unit was packed with quite a bit of review, decimal computation and coordinate grids.  Students were able to play Hidden Treasures, use a 1,000 base-ten block (first time they’ve seen this), and create polygons using points.  One of the tasks that stands out to me for this unit involved a coordinate grid problem.  The problem caused the majority of my class to struggle. It was a great learning experience.  Many students thought they knew the answer initially, but then had to retrace their steps.  I modified the questions a bit from the resource that I used in the classroom. Here are the directions:

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So, many students read this and thought twice as wide meaning they’d have to multiple the width by two.  Even without looking at the diagram they assumed that this was going to be a multiplication problem involving two numbers.  Below the directions came the house and grid

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Even after seeing the diagram, students were fairly sure that they just needed to multiply the width (4) by two.  When digging a bit deeper into what they were thinking I found that students were looking at the height where the roof started (0,4).  Many were absolutely certain that 8 was the correct answer.  I had the students think about the direction and discuss in their table groups what strategy and solution makes sense.  Students had about 3-5 minutes to discuss their idea.  Students reread the directions and then started to gravitate their attention to “as it is” high and then started the problem over again.

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As I walked around the room I heard students say:

What is the actual height?

Does the original width matter?

You have to multiply it by three?

What do you have to multiply?

This is confusing.

I’m not sure about this.

Is this a trick question?

I’ve got it!

How do you know?

I thought the conversation that students had was worthwhile.  I probably could’ve spent a good 15-20 minutes on just the conversation. Fortunately, at least one person at each table starting to think that multiplying the x-coordinate by two wouldn’t work.  When I brought the class back together, I started to ask individual students how their thinking has changed.  Some students were still unsure of what to do.  I brought the class back to the directions and then more students started to make connections.  Eventually, one students mentioned that because the height is six, the width has to be double that, which is 12.  Another student mentioned that the x-coordinate needs to be multiplied by three to equal 12.  More students started to nod their heads in agreement.  The class then moved to the next part of the task.

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Students said that we should write down “multiply the width by three.”  I wanted students to be more specific with this, so I asked the students if that meant that you could multiply and part of the width by three.  The students disagreed.  A few students mentioned that you need to multiply four by three and that could be the rule.  Again, I went back to the directions, which asked students to create a rule.  After more discussion, the class decided that you needed to multiply the x-coordinate by 3.   Students were then asked to fill out a table to show what the new drawing would look like.

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Students were drawn to complete the third row above.  That made the most sense to them since the first two started with zero.  Multiplying the x-coordinate by three would create (12,0).  Most kids were on a roll then and were able to fill out the rest of the table.

On Monday the class has a test on this unit.  Even after the test I’m thinking of spending some time reviewing coordinates and having students actually re-create this house on a grid.  I believe it’ll be useful as later in the year students will start to look at transformations.  This may be a good entry point to that topic.



Being Mindful of Math Manipulatives

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I’ve been thinking about this blog post for a while, but haven’t had a chance to type it up.  It’s been sitting in the forefront of my mind and I haven’t fully made up my mind about it.  So, what you’ll find below is a free-flow thought of ideas related to math manipulatives and students’ K-5 math experience.

In my current position I get to work with kids in grades K-5.  I’m able to see certain grade levels longer than others, but one my goal is to provide students with meaningful math experiences.  I use district’s adopted-texts and other resources throughout the day. Working with students K-5 is unique and gives me a different perspective as I notice how instructional strategies and resources evolve as students progress from one grade level to the next.

One thing that I notice in my current position is that math manipulatives are used in every grade level.  I believe that’s a good thing.  If you prescribe to the CRA model, then this tends to make sense.  The amount of math manipulative use significantly decreases as students progress through the grades.  There are a lot of factors to consider as in why this happens. At the kindergarten level I find that students use manipulatives (mainly counters) for a decent amount of time.  If I had to pinpoint it, I’d say these types of physical representations are being used in the classroom just bout everyday.  Counters, ten-frames, pattern blocks, dice, and other tools are used in math stations and small groups throughout the day.  What’s great is that students know that these are math tools and they’re given time to explore how they connect to math concepts.  Students are given time to “play” and make connections. Worksheets are also evident, but are generally used as students circle or cross out counter-like representations.  The worksheets are generally not more than one page, front and back.  The staple rarely comes out for worksheets at the kindergarten level.

The first and second grade math manipulatives are still used, but not as frequently as in kindergarten.  There seems to be more of a focus on using workbooks and paper-based assessments.  Part of this is systematic as more academic student data collection is used at this level.  Unifix cubes, clocks, 100 charts, base-ten charts, coins, and fraction pieces are all used with this group.  Counters are still part of this as students group numbers together more fluently.  Students take-away and add-on as needed.  Odd and even are emphasized.  There’s also a larger emphasis on data and charts.  The consumable math journal is used daily at this level.  The rise of the consumable math journal sometimes takes time away from using physical math manipulatives. This is especially evident when grade-level teams are asked to stick close together when completing lessons and assessments.  That means that teachers need to ensure that they’re at the same pace as their colleagues.  The assessments need to also be given around the same time.  Sometimes the assessments that are used are multiple pages, so the stapler definitely comes out for these grade levels – more so at the second grade.

For third and fourth grade, students start to move away from a worksheet counter to more abstract-like representations.   Multiplication and division facts tend to move from arrays with counters, to the horizontal and vertical representations that are often associated with timed-tests. Multi-digit multiplication involves using the standard-agorithm at the fourth grade level. Polygon blocks, card sets, base-ten blocks, place value charts, square counters associated with perimeter/area, and fractions are all used.  There’s a heavier emphasis on transferring students’ understanding of a representation on paper to abstract text.  Similar to first and second, assessments are all paper-based, although students are required show a visual representation to communicate their math reasoning.  Many more word problems are involved at this stage.  Teachers often have manipulatives on hand if students need to use them.  I also find that mini whiteboards are a precocious commodity at this level.  From what I see, students enjoy creating the models on the boards and then transfer their answers/work to a paper-based assignment.  The stapler is definitely used as this level.  Sometimes the assessments are 3-5 pages long and require a heavy dose of time to complete.  Grades are also emphasized at this level, which brings in a heavier focus on assessment points and growth indicators.

The fifth grade level includes a large amount of math manipulatives related to fractions. Fraction computation is heavily emphasized.  Base-ten blocks are also used for decimal concepts.  Counters are brought out to discuss proportional reasoning.  Similar to third and fourth grade, students are expected to explain their mathematical reasoning with visual models and in written form.  Sometimes I find that students work in groups together and  report out answers to open-ended tasks.  These tasks involve multiple answers with an emphasis on explaining their math reasoning.  I find that this level has more problems involving abstract problems more than any other.  Students complete most of their work in a consumable journal. The journals have increased in size since third and fourth. Math manipulates are often readily available, but they tend to be used with students that are struggling with current math concepts.  Assessments are all paper-based and are multiple pages.  As students prepare for middle school, some teachers introduce students to the idea of equations.  Mobiles and Hands-on-Equations manipulatives are sometimes used in those situations.


The above is not an all-exhausting list and include my observations.  As I write this, I’m also remembering that I forgot to include the use of number lines.  Number lines are heavily emphasized throughout K-5.  They are found in all of the consumable math journals.  Students are also expected to include number models at every grade level.

I forgot to include the role of technology with math manipulatives.  I’ve seen and used technology versions of math manipuatlives at all of the levels indicated in this post.  A digital math representation can be used as a powerful tool.

As I finish up this post, I’d like to bring up one issue that I’m continuing to observe. Across the board, I’m a bit concerned with the reluctancy to move out of the consumable math journal from time to time.  An over-reliance on using a consumable math journal isn’t the only options when it comes to engaging students in powerful math learning experiences.  I’ve always thought that math manipulates are put away too quickly.  I think they have a role at every grade level, but in an attempt to appease systematic policies, they’re occasionally sidelined and consumable journals take their place.  In my ideal world, I’d have every elementary teacher observe how math manipulatives are used in kindergarten and first grade classrooms.   I think it would give teachers a different perspective on the use and purpose of math manipulatives.

That’s just my two or three cents.

A Fourth Grade Math Routine

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I’m a fan of routines.  From waking up at a specific time during the week, to my classroom preparation – routines are part of my life.  Like most teachers, routines are a major part of a classroom ecosystem. These routines can happen anytime, but in my case they generally occur during the beginning and end of class.  I teach concepts from first to sixth grade this year and each classroom has their own math routine.

What do I mean by a math routine?  Well … my version of a math routine is the time that’s spent when students first enter the classroom.  That golden first 10 minutes of class is when I have students work on bell ringer work – aka: my version of a math routine.  Without the routines in place, I find that students are more likely to catchup with one another another and/or I get into conversations with students and time flies.  I’m all about creating a collaborative classroom and touching base with students.  Even with that said, sometimes time gets out of hand when I’m telling stories or  having a whole-class conversation. I’m then redirecting and spending time getting the class back on track.  That causes anxiety and then I feel like I’m playing catchup. Educators know how precious class time is and using it more effectively is forefront on a lot of our minds.

My routines look bit different for each grade level.  Some of the processes are standard and others aren’t.  All my classes enter the classroom, pick up their folders, and check out the agenda that’s posted .  Keep in mind that students trickle in the classroom as I take students from multiple classes.  Some students come early,  while others drop in after a band or orchestra lesson. Then, depending on the class, they have different procedures.  The procedures have changed a bit since I last wrote about this back in July.  This year I’ve started a new procedure for my fourth grade class.

Earlier in the year my fourth students worked on a Dynamath magazine and I’d review the solutions with the class.  This year I decided to change up the process since students were finishing up the magazines at such a quick pace.  So, while perusing the always great  solveme mobiles, I noticed an addition to their website.  Specifically, I noticed a new puzzle section that looked useful.

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After exploring around 10 different puzzles, I started to think of how this could fit into a math routine.  I put together a short student sheet template and introduced the students to it on Monday.

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Similar to my third grader’s Estimation 180 routine, my fourth grade students are completing one question per day.  The questions are like puzzles and involve place value and pre-algebra skills.

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I’m looking forward to using this as new daily routine with my fourth grade classes.  I’ve also been exploring the section involving coding and contemplating whether students will create their own puzzle at some point.

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This might be perfect for the Hour of Code next week.

Fraction Progress

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My third grade students have been exploring fractions.  For the past month, students have been delving deeper and constructing a better understanding of fractions. Last week, students cut out fraction area circles and matched them to find equivalent fraction pairs.

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For the most part, students were able to match the fractions to observe equivalency.  Afterwards, students discussed how to find equivalent fractions through different means.  Some students made the connection between doubling the numerator and denominator, while others noticed that they could divide to find an equivalent fraction.

Early this week, students started to place fractions on number lines.  They used the whiteboard and a Nearpod activity to become more accurate when identifying and labeling fractions on a  line.

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It was interesting to see how students showcased their understanding as the number line increased from 0-1 to 0-2, and beyond.  Giving an option for students to decide which number to use seemed to encourage them to take a risk with showing their understanding.

On Wednesday, students started a fraction task related to computation.  Students were asked to color each fraction bar, cut them out and organize the fraction pieces to complete given number sentences.  Students had to rearrange the fraction pieces and found that there were leftover pieces, which makes this a more challenging task.  You can find more information about this activity here.

This task took around a day to complete.  Students struggled at first and they used a lot of trial-and-error.  Students compared the fractions bars and switch the pieces around quite a bit before taping down the sum.  A few students needed a second attempt to complete this.

On Friday, students used polygon blocks to show their understanding of fractions.  Using polygon blocks, students were asked to take one block and label that as 1/4, 1/2, 1/8, or 1/12.  They then combined at least three different blocks to find a sum of 3 1/2.

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Students used whiteboards and geometry blocks to combine the fraction pieces.  I observed students using different strategies to combine and then take away blocks to find the sum of 3 1/2.

Next week, students will investigate the relationship between fractions and decimals.



Area and Volume Skills

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My fourth grade students finished up a unit on volume about a month ago.  This past week I gave the same group a cumulative assessment on the first two units.  After grading the assessment I started to notice trends.  Many students had issues with converting square units to cubic units.  Students also mislabeled units related to measurement.

This is my first year using a new version of a district-adopoted math resource.  This year’s scope-and-sequence had students encountering area first and then volume was discussed in a completely separate part of the unit.  I believe that isolation made students think that problems in the different sections were either 1) related to area or 2) related to volume.  The assessments that I graded indicated that students needed some bolstering in applying area and volume.  Combining them would be a bonus.

Early this week I came across Graham’s Tweet about test questions.

I clicked on the article and found some amazing questions.  I definitely geeked out after trying out a few. These types of  questions made me think beyond one math skill or idea and I thought it would move students in that direction too.  I decided to use the area question with my students.

You see, in the past students have been given the length, width, and height, and then asked to use a formula (often given to them) to find the volume.   In this case, students were given the area and had to use that to find the side lengths.  This type of task in Graham’s Tweet was definitely different problem for them.

I gave each student a copy of the sheet and had them work on it individually for about 10 minutes.  Students initially thought of adding all the area sides together, but then they realized that adding them wouldn’t help in the process.  I redirected students to look at what the question is asking.  There were a few minutes of frusturation as students were looking for ways to find the length, width, and height.

Students were then put in groups to work out the problem.  Eventually, students started to think of factors as they started to investigate numbers that  work for the length.  Some went the route of using a factor rainbow, while others used a trial-and-error method.

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More frustration ensued, but students still moved forward.  A few groups were confident that they had a solution.  I briefly looked over all the responses and saw that no one had the correct answer, but I pokerfaced it and had the students work it out at home.  That evening some of the students tackled the problem and came back with a solution.  I was impressed with the perseverance and also how these students applied their understanding of area to find volume.

Conjectures and Math Arguments with Fourth Graders

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My fourth grade class just started a unit on division and fractions. Students started out the unit by observing the relationship between fractions and division. This was a realization for some students as they perceived fractions as an isolated topic, separate from the division operation.  From there, students moved to adding and subtracting fractions. The class completed an Open Middle problem on Wednesday.

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Last year, students were introduced to this computation skill and many were comfortable transitioning that skill to more complex fractions.  I’d say the majority of my students completed problems using a common denominator strategy.  Very few used a visual model.  For the most part, students were able to find solutions to the computation problems.

The next day, students experienced the idea of using mathematical reasoning to create conjectures.  This was brand new for the students and I recently read about how to use conjectures in class from Tracy Zager’s book.  The class discussed very simple conjectures, claims, and arguments.  After about ten minutes, a student mentioned that someone is being a “math lawyer” when they go through the conjecture, claim, and argument process.  I think the whole class laughed at that response, but agreed.  I then helped model a few different conjectures and clarified the definition.  Students then worked in table groups to role-play how to defend their claims.

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Near the end of class, I was able to have students work on a page related to fractions and conjectures.  Each table group 1) looked at a specific fraction computation problem 2) each student added their own view point of whether they agreed or disagreed 3) the groups came to a consensus and wrote their argument to prove their case 4) students wrote their arguments on the whiteboard.

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I was impressed with how students started to improve the clarity of their mathematical writing during this process.  This is an area of growth for all of my classes.  Writing in math class doesn’t happen as often as I’d like.  I’m hoping to complete more conjecture, claim and argument activities as the year progresses.

Blueberry Cereal and Ratios

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One of my classes is in the midst of a unit on ratios.  Students have been exploring equivalent ratios and using visual models to compare them.  For the most part, it has been a great journey and students are making connections.

One of the common themes of this unit is for students to apply their understanding of ratios in real-world settings.  On Thursday, I introduced an activity about cereal and flavors.   This was my first time using this task so I wasn’t quite sure of what to expect.  Basically, students are given four different cereal recipes.  Each recipe has a certain number of cups of blueberries and cornflakes.

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The ratio is different for each cereal.  For example, cereal A has 1 cup of blueberries and 2 cups of cornflakes.   Students are then asked questions related to doubling, halving, making additional cups, and finding equivalent ratio recipes.   Students seemed to struggle at first and then created models, mostly tape diagrams, to find solutions.  Questions about making a certain cereal taste more like another were also part of this task.  Students had to stretch their understanding of ratios and apply it here. What I found was that some students struggled through this process.  Through the struggle, they moved towards using pictures/models to help themselves.

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This was where I stayed back and let students work.  That was tough. At times, I asked students questions to help move them towards validating their reasoning.  The application piece of taking ratios and using that understanding to see whether a cereal would taste more like blueberries or cornflakes was challenging for students.  They had to perceive the problem in an abstract sense, and then bring justification to why it would taste a certain way.

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One of the more open-ended problems involved a taste test.  Students were asked to create a cereal with a  stronger blueberry taste than all the other cereals.  The kicker was that the new cereal couldn’t have more blueberries than cornflakes.  So, students were required to analyze all of the ratios and create a new ratio where the blueberry taste was more evident.  I observed that students tackled this problem in a couple different ways.  Some students perceived the problem as adding more cups of blueberries than any of the other cereals.  This caused an issue as the ratio wasn’t dependent on the amount of cups, but the actual ratio.

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Compared to what?

This was a red flag to me as students were looking at the value, not necessarily the ratio.  Other students took each ratio from the initial cereals and converted them into fractions.  They then created a fraction that was closer to one compared to the other cereal. This was used in their justification.

This task seemed like a worthwhile activity.  I’m looking forward to what tomorrow brings.