A Week of Equation Exploration

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My fifth graders started to explore equations this week.  They’ve created number models and solved for the unknown, but most of their experience has been using one method.  They tend to substitute a number in for x and then check their answers.  If it doesn’t work then they guess a different number.  This guess-and-check type of of strategy has worked well in the past with 1-2 operations and with x on one side of the equation, but this unit that I’m teaching starts moving students towards using a more formal substitution method.

So, in an effort to improve students awareness of equations I decided to use a few specific activities.  My intention was to give students an opportunity to see equations in many different settings.  I started off the week with a few Nearpod review questions related to order of operations.  The class worked in groups of 2-3 to solve the problems.  Students definitely needed a review on this topic because it seems like forever since they’ve completed problems like this.

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The next day students used SolveMe mobiles.  I drew a balance on the board and the class completed a few different examples.   Students worked in groups to find out what each shape represents.  This class used these types of mobiles earlier in the year with a certain degree of success.  This particular math unit will put the reasoning behind these mobiles into a better context.

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How much is the triangle worth?

Students were given homework that night related to equations.  After checking it over I noticed that students needed additional practice with the properties of numbers.  Specifically, students were struggling a bit to identify the correct property.  Students completed a few different problems involved with properties the next morning.

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I reviewed the terms with the class and connected them to what happens when a variable is substituted for a number.  Students were making progress.  They were continuing to use the guess-and-check substitution method and checking their work to see if they’re correct.

Next week, students will start to investigate inequalities.  This is one of my favorite lessons as students observe that there can be multiple answers for an equation.  While some students are  stoked to learn about this, others get confused.  At this point, many students have been conditioned to look at equations as problems that have one solution.  Having multiples solutions, or solutions with a specific range of numbers isn’t usually the norm at the fifth grade level.

While looking for a few new ideas I came across Always, Sometimes, Never.

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I’ve heard of ASN, but haven’t had a chance to try it out in the classroom.  I paired up students and modeled one of the solutions.  Students were off to the races to think about statements and label them as always, sometimes, or never true.  The discussions about numbers were fantastic.  I went to each group and asked questions to help direct students towards possible solutions.  While this was going on I could tell that students continued to have questions.  These questions impacted whether a statement was sometimes, always, or never true.

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The discussion that stemmed from the above questions provided an opportunity for students to discuss their understanding of numbers.  Overall, this discussion, along with the previous activities will help set the stage for students as we continue to discover and solve equations.

Next week, students will  use the distributive property to solve equations.  They will also delve deeper into a study on inequalities and how they’re represented outside of the classroom.

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Moving Towards 2018

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It’s the last day of 2017 and I’m in reflection mode.  It’s around two degrees outside right now and I’m planning on staying inside with a warm cup of coffee.  By the time I’m done with this coffee, I’m hoping to have finished up this post.  This is allowing me some time to look at what I can change and keep the same for next year.  It’s been a year of ups and downs and many different commitments along the way.  I’ll be splitting up this post into ideas that I’d like to continue and some possible additions for the new year.

Continue

  • I’d like to keep my arrival and flow chart process in place.  I’ve been using this process with K-5 this year and so far it has been working fairly well.  Over the years I’ve noticed that the first five minutes of class are golden.  Getting the students thinking about math quickly after they enter the room is an important piece and is also helps get the class moving in the right direction from the get-go.
  • Using WODB, Estimation180 and Scholastic math magazines for my bell-ringer work.  I believe these daily tasks are beneficial to students and gets them thinking about math in different ways
  • I’ve used a daily agenda this year.  It’s visible to students as they enter the classroom.  I’ve used a slide made in PowerPoint or Keynote and it includes all the daily activities for the class.  I believe it has eliminated a lot of the “what are we doing today” questions that I’ve heard before.  It doesn’t eliminate all of them, but I’ve noticed a huge reduction. Students can take a brief look and get a general idea of the tasks that are planned out.  I’d say that the class rarely makes it through everything on the agenda, but it helps keep the students (and teacher!) aware of today’s happenings.
  • I want to continue to have a balanced instructional approach in the math classroom.  I tend to use bell-ringers and have certain math routines that stay the same, but changing up the lessons and tasks has benefits.  Designing lessons throughout the week that has students working with partners, group conversations, including technology components, and having whole-class conversations tends to help students encounter math in different ways.  It also adds an unexpected element that students sometimes need at this level.
  • I’ve been using a digital planbook this year.  It’s been a great way to plan out lessons  away from school.  Also, it has helped me leave school at a decent time this year – a struggle many teachers have.  Being able to create the lessons and then copy and paste the lesson into my agenda slides have been an efficient process year.

Changes/Additions

  • I’d like to be more intentional in planning out my math questions during lessons.  Creating questions that are open-ended, yet give students time to truly think about mathematics in multi-faceted ways can be challenging.  Depositing a question in a specific place within a lesson can yield dividends later on in the lesson and throughout the year.
  • I’d like to actually use my planning time for planning purposes.  That sounds odd while I’m writing this down.  Like most teachers, I have a certain amount of time that is deemed for “planning”, but I tend to not use it for that purpose.  Generally I use it to check emails, copy, call parents, or check-in with other teachers.  Ideally, I’d like to use it for planning out or modify my upcoming lessons.  I think this is more of an effort on my part to use this time for actual planning.
  • When planning out my lessons I’d like to add more of a cyclical design.  Lessons are usually designed to meet one specific mastery objective.  This is often required at certain schools/districts and is part of the evaluation. The assessments and tasks are related to that one objective.  I’d like to include more opportunities for students to review past concepts.  This also moves students away from thinking that “fractions are done” since we finished a unit on that particular topic.  Having a revision review is such an important topic and I feel like I could write an entire post on just that topic.   I’m continuing to look for ways to make a 2-3 times a week commitment for this purpose.
  • I’d like to commit to being more aware of what is being taught to students after they move on from my classroom.  It shouldn’t be a mystery to what I’m preparing students for, although there’s sometimes a disconnect between what happens at a 5th grade level and in middle school classrooms.  I’d like to check out how the standards that I teach connect to what students will experience in 7th and 8th grade classrooms.

Side Notes:

  • I’m currently in the process of getting through module 3 of my NBCT certification. Watching yourself teach is a bit cringeworthy, but I’m making progress with the editing.  If everything goes well, I should be getting my credentials next December.  This seems like a long way off and I know that there’s a lot of work that needs to be done before then.  Also, I’m hoping to connect, learn and share with my PLN during the NCTM conference in April.

I’m looking forward to 2018.  See you next year!

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.

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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.