Angle Relationships


My fourth graders are just about finished with their unit on geometry and measurement.  They classified angles earlier in the week and are now looking at angle relationships.  This is one of my favorite topics to teach as it involves logic and an understanding of basic geometry.  I’m finding that students are becoming better at measuring angles using a protractor.  Using Angle Tangle has helped in that process. They’re able to identify and measure acute and obtuse angles comfortably.  Reflex angles still give them issues, although this is improving as students are able to subtract an acute or obtuse angle from 360 to find the measurement.

Students then moved on to angle relationship skills. When asked to find the missing angle in a triangle they immediately started to look for their protractor.  Students wanted to find the actual measurement without looking at what types of relationships actually exist and if a protractor is needed.  So on Tuesday the class reviewed interior angles.  Students found through patterns that they could split a convex polygon into triangles and find the sum of angles.  This was eye-opening for some students and you could tell that they were relieved in seeing that they wouldn’t have to measure all of the interior angles.

One of the assignments called students to create  polygon and find the sum of angles without actually measuring each interior angle.  Some students were stumped while others students looked at how a triangle’s sum can aid in finding the sum of other polygons.  The student projects turned out well, although some had to redo them as the drawing actually started to get in the way of creating triangles.  This is one of the better projects.



I could tell that students needed a bit more practice with using angle relationships to their advantage.  On Thursday I asked students to create a qudrilateral using a straightedge.  Students drew arcs to indicate the angles on each vertex.  The quadrilaterals were cut out and the sides of the shape were torn off.  Students lined up the sides and the class had a brief discussion on what they noticed.


Right away, some students noticed that the arcs didn’t line up.  They also noticed that the four corners actually created a circle. Some even said that the total was 360 degrees. Students checked their work by using a compass to add all of the angles together.  Their prediction rang true.  This was a winning moment as I could tell that students were starting to grasp this concept better.  I gave each student some tape and they tapped together their circle to their folder.  I’m hoping it stays on their folder and in their memory banks.




What’s My Rule?

My third grade class ended their unit on data analysis and computation last week.  We’re now onto our next adventure of exploring patterns and number rules.  This last week the class started to identify number patterns.  The class observed how they could develop rules to find the perimeter of connected squares.  This was a bit of a challenge because students had to combine two different operations to find the actual rule.

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What’s the perimeter for three connected squares?

We used this activity that I discussed a bit more in detail last year.  They looked for consistency and investigated with trial-and-error what the “rule” might be.  The class used a Nearpod presentation to see how a function machine transforms numbers.

Click for presentation

Eventually the class moved towards creating their own rules using dice and a whiteboard.  It was during this time period that students started to dig a bit deeper into how rules impact a table.


One issue came up with the consistency of the numbers on the “in” side of the table.  A few students were confused with the idea that numbers didn’t necessarily have to be in order on the “in” side of the table.  A few examples helped address the issue but I thought it was interesting as most students are so used to a specific 1:1 scale.  I wonder if this is something that’s emphasized more at the second grade level and it just continues with our third graders.

Later in the week I brought out a digital function machine.  The kids had a great time placing numbers in and watching at they transformed into something different.



I highly recommend the PheT simulations. Feel free to check out other simulations that they’ve developed.  Next week the class will be working on creating and identifying true or false number sentences.

Data Landmarks and Spreadsheets


My fourth grade class reviewed data landmarks this week.  On Monday the class explored examples of the maximum, minimum, median, mean and mode.  I had to review the terms multiple times throughout all of Monday.  Kids kept on asking about the difference between median and mean.  During this process I was finding that students needed additional practice with the terms.  They seemed to need another way to remember the difference between the data landmarks.  After contemplating a few different review lessons I decided to check out my school’s laptops.  I vaguely remember reading about a teacher that used spreadsheets to reinforce math terms.  I decided to go that route for Tuesday.

So Tuesday arrived and students received their laptops.  I modeled the different components of Excel.  This took more time than I thought it would.  I reviewed the idea of a cell and the components of a spreadsheet.  During this time I had a lot of hands fly up in the air with questions.  The questions revolved around how to change the column/row size, what a cell is, where’s the formula bar and many others.  To get the ball rolling I had the students take some personal data and use it for this project.  The class formatted the spreadsheet and we were about ready to start putting in formulas and then … class ended.

We started back up on Wednesday and began the lesson by explaining how to use formulas in Excel.  I modeled the first formula of how to find the maximum of the data set =maximum(b2:b14).  Students followed the example with their own data.  We then moved on to minimum, which they easily constructed.  Median and mean were a bit more challenging but the students explored and found the formulas using the first example. The magic started when students were asked to manipulate the data in the non-formula cells.  Students started to observe how the data landmarks change when the data changes.  This sparked a classroom conversation on the difference between the mean and median and which indicator might represent the data better.


Afterwards, students were able to print out their creation and take it home.  The class will be discussing this in more detail next week.


Meaning and Algorithms

My third and fourth grade classes are exploring computation algorithms this week. Taking apart and putting together numbers is one of the skills needed to tackle higher level math concepts. I find it interesting when this topic is brought up as I sometimes hear students commenting that they already know how to add/subtract. Many of my students have been instructed to use the traditional algorithm to add large numbers.  After digging a bit deeper I started to see students falter with their explanation of the traditional method.  I asked them to explain their thinking. I found that their reasoning started to turn into comments about the process. Students explained the steps involved to find the product.  One student told me about the example below.


The process, although important, doesn’t reflect mathematical understanding. Students didn’t mention place value at all.  Place value within the traditional method is evident, it’s just that the students weren’t able to explain it with confidence.  It reflects more of an understanding of the procedures required to find a solution. Most of the students that I see are used to adding/subtracting numbers on a number line. They can hop up and down the number line with ease.  Once they feel comfortable with that, parents or others start to show students the traditional algorithm. Students become familiar with the terms carry, borrow, cross-out, add a zero, and others without necessarily knowing the reasoning behind the process.

This past week I introduced the partial-sums algorithm. This is one of my favorites as students start to see the reasoning behind the process that they’ve been following for years.  Although at first it can seem clunky, students started to see how place value can be a key indicator in determining the sum. That’s an #eduwin in my book.


Later that hour students were given an assignment where they had to find the sum using the partial-sums method. Some students struggled a bit to move out of the traditional algorithm process and move to a different algorithm. Afterwards I gave student the option to use whatever method best meets their needs. It’ll be interesting to see the progression that students make as they decide on an algorithm. When asked about how and why the algorithm works I’m hoping they’re able to confidentially create their own proof.

We’re making progress!

Math Puzzles

My students enjoy puzzles. It often doesn’t depend on the type of puzzle. They like the trial-and-error of attempting to find patterns and eventually solutions. In math class these puzzles take on many different forms. I believe that patterns and puzzles play an important role in the math classroom. Some skills are more aligned with using puzzles than others.

Puzzles offer learners multiple entry points. Students have the option to look at a puzzle and decide to start in one section while another student decides to start in a different section. One puzzle that’s been thrown around the Internet is below.


I gave this puzzle to my fourth grade students. Students started to calculate how much each horse was worth and worked from there. I had other students that immediately looked at the horseshoes and found that the total worth would have to be divided by two. After working with each other the students then conversed in groups. The discussion was fabulous. Students started to identify where a mistake was made and corrected their papers as needed. During this time students were engaged, using math vocabulary and practicing skills that they will see again throughout the year. What I found interesting was that zero students had the correct answer the first time. It took perseverance as well as a thorough amount of collaboration to get to a consensus. The class had a conversation afterwards.

The key I find is connecting the puzzle to the skill/standard. Afterwards, students understand that this was a fun problem, but was the puzzle connected to a certain skill? Connecting the puzzle to prior skills not only shows how this fits into a continuum, but also gives students a picture of what skills are being addressed. Maybe the skills can be introduced after the lesson. I know that the exploration that students participate in is a valuable piece in the learning process. What are students exploring as they unpack the problem? The objective isn’t to just solve the problem. The bulk of the student learning experience is using the substitution process to find a solution.


After discussing the solution I drew the picture above on the whiteboard. Through this I attempted to bridge the puzzle to more of an abstract model. This made sense to some students while others were debating on whether this matched the earlier puzzle. Regardless, the transition seemed beneficial in having students use substitution to find a solution.

Next week the class is tackling a different problem.


I’m already looking forward to using this with my students.

Evolving Teaching Strategies


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This past week I was traveling and was taken off guard when I saw a phone station. At first glance I thought this was a modern day phone booth. I looked for the phone, credit card reader, directions and buttons but couldn’t find anything. I actually looked a bit closer at the parts. There were only two.  There was a small partition and place to put your phone as you speak with people.  After a few weird looks from commuters I concluded this was a modern day phone station/booth.

It had me thinking of how the phone booth has evolved over time.  It had a purpose back in the past and seems it still has one now.

Examples of how products have evolved over time can be found just about everywhere.  This can be true for products as well as strategies/processes.

I believe the same can be said about instruction in schools. New research can impact the strategies that teachers use in the classroom. These strategies have also evolved over time. Marzano and Hattie are just two names out of many that have impacted the field of instruction and teaching strategies.  Some strategies have been proven to be more efficient than others.  Books, articles, administrators, coaches and other professionals often impact what new techniques educators utilize. How students respond to those strategies is important. Some of the strategies I used when I first started teaching continue to work and others were cut after the first year. Educators reevaluate tools and techniques in their classroom. I believe this reevaluation is a form of evolution that comes with experience and betters teachers and their students over time. Some of the strategies that I use stay the same from year to year while others change. I question some of the tools and strategies that are used or given to me. Are they efficient?  Do they provide opportunities for students to make meaning? Is this the best strategy for my students? Educators often adapt and evolve their teaching strategies to meet their students’ needs. Teachers evolve over time and this is a driving force that can impact students for the better.

Learning Number Concepts

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This summer is moving by quickly and thoughts of the upcoming school year are in view.  I’m in the process of preparing materials and the first few lessons.  This year my math sections are larger than before and it changes some of the ways in which I prepare. While researching a few ideas I came across a Tweet from Jamie.


I’ve heard of Kathy Richardson and her role in the math education world, but haven’t delved too deep into what she’s written.  This book interested me mainly because of the use of Kathy’s critical learning phases.  As students progress in school they visit different stages of mathematical understanding.  It’s not always linear, but these stages tend to follow each other. I wanted to learn more about this process.

So the book arrived yesterday and I was able reading through the first section. Kathy states that children learn number concepts in different learning phrases.  She coined the phrase critical learning phases to describe the stages that kids pass on their way to making meaning of the math they encounter.  The first section of the book focuses on understanding counting.  On first glance I thought this would be very basic.  Advertised and delivered.  It’s basic, but also intriguing and gave me a few takeaways.  After reading this section I started to draw parallels to how my own students make sense of numbers.  I then wanted to reach out to my colleagues to direct them to this section. Understanding counting, although basic, is a foundational skill that is built upon.  My paraphrased version of Kathy’s learning phases are below.

Counting objects

  • Counts with 1:1 correspondence
  • Knows “how many” after counting
  • Counts out a specific amount
  • Spontaneously adjusts estimates while counting to make a better estimate

Knowing one more/one less

  • Knows one more and one less in a sequence without relying on counting out
  • Notices if counting pattern doesn’t make sense

Counting objects by groups

  • Counts by groups
  • Knows quantity stays same when counted by different sized-groups

Using symbols

  • Uses numerals to describe amount counted.  Connects symbols to amount counted.


As I read through this I started looking through my school’s teaching materials for grades K-3.  Some of the materials follow a linear progression while others tend to favor spiraling.  I have a few takeaways after reviewing this first section of the book and looking at my district’s materials.

  • If counting is important than students will start to see why keeping track/organizing numbers is important.  If it doesn’t matter students won’t care whether they come up with a different amount when a specific amount exists.
  • Being able to “spontaneously adjust estimates while counting to make a closer estimate” is a skill that’s found as students progress through the elementary grades. This plays a major role in measurement as students are asked to estimate using non-customary  tools.  I also see a connection to Estimation180 as students might use proportional reasoning after initially counting to find an estimate.
  • Moving from 1:1 correspondence counting to group counting can be confusing for students. Understanding that counting by 2s, 5s, or 10s is a strategy that’s more efficient and that may be learned in time.  Students might not initially come to this conclusion and replace 1:1 with 2:1, 5:1, or 10:1 if they don’t understand the reasoning and if they’re taught in the same period of time. For example, I find this evident when there’s a disconnect between counting by 10s and grouping numbers into 10s and then counting the groups.