Area of Complex Shapes

One of my classes has been exploring area lately. They started by counting squares and differentiating between what is considered area and perimeter. Students were able to add halves and reasonably estimate what the area of a rectangle, parallelogram and triangle would be based on a brief observation. Becoming precise was not valued early on in the process but proved to be a tough transition as students were expected to use formulas later in the unit. Late last week students were asked to find the area of the shape below.

At first students were fairly confident in being able to find the area. They quickly counted up the squares that were fully visible. Then added the halves or what they perceived as half.

Students knew that there were at least 15 full squares covered and then added the halves. Estimates were given based on the full squares visible and ranged from 20 to 45. Confidence waned during this time as some students erased the numbers and started to deconstruct the shape into smaller shapes.

Earlier in the unit students made the connection that the area of a triangle can be found by using a rectangle method. Students also explored how parallelograms can be modified and rearranged into a rectangle.

Using that understanding, a number of students tried different methods to find the area of the shape. Students worked in groups to find a common understanding of where to start and how to dismantle the shape into parallelograms, triangles and rectangles.

This group decided to split apart the shape into triangles and rectangles. They specifically used the rectangle method to find the area of the triangles and counted the middle.

Another group tried a hybrid approach with mostly triangles and two parallelograms. The problem that this group had was trying to decide what constitutes the base and height of each triangle.

The other group decided to split one side of the shape into triangles and the other side into parallelograms. When I showed this to the class I received a few shocked looks. They were amazed at how simple this looked and yet they came up with the correct answer.

Overall, this was a time consuming task, but I feel like it was worthwhile. Students were able to think about math and measurement a bit differently. There are more efficient ways, but not one right way to complete the task. I am hoping that students remember this task and build upon their understanding as we move towards additional measurement concepts next school year.

Stock Market Game Simulations

My fifth grade math class has been participating in a stock market game simulation this year. In years past, this has been a culminating math extension activity for students where they can see how math and economics are related. Students use spreadsheets, gather data related to revenue/expenses, use math terms such as interest, rate, and explore world events that impact markets. All-in-all, it is a fun session that students tend to remember as they move onwards towards middle and high school.

Each year I have 5-6 teams consisting of 3-4 students on each team. Each team is given 100k and asked to invest at least 30% of their money in equities. The game occurs January – April. There is a brief introduction to the stock market and the metrics used to determine whether something is a good buy or not. My teams are only able to purchase/sell during class time and after a consensus is made.

This year the stock market game has been a wild ride. The invasion of Ukraine has directly impacted markets and students’ portfolios. Some of the teams are near 120k while others are hovering around 80k. Teams are getting their information from a variety of sources. Hot stock tips from someone at home (this happens every year!) or carefully researching and then deciding on what to purchase. The decision tree in what to purchase runs the gamut. Once students purchase a stock the emotional highs and lows are quite significant – especially this year.

At the end of the game there is usually some type of reflection. Students analyze their holdings and trading history. They reflect on what could have been done differently to optimize their overall equity in the end. While doing this, I tend to also reflect on how the game was organized and decide on what changes might be needed for the next session.

Even before this session ends I have come to the conclusion that a change is needed. Although I believe school stock market game simulations are fun and applicable, the game itself does not encourage students to look long-term. While reading students reflections in past years, I rarely hear comments about long-term investing because it is not part of the goal. Usually the comments involve regretting not buying at the right time or selling too early. There is generally a lot of emotional buying and selling going on during these simulations. I would say it is much better to do this with a fictional 100k and at 10 years old compared to 30. I have to wonder though what is being taught indirectly during the stock market game simulation process?

I would like to see simulations last longer than a few months and involve applicable situations. This year I heard the terms Bitcoin, cryptocurrency, Meta, Apple, and Netflix multiple times. I did not hear mutual fund, index fund or fees once. When is it appropriate to invest or save? How does investing look depending on these situations?

  • Plan on making a downpayment on a house in 5 years
  • Create a college fund for a daughter that is currently 7 years old
  • Plan a retirement fund for someone that is 35 years old

There are plenty of other situations that could be used. This adds a different dynamic to the game, but also allows students to see how investing involves planning depending on the situation. Instead of going with a gut-feeling or gambling, students could look at the risk involved in the time horizon and manage their investing accordingly. This type of simulation would involve more up-front time and education. I think it would pay off though as investing is not as simple as what is currently being used during stock market game simulations. I assume that students would see that investment risk depends on the context and that would influence their decision making process.

Fraction Division Strategies

My classes have been recently exploring fraction division. Students completed word problems involving dividing fractional pieces and they were finding the idea challenging. In order to gain clarity, I worked with students in small groups to determine where the trouble spots seemed to developed. I started to notice a couple things: 1) students were relying on a fraction division algorithm without context 2) students were not sure how to determine the dividend, which made creating a number model problematic.

Relying on the traditional fraction division shortcut ended up causing problems for more than a few of my students. Students were not able to explain their reasoning for flipping the second fraction. This become even more apparent when students attempted fraction division word problems. Because you have to “flip” the second fraction students were not sure how to identify the dividend. This caused confusion. I planned out a small fraction bootcamp for students to explore fraction division through visual models. Students started out with problems like 2 ÷ 1/4 and progressed to where a fraction is in the divisor and dividend. Students were making progress and relying less on the shortcut method, although some used that to check their work.

After our mini camp, students were given prompts to show their understanding of fraction division.

1.) Juliane has 12 bags of confetti to spread on 16 tables. She wants to put the same amount of confetti on each table. How much of one bag of confetti should she put on each table?

This was the first problem and achieved the highest accuracy. Students drew out the 12 bags and spread it on 16 tables, finding the answer to be 12/16. Some showed a number model of 12 ÷ 16 = 12/16 and others drew a picture.

2.) Write a number story that can be modeled by 4 ÷ 5 = 4/5

This was more challenging. The number stories indicated whether a students could determine what was being shared and in how many pieces. It was interesting to read the responses and revealed an understanding of what is being split equally. Here are a few response:

There were 4 candy bars and 5 children. How much of the candy bars will each child get?

I have 4 boxes of apples and I wanted to put them in 5 bags and all the bags have the same amount of apples. How much of the box of apple go into the bags?

Tyler has 4 rats and 5 carrots for his rats to each get equally fed how much will each rat get?

There were 4 oranges jamal and his four friends wanted to spilt the oranges to a even amount how much of and orange does each person get?

3. Explain using words and the process you would use to complete the problem 5 ÷ 1/3.  Give the reason why you completed each step.

This problem caused a few student headaches – but in a good way. Students that relied on the shortcut were confused in how to explain the reasoning for flipping the second fractions. Out of all of the problems, this one highlighted the conceptual understanding of fraction division the most. Some students sent in pictures with written explanations while others created number models. Here are a few of the responses:

First I would do 5 ÷ 1/3 This works, because it is the same question just written in a different way. Next I would see how many 1/3 can fit in 5. To do this  I would  do 5*3. This works, because there is 3 1/3’s I one. And there is 5 ones in 5*3 = 15. So the answer is 15. (appreciate the thorough thinking behind this response!)


First I switched 5 to 5/1 and then 5/1 to 15/3. Why I did this is to make the denominators the same same number. Then I divided across numerators and denominators to get 15/1 then I simplified 15/1 to get 15. Why I divided across numerators and denominators is to get the answer. Why I simplified to make the number a whole number.

I think the answer is 15 because you can think about how many 1/3 are in 5 and that answer is the answer to your problem. 

First I converted 5 to 5/1 then I did 5/1 divided by 1/3 to get 5/1/3 then I did 5/1/3 X 3/3 to get 15/1 which I simplified into 15

I was pleasantly surprised to see the improvement in being able to navigate fraction division. Being able to conceptually understand fraction multiplication/division can sometimes be a roadblock for students. I am hoping to break that and looking forward to discussing and highlighting a few student examples with the class next week.

Math and Multilingual Students

This year I have been trying to intentionally read more books. Some have been educational while others have been more non-fiction wonderings. During the last couple weeks I have had the opportunity to read Teaching Math to Multilingual Students with a group of Illinois educators brought together by the Metro Chicago Mathematics Initiative. We read a few chapters and meet online to discuss our thinking. We are about halfway through the book right now and this post will document some of my takeaways as I think about math through a different lens.


“Contrary to popular belief, student silence is often the result of unfair or inequitable positioning in content classrooms” p. 27

To be honest, the idea of positioning multilingual learners as classroom leaders has not been at the forefront of my mind. Positioning is is a concept that involves identity and access. Teachers are required to make many decisions lesson by lesson and they impact positioning within their classrooms based on what is being communicated and who is being a spectator. Positioning can have students’ competencies recognized or ignored by highlighting certain work/strategies and dismissing others. Intentionally planning out phrases that can be used might be one way to think about positioning differently moving forward. In the moment this can required a large amount of patience as the pace of the class has the potential to be disrupted. Hello wait time! Teachers should refocus students’ attention if disrespectful behavior occurs. It might be helpful to revisit norms to ensure everyone is on the same page.

Encountering Unknown Contexts

“How will you identify factors that hinder participation for multilingual learners in your mathematical classroom?” p. 43

Teachers tend to engage students in learning through contexts that are understandable. Many of the problems in district-adopted resources involves a few problems related to sports. From what I see, those sports at the K-5 level in math class are primarily basketball, football, baseball and occasionally soccer. Understanding the games themselves is a prerequisite to answering the question. These may be unknown to multilingual learners. Put the shoe on the other foot. I doubt many students in my class would be able to complete a math word problem about the game cricket without understanding the game first. This also applies to the vocabulary terms used to describe the game.

Group Work

“… One student grabbed Julia’s pencil out of her hand to complete her mathematical work for her.” p. 45

Many math classrooms are instructionally moving in the direction of having students work together to discuss their mathematical thinking. Communicating understandings and having to defend them is an important tasks and group dynamics play a role here. Teachers should discuss with their class what productive partnerships look and sound like. This might also be an important time to revisit math station norms. I have noticed that groups may sometimes show that patience is lacking and a particular students will complete the work for the entire group. I am assuming most educators have seen this type of behavior. I have also seen students take pencils out of the hands of others to write the answer. This is an act of positioning and the behavior should be addressed. This year has been trying in having consistent quality discussions in small groups. The last couple years of elearning and hybrid instruction has significantly decreased the amount of opportunities students have had to work with others outside of a Zoom breakout room. Getting back into the groove of being able to facilitate a conversation and possibly encouraging students to use sentence starters can go a long way in helping.

I am hoping to learn more as the book study continues.

More than Numbers

Before winter break 2021, my 3-5th grade students started an isometric name design project. I found this idea a few years ago on the bird app and was reminded after taking a look at Adrianne’s Desmos task. Since most students that I teach are in-person this year, I thought it’d be beneficial to expand on my geometry and measurement unit by having students explore the connections between math and art. To introduce the activity I showcased isometric art and grid work. Students were especially fascinated with optical illusions. Students were given directions.

Students were handed two pages of isometric paper and an example letter page.

After reviewing the examples, the students were off to work independently. Some students created draft drawings and other immediately started on the isometric grid. There were errors – many as expected, and the students took it in stride and persevered. I heard a few comments related to how this was definitely different than “regular math” and some students even brought a few pages home to practice. I’d say most students used 2-3 pieces of isometric grid paper. The shading was key to make the letters pop. If I was doing this project again I’d probably spend additional time having the students watch this video. Students needed to look at the 3d letters and pick which side to highlight to show perspective. This took a different type of thinking. Students also were asked to find the volume of their name using cubic units.

After the projects were posted, one student mentioned that math is more than just numbers. I’m more than inclined to agree!

Box Plots and Spreadsheets

One of my classes has been exploring box plots and data landmarks lately. Earlier in the year the class created histograms and found data landmarks on line plots. Box plots was not as easy as a transition as anticipated. There were a few roadblocks as students analyzed and created their own box plots while determining Q1 and Q3. Some students picked up on the concept quickly while others took more time. To help reinforce the concept I thought about bringing in a spreadsheet activity. I have been using spreadsheets quite a bit this year and it has been another medium in which students can experience statistics.

Students were first asked to create a question that they would be asking the class. The numbers could range between 1-51. I gave students free rein on what questions to ask and held my breath.. Here were a couple of the survey questions:

  • What is your favorite number between 1-51?
  • How many hours of sleep do you get per night?
  • On a scale of 1-50, what do you rate a cheese burger?
  • How many movies have you watched this year?
  • On a scales of 1-50, how well do you like dogs?
  • How many digits of pi can you recite?

Once students created questions they went around and surveyed everyone in the class. I gave each student a roster list so they could check-off who answered This took a good chuck on time – 10-15 minutes. Once the data was collected students grabbed a Chromebook and copied a spreadsheet that I had pre-populated.

Students took the data from the survey collection sheet and transferred it to column A. The data landmarks in row three were placeholders and awaiting formulas. Students then entered the minimum, median, maximum and mean formulas. They were familiar with those formulas as we explored them earlier in the year. I discussed with the class about quartiles and we put together a formulas for Q1 and Q3. We made predictions of what the vertical box plot might look like before finalizing. Students then entered the formulas for the quartiles and analyzed the box plot to see if it matched the data.

It was interesting to hear the conversations that students had as they compared the data to the box plot. The class had a discussion about interquartile range and variability. It was time well spent. From there, students shared their spreadsheets with me and I took a closer look to see how the data matched and if the correct formulas were in the appropriate places. Students seemed to grasp the concept fairly well. Feel free to use a copy of the spreadsheet by clicking here.

During the next day the class reviewed box plots and the spreadsheets that were created earlier. Students then complete the Desmos task Two Truths and a Lie. This is one of my favorite tasks for students to discuss box plots and use math vocabulary while doing so.

The spreadsheet and Desmos task took about 2-3 days to complete. The class took a unit assessment on Friday and I will be checking out how they did over the weekend. I put these two activities in a digital folder for next year.

Dividing Decimals and Spinners

My students have been exploring decimals for the past week and a half. The class identified place values, rounded and placed digits up to the thousandths on number lines. While looking for ideas I came came across Erick’s response to my Tweet about decimal division. After reading Erick’s post, I delved a bit deeper into how to connect the activity with my upcoming standards related to decimal addition/subtraction and long division. I did not have the connected blocks needed for the activity so I asked the kindergarten teachers. Fortunately they had a few boxes that I could borrow. Some of the already connected blocks were stuck together. Trying to pull them apart the first time took Hulk-like strength (as one of my students mentioned). I then put together a few questions for students to follow as they progressed through the task. The questions were placed in Canvas and formatted as a quiz with image uploads.

I randomly place students in groups of 2-3 . Each group was given a bag with 15-30 blocks.

The groups were given around 10-15 minutes to create a prototype. Groups used trial-and-error to figure out what helped the top spin more effectively.

I found out quickly that not all snap-cubes are created equal. The one on the left in the picture spun longer (I think it was 5x) than the right. Since this was not considered a competition I do not think it mattered to much, but this is something to consider moving forward. Students then went into the hall to find a flat surface and timed the spins and recorded it on their sheet.

Groups then added the trials together to find a total.

Students were then asked to find the average time for the trials using the long division algorithm. Based on the student responses, this seemed to be the the most challenging part of the task. Most groups estimated the quotient first and used that as a baseline. Students then used long division (they are used to using partial-quotients) to find the quotient and remainder in decimal form. They were required to round to the nearest hundredth during the process.

Some groups were required to round a repeating decimal, which was a new skill for them. Groups then shared their spinner with the class and the strategy that was used during the creation process. I was impressed with the different models and the teamwork that was demonstrated by most groups. This is a task I am planning on trying out again in the future.

Major kudos to Erick, Jen and Mike for the ideas!

Texas and Electricity Rates

One of my classes has been exploring rates and ratios. We started off the lesson sequence by using tiles and eventually moved towards rate tables. The class used simulations and the paint Desmos deck. The class progressed nicely through the different ratio/rate models and late last week we began our final task of the unit. This task was adapted from the Chicago Everyday Math resource and I thought it was a nice blend between current events and rates.

In 2021, Texas was hit with a record winter storm. The storm knocked out power supplies across the state causing a shortage of electricity. Electricity is measured in kilowatt-hours. Customers are charged according to how many kilowatt-hours they use. An average household uses just over 30 kilowatt-hours per day.

Before the storm hit, customers who had a variable rate were paying on average about 12 cents per kilowatt-hour. Because of the shortage caused by the storm, some customers had their variable rates go up as much as 9 dollars per kilowatt-hour.

How much would a typical household on a variable rate contract pay for electricity for five days without a storm?

How much would a typical household on a variable rate contract pay for electricity for five days at 9 dollars per kilowatt-hour?

Why might some customers claim their bills are not fair?  Make a mathematical argument to justly your claim.

This was a challenge for students. Students read through the directions at least a couple times and still had questions. The questions dealt more with the significant difference between $9 per kilowatt hour compared to $0.12. They asked how that could be possible? Is that even legal? Why was it so cold in Texas? Is it because of climate change? I appreciated their curiosity and willingness to think about this as a fairness issue. This discussion lasted around 15-20 minutes. We then dove into creating a rate table.

Students first found out how many kilowatt hours a typical family uses in five days.

Once students put together their rate tables they started to work on the written response.
The students were elaborate with their written responses. One of the more challenging aspects of this task was that students needed to create a mathematical argument. Students are not used to that type of questioning at fifth grade and the strategies involved in finding a solution.

I am looking forward to using more tasks like this throughout the school year.

Math Stations and Expectations

Last week I was paging through Building Thinking Classrooms in Mathematics by Peter Liljedahl and thought it was time to revisit math station norms. I’ve been using them more this year than ever and for the most part, the students have reaped benefits from being in them. Last week I walked through the classroom to find some groups on-task while others were talking about non-math topics. I really don’t mind the social aspect of the math stations, but I also want to make sure that time is being spent wisely seeing that I only see students for 50 – 60 minutes.. I find that the math conversations and strategies that that occur at these stations pay dividends later on throughout the school year. I remember briefly discussing the math stations back in August and I thought a refresh was needed. My intention was to start off the week discussing math stations and then have students work in partners keeping in mind the expectations that were discussed that day.

I ended up using Desmos to collect student information about the environment, attitudes and behaviors occurring during math station work. Students first started by self-reflecting on their beliefs during math stations and then rated their group’s actions.

The class then reviewed overall results. This helped spur on conversations about math stations and group work. This also reinforced the notion that math station groups are meaningful and intentionally used in the classroom.

The conversation was essential in my mind to get students to think more critically about what makes a great math station. Students were then given the following slide with a text box.

This was also followed-up by:

What does a great attitude for math station learning look/sound like?

What does great behavior for math station learning look/sound like?

Every student added their response to the list. The class reviewed the results together and we created a notable list of the highlights. Students agreed to what was written down and then we categorized them into groups.

The answers were put together into a document and printed out.

Students then went to math stations for a group task. I’m looking forward to referring back this day to reinforce what math station groups should look/sound like moving forward.

You can find the slide deck for this activity here.

Four Weeks Down

Students finished their fourth week of school yesterday.  Routines are fairly established although there have been interruptions with students quarantining during the past two weeks.  Flashes from last year have been making appearances in classrooms as teacher navigate working with Zooming and in-person students at the same time.  I’m hoping this is temporary but no one has the confidence to say that’s the case.  When students remote into a classroom it changes routines and impacts more that what I can write here.  I’m moving forward and attempting to find lemonade in the situation. Looking back at the last month I’ve found ways to engage students differently this year compared to last school year.  This post highlights two of those instances.

Fortunately, this year my students have been able to work in groups. Words can’t express how big of a game changer this is and what a loss it was last year. Breakout rooms were a poor substitution. I’ve been utilizing whiteboards and math stations throughout the classroom.  While students work on tasks I bounce from one group to another to ask questions and to gain an understand of students’ thinking.  During the last few weeks I’ve been reading through Building Thinking Classrooms have been using some of the strategies found within. Being able to give feedback through questioning at the stations and hearing the students’ responses impacts my next steps as a teacher. I’d like to expand the time at stations a bit more as the year progresses and as social distance policies evolve.

Another strategy that seems to be working this year relates to how students interact during brief math conversations. Students are often given a daily math task or question that’s designed to encourage dialogue. Students take turns discussing the strategy or steps involved in attempting to solve a problem.  While one student is talking the other student is giving non-verbal cues that they’re actively listening.  Students are then brought back to the class as a whole group.  I visibly randomly pick students to share what their partner said during that time.  The student that is picked doesn’t offer their opinion about what the partner stated although the strategy is discussed as a class. I’ve used this at least twice every week since school has started and have noticed that students are listening better in their groups. Another bonus is that students are using the strategies that they hear from their partner/class.

I’m hoping to carry both of these strategies forward as the year progresses.  

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