**Rope Polygons: Body-Scale Exploration**

Last Friday, I decided to implement a Rope Polygons lesson from Whole-Body Math Lessons developed by Malke Rosenfeld. I was hoping that my students will have an opportunity to refine their understanding and to develop their vocabulary by working collaboratively; communicating with each other should create a need for more precision and justifications. Malke shared a very detailed lesson plan here.

After students investigated the ropes and noticed the knots, I asked them to create as many regular polygons as they could. Eventually, all the groups started using the knots and created squares, triangles, hexagons. I moved some students holding the vertices of a square apart, and they informed me that it is not a regular shape anymore but struggled to explain why. A few mentioned angles.

After half an hour of building, I asked my students to reflect on the activity. What strategies did they use? What helped them to be successful? What were the challenges? And how would they explain what regular polygon is to someone who doesn’t know? Give me the definition.

**Definitions: Polygons Are Big**

Here are the (non-exhaustive) list of definitions I was surprised to read after school.

*A regular polygon is a shape that has edges.*

*A regular polygon is even.*

*A regular polygon is the shape whose angles are the same.*

*A regular polygon is straight line and symmetrical.*

*A regular polygon is like a square or triangle.*

*A regular polygon is the shape that has same perimeter and area.*

*Polygon is the shape with parallel lines.*

*Regular polygons have the same angles.*

*Polygon is the shape that is really big.*

Students described all the properties that they noticed while building regular polygons with the rope, including the size. How do I zoom in on the defining ones?

**Attack and Counter-Attack: Refining the Definition**

I got the idea from the blog post Attacks and Counterattacks in Geometry by Sam Shah. There are still a lot of ideas in this post that I would like to try this year, like finding counterattacks for the altered textbook definitions. I made it into a whole class activity with students working on the small whiteboards.

We started with this example: “A circle is a shape that has curves. Counterattack! Draw a shape that fits this description, has curves, but is NOT a circle.”

Then we moved on to regular polygons. Here are some counter-attacks.

We made a list of properties on the board. After each counterattack we had a discussion if this property applies to all regular polygons and if it is essential. I think I overdid it a bit with circling and crossing, so I pretty much had the definition on the board by the time I asked my students to go back and to revise theirs.

**Thoughts: What’s Next?**

Something that naturally appeared in the discussion was “Never, Sometimes, Always” format. Regular polygons never have curves, sometimes are big and always have edges of equal length. I might pull a few statements for Talking Points (I learned about Talking Points here). I’ve been thinking about Van Hiele levels. How do I support my students in moving from the shapes as objects of investigation to the properties of shapes?

]]>I spent a large part of my summer getting my head around grade 5 topics and outcomes and trying to plan. My google drive notes progressed from “Grade 5 Math Miscellaneous” to “Math Weekly”. For the first few weeks I tried to choose a range of activities that would allow me to set up norms and routines as well as get a sense of how my students solve problems and work with numbers.

**Observations: Are You Good At Math?**

On our first week, I asked my students to share with me if they think they are good at math, explain how they know, and tell me if other people know about their math abilities. Grade 3 responses last year revolved around calculations, but the majority of the students had good confidence about their math skills. Apparently, something changes by the time they reach grade 5.

It sounds like some students are trying to give up already. Something to be mindful of.

**Successes: Visual Patterns**

I love starting a year with Visual Patterns: A perfect example of low floor/high ceiling routine. I get inspiration from visualpatterns.org and eventually move on using the patterns that students have created as prompts.

We started with these patterns.

Students created their own and used Desmos to graph them. I plan to continue working with patterns throughout the year. Building more patterns. Writing expressions. Building patterns for expressions. Checking their predictions with graphs.

**Challenges: Subtraction**

Grade 5 curriculum in Alberta doesn’t mention addition or subtraction of whole numbers. I guess the review is assumed though I wish it was explicitly mentioned. I wasn’t about to dive into decimals operations and multi-digit division and multiplication without ensuring addition and subtraction skills are ready. We started our number talks with subtraction strategies, but I also gave my students some questions to answer that would require them to use subtraction: a mix of word problems and pure arithmetic questions, spread around a few days. Here are some examples. What do you notice and wonder?

I see a pretty vast range of different understandings of numbers, place value and operations.

- Understanding of one-digit operations however without understanding of place value
- Confusion with the algorithm
- Understanding of place value however uncertainty how to apply this understanding to subtraction
- Understanding the relationships between addition and subtraction
- Flexibility with different subtraction strategies and confident understanding of place value

I tried to choose the prompts that would call for particular strategies like adding up or making friendly numbers. About 90% of my students chose to use the standard algorithm for all the prompts, and 50% were getting lost in it. Here are some strategies that students generated. Where do I begin?

**Closing Thoughts**

It seems like the map of understandings is expanding the older the students get, and I find students in very different places. I wonder how grade 7-12 teachers manage to navigate it. How does differentiation look like in different classrooms? Yes, all my students were able to access visual patterns lessons at their level, but they all need to master subtractions skills as well. How do I support my students in getting access to the skills and problems that are currently very challenging for them and the challenges are different for everyone?

]]>**The Cookie Fiasco**

The Cookie Fiasco is a wonderful funny book in which four friends are trying to share three cookies. We read the first half, stopped and I asked my students to help the friends with the cookies. Apparently, they could get 1/2 and 1/4 or 3/4. Some students noticed that these must be the same. I noticed that while everyone was successful helping the friends, some students were calling any equal parts halves.

**All About Fractions**

After my students agreed that the friends had to use fractions to share their cookies fairly, I asked them to tell me all they know about fractions.

Some students also added their wonders.

We spent some time working on notation afterwards. And I asked my students to brainstorm what numbers are to see if fractions fit in. I don’t think everyone was convinced though. Yes, you can count them and they fit on the number line. But they are smaller than numbers. I am still unsure how I should have continued the conversation about fraction being numbers, or parts of numbers, or some new weird numberland creatures.

**Number Talks Images and WODB**

This image from Number Talks Images website appeared to be a great prompt, intriguing and accessible. My favorite solution: put all halves together, cut one corner watermelon into three quarters, and give these quarters to the other watermelons.

Which One Does Not Belong helped to focus once again on the idea that the same fraction of the same whole can have different shapes.

**Half of a Square**

I love the paper folding challenge from youcubed website, and we did it this year again. I simplified it a bit. I asked my students to make a triangle that is half of the original square, a quarter of it, then make a square that is a half and prove it to their classmates and to me. “These 2 squares fit on top of each other, they are the same.” I tried to talk about Plato but the audience didn’t seem excited.

**The Sad Bunny Story**

Once upon a time there lived a lazy bunny who went to visit his friend owl. They had a sleepover, and in the morning lazy bunny was very lazy to go home. The owl suggested that the bunny shouldn’t push himself too hard. He should go half the way on the first day, half of the remaining way on the second day, then half of the remaining way again. How many days will it take for the bunny to make it home?

I don’t remember where I got this story but the scenario worked well for eight-year-olds. We modeled the first couple of days on the clothesline and did the initial estimation.

I offered meter sticks, strips of paper, poster paper, and then in a stroke of insight I taped a couple of number lines on the floor. We ended up getting together and doing a proving part on the large number line. “Three days? Can you show me on this line?”

My stories about Zeno were met with much more enthusiasm. Especially the Arrow paradox. My students did a bit of experimenting with throwing non-sharp objects to prove to me that they do indeed hit the target. “The Paradox!”

**Clothesline Fractions**

Building on our work with the fractions on the number line, I asked my students to put these fractions in order in their table groups, and then we discussed the order and found a correct place for each one. One half and two quarters caused a debate. “Can you have two numbers at the same place on a number line? Is it the same number or different?”

**Quarter the Cross**

I was really excited to try this challenge in my classroom. For the first lesson, I followed the lesson outline that David Butler described in his blog. I gave out the templates with the crosses, my students sketched down some ideas, did the gallery walk, shared a few with the class, and then everyone had another try.

I’ve decided to end the year with the quick art project. Quarter the cross. Same fraction. Different shape. Two colors.

**Questions and Thoughts**

I wonder at which point fractions stop being a wonderful new mathematical concept and start being frustrating and challenging. Is it when students need to start adding, subtracting, multiplying and dividing them? Is there anything I can do to prepare them for this? What intuitions would be helpful?

]]>**Introduction**

As the school year is coming to an end, so is our Mount Holyoke College Course. During the most recent session with Kaneka Turner, she shared her call to action with us.

- Interview a set of student you serve using the following:

- Are you good at math? How do you know you are? are not?
- Do other people know you are good/not good at math? How do they know this?
- Is there anything about math class that causes you to be good/ not good at math?
- If you could change one thing about math class what would you change and why?

- Consider the results of this interview and the implications for your math classes.
- Plan to or actually make 1 change as a result of the results.

I decided to start with taking two questions and asking my fifty grade 3 students to write an answer. I told my students that I need critical feedback so me and other teachers can be better math teachers, and I trust my kids’ honesty.

**Question #2 (we are moving backwards):
**

**If you could change one thing about math class, what would you change and why?**

There were some interesting ideas, but a found that a lot of the answers were very personal and focused on what students find hard or easy, and then they want things to be either harder or easier.

**Students want:** more problems, more time, more division because multiplication is too easy, more geometry, more multiplication bingo, more Desmos, word problems instead of “just numbers”, learning more strategies, harder, easier, more times tables because I know them, more tests.

**Students don’t want:** subtraction because it is too hard, multiplication because it is too hard, too hard math, too easy math, no standard algorithm, no tests.

This was all alarming, but it wasn’t the worst.

**Question #1**

**Are you good at math? How do you know?**

Most of my students felt they are good at math. And they should. But the reasoning stunned me.

Some responses I hoped to get more of.

- I am good at math because I am good at finding patterns.
- I am good at math because I can find patterns.
- I am good at math because I know good strategies and know how to use them efficiently.
- I re-think what I learned when I get home.
- When I get to solve problems it’s actually kind of fun!

That’s it, five responses, the rest I hoped I wouldn’t get.

- I am not good at math because I don’t understand how standard algorithm works.
- I am good at math because I learned division and multiplication before grade 3.
- I’m ok because I am good at adding but subtraction is confusing.
- I am kind of good because I remember my times tables up to ten.
- I am good at math because I can times big numbers.
- I am good at math because I am fast at multiplying and also correct.
- I am good at math because I can add, subtract, multiply, divide and my brain works fast and smart.
- I am good at math because I get more answers correct.

And the nail in the coffin.

**Emotional moment # 1**

I went through…

**Denial:** They didn’t understand the question. They just tried to give the “right” answer.

**Anger:** Why didn’t they get it? All year was for nothing!

**Bargaining and looking for excuses:** If only program of studies didn’t put that much emphasis on numeracy in elementary. I only have one outcome for geometry and two pages for operations. If only I have asked these questions earlier.

**Depression:** I am a terrible teacher and now it’s too late to do anything. I failed at one thing that I believe really matters. I’ve been leading workshops about math teaching all year and I’m a fraud. I’ve ruined my students’ math education and they will hate math for the rest of their life.

**Acceptance:** Now, what can I do about it? Because I can’t just leave it.

**Continuing the Conversation**

We started by getting together with all fifty kids and my teaching partner and brainstorming what it means to be good at reading. I hoped to get a T-chart and compare reading and math. We did make a T-chart, but a result was surprising. What do you notice if you compare this chart to students’ personal responses?

I noticed that there were more answers that I expected to get in the first place. Were my students just telling me what they knew I wanted to hear now? Did they get more ideas from each other? Whatever it was, my original plan to compare and contrast it with reading was not turning out to be dramatic enough. We noticed similarities.

Then I asked my kids what we did in math this year. They told me about addition and subtraction. I put it on the board and circled it. And then we kept going. We kept going for a while, and when the bell rang I asked the students who still had more ideas to add them on the post-it notes. Here is what we came up with.

And more from the notes: Is 5 closer to 0 or to 10, patterns in a triangle, polydrons, odd and even numbers, egg experiment with weight, paper (cardboard) SOMA cubes, multiplication gummy bears, Euler’s formula and making geometric shapes.

After lunch, I asked students to look at everything we have on the board, on the ideas on the post-its, and put a thumb up if they felt successful at something, did good at it. My students are used to showing the number of strategies they came up with on their fingers, so I saw how more fingers were following the thumbs, kids reminiscing on the year of math and counting their successes. They were beaming. Two girls quietly moved to writing more questions.

**Emotional moment # 2**

I told my students that I am writing their report cards now and no one is getting any 1s, and if they worry about it they shouldn’t. I told that asking good questions is sometimes more important than getting right answers. I told them how I’ve been sharing their work with other teachers who always said what amazing mathematicians they are. I stopped because I realized my rambling might actually spoil the moment. I left all their notes on the board. I want my students to keep seeing it and thinking about their successes. I want them to remember that arithmetic is just one thread in the math tapestry.

**Final thoughts and questions**

I am glad I did not just end my year with the assumption that I know what is going on through my students’ heads as they enter and leave my math classes. I still wonder what I did wrong, what I need to change next year so that my students have this realization about the nature of math as a subject earlier than May. I wonder if something in my words, my lesson design was reinforcing the stereotypes. Do my words and my actions and choices always align? I realized that I want to change how my students feel about mathematics, I need to explicitly design my lessons with this goal in mind. I need to be more careful and more reflective.

I wonder what’s one thing I should change in my math class?

]]>There is a lot of reading, writing, questioning, inquiring and thinking critically that absolutely has to happen in a math class for students to learn math in a meaningful way. And I also think that often math topics can lead into some great creative writing, art and discoveries across the disciplines. This post is about the latter.

I have blogged extensively about my geometry lessons this year (here and here). I am working with an amazing teaching partner this year, who is taking care of the language arts teaching part of our fifty students’ community. Working with Megan allowed me to regularly cross the imaginary boarders between the subjects without any teachers being hurt in the process. She developed and implemented the writing/reading part of this polygons project.

**Polygons Pen Pals**

Students have been exploring, classifying and creating their own polygons from tangrams. I noticed some started drawing eyes and hands, and that’s how the project was born. After identifying the properties of their polygons, students gave them names and considered their personalities and life stories.They wrote letters to their unknown polygon pen pals, and next week we hope to exchange the letters and to write responses. I will leave the rest of this post to my students’ work.

**SKITTLES**

**LARRY’S SWORD**

**LARRY**

**SHIRO**

**LARRYO**

**BOB**

**LINDSAY**

**CENAMAN**

**QUAD QUADRILATERAL**

**KASEY**

**Thoughts and Questions**

I know that students enjoyed giving “life” and stories to their polygons. I admit my worries that it might have been a superficial connection. There was no mathematical need for written communication and there are still many questions that I wonder about.

Did moving towards creative writing still support my students’ mathematical thinking in some way?

What is the value in putting mathematical objects and relationships into non-mathematical context?

Months later, my students keep bringing up our infinity art/writing/reading lessons; they keep asking mathematical questions and making mathematical connections. My students spent time thinking about dragons and favorite food of their polygons, but they were also very careful making sure they identified their polygons’ “physical” features correctly. Mathematical context created motivation for creative writing which in turn created motivation for mathematical precision. Maybe we do need the whole range of experiences to make sense of the whole range of things and our literacies can be a bit more interdisciplinary.

]]>“I think, it is closer to 200, because you count 250 first.”

“Because a teacher told if it’s 50, you round down.”

This made me pause, “What about numbers from 0 to 10, Is 5 closer to 0 or to 10?”

“Closer to zero.”

“Raise your hand, if you think it’s closer to 0.” – More than half of the hands go up.

I did not expect this at all.

“I am not sure. Is 5.1 still 5? Where does the number begin and end?”

**Developing the Question and the Claims**

In the next few days I’ve been obsessed with this question and with the possible ways to approach it, and I think this question made it into most of the twitter chats. Kent Haines wrote about the rounding conversation in his blog.

In another conversation about natural numbers, David Butler pushed my thinking in an unexpected direction when we started talking about calendar.

I wondered if that might have been the reason for my students’ uncertainty. Ordinal and Cardinal number duality causing a lot of confusion!

I was trying to figure out how to approach it when Malke Rosenfeld’s book, Math on the Move, finally arrived in the mail. The next day, I made four large number lines on the floor of my classroom.

Students explored the line in groups to test their ideas and to discuss them.

S: “You start here (points at 0), you count 1, 2, 3, 4, 5, 6. But when I go to 10 if I count from 5, it’s 1, 2, 3, 4, 5.”

Me: ”Why does it change?”

S: “You don’t count 5 because you already counted it.”

Me: “Do you count your starting point?”

S: “Yea.”

Me: “So what happens if you count from 10?”

S: “It will still be 5!”

Me: “Try it.”

S: “1, 2, 3, 4, 5.. (stops)…. 6? It changes. When you walk up, 5 is closer to 10. When you walk down, 5 is closer to 0.”

We were thoroughly confused.

This time, Malke Rosenfeld, Sarah Caban, Simon Gregg and Kristin Gray joined the conversation and shared some ideas on using kid-sized math tools. The calendar also reappeared.

I didn’t want to stop just yet. The discussions were great, and I was looking for some structures to prompt students to verbalize their conjectures and to listen to each others’ arguments. After the recent online session with Kristin Gray at Mount Holyoke College, I’ve decided to start with the Talking Point and this time limit it to one statement to focus our conversation. Talking points helped students to find like-minded peers to do more proving together.

**Developing Arguments and Revisiting Claims**

We ended up making four teams as there was always one more popular claim. Each team got their own large number line and any other manipulatives or measuring tools that they could find in the classroom. Our most popular choices were meter sticks, snap cubes and balances. I also handed each group a set of cards with numbers 0 to 10 and some tape to arrange their numbers on the number line when and as needed. Some added fractions. Some took rulers to be precise.

Groups presented to each other and asked questions. Many students changed their opinions in the process.

**Counting Argument: Five is in the middle.**

S: “You go 1, 2, 3, 4, 5 then 1, 2, 3, 4, 5.”

Me: “Do you count zero?”

S: “No.”

Me: “Why no? Shouldn’t you?”

S: “You don’t count zero because it’s a number that you don’t use in counting.

**Semiotic Argument: Five is closer to zero.**

**Calculations argument: Five is in the middle.**

**Change argument: It depends.**

**Snap cubes argument: Five is in the middle.**

“The number of blocks between zero and 5 is 4, and the number of blocks between 5 and 10 is also 4. The number of blocks stays the same so it is the same distance.”

**Hotel argument: Five is in the middle.**

“Say you wanted to stay for 5 nights and then go to a different hotel for 5 more nights. And then you leave. Days are spaces and nights are numbers.”

In my final move, I asked students to take the numbers away, then find the middle and stand there. Then together we put the numbers back. Five was landing on the middle spot. We proceeded to our journal reflection.

**Closing Thoughts**

This question appeared to be much trickier than it sounded at first and it brought forward more questions. What is the middle? Are numbers “spaces or points?” Are fractions numbers? How do you count? Do you count zero even if it is nothing?

I thought about a difference between the models of number line and a hundreds chart. I never considered that essential difference before. I wondered if developmentally ordinal understanding of numbers comes first and it is a an abstraction leap to the cardinal numbers. I also wondered about historical development of number concepts and at which point numbers stopped being counters. My twitter and classroom conversations made me think about our zero-less calendar. We count years, days and months as they begin. Our whole calendar system is based on the intuition of ordinality. Zero is hidden in the moment of midnight, too dark and fleeting to count it.

And a large number line is a keeper.

]]>“What is a pattern?” – “It is when something is repeating like ABAB.” I found at the beginning of grade 3 that my students could build repeating patterns. We’ve been working throughout the year on identifying rules in increasing and decreasing numerical and visual patterns, and I wanted to find a way for all this scattered practice to come together.

The credit for the idea goes, yet again, to Simon Gregg who shared how he used Desmos with his students to graph the visual patterns they created. He shared his blog post with the details of the lesson and students’ work. I wanted to adapt it for my classroom and I hoped that my students would notice the connections between the model, the symbolic numerical representation and the diagram/graph.

**Questions and Thoughts: Before**

I was anxious about the logistics of the lesson. Thank you, Sarah, for moral support!

Will everyone be able to build an increasing pattern?

Will all students identify which step is which when translating the pattern into the table?

If we work in math journals, will someone spend an hour creating a table to fill out?

Will the students be able to handle the technology part independently?

**Preparing for the Lesson: Getting Ready to Make a Pattern**

I used one of the three-act math prompts from Graham Fletcher’s website to get students thinking about the growing patterns. After watching the video, I asked my students to build steps 4 and 5 of the pattern, label the steps and the number of tiles in each step and write down the pattern rule. Then students had to figure out how many tiles would be in step 10 without building or drawing all intermediate steps. This part has gone smoothly enough to proceed to the next lesson.

**Preparing for the Lesson: Getting Ready to Make a Table and a Graph**

Many of my students used the table to organize the numerical information about the pattern to find the number of tiles for step 10. We created and extended the table together with the class. I decided to use teacher.desmos.com to utilize the dashboard feature that would allow me to keep track of the students work and to bring it on the interactive whiteboard for future discussions. Here is the activity screen I used to for my lesson. We graphed the pattern together and discussed the behaviour of the graph.

**The Creation**

The lesson went much smoother than it did in my nightmares. I made a table template to simplify the logistics. Most of the students were successful in creating an increasing pattern that follows a specific rule. Some students created patterns that were increasing, but they needed support to identify that their rule was not consistent.

While most of the students worked with addition, some students tried to triple or double the number of cubes they use in the pattern.

Pattern 1: A Pyramid

Pattern 2: A Fire

Pattern 3: Plus Four

Pattern 4: “It’s a bit up and down but still increasing.”

Students had some great notices and wonders.

The really exciting things started happening when students crowded around the interactive whiteboard and started analyzing their own and each others’ graphs. Someone noticed that if there is an unusual change in behaviour in a graph, there might be a mistake somewhere in the calculations. One student said, “If the number of cubes is the same on every step, I think the graph will be just the straight line, not tilted.” Someone else asked, “How can you make a circle if there will be more than one Y for one X?”

I had a moment of insight.

**Making Predictions**

During my lunch break I threw together my template for the afternoon reflection which included three questions and three spaces for the sketches.

-Without numbers, sketch how the graph of your **increasing** pattern looked like.

-Sketch how you think a graph of a **decreasing** pattern might look like.

-Sketch how you think a graph of a **repeating** pattern might look like.

I asked students to try and explain their intuition.

There were a few interesting ideas considering the behaviour of the graph of a **decreasing pattern**.

Many students tried to look at the graph differently rather than switching the direction of the slope. As if they re-located their point of view rather than changing the orientation of the graph to represent the decrease. I wonder why and I currently have no explanation.

This reminded me of a graph of an absolute value function, and it seems to me that the student’s logic was also describing something similar.

As expected, **repeating patterns** had the largest variety of predictions. Most of the students went with the equally spaced bumps. A few sketched a graph of a constant.

There was an attempt to express a repeating pattern as a periodic event.

And I need to ask some questions about this graph: How far do you think these lines can go up and down?

**Questions and Thoughts: After**

My students need more Desmos in their lives. The interface appeared to be 8 year old friendly. I did not need to explain anything about the coordinate grid. I think, if we try graphing decreasing patterns, there will be more clarity about the structure. I loved the teacher dashboard with all the functionality of tracking my students’ work and the ability to pause the lesson and to control the slides. I would love to have the function to print students’ work from the dashboard implemented because printing 50 clipped screenshots was somewhat time consuming.

I wonder if showing all three steps of the process before engaging with it pushed many of my students to work numerically rather than visually. Many of the students were just building the “rods” with the cubes instead of going for more creative shapes. Would the outcome have been different if I asked them to build the patterns first, took the pictures, and then introduced the next two steps? If anyone tries, please let me know how it goes.

I also wonder what my next step(s) should be. I am considering giving kids their graphs and pictures of their patterns and finishing their notices and wonders first. Some of the students wanted to test their predictions for decreasing and repeating patterns. Some started experimenting with creating pictures with their lines. I am also thinking about reversing the process; giving my students a graph and asking to create a table and to build a pattern. We will start with wonders and then I will make my decision.

]]>

When I started teaching grade 3 last year, I printed out the Program of Study and went over it with three different colors highlighters. One for the concepts introduced or explored deeper in grade 3, one for the numeracy skills my students need to master (e.g. subtraction of three-digit numbers), and one for applications (measurement, data analysis and representation). Oftentimes, I intentionally integrate my “applied math” lessons with other subjects, sometimes the need to measure something arises in a context. I try to be sensitive to these contexts to catch the teachable moment and to turn it into a lesson. Last week, we ended up learning about measuring length.

**Tuesday: Counting**

About a week ago, Christopher Danielson wondered about one of his prompts for his future book *How Many? *How many shoelaces will the students see, two or four?

On Tuesday, I gave the prompt to my class; we noticed, wondered, and counted everything from shoelaces to letters on the box, to dots inside the shoes. It was wonderful. One girl said, “I have these kinds of shoes at home, I’ll bring them tomorrow and we can check how many shoelaces are inside them.” I said, “Sure”.

After school, we had another Mount Holyoke College session, this time with Dan Meyer. The big question from the session: Can you involve your students in the co-development of the activity, rather than just assigning it? I didn’t have an idea yet, but it put my mind into the “search” mode.

**Wednesday: Estimating and Measuring**

The girl handed me her shoes first thing in the morning. We were about to take the shoelaces out, when someone asked, “How long are the shoelaces?” This was perfect!

Here is the Outcome from **Alberta Program of Study Mathematics**:

*Students will demonstrate an understanding of measuring length (cm, m) by selecting and justifying referents for the units cm and m, modelling and describing the relationship between the units cm and m, estimating length, using referents, measuring and recording length, width and height.*

First, we recorded what else we might be able to measure: shoe size, width, weight, how tall the boots are. We discussed what units we need to use to measure the shoelaces. The moments like this I appreciate that I teach in Canada and I don’t need to figure out Imperial units. I brought in the meter stick to use as a referent point.

Then, we gave our too high, too low and just right estimates.

After students used post-it notes to give their just right estimates, they tried to figure out the length of one shoelace by measuring anything they need to measure without taking the shoelaces out. Someone asked if they could use the rulers. “Yes, if you think it’s helpful.” The math notebooks were used to document and explain students’ thinking and to write down their final estimate. Some students tried to get the answer out of me. “I have no clue, these aren’t even my boots.”

The reveal was truly grand with lots of cheering and jumping. The shoelaces had the total length of 125 cm! We put some estimates on the number line to see who managed to get the closest estimate.

We discussed the possible sources of error.

-”Shoelaces go inside the holes.”

-”Centimeters are different on different rulers.”

-”Everyone was touching the shoes, pulling and stretching the laces.”

-”Some people would start a little bit ahead, not at zero.”

We had to pause for a discussion of “centimeters are different” to compare personal referents to standard units of measurement.

Then I had another moment of insight. “Let’s have a challenge. We can measure the shoelaces in my boots on Friday.”

**Friday: Challenge**

Here is why it was a “challenge”. What is your estimate?

When my students saw my boots they looked at me like I am a magician who just pulled a rabbit out of a top hat. We went through the same routine, but this time we had a new referent point. The knowledge of the length of our Wednesday shoelaces. I honestly had no clue again how long the shoelaces might be, and my own estimate was quite a bit off. I loved the range of the strategies that my students employed to support their problem-solving.

**Saturday: Thoughts and Questions**

I do estimation challenges with my students somewhat regularly. They know the routine which is helpful when your audience is eight years old and the logistics can interfere easily with the flow of the lesson. Dan asked us to think about the verbs in our lessons. With my students, we noticed, wondered, selected tools, compared, modeled, estimated, discussed, measured, and some of us jumped and made not classroom appropriate sounds when our estimation was close. Most of us came up with reasonable answers, and all of us tried. Some students estimated that the shoelaces in my tall boots will be shorter than in student’s boots. I had to explicitly point at it for them to notice the disconnect. I wonder what I could have done differently to encourage them to make sense independently. I think that during our next estimation, we should have spent more time comparing our strategies and reflecting on our work.

I have previously done one week estimation challenges and I am considering bringing this format back . I would introduce the challenge on Monday and do the original estimation,(usually some kind of physical prompt), then leave it around for a week to allow students to get their ideas together. Finally, on Friday we would do measuring, calculating, summarizing, checking and reflecting.

I wonder what we can measure next?

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**Which One Does Not Belong?**

After realizing that there are, indeed, more hexagons than you think, I wanted to prompt my students to start focusing on particular properties. Not surprisingly, Simon already had a shapes WODB ready for the cause with four not obvious hexagons. I could anticipate some of the responses (like identifying a regular hexagon), but I was not sure what my students will make of the red and yellow shapes. I was underestimating my kids.

**Late idea **

At some point, students started comparing the shapes to the real objects and getting really sidetracked with it. I asked them to get back to focusing on properties. Things I should’ve asked instead: *What makes it look like a hammer? What makes other shapes not look like hammers?* Those observations that students made were not irrelevant, but I failed to connect their relevance to the goals of my lesson.

I’ve noticed that students had a lot of observations around composing and decomposing shapes, and after discussing it with Simon, I’ve decided to continue with tangrams hexagons challenges.

**Tangrams the Hexagon Game**

The Rules: Build a hexagon with 2 tangrams. Call a teacher to take a picture. How do you know it is a hexagon? Proceed to 3, 4, 5, 6, and finally 7 tangrams. Is it possible to build a regular hexagon with tangrams?

I also decided that it was the right moment to introduce some formal vocabulary like concave and convex to define “tucked in” vertices more conventionally.

Here are some of our hexagons.

And then there were some different ones.

Students kept building these funky shapes, and I had to admit that I am not certain how to count the edges and vertices on them. My students weren’t certain either. In our table groups, these shapes caused a lot of discussions. I took a few pictures and asked for now to stick to the polygons that don’t look like their edges self-intersect. Then I called Christopher and Simon for help.

**How do mathematicians agree on definitions?**

After looking through a chapter in Christopher’s teacher’s guide to his WODB shapes book, I decided to revisit the conversation about the definition of polygons. I brought back the pictures of shapes in question. We still did not have the agreement on what these shapes are. I told that we all had different ideas, and mathematicians have different ideas too. But the definition that many mathematicians agree on is that polygons are not self-intersecting.

Then one of my students asked this question: **“How do mathematicians agree on definitions?”** – “What do you think?”

I did not expect that almost every student had an opinion on that.

*“They have a box, they write their arguments, and they try to convince each other. Then they agree on some idea. But the ideas might change later. Or maybe some parts of the idea.”*

*“They vote, but they have a debate first. People give proof and evidence to support ideas.”*

*“They write their theories and talk about them.”*

*“Someone comes up with the name and has an idea what this name means, then others discuss it and agree or disagree.”*

*“They have an argument and they prove to each other why their theory works.”*

*“There are lots of Greek words in math, maybe people in Ancient Greece invented them and then we borrowed them in English. So we borrowed the words and we borrowed what they mean.”*

*“In olden times in Ancient Greece people would come up with an idea, and then they would invite other people to discuss it and try to convince others and give proof.”*

*“They mix ideas together and make a new idea.”*

I promised to interview some professional mathematicians and to share their responses with my students next week. Daniel Finkel and David Butler agreed to help and to share their thoughts on the matter, and on Tuesday I plan to bring their ideas to our classroom conversation. In the meantime, Simon Gregg made two more hexagons WODBs that can also be put to a discussion ( WODB#1, WODB#2). Also, I think I am ready to face “square is not a rectangle and if you turn it around it’s a diamond” challenge.

**Final Thoughts**

I struggle with giving definitions. There are always a lot of questions. There is always some disagreement on the properties that are deemed important. Where does the classification system come from and how do I justify it? My students offered me a solution. “After some discussions, many mathematicians agreed to call the “closed” 2D shapes with straight not self-intersecting edges polygons.” Then we can reason why. Sounds much more reasonable to me than “Polygons are…”

And thank you again to everyone who helped me and my students to make some new ideas.

]]>This lesson happened some time ago. The pictures and notes from it has been sitting in my folder for a while. Sometimes I come back to look at them because they amaze me. I’ve been sharing bit and pieces with my colleagues and on twitter, but I wanted it all in one place, and this post will be the place.

**Can Number Line Be a Number Circle?**

We have just had a few clothesline lessons on integers which extended our number line to the left of zero. Students were gathering on the math carpet, and I overheard a few boys arguing about infinity on the opposite sides of the number line. Will it be the same, or will the other one be negative? If it is the same, can’t you just connect infinity with infinity, and then make a number line a number circle? More students were joining the conversation. I grabbed an anchor chart and started putting down some questions that were happening.

Based on the idea that numbers are creatures that live on the number line, we tried to figure out if infinity is a number.

Someone said, “I saw a video about an infinite hotel, but I don’t know how it works, can we do this puzzle?” My original lesson plan would have to wait until tomorrow.

**Infinite Hotel**

Here is the link to the paradox in question.

So, I told my students a story about a hotel manager, and a lonely traveler arriving at night, and asked what room number will he get if the hotel is full but infinite. At first, everyone jumped with “I get it!”.

S: “There is infinity of rooms, so he will just go to the next room!”

Me: “But what number is that next room?”

Ss: “Infinity.” “Infinity is not a number!” “Maybe it is!”

Me: “And what room number is right before infinity room? How will the guest find it?”

Ss: “Is it an actual normal number?”

Me: “It is a normal number that you can count, so you can give the guest a key.”

Students got together with their table groups to discuss their ideas.

“There are rooms with fraction numbers. So he can go into 1 ½ room.”

“He can quickly build another infinity hotel”.

“He can give the guest his own room.”

“There was a secret room 403 that was left empty.”

Me: “Nope, it was really full”.

S: “People can share a room.”

Me: “Nope, hotel’s motto is *We Have a Room Just For YOU.”*

**Infinity And Us**

By the afternoon, I was curious to know how my students would describe infinity. I tried to come up with an art project, but nothing really came to mind. So the proper art lesson did not happen. Instead, we read the beautiful book *Infinity And Me *by Kate Hosford, and then I asked students to write how they imagine infinity. Then draw it on a square of black paper. Not a full blown art lesson, but it worked with the writing.

*“I imagine infinity as millions of dust in the air.”*

*“I think infinity is like a rainbow because it disappears and comes back.”*

*“Infinity means to me is freedom.”*

*“I imagine infinity as my friendship and love for my family and friends.”*

*“I imagine infinity as a bunch of rocks cause you can always break rocks smaller and smaller and smaller and smaller and… a lot of rocks.”*

*“Infinity mean to me a color wheel with many colors and the wheel spins around and around forever.”*

*“When I think of infinity I thing of earth because it is a circle and circles are endless,”*

*“I imagine infinity would look like a ball rolling around a cup for forever.”*

*“I picture infinity as a dragon going around and around looking for something.”*

*“I think of infinity as the space because if it ends what is next?”*

*“I imagine infinity like someone walking through time without even noticing he is.”*

It will be interesting to look at a different kind of infinity when we get to fractions. I wonder what images will come to my students’ minds when infinity will appear between the familiar numbers, when it will be close, not at a far distance, not vast, but unimaginably small. Sometimes I wonder if diversions like this are justified from the point of view of learning intentions, assessment and outcomes. I think, sometimes it’s worth to stop and look at mathematical landscape and just admire the beauty.

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