After several years in the classroom, I've decided that I want to be a better educator. These are my exploits and thoughts as I strive to rediscover my passion and purpose.
When I first decided to use my Integrated Math class to work on Project-Based Learning, I knew that that there were going to be perils and pitfalls. I also knew that most of them were going to be mine.
Over the past month of so, my students have been working in groups to develop proposals, blueprints, budgets and presentations for a greenhouse on which the school and community would vote. The winning proposal would be built and filled with flowers, vegetables and herbs, and tended by students.
The presentations were supposed to be completed before Thanksgiving break.
They were not. I let them put the finishing touched on them during the long weekend and they presented today.
My main criticism for them was none of the presentations were of a quality that I would feel comfortable taking to local businesses to ask for donations.
I shoulder a HUGE amount of the responsibility for this. I made many assumptions about student time, ambition, drive and motivation going forward in this project and very few of them were accurate.
Things I should have done:
Organized our guest speaker MUCH earlier in the project (like day 3)
Held regular class meetings to talk about progress and problems
Held regular meetings to discuss expectations
Provided examples of business plans/proposals for them to model
Broken the project into smaller, bite-sized pieces
So...now what?
Do we take what we have, give grades and move on?
I think not. I think I'm going to use this as a growth mindset experience for them and for myself.
Tomorrow, we are starting over. Sort of...
Their assignment tonight was to brainstorm every aspect that would be needed to make this project a reality. Tomorrow, we're going to discuss what needs to be done, how we can achieve it, and break those tasks into smaller pieces.
Small, self-assigned groups will tackle each piece, including 3D modeling, research, design, presentation, budgets, etc.
We will move forward, we will continue to work and we will do so together.
I believe in this project and the students have shown great enthusiasm and interest in it. I will practice what I preach and learn from my failures.
We spent all of my classes today talking about a single pattern from our warm-up.
In the Math 7, the discussion was about how we visualize this pattern and how it grow. In Pre-Algebra, I told the story of Gauss and finding the sum of the first 100 integers. As is usually the case when I tell a story, the students were highly engaged.
In Integrated math, this pattern moved us into a discussion of the math electives that we are trying to develop for the school. We ended up talking about Pascal's Triangles and polynomial expansion.
In first period, we talked about code-switching.
In second, we talked about the Alt-Right and the aspects of the election that the students didn't know about.
In third, we examined some of my favorite recreational math.
Overall, I very much enjoyed today and, while not directly related to the curriculum, we were solidly focused on Standards of Mathematical Practice 2 and 8.
I can think about numbers in many ways.
I can solve problems by looking for rules and patterns.
Also, I finished the last layer of polyurethane on my first bowl and turned a new handle for the crockpot!
Today was designed to be a practice day for the Math 7 kids. We've been working on fraction division, which is admittedly, a difficult topic.
In the first class, I had a worksheet of practice problems.
They were a bit beyond reach. They would have been fine had we been doing "keep-change-flip" or the butterfly method or any of that nonsense, but since we've been using physical counters or pictures, they were a bit ugly. The kids got bogged down with the numbers themselves instead of the operations.
So I revised it. For the second class, I focused on word problems.
"Peter wants to make cupcakes. They take 1/3 cup of flour per batch and he has 7 cups of flour. How many batches can he make?"
We talked about the problem for a little bit before even bringing up the idea of division. We talked about it as a practical measure. If you want to make a bunch of batches of cookies or cupcakes, you need to make sure you have enough ingredients.
They got the solution to the problem without using fraction division and we talked about how that was fine.
The math should fit the problems. Problems shouldn't be invented to fit the math.
"You got the right answer and all of your work is valid, but I wanted you to use a different method, so you get an F" is one of the MANY reasons why so many kids hate math.
The sheet that I gave them had 10 word problems. I told them to read all 10 before deciding where to start. I told them that the problems lent themselves to fraction division, but however they solved them was fine with me. I told them they needed to show their work.
The woman who teaches the Pre-Calc and Algebra 3 classes next door to me was out today. Her students kept coming to my room to ask me for help on their assignments. I eventually gave my own students an assignment to work on and just traded classes with the sub who was next door.
I had a blast!
There is something invigorating about running back and forth between two groups of students, one of which is working on function composition while the other is dividing fractions. It's interval training for the mind.
I've been thinking more and more about using my time in the wood shop to make fraction bars. The teacher who was in this room last left a game called Fraction Stax which has some pretty awesome pieces, just not enough of them.
I need to decide if I want to make mine like these, or more like the paper ones that my students have been using.
I also have a considerable amount of grading that I have to do this weekend. Using Standards-Based Grading means I have fewer grades that go in and I've been forgetting to actually do them. I also need to make up an assessment for the Pre-Algebra kids.
That's all for me this week. My daughter turns 7 today, so I'm going to spend some time with her and eat pizza and cake.
Today was filled with an odd assortment of great things and deeply frustrating things. I'll write about the frustrating ones first so that I can end my writing on a high note.
Many of my students are struggling with timely task completion. I know that failing to understand certain concepts has a tendency to slow down work, and I'm alright with that. If you're having difficulty, I want you to work more slowly and carefully, to allow me to help you as you need.
At the same time, if I assign 20 problems to be done in 40 minutes and you get to number 6, we are having an issue that needs to be addressed. Similarly, I try to make my class as hands on as possible, but that still requires students to follow directions for an activity.
Several of the students in Math 7 and Pre-Algebra have been complaining about how they don't understand what's going on, but when I try to explain, or answer their questions, they are busy touching other kids or throwing things around the room.
Today's warm-up was mental math: 104 + 97
"What strategies did you use to solve this in your head?"
I was getting increasingly frustrated with numerous iterations of the following conversation:
"I just added it."
Me: "How did you add it?"
"I put the things together."
Me: "What does that look like in your head."
"It looks like adding."
Me: "Can you show us on the board?"
**write standard vertical column algorithm**
Me: "You did this in your head you visualized columns?"
"No. I didn't need to. I just did it."
So I put up a few examples of what I do in my head.
"I start at 104 and add 90, which brings me to 194, adding 6 more brings me to 200. I have one more left over to put me at 201."
They stare blankly at me.
"Or, I know that I need 3 more to go from 97 to 100, so I borrow it from the 104, leaving it with 101 on that side. 100 + 101 gives me 201."
At this point, I hoped the pump was primed and asked them how they had done.
"I just added them."
On the upside, some of the kids in Math 7 gave me amazing ways to think about this question.
One student said he pictured a square with 100 pieces and four on the end. Then he pictures a square of 100 with 3 missing. He moved 3 of the four to fill in the unfinished block, making one block of 100 and one of 101.
Fantastic!
On top of this, we continued working with our fraction bars on fraction division and it seems as though a large number of the kids are actually getting it, and more importantly, really enjoying it! I'm thinking that the next woodworking project will be to make a wooden set of fraction blocks.
I see much cutting and sanding in my future...
The other amazing thing that happened today was that my Integrated Math class was able to Skype with a landscape architect to talk about their gardening projects. Our superintendent let me know that she won a $500 grant to build a garden at the school, so the pressure is on. The groups were supposed to present today, but I think the Skype call made them rethink the way they were working on their projects.
I foresee much revision and hard work in their futures...
They were supposed to be presenting yesterday and today, but none of the groups were ready. I would rather take more time and have the products be of better quality.
YAY! I did it right! I wrote about the bad stuff first and the good stuff last and I feel great now!
I've been tired for a week now. My patience with my students has been wearing thin when they don't do the simple tasks that I ask of them. I'm not pleased with how short I'm getting with them. I need do something to recenter myself.
I am, however, quite pleased with the lesson we are doing in Math 7 dealing with division of fractions.
For the last month or so, I've been working on using very specific language when it comes to these operations and, hopefully, it will start paying off in this lesson.
Screenshot from my Promethean Board. I made this thing!
The fraction bars can be dragged down to the paper to look at different types of division. So here's how it works:
"What is 2 divided by 1/2?"
We put two "whole" bars at the top and a "1/2" bar on the bottom.
"How many groups of 1/2 can we get out of 2 wholes?"
The students then convert the bars into other bars of equivalent value (2 wholes become four 1/2 bars) so we can makes groups of whatever value we have at the bottom.
"There are four 1/2 bars that can fit into 2 whole bars. so 2 divided by 1/2 is four. We can find 4 groups of 1/2 in 2 wholes."
We spent a bit more time that I would have liked in just cutting the pieces out so each pair of students could have a set. I may get them printed on colored card stock. I also need to find a way to get them to actually do what I ask instead of rolling on the floor because they dropped a tiny 1/12 bar.
We will be working more on problems with whole number answer for a few days before moving to ones with fraction answers. I want them to be as comfortable with the system as possible before tossing them into the deep end.
I have a good feeling about this particular lesson/activity.
In some of the classes, this was a basic question for our warm up, but in a few others, it took on a deeper exploration, particularly in how students saw the pattern itself.
"In the 5th step, I had a line of 5 and put 4 more going up from the last one." (n + (n-1))
"I had a line of 4 on the bottom with a column of 5 on the end." ((n-1) + n)
"I had a row of 5 and a column of 5, then I took one off for where they overlap." (2n-1)
"I saw a line of 4 and a column of 4 with another block in the corner to connect them." (2(n-1) + 1)
I saw a 5x5 square with a 4x4 square taken out. (n^2 - (n-1)^2)
I really enjoyed taking whatever formula they developed for the number of blocks in the nth step and showing them that they were all the equivalent.
Each time we do an exercise like this, I make a conscious point to remind them that I'm looking for them to think about numbers in many ways. I want them to be able to manipulate numbers and figures into configurations that are most comfortable for them, provided they are valid methods.
The Math 7 students took an assessment today. It was supposed to be on Friday, but I didn't think they were ready. Then I was out with a sick child yesterday. I'm still not sure they were ready.
The quiz had 9 questions: 4 problems of "convert these decimals to fractions," 4 problems of "add or subtract these fractions" and 1 "here are the dimensions of a table. If I add a border, what are the new dimensions?"
A large portion of the students didn't finish the assessment in 40 minutes. I told them again that their not being able to finish in that time meant that they were not familiar or comfortable enough with the material. Several students after school to finish and, in 25 minutes, managed to complete 1 addition problem. I'm not really sure what's going on or how to address it. Much of what I see deals with lack of focus and concentration to the task at hand.
I also started one of my classes by telling them that, regardless of the election results, my expectations for behavior were unchanged. The good or bad behavior of adults is not an excuse for students to behave poorly.
I told them that there were strong emotions on all sides and, if they want to talk about any issues, I would be happy to do so.
I tried to give this talk with dispassion, but the students know how I feel and where I stand. They know I won't stand for incivility of any kind.
I need more eyes. My 7th and 8th period do an amazing job, provided I am making eye contact with them. As soon as I look away, they are off to something else....
I was going to post a picture of the Boo Buddies from Super Mario until I realized I had already done that in reference to this exact problem.
I'm exhausted from this week.
We had some interesting discussions about our Find The Flub warm-up and I found that none of my 8th period remembers how to add fractions.
I started new warm-ups this week and, as a result, am spending more time than normal getting the students used to them.
Today was "Think Through This Thursday."
When students arrived, they saw an expression on the board: 8*6
I told them that I purposely selected an easy problem because the answer wasn't important. The point of this warm-up is to gt them to think about their mental strategies for solving problems. Since this is the first time we did it, I got the answers I expected.
"I added 6 to itself 8 times."
Me: "Really? When asked to do 8*6, you sat there and went '6, 12, 18, 24, 30, 36, 42, 48'? I find that hard to believe."
"I made a box that was 8 long and 6 tall and counted the boxes inside."
Me: "...riiiiiight."
"I used mental math" and "I just knew that 8*6 was 48."
So I started putting my methods on the board.
"For me, multiplying by 5 or 10 is much easier, so I would multiply 8 by 5 to get 40, then add in the last 8 to get 48."
This started the ideas flowing and soon the board was full or great methods, including several that I hadn't seen before.
I was very pleased with how well they took to this activity and I'm looking forward to next week, especially now that they have a solid idea of what we're doing.
In the pre-algebra class, we continued our discussion of transformations by having the kids develop their own tessellations. They were very excited to be doing something creative and several had to restart because they didn't have a repeated pattern, or hadn't planned it well enough. Overall, I was very proud of the work they were doing and we will continue it tomorrow.
I also had a formal observation today for the third time since 2006. I'm very much looking forward to the feedback.
On June 3, 1888, the Daily Examiner in San Francisco published a poem by Ernest Thayer entitled "Casey at the Bat: A Ballad of the Republic Sung in the Year 1888." The poem details a fateful afternoon for a small town baseball team, down by 2 runs in bottom of the 9th inning. In the first stanza, two batters are called out at first and it's looking grim. The team and crowd believe that if Casey can make it up to bat, he can deliver the hit that will send them to victory. Before he can get up to the plate, however, there are two other batters, both seen as weak hitters and hope flags. Against all odds, both men get on base and it's up to Casey to bring all three home, winning the game. Casey approaches the plate with the entire crowd cheering. Confident in his abilities, he literally watches the first pitch go sailing past, claiming "that's not my style." When the strike is called, the crowd screams death threats at the umpire. A riot is only prevented when Casey, so graciously, raises a hand to silence the stands. A second strike is called when Casey ignores the next pitch as well. This time, he sneers at the fans who are jeering the umpire and they quickly fall into an awed silence. Casey lines himself up for the third pitch and, well...
The sneer is gone from Casey's lip, his teeth are clenched in hate; he pounds with cruel violence his bat upon the plate. And now the pitcher holds the ball, and now he lets it go, and now the air is shattered by the force of Casey's blow.
Oh, somewhere in this favored land the sun is shining bright; the band is playing somewhere, and somewhere hearts are light, and somewhere men are laughing, and somewhere children shout; but there is no joy in Mudville — mighty Casey has struck out.
This poem has been re-written and adapted countless times in the past 128 years. My first memory of it was a Disney version from 1946. I have no clue how or where I happened to see it, but it's firmly planted in my mind.
I also saw a version on Tiny Toon Adventures, where the role of Casey was played by Buster Bunny. In this version, however, along with many other adaptations, Buster actually hits the ball and scores the winning runs. A similar one was featured on Animaniacs.
While fun and entertaining, these parodies and remakes miss, what I feel, is the most important aspect of this poem.
Aristotle claimed that stories could be broken in comedies and tragedies. This wasn't a function of whether they were funny or sad, but rather of how the conflict arose. Stories where conflict happens as a result of circumstance or timing (earthquake, cancer, friends missing each other on separate trains, etc.) were comedies where those where conflict arose due to the actions of the main characters (lust, hate, love, ambition, etc.) were classified as tragedies.
Casey has a tragic flaw. His arrogance at the plate leads him to a place where his missing a single swing ended in his own ruin and the downfall of the team.
There is no epilogue to talk about the reaction of the crowd to the collapse of the titan. The only reactions that we see are fury at the umpire for completing the task assigned to him.
This fallen hero leaves the entire town of Mudville under a cloud of depression and sadness.
It wasn't until much later in my life that I realized the other piece that had been left out.
Mudville had an opponent.
As vocal as the crowd was in favor of Casey and against the tyranny of the umpire, there was an entirely different group who were rooting for Casey to fail. They may have been singing ballads of the pitcher who struck out the last batter in that crucial game.
More than ever before, this poem speaks to me. I feel for the crowd, who placed all of their hopes in a single person. How many of those people went home, cursing the umpire for doing his job, cursing the pitcher for playing as best as he could, or cursing the name of mighty Casey himself.
I don't see Casey as the villain. I see him as a product of the environment. Without a crowd to place him on a pedestal, would his arrogance have reached the level of allowing the pitches to fly by? Probably not. No one develops hubris in a vacuum.
That crowd went home, angry and disappointed, with no notion of their own culpability in their emotions. When your team loses, you blame the team, the umpire, the coaches, the managers, the weather, etc..
Only this isn't a game. Instead of disappointment and sadness, the emotions that run through many of my students, both present and former, are more akin to terror.
We are Casey and we are the crowd. Our arrogance allowed this to happen. Just hours after the election, the Klan was marching in celebration.
Several of my students came to me to express their hatred of Mr. Trump and I told them that while I sympathized, he wasn't the problem, merely a symptom of a much greater disease.
He is the spot on the otherwise pristine apple that belies that rot beneath.
My deep and fervent hope is that with this rotten center laid bare, with the attitudes of hatred, racism, misogyny and bigotry, now clearly in the open again, we can find a way back to where we need to be.
As a white man, I am deeply fortunate. I am, however, deeply afraid of what we have become. I fear for the message that we have just sent to our citizens as well as the rest of the world.
"Mr. Aion, what do we do now?"
We do what we can to spread love and tolerance. We make it clear that we will not accept hate and fear as the norm. We make ourselves available for those who need us. We use our voices to shout when we see injustice. We use our hands to raise up those in need. We speak for those with no voice and carry those who can't walk.
We stand with our brothers and sisters, regardless of creed, color, religious affiliation, gender, sexual identity or political leanings.
When they offer hate, we offer love. Where they build walls, we build bridges.
When they go low, we go high.
But we never, ever stop fighting for what is right.
And we never let our arrogance allow us to watch the pitch go by.
She has been spending the last few weeks talking about how stupid she is and how she doesn't understand anything. At the same time, she often refuses to write anything down and when given a task to work on, she stares blankly into space, holding still and hoping not to be seen.
She thinks I'm a predator.
I sat down in the desk next to her and painstakingly worked through the problem with her. I made sure to keep my voice low and calm. I tried to be encouraging without being placating. In 20 minutes, we finished the problem. It took that long not because she's stupid. She's not.
It took that long because I insisted on waiting for her to give me the next step and she insisted on waiting until I gave her the next step.
As she sat there with her face getting slightly red, hands clasped to the sides of her head, answering me in one syllable responses, I had a flashback.
I'm in middle school again, sitting at the kitchen table while my mom makes dinner. My math book is open in front of me, but I can hardly see it. My vision is blurry with frustration and colored by anxiety.
I'm crying and literally tearing at my hair, wringing my hands and gritting my teeth. There is nothing that I want more in the world than to not be doing math homework. I don't understand what's happening and have no idea how to be done other than completing it, which I don't know how to do.
As awful as it was at the time, my parents never let me give up. I would roar my terrible roar, and gnash my terrible teeth, and roll my terrible eyes and show my terrible claws, but I still had to do the assignment.
Somehow, I got past it, but not without considerable stress on myself and all of those around me.
I knew where she was coming from. I knew what she was feeling.
I knew that there was nothing that I could say to get her over that until she was ready to be.
So I sat with her and patiently worked.
She got the answer right, but with such prolonged agony, that I'm not sure it was worth it. How do I relieve that math anxiety? Is it ACTUALLY math anxiety, or some other form?
Was she upset because it was math, or because it was SOMETHING?
She is by no means alone in this. I'm starting to believe that the rambunctious nature of many of my students is less about maturity and more about comfort. It seems like a defense mechanism.
One of the many ideas that I collected while in Phoenix last week, was the idea of changing and formalizing my Warm-Up routine.
Up to this point, students have entered the room and there was an estimation question on the board from Estimation 180. The wrote their estimate and rationale. Then we read our Pledge to Improved Mathematics and discussed their estimates.
Over the last few weeks, I've been noticing that I've been slacking off in terms of the formalization of those tasks. I know, for my own self, habit building is vital to having a consistent classroom. To this effect, I took a page (literally) from Lisa Bejarano.
She has a form sheet where students complete their daily warm-ups for the week.
Monday: Which One Doesn't Belong
Tuesday: Visual Patterns
Wednesday: Estimation
Thursday: Mental Math
Friday: Error Analysis
For more specifics, check out her blog!
Today, we started with WODB.
I put the following image on the board and asked them to identify as many qualities as possible that make each shape different from the other three.
I gave them a few minutes to silently work on it and then we talked. I went through and asked what properties they had identified. We found several for each object and when they ran out of suggestions, I tried to offer one that they hadn't thought of.
What was most interesting to me was how almost every student identified the shape in quadrant 2 as being the only one that was "upside-down."
This sent my mind off in a strange direction. Can shapes be upside-down without a designated "top"?
Isn't "upside-down" a relative term anyway?
In any event, they seemed to like this new warm-up. I anticipate a little difficulty for the first few times, but I'm looking forward to the change.
Over the last few days, the Pre-Algebra students have been working with transformations. On Wednesday, they looked at the properties of various shapes that tessellate when only using translation.
Today, in 2nd period, we extended that and talked about reflection. This was a mistake. If I had planned it better, I would have moved to rotation first.
In anticipation of the difficult discussion that came with reflection of triangles, I built an applet in GeoGebra!
It allows the user to modify the angle measure and the lengths of two of the sides to create any triangle they want. It also reflects those triangles over the sides multiple times, creating a kaleidoscope
Translation is pretty limiting in terms of the patters that can be developed. Rotation is the next logical step because a clever person can use well placed mirrors to create a rotation, but reflection can't always be found through translation and rotation alone.
So I planned better and changed my lesson for my afternoon classes.
Thankfully, the class that's now on a slightly different schedule only has 5 kids in it and it will be easier to get them back on the track.
Not because it was necessary for the current discussion on rotation, but because later, including circles into the conversation will help students to better grasp the concept of rotation, I built another applet!
At NCTM this week, someone mentioned how when we teach rotation of objects, we only ever talk about 45-, and 90-degree rotations and I realized that he was right. He put up a picture with the rotation circles visible and it made so much sense that I wondered why I hadn't thought about it before!
Even having only had 3 work days this week, I'm exhausted and deeply glad that it's the weekend.
I'm going to go to dinner and watch Doctor Strange with some friends!
I love the white boards. They are versatile, fun and the kids love writing on them. When we use the boards, they get to be out of their seats and moving around.
They are, however, by nature, temporary.
I like the idea that students can go back and look at examples that we've discussed, but a harsh bit of reality intrudes on that.
They don't.
They NEVER go back and look unless specifically directed to do so. Part of this is because, as someone who never did well as a note-taker, I haven't put emphasis on it as a teacher. I also think that often the process of taking note for the sake of notes has a tendency to ruin the flow of learning.
Although that may just be an excuse to justify my own habits.
In Math 7 yesterday, we had begun a truly interesting discussion, using open number lines to talk about rational operations. A few kids came up with their own methods and were incredibly excited about it.
We had done our work on the whiteboard.
Today, when they returned and I attempted to continue the discussion, they had no record or memory of it.
What they did remember, however, was the feeling of excitement from yesterday. As a result of that, they were willing to go with me and took the time to focus. The discussion was fantastic!
We took the entire period to work on 2 problems, but we explored those two from multiple angles and approaches. Kids who normally are spaced out were involved and engaged. I'm honestly not sure exactly what the difference was, but I'm going to do what I can to recreate it.
I am incredibly proud of them.
Hopefully, their minds will not be wiped clean of this experience, like so many white boards.
In addition to all of this, the students received their report cards yesterday, which paved the way for tears and blame today. Several entered my classroom today and immediately asked about extra points.
"We've discussed this. Your grades are based on the mastery that you've demonstrated on the skills in class. Your grade will go up when you demonstrate the mastery."
"I don't know how to do that!"
"That's mostly because of the decisions you make in class to put your head down, to not complete assignments, to not write down the examples or the directions. On top of this, you are constantly telling yourself that you're stupid and the moment you have a tiny bit of difficulty, you give up."
This moved us into a 30 minute discussion about the choices that they make and the effort that they put towards their education. I made it very clear that while I will try to help them to the best of my ability and I respect the decisions that they make, I will not be held responsible for those choices.
The grade book remains open and I had a student stay after today to get extra help in preparation for reassessment.
The students who were most vocal, however, were nowhere to be found.
I will continue to encourage them to grow and learn and make myself available when they are ready.
A new student transferred into the district this week and today was his second day in that class. After class, I told him that we do normally do other things and he said he loved what we were doing.
"This stuff is great. What you're saying makes a lot of sense!"
My return from the regional conferences of the National Council of Teachers of Mathematics has left me physically drained, intellectually stimulated and emotionally energized.
I've been teaching the exploding dots to everyone who has been willing to listen. Robert Kaplinsky's talk on Depth of Knowledge has given me much to think about and I started implementing a bit of it today in Pre-Algebra.
Before I left, we started talking about transformations and congruent figures. The activities in the book fall under the umbrella of "determine if Figure B is a translation of Figure A."
I can do better than that!
"What do you think of when you hear the word 'translation'?"
In an effort to try to please me, they started trying to remember the technical definition. I stopped them.
"All of that is great, but it doesn't answer my question. The regular person on the street, when they hear 'translation,' what are they thinking of?"
Students: "Something about language."
We had a brief talk about how translating doesn't change the meaning of a word or phrase, but simply moves it to a new language. In the same way, geometric translation slides a shape, but doesn't change it's orientation or size.
They were familiar with the idea of tessellation, but we solidified the definition for them:
To tessellate is to cover a plane with repeated use of the same shape with no gaps or overlap.
We actually had a rather interesting debate about this definition, concerning whether it was acceptable to have "gaps" provided that those gaps were an integral part of the pattern.
Half of the class said both of these are tessellations while the other half said no.
Instead of putting shapes on the board and asking "are these translations?" I thought about putting the shapes up and asking if they could create tessellations from translations.
Instead, I put a giant box of shapes on the desks, along with a stack of graph paper, and had them determine which ones would tessellate ONLY using translation. They had to prove it either way.
"These tessellate, but I have to rotate them to fill in the gaps."
"So do they tessellate from just translation?"
"No."
**looks meaningfully at student, walks away**
As the groups finished, I had them think about, discuss and write about what they had discovered.
Back at the beginning of the year, I asked them to come up with a definition for a "sandwich" based on the properties. This time, I asked them to classify the properties of the shapes that tessellated using only translation.
Later this week, or early next week, we will do the same exercise, but add in rotation and reflection, and come up with rules. We also will be constructing tessellations that are more complex near the end of next week.
