Minecraft and The Coordinate Plane

I explained to my students today that my son forces me to play a game called Minecraft and sometimes we bury treasure chests for each other to find. I pulled up the map below and asked my class how they would describe the location of the treasure.

Students suggested a bunch of very vague directions:

• It's in the desert.
• It's where the snow and the desert meet.
• It's next to the large pond.
• No, I didn't mean that pond. The other pond.
• Go northeast, then dig.

None of these directions were that helpful. While some of the more detailed ones could have gotten me closer to the treasure, it's still difficult to find it unless you have the exact location.

Enter the coordinate plane. Some students were familiar enough with the game to know that x-, y-, and z-coordinates are given to you on the map. (They were cut off on my original picture.)

Of course, my students weren't exactly sure what those numbers meant, but it didn't take long for them to see that these values were simply directions from the origin of the map (white crosshairs) and they would provide the exact location. 

I particularly liked this introduction because it created a need for the coordinate plane (Dan Meyer did something similar here).

I'm Crushing Your Head

Yesterday, I e-mailed my favorite estimation guru, Andrew Stadel, a question about estimating and collecting data. He said I should share my insights with the rest of the world. So, for the dozens of you who read my blog, enjoy!

The other day, I wanted to start easing my sixth graders into estimation (before diving into Andrew's estimation180.com), so I put this up as a warm-up:

For most of my students, this problem caught them off guard. It seemed as if no one has ever asked them to guess the length of something. Some were confused about what I was asking and it was apparent in their answers. I made a line plot for each class and noticed that about 80% of each class thought that side B was 24 inches...as if I was referring to some archaic property of rectangles that says that the longer side of a rectangle is twice the length of the shorter side. Only a few students in each class even got close to the right answer (which I've put at the bottom of this post).

After we talked about some estimation strategies such as using your hand as a guide (see picture below) and identifying lower and upper limits of reasonable answers, many were eager to try another problem. As each of my classes is only 37 minutes in length (crazy, right?), I told them that we could try another one the next day.

"I'm Crushing Your Head!"

So, here's the problem I gave them the next day...

And sure enough, their guesses were much more informed. As with yesterday's estimation, I made line plots for each class's data and we could see that many more students were closer to the right answer. As a class, we felt that progress was made.

And then came the beauty of the line plot itself. For every class, I asked: what do you notice? In one particular class, we noticed that the data points were spread out. In another class, we saw that we had outliers. In another class, we saw that somebody guessed 18 inches, so they really must have been thinking that the rectangle was a square. In another class, we noticed that the data was skewed to the left or closer to a bell curve. In many of the classes, we noticed that students typically underestimate (which I'm very interested in understanding why, but I'm not going to delve into that here).

Later in the day, I noticed that the data from one class was very similar to a previous class. So I put both data sets up, and all of a sudden, we weren't just evaluating different students' guesses, but two different data sets. Finally, I added a third set, and we started having discussions about which class guessed the best. And the kids were really into it and coming up with some interesting ideas about how to determine the best class.

And I thought, this is awesome. Not only are my students driven to become better at estimating, but now they're looking at using math to help figure out if they're getting better at it and if they're better than somebody else. (They're downright vicious when you throw a little competition their way.)

By the way, the answers to the two estimation challenges are: The first rectangle is 12 inches by 32 inches. The second rectangle is 18 inches by 26 inches.

Every Math Teacher in the World Should Do This...Right Now!

Yesterday, I was teaching students how to find the greatest common factor of two numbers. We start this lesson by using easy numbers to work with (like 10 and 14), list all of the factors, circle the common factors, then determine which of these common factors is the greatest. No big deal.

Next, we moved on to bigger numbers (48 and 84), and it became much more challenging. Some students just don't know their times tables that well, especially past ten. 3×16 equals 48? Even I'm a bit sketchy on that one.

I showed the students how to write the prime factorization of 48 and 84 using factor trees (which they've already learned), how to identify the common prime factors, and finally, to multiply them to find the greatest common factor. I then immediately sent these students to the whiteboards surrounding my room, so that they could practice finding the GCF for a different set of numbers. As you can see in the picture below, every student has their own space to work.

What happened next? Only the greatest damn thing ever! When students are working on the whiteboards, I can see everything happening at once. It's like I'm looking at the freaking Matrix. With a quick glance, I can see which students got it, which students are making minor mistakes, and which students have no idea what's going on. I can quickly identify errors for students. I can ask a stronger student to help a struggling one. Once a student has the correct answer, I yell, "Great! Erase it! Next problem!"

And the kids love it. As soon as the kids walk into my classroom each day, they ask "are we working on the whiteboards?" As soon as I say, "Go to the boards!", they rush out of their seats potentially harming each other as they make their way there. As soon as I put a problem up, they quickly get to work, Even the students that I know would typically struggle in math class, love the whiteboards and are learning much more because of them.

Now imagine what would happen if these same students were doing this work in their notebooks at their desks. Would they be enthusiastic? No. Would I know how much my students understood about the lesson? No. Would I be able to help students in a timely manner? No. Would they learn as much? Probably not.

I cannot stress enough how much these whiteboards have transformed my students' growth. If you do not have enough whiteboard space on the walls in your classroom, install them as soon as possible. It is the most important thing you could possibly do.

If you'd like more information about this, visit Alex Overwijk's blog post on it (who I give credit for teaching me about Vertical Non-Permanent Surfaces). I believe Peter Liljedahl deserves credit for bring the research on VNPS's to light.

Real World Math

My school recently instituted an end-of-the-day program where almost all of our students partake in different activities. There's a journal day, a current event day, a homework day, a silent reading day, and a real world math day.

And there it is. Real World Math. I see it over and over again. It's almost comical. People love hearing that students are learning real world math (as opposed to all of that other crap that is typically taught in math class). The other day, a colleague was writing some horrible thing called a Student Learning Objective, and he was asking me what he should write. I told him to just throw a couple of buzz words in there like "real world". People eat that stuff up.

Why does all of this bother me? Because when we keep putting real world math on a pedestal, it marginalizes everything else I try to do in the classroom. It says that there are really only a few things worth learning in school, so when something doesn't sound like it's "real world", go ahead and give up. Tune out.

Yesterday, my students and I watched Vi Hart's video about doodling stars.

The kids were entranced by this. I stopped the video to show them what Vi was saying, because, let's be honest, she does talk way too fast. I then showed them how to make one of these stars by picking a random number of points (P) and a random skip number (Q). And they thought it was awesome. At this point, I pointed out that they will probably never use this in life. But that doesn't make it any less relevant. It is beautiful and fun. And if you get any kind of reaction out of it, then it was worth your time. Not everything has to be "real world".

Side note: Throughout the day, I worked on my own star in the back of the room. Pretty damn cool, right?

When am I ever going to use this?

As a teacher, I hate this question. For years, I would stumble with the answer, especially when I taught Algebra 1. As a former engineer, I could typically think of ways that I used math, but how does a lawyer, a nurse, or an animal shelter worker use algebra? I have no clue. And like an idiot, I would always try to construct some kind of answer that would never be satisfying to the student.

The real issue with this question is that the student only wants one answer. They want to hear you say, "You know what? You're right. You'll never use this. I've been wasting your time with this nonsense. Maybe I should just teach you how to pay your bills and call it a day."

Students don't want to hear about how every single profession uses box-and-whisker plots. The reason they ask this question in the first place is because they are frustrated. They don't get what you're trying to teach. And they're just looking for an excuse to give up. If the same students were learning FOIL, and could produce a right answer every time, they probably won't complain about never using it (even though they probably never will).

If they don't have to struggle very much to learn something, then they don't need excuses not to learn it.

---

Neil DeGrasse Tyson is a hero of mine. I even got a print of him to hang in my classroom. You can buy it here.

I've heard him talk about how students will often lament about how they will never use some of the things they've learned in school. Here is a panel discussion where he talks about this. In this video, he goes on to say how working on problems in physics (and math) helps rewire the brain and prepares it to solve other problems. And understanding how things work will lay the groundwork for innovation. This is exactly what most business owners want from their employees. They need problem-solvers. They need innovators.

On the first day of school, I talk about this with my students. I explain that the jobs of the future require us to be innovators and inventors. I then show them this newspaper clipping from the local newspaper:

Each year, kindergartners are asked what they'd like to be when they grow up. I read each of these responses with my students. I don't hesitate to tell them that I also want to be Elsa from Frozen. And then I point out Emmett's entry. Emmett wants to be an inventor. I explain to them that I'm really excited about this because Emmett happens to be my son (which would help explain this child's fascination with Back to the Future). I tell them I'm excited because, even at an early age, Emmett wants to learn about math and science. The motivation is there. He is already inventing things and experimenting with electronics sets.

There is only one problem with Emmett. This summer, we went to Disney World and one of his favorite attractions was the Jedi Training Academy. Basically, they give you a light saber, throw a brown robe on you, and after some light saber "training", you face off against Darth Vader. When it was Emmett's turn to fight Darth Vader, he seemed very reluctant to fight. I wasn't sure what was wrong, but after it was over, he explained that he didn't want to fight Darth Vader. He always sympathizes with the villains in movies. He wants to join the Dark Side. What's troubling is, I fear that some day he will take his love of invention, and use it for evil.

So, while I would like all of my students to be intrinsically motivated to learn about math and science, I am really worried that Emmett may someday destroy the Earth. We need smart people to stop him. And that's my new rationale for why students need to learn everything in math class.

Visual Patterns

I'm rehearsing for a play called August: Osage County, and in it, my character abruptly stands up and announces, "I have a truth to tell!"

I also have a truth to tell. I never used Fawn's Visual Patterns website. I've known its existence, but I try not to do too many new things each year because I have a hard time following through on everything. I have been using Andrew's estimation180.com, which will probably infuriate Fawn even more. Sorry, Fawn.

Fawn wrote about her opening day activities, and one thing she included was this visual pattern:

After reading her post, I decided that this is the year I'm going to give her website a shot. On the second day of classes, I asked students to draw the fifth diagram. Aside from having difficulties with drawing cubes, most of the diagrams were fine. I then explained that we would be looking at more patterns throughout the year, and they would develop a better understanding of algebra because of this. I explained that one of the things I would like them to learn is how to figure out how many blocks would be in any diagram, such as the 43rd. As soon as I said that, about six kids started frantically scribbling in their notebooks. My first thought was, "Crap! I was going to do something else now and I've just distracted you with a math problem!" And then I thought, "Hey, they're distracted by a math problem. Let's go with that." So, as I would normally do in this situation, I let them try it. And sure enough, quite a few of them figured out the correct number of blocks in the 43rd diagram. There were even some great variations on the process that we were able to share (none of which I've captured here...sorry).

This was the first activity of the year that challenged my students. I'm pretty sure that Visual Patterns (along with estimation180!) will continue to be a part of my classroom routine.

Circles and Dry-Erase Boards

This summer, at Twitter Math Camp (TMC), I met Alex Overwijk, the World Freehand Circle Drawing Champion. Check out his video on youtube. It's amazing.

Alex is a great guy, and while this circle thing is a pretty cool gig for him, he knows a lot about good teaching. He (and another great friend, Mary Bourassa) did a great presentation at TMC on spiraling curriculum. He also gave a presentation on Vertical Non-Permanent Surfaces (which is just a fancy way of saying chalk- and dry-erase boards). The research comes from Peter Liljedahl, and Alex does a great job summing it up in his blog post.

It was clear to me that I needed more dry-erase boards. Big ones. All over my room. Wherever I could put 'em. So I made a trip to Lowe's, got a bunch of white panel boards cut up, and fastened them to my windows and to one of my bulletin boards.

With so much white board space, I can have every student working on the boards at the same time. With a quick glance, I can see what every student is doing. I can find mistakes faster. I can see who needs the most help. If a particular student has some great way of organizing his work, I can take fifteen seconds to point this out to all of the other students. We can share more easily and compare different solutions.

Compare this to how I typically had students working over the past eight years. They sat at their desks, working in their notebooks. I would walk around, constantly trying to check work one student at a time, struggling to see what they scribbled on their paper. It might take five minutes for me to make my way around the entire room, only to find one student has nothing on his paper because he spent this entire time trying to fix a mechanical pencil.

After just one week, I am convinced that installing these boards was the right move. I can't wait to see how this affects my students' learning this school year.

Side note: I mentioned Alex to my students and tried to demonstrate how he drew his circles. Here's my first attempt:

So I started practicing a little bit throughout the rest of the day, and before I left the school, I was able to produce this:

I still have some practicing to do. The upper left parts (of both circles) extend out a little too far. Maybe someday I'll be good enough to challenge Alex.

Van Gogh in Post-Its

It seems that every year I try to do something ridiculous in the name of mathematics. Last year, my students and I created a humongous triangle out of toothpicks. This year, I wanted to do something a little prettier, and so, I decided to create a Post-It mural of Vincent Van Gogh's Starry Night on my classroom windows.

Students were tasked with calculating how many Post-Its would be required to cover the windows and approximately how long it would take me to finish. They were supplied with the following information:

• Each Post-It is 3 inches by 3 inches.
• There are seven windows. Each window is 45.5 inches wide and 80.5 inches tall.
• I can place 17 Post-Its on the window in 2 minutes.

After school that day, I got started on the windows. I started at 3:45 and finished at 9:15 that night. I placed 2,730 Post-Its on the windows. When I got home, I created the time lapse video below to show the students the very next day. (This was created using an App called "Lapse It". It's very easy to use and costs only $1.99.)

Van Gogh's Starry Night in Post Its from Nathan Kraft on Vimeo.

Many people have asked how I created this. I took the original painting and fit it to a grid I made on Excel. I then looked at each individual grid space and decided what color each should be. This was a little tricky as it is not easy to show good definition in Post-Its. You can see the side by side of what I created in Excel below.

The other challenge was trying to pick the right color for each space, as Post-Its are only available in so many colors. There are nine colors shown here: black, white, gold, yellow, dark blue, light blue, lavender, orange, and hot pink.

The best part of this project was that the Post-Its gave the windows a stained-glass effect. The two pictures below are taken inside with the lights off and outside with the lights on at night. And after a month, the mural is still intact. Not one Post-It has fallen down.

Finally, I'd like to thank Andrew Stadel who was partly responsible for inspiring me to do this through his File Cabinet lesson. And a special thanks to Blair Miller who tweeted that mine is better.

Guessing Percents

This is my first year teaching sixth grade math, and one topic that students have to understand is how to construct circle graphs. This is a part of the old Pennsylvania math standards which are currently being phased out because of the Common Core. (Note: Circle graphs are not mentioned in the Common Core standards (prove me wrong), though finding angles and percents are.)

These sixth grade students have some familiarity with common percents and how they relate to fractions (50% = 1/2, 25% = 1/4). They have experience measuring angles and can identify straight and right angles. They can convert fractions to decimals. But how all of this relates to circle graphs is a mystery to them.

I'm preparing them to create these circle graphs, and before I explain to them how to do it, I present them with a few simple ones to see if they can guess the percents. Here is the first one:

Me: Ok, can someone pick a sector and tell me what percent it represents?

Ray: The green one is 50%.

Me: How do you know?

Ray: Because it looks like half of the circle, and half is the same as 50%.

Me: Ok, how many agree with that?

(The majority of the class raises their hands.)

Me: Great! I'll write 50% in here. Now, what else do you know?

Dana: The purple one is 25% because it is one fourth.

Me: Ok, how many agree with that?

(Again, just about everybody raises their hands.)

Me: Alright, we'll put 25% here. Hmm. I'm guessing these last two are going to be tougher. Can someone tell me something about these last two sectors without telling me the value of either sector?

Winston: They add up to 25%.

Me: How do you know that?

Winston: Because the circle represents a whole or 100%, and all of the sectors' percents need to add up to 100%. So if we know the other sectors add up to 75%, we can subtract that from 100% to find the remaining percent.

Me: Great! Does everyone see that? In fact, when you look at red and orange sectors together, they look like they're the same size as the purple sector. So a total of 25% makes perfect sense to me. Can anyone tell me anything else about these two?

Janine: The orange one is bigger than the red one.

Me: So?

Janine: So the percent for the orange one should be bigger than the percent for the red one.

Me: Ok, do you have a guess as to what those two percents could be?

Janine: Well, I'm guessing that the orange one is 15 and the red one is 10.

Me: How many people agree with Janine?

(Again, a bunch of hands go up, but not as many. Some students are squinting at the board, trying to come up with a better guess.)

Me: It doesn't seem like we're too sure this time. (Peter is eagerly waving his hand in the air.) Yes, Peter?

Peter: I'm pretty sure that Janine is right, because it looks like you could fit two red pieces inside the orange piece.

Me: Can you come up to the board and show us what you mean?

Peter: If I use my hand to measure the orange piece, I can fit two hands along the edge of the red piece.

Me: How many agree with Peter and Janine then?

(More hands go up.)

Me: Ok, then I'm going to write Janine's original answers in. Now, does anyone know a good way to make sure that our numbers make sense?

Egon: We could see if they add up to 100.

Me: Do they?

Egon: Yeah.

Me: Great! Ok, I guess we have to check to see if we're actually right.

(One at a time, I remove each white box that is covering each answer, and everyone is relieved to see that our guesses were correct. Ray looks skeptical.)

Me: What's wrong, Ray?

Ray: I don't think that's right. To me, the orange one looks bigger...maybe 16 or 17%.

Me: Well, to be honest, I got this picture off of the internet. And we all know how reliable the internet is. Maybe I shouldn't be so quick to assume that all of the numbers are right. Do you have any ideas about how to check to make sure that it's right?

Ray: We could measure the angle.

Me: Actually, we're in luck. I think there's a protractor tool in Smart Notebook that lets us do that. (I pull up the protractor and quickly measure the angle.) It looks like it's about 55 degrees. It's a little hard to see on here. So, what does that mean?

Dana: That's in degrees. We want percent.

Me: Yeah, so how do we change the number of degrees into a percent?

Ray: We could turn it into a fraction.

Me: Yeah, but...we know the part is 55 degrees. What's the whole? How many degrees are in the whole circle?

Ray: 180. No, 360!

Me: Is that right, class?

(I get some nods.)

Me: Alright, you guys tell me. 55 out of 360. What is that as a percent?

(Some kids punch some numbers into their calculators.)

Louis: I got it.

Me: What is it?

Louis: 0.152777...

Me: Ahhhh! Stop! Round it off to the nearest hundredths.

Louis: Umm, 15 hundredths.

Me: And what is that as a percent?

Louis: 15%.

Ray: Well, I was kinda right. I did say it was bigger than 15, and it wasn't exactly 15.

Me: Yeah, maybe. Or maybe I made a mistake when I measured the angle. To be honest, I think guessing these percents and only being off by a degree or two is really good. Let's try some more...


We then go on to try another one on the board (see picture below) and we have some more great discussions/arguments.

Once the students started to get the hang of it, I gave them two graphs to try (one easy and one hard). They were on paper and I encouraged them to use whatever tools they thought might help them get the answers (rulers, protractors, calculators).

They worked with partners and I walked around to check that each group had the right answers for the easy circle graph. With only a few exceptions, everybody seemed to have a pretty good feel for how the different sectors related to each other and what the percents should be.

The tough circle graph certainly proved to be more of a challenge, and it was great to be able to talk to the students about their reasoning and question them when something didn't make sense. The student below thought that sector C was 10 and sector D was 5, probably in order to make everything add up to 100. I pointed out that this wouldn't work because it means you could fit two D's into one C, or in other words, C is twice as big as D.

This student used his ruler to cut the graph into quarters and make better guesses about each percent.

This student assumed that sector B was about 30 degrees, then chopped the entire graph up into 5 degree segments.

Though it's not obvious, this student used the protractor to measure the angles and converted them to percents. He had the closest, though not the exact answer.

After the students were finished these graphs, I had a spreadsheet ready to go that would calculate the total error in degrees for each graph. Students volunteered their answers and, after all five percents were entered, the spreadsheet showed their error score. It was fun to see the students get excited. After seeing their error and some of the other students' thinking, I allowed them to revise their answers and try again.

I loved seeing all of the different approaches used and listening to students argue with each other. As a bonus, there were definitely some bragging rights for the kids who got the closest. While many students heavily relied on using other strategies, they now saw the benefit of using a protractor and were eager to know how to convert those fractions into percents.