Writing in Math Class: Greatest Common Factor

I'm trying to find more ways to get students writing in math. I know that the process of writing helps clarify and consolidate thoughts. It also is a great way to have students engage with the vocabulary.

After teaching three different ways to find the greatest common factor of two numbers (list all of the factors, use prime factorization, simplify fractions), I split the students up into three groups and asked each group to solve the problem a different way.

As they solved it, I took note of which groups finished earlier, which groups made more mistakes, which groups were more confused, etc. We reviewed each of the three solutions on the board and I then asked everyone to write one good thing and one bad thing about each method. I then asked students to share those thoughts and I summarized them on the board next to each solution (see picture).

Not only did creating this pro/con list help students decide which method they preferred, but it also clarified some misconceptions about why each solution works. They also saw some similarities between the three methods (the numbers 5 and 7 keep showing up). Incidentally, most students did not like method #1, but I warned them that, because it is so intuitive, it would be the method they remember the best.

An Alternative to "Add the Opposite"

I've always been a little bothered by how textbooks (and presumably, teachers) explain subtracting integers on a number line. Here's an excerpt from a recent Pearson textbook which has been aligned to the Common Core:

From this, we see that 9 - 5 = 9 + (-5), and from that we conclude that we can always subtract numbers by adding the additive inverse. This makes sense, but what about subtracting a negative? We're just supposed to accept that it is the same as adding a positive? Or what if we are subtracting negatives from a positive? How do you take something away when it's not even there? (I know...zero pairs.)

So how do you explain this without simply telling students to "add the opposite"? Wouldn't it be better if students were comfortable with subtracting negatives?

I teach adding and subtracting integers by having students locate the first number on the number line. You then have two options...you're either going left or right. To do this, they look at the operation. If they see +, they think that they need more of something. If they see -, they think that they need less of something. If we see plus a positive, we need to go in a more positive direction (right). If we see plus a negative, we need to go in a more negative direction (left). For minus a positive, we go less positive (left). For minus a negative, we go less negative (right). And that's it. It makes sense to them and we don't have to be afraid of the subtraction sign.

From here, students use number lines to solve addition and subtraction problems, and eventually, they start to make their own connections. They see that subtracting a negative has the same effect as adding a positive. They see that subtracting a positive has the same effect as adding a negative. As we work with larger numbers, students become less reliant on the number line and use their intuition.

One of the best things about teaching this way is that some of my struggling students can always fall back on the number line. Don't get me wrong, it can be painful to watch a student solve -27-1 by extending a number line far out to the left. I let them do it and then ask them to try a similar problem without writing anything down. Over time, they learn to trust themselves and do it mentally.

Another nice thing about teaching this way is that you can easily extend these ideas to multiplying integers. Positive times negative means more negative. Negative times negative means less negative. You can show how this works with repeated addition/subtraction: -3(-4) = -(-4) - (-4) - (-4).

I hope this provides you with a better alternative than the standard textbook explanation. If you try this, please leave a comment below on any insights that you have.

Developing Student Intuition for Mean Absolute Deviation

                For some time, I’ve been considering a new approach to teaching mean absolute deviation (MAD). This is a new concept for 6^th grade as it is in the Common Core standards (CCSS.MATH.CONTENT.6.SP.B.5.C) The lesson in the student’s textbook is not terribly helpful. It doesn’t give any purpose for finding the MAD for a set of data and the directions for doing so are somewhat intimidating. It is my hope that I can help students intuitively derive MAD on their own, or at the very least, give them the motivation to learn MAD to identify which set of data has more spread.

                Last year, I had the same hopes of creating this intuition by having students create equilateral triangles. This idea was borrowed from a similar activity I worked with Dan Meyer on where students had to identify which of four triangles was the most equilateral. I had students create their own triangles and measure the lengths of their sides. We compared the triangles and their measurements to determine which was the best.

                It was my hope that students would see the data and have some basic understanding of what to do with it. Unfortunately, I only had one student in my five classes really figure it out without a lot of assistance from me. It was obvious that, if I was going to do this lesson again, I would have to find some way of creating an easier path for my students to find the MAD. To build investment and help find meaning, I would again need data that was student generated, but easier to work with. Thinking about absolute deviations would have to come naturally and the mean of those deviations the obvious answer to comparing data sets.

I created a game for students to play that would require the MAD to determine the winner. Of course, I couldn’t tell the students that this was how the winner was determined. They would have to come up with this method on their own. I called for two volunteers to come up to the front of the class and explained that they would be rolling two dice. Whoever rolled a sum closest to seven would be the winner. One student rolled a five and the other student rolled a ten. I placed their sums on a number line in the front of the room for everyone to see and asked who won and how did we know.

There were a couple of variations in answers, but the general idea was that one was closer to 7 than the other. One student was more specific about how five is two away from seven and ten is three away from seven. Therefore, five is better. I tried to impress upon my students that quantifying how far each number was away from 7 would really help them as we worked through these different scenarios.

                I asked the students to roll again, but this time I wanted them to roll twice. The boy rolled a seven and a four. The girl rolled a twelve (already losing) and a ten.

It seemed obvious who won, but I asked students to write down a sentence or two telling me who won and explain how they know. There were a couple of ideas about this, but no one was really thinking about mean absolute deviation at this point. To their credit, it would not make sense to do it here. There are much easier ways to compare these sums. What I did want students to see is that the boy’s two sums deviated from seven by three and zero. The girl’s two sums deviated by three and five. The sum of those deviations was enough to determine the winner.

                One girl said that she determined who won by taking the average of the sums. I thought this was a neat idea and it didn’t occur to me to think about it this way. The boy’s average was 5.5 (1.5 away from 7) and the girl’s average was 11 (4 away from 7). This seemed to validate our belief that the boy won. I asked the two students to roll again and again had the students write about which person won. The girl with the averaging method used it again, and again it seemed to work. I then created a hypothetical situation where the girl would roll two seven’s (best case scenario) and the boy would roll a two and a twelve (worst case scenario). I asked, “Who won?”

Before anyone even answered, I could see some students making the connection that the average was not going to work every time. In this case, the sums both averaged out to be seven, indicating a tie, but the boy’s sums were obviously worse than the girl’s.

                I explained that the students would now be placed into groups and creating their own data. With one student rolling the dice for me, I showed students how to record their results. I rolled the dice ten times and when finished, I had a line plot that looked like this:

After students finished creating their own line plots, they brought them up to me and I recreated them on Microsoft Excel:

                With this data, I asked students to rank the line plots from best to worst. Three groups volunteered their rankings:

We noticed that we were in agreement about ranks 1, 2, 3, 7 and 8, but we had trouble figuring out how the middle groups performed. I placed two of these groups’ line plots on the screen and I asked all students to figure out, mathematically, which one was better.

                From here, I got a lot of interesting ideas from the students. One girl tried making box and whisker plots of the data. This made sense because we’ve been using box and whisker plots lately to describe spread by looking at the range and interquartile range. (The following day we had a conversation about how box and whisker plots can be misleading when trying to understand spread.) Another student had an idea to compare the sums from each side. Another girl tried to develop a point system where a sum of 7 would be worth 7 points, 6 and 8 would be worth 6 points, 5 and 9 would be worth 5 points, and so on. The point values were somewhat arbitrary, but she was really developing a good way of quantifying the spread. After sharing this method with the class, another girl suggested using the distances to seven instead, just like we did in the beginning of the class. Rolling a 7 would be worth zero points, rolling a 6 or 8 would be worth 1 point, and so on. I didn’t mention this to the class at the time, but this girl was describing the absolute deviations.

                I wrote down all of these deviations with the class and asked, “What’s next?”

Box and whisker plot girl asked if we could add all of these deviations together and compare. So we did. We found that Amari’s total sum of these deviations was 37 and Avarey’s was 28. Most of the students felt that Avarey was clearly the winner. Amari quickly raised his hand to protest, “But I rolled more times than her! That’s not fair!” At this point, many students suggested that either Avarey’s group be forced to roll an equal number of times, or we remove some of Amari’s data. I asked them to consider how we compare different hitters in baseball. If one player gets 78 hits in 100 at bats and another player gets 140 hits in 200 at bats, we don’t force the first player to take 100 more at bats to even things up. After a couple of students made guesses about how to do this, a girl suggested we find the mean of these differences. We quickly divided each value by the number of rolls each group made and found that, on average, Amari was 1.85 away from 7 and Avarey was 1.87 away from 7. We can say that Amari’s rolls were closer to 7 (less spread), but just barely.

                We then reviewed how the students ranked each of the line plots and compared this against the mean absolute deviation for each (picture below). It was interesting for students to see how some of their predictions came true and how they were completely wrong for others. Nevaeh’s data is a good example of this – students overwhelmingly thought that her group came in last place, but her score indicated that she was actually in 3^rd place. This misplacement had more to do with students thinking less about spread and more about total number of rolls in the 6-8 range. Because Nevaeh didn’t roll as often as the other groups, it was assumed that she lost because she didn’t roll very many 6’s, 7’s, or 8’s. However, she only had one sum that was far from the center.  (There is probably a good lesson here about how the amount of data collected affects comparisons of data sets, but there was no time for me to discuss it.)

                Now that we had some way of comparing the data, I asked students to collect one more data set. Again, they had to roll their dice and write down the sums. The only difference is that they had to find the absolute deviation from 7 for each roll and take the average of those deviations. Students turned their data in to me and I quickly checked that they calculated the mean absolute deviation correctly. Again, we compared line plots and checked those comparisons against the MAD of each data set.

                During the next class, we took some quick notes on how to calculate the MAD (this time using the mean of the data set as our central point), constantly referring back to the work we did the previous day. Students practiced by finding the MAD for a made up set of data. Finally, they calculated the MAD for average high temperatures for different cities in the U.S. (This came out of necessity. I explained that the temperatures in Pottsville, PA varied way too much and I needed to move where it’s warm all year round. As they were anxious to see me go, they had quite a few suggestions.)

                Overall, I’m pretty happy with how this lesson went. I think it was worth building the context over time and it pushed them to really connect the visual (line plot of the data) with the statistic. When we calculated the MAD for the different cities, students already had an intuition about which cities would have a low MAD and what that number actually means. I feel confident that I will keep this lesson for next year with some minor adjustments.

                Special thanks to Bob Lochel and Tom Hall, two math teachers who were nice enough to exchange ideas with me about this through email. Also, I'd also like to thank Stephanie Ziegmont for helping develop some of the writing components of the lesson.

Can you remember more than 7 digits?

The other day, I came across this website that tests your ability to remember digits.

I thought it was interesting that, according to the website, the average person can remember 7 numbers at once. I've heard this before. This is supposedly the reason why telephone numbers are 7 digits long. At this point, I'm sure you're wondering if you are an "average" person. So, go try it...http://www.humanbenchmark.com/tests/number-memory.

Did you do it? I did it a few times myself and the farthest I got was 12 digits (my worst was 10). This probably means that I'm a superhuman or I have evolved past the rest of you. I'm sorry, but your days are numbered. (Numbered! Get it? No, of course you don't.)

I was still curious about this 7 digit claim, so I posed the problem to my students. Can the average person really only remember 7 numbers?

I had all of my students load the website and play along. After everyone was finished, I recorded the results and made a line plot with the data.

I asked the students to talk to their neighbors about whether or not this data confirms that the average person can remember 7 digits. Overwhelmingly, they felt pretty good about it, especially since the median of the data was 7. (I should note that sixth grade standards are all about analyzing distributions.) They were also able to see that more than half of the students were able to remember at least 7 digits, but less than half could remember 8 or more. Another reason to believe the claim that the average person could remember 7 digits.

We then discussed strategies for memorizing the numbers. Some students mentioned that they chunked the data...remembering 62 as "sixty-two" instead of "six-two". Some of them would practice typing them to build the motor memory. 

I also shared a couple of my own strategies...sometimes I could associate a number with something. For instance, once I saw a 53 and, for whatever bizarre reason, I remember that as Bobby Abreu's jersey number. Once I had that image of Bobby Abreu in my head, I stopped worrying about remembering 53. For the longer sets of digits, I would repeat the second half of digits over and over again while staring at the first half of digits. This way, I was relying on both my visual and auditory memory.

Now that the students had some new strategies, I gave them another chance to increase their digits.

As you can see, the data changed, but there really wasn't much improvement. Many students did worse while a few did marginally better. We couldn't make much sense of it, though we suspected that some of these strategies need to be practiced before we could see some results.

At this point, it would have been nice to keep practicing to see if we could improve, but my period is only 37 minutes long. I also had a couple of situations where students figured out they could copy and paste their answers. Cheating would be difficult to monitor.

Side note: Some of my students with IEPs could only remember three digits. This was consistent each time they made an attempt. This was eye-opening for me...when short-term memory is so weak, learning anything must be a huge struggle.

Why, Common Core? Why?

The other day, I was checking students' work on mean, median, and mode. One of the problems involved finding out what grade you would need to get on a fourth test to have an average of 85 for the class. It's basically a mean problem in reverse, and for students who have never solved this problem, it can be challenging.

One of my students was struggling with this and wrote in her notebook, "WHY COMMON CORE WHY". I laughed and assured her that this problem has been around a lot longer than Common Core. What I really found amusing was that, in terms of content, this sixth grader really hasn't been exposed to some of the more unique things about Common Core. Most of that is happening in elementary school and Pennsylvania only switched over last year, when she was in fifth grade.

In all likelihood, this girl's hatred towards Common Core probably stems from something she overheard her parents say. And now, every time I present her with a challenge, a little voice in the back of her head is going to tell her that this problem is Common Core and it's not really important for her to figure it out. And that's all she needs...another reason to give up.

Warm-Ups with a Purpose

Warm-ups last year:

I would display four or five review problems on the Smartboard for students to work through as I took attendance. I would then walk around the classroom to see how students were progressing, but would often struggle to help very many of them, nor would I have a good sense of how the class did as a whole. We would then review every problem which was time consuming and not always helpful. The next day, I would create a few more warm-up exercises but I never had a clear picture of what my students were still struggling with or why.

Warm-ups this year:

I was asked to move into a new classroom where every student would have his or her own computer. Over the summer, I looked at several websites that would help me use formative assessment on a daily basis. I was happy to find Socrative (which is FREE!) and I use it everyday for my warm-ups. Students can quickly log in and start working on the exercises. I can create multiple choice, true/false, or short answer questions, and as students are answering them, I can see their responses live! It looks something like this...

This is kind of a big deal. As soon as a student gets something right or wrong, I know. And there's a lot I can do with that information. During those exercises, you'll routinely hear me say things like...

"Mary, awesome job on that last one. Everyone's having trouble with it."

"Almost everybody's getting #1 wrong. Make sure you read it carefully!"

"Sheri, that last one...how are you supposed to set up an addition problem with decimals?"

"Fawn, you seem to be having trouble with greatest common factor. Can I see your work for that last problem?"

"Hey, Andrew. Where's your notebook? Stop trying to do the work in your head. You're not Rain Man!"

After the students finish the exercises, I share the results with them and I let them tell me which ones we need to review (and which ones we don't). We look at commonly selected wrong answers and think about what mistakes students were making.

At the end of the day, I can throw this data onto a spreadsheet (shown below) and decide which topics/skills students have a firm grasp and which need further review. I can see how students progress in some skills over time and share that as a model of learning.

I love that students are getting instant feedback. I love that I have evidence of their growth. I love that we can review results as a class and, rather than students only focusing on their own mistakes, we can ask ourselves, what are we, as a class, doing wrong? What are we, as a class, doing right?

Quizzes without Grades

A few weeks ago, I blogged about how I was going to stop putting grades on quizzes. This decision was heavily influenced by Dylan Wiliam's ideas from his book, Embedded Formative Assessment. I also need to mention that Ashli Black has very helpful explaining how she does comments-only grading and pushing me to design a system of grading that works.

This past week, I was finally able to test-run this idea after the students took a quiz on the Order of Operations. I explained my reasoning to the students and, for the most part, they seemed to be okay with it. I told them that this creates a better working environment where students can feel less embarrassed about their performance and work together to identify and correct their mistakes, no matter how well they did. I marked the quizzes by circling the problem number for every wrong solution and then color-coding three problems that I wanted the student to correct. If a problem had a pink mark, they had to identify their error. If there was a purple mark, they had to rework the problem. If a student did not get anything wrong, I gave them a more challenging problem to solve. Finally, while grades were not written on the quizzes, they were calculated and recorded into the online gradebook so parents and students could see them at home.

Overall, I thought it went really well. The students had about 10 minutes to work alone or together on their mistakes and handed the quizzes back to me. Those who did not finish had extra time overnight to do so.

The next day, I used socrative (an online quizzing tool) to ask my students how they felt about my "no grade" policy. The good news is that 70% of my students either liked it or didn't care. More students liked it than didn't like it. However, there is still 30% of my students that didn't like it. While it was not obvious in their responses, I believe that this frustration comes from not having that instant gratification of knowing what your grade is. This impatience isn't unexpected. Many times students will ask me if I graded their quiz ten minutes after handing it in.

In the end, I think the benefit of students revisiting their work and working together to fix mistakes outweighs the annoyance of not getting their grades right away. I'm hoping that, over time, students will begin to also see that benefit.

As a side note, I should say that I'm not really doing "comments-only grading". I had considered writing out comments, but it occurred to me that most of what I'd be writing could later be discovered by the student upon more reflection or figured out with help from a classmate. I believe that writing comments on every wrong answer would have been extremely time consuming and would have deprived my students from discovering their own mistakes.


Update 2/15/15:

Carolina Vila (@MsVila on twitter) asked me if I have kept up with this system. As with anything I experiment with, I look for more efficient ways to do things. (Okay, maybe I just got lazier.)

I mentioned that I color-coded problems in the beginning of the year and that these colors would tell students how I wanted them to reflect on each problem (identify the error/explain what they did wrong or rework the problem). After doing this a few times, it just seemed to make more sense to have students do both things. On a separate piece of paper, they would have to tell me which three problems they chose to rework, tell me (in sentence form) what they did wrong, and finally, rework the problem.

For students that got everything right, I backed away from trying to give them a more challenging problem, and instead, asked them to help other students make their corrections.

Students would turn in their corrections along with their quiz, I would check to see that it was done, AND THEN, I would write their grade on the quiz to give back to them the next day. When I first started taking grades off of the quizzes, I had hoped that I could just put their grades online for them to check, but I ran into too many issues where students and parents couldn't check the grades online because they lost their passwords or didn't have internet access at home. By finally putting the grades on the quizzes, students complained less and respected the correction process more.

On the student side, one of the biggest misconceptions was that making quiz corrections would improve their quiz grade. I explained that they would get credit for making the corrections (similar to a homework grade), but that their quiz grade would remain the same. The only way their grade would improve would be to retake the quiz, and the only way a student would be allowed to retake a quiz is if he or she made the corrections on the first quiz. Altogether, there is plenty of incentive to make these corrections.

Spaced Practice and Repercussions for Teaching

I've been reading John Hattie's book, Visible Learning, in which he ranks the effect sizes of different strategies that help student achievement. One of the strategies that is pretty high on the list is that it is better to give students spaced (or distributed) practice as opposed to mass practice. In other words, rather than having a student practice something over and over again in one day, it is much better to spread that practice out over multiple days or weeks. (You can read one of these studies here.) The main benefit is that spaced practice helps with long-term retention.

While this research certainly gives some justification for providing students with multiple opportunities to revisit older topics, I am left to wonder if this should change how I structure my lessons and assessments. I, like many others, teach by units. My students might spend a month on fractions followed by a test. They then get a month of algebra followed by another test. We, as teachers, create this span of time when all learning about a particular topic must happen. We don't always give students the time to practice these ideas, particularly the more challenging ones that almost always happen at the end of the unit and right before the test.

Based on what I've read about spaced practice, I would propose that teachers shouldn't give tests at the end of a unit. Perhaps students need time to practice these skills over several weeks before you should assess them. This is something I'm going to explore this year with some of the concepts that were challenging for my students last year.

Note: This is probably not an original idea and I'm sure someone else out there has probably explored it. If you have any resources to share on the subject, I'd greatly appreciate it!

Another note: I do allow my students to retake quizzes which I had hoped would send the message that learning doesn't stop after the quiz is taken. However, very few of my students have taken advantage of this in the past. I am hoping to correct that this year with some ideas from Dylan Wiliam, Ashli Black, and others.

Update: Henri Piccioto has written about this and calls it "lagging homework". He also reinforces the idea that quizzing should happen much later then when the material was taught. Thanks to Mary Bourassa and Chris Robinson for helping me find his work!

Movie Popcorn

I ordered a small popcorn at the movie theater and the cashier asked me if I'd like the large size for only $1 more. I knew that this had to be the better deal, so I took it. I mean, what if I had gotten the small popcorn and ran out during the movie? That would be unacceptable.

However, as I left the theater, I noticed that I didn't actually eat all of the popcorn. There was about two and a half inches of popcorn left at the bottom of the bucket. I could take it home with me, but stale popcorn doesn't sound too appetizing and I decide to throw it away. Did I just get ripped off? Should I have just bought the small popcorn?

There's a couple of ways of modifying this task to address the needs of different grade levels. It all depends on what information is given to the students. If you can just give the students the number of cups of popcorn in each bucket, then this is a fairly simple unit price problem. If you just give dimensions of the buckets, you will need to derive and use formulas. It would also be extremely helpful to use a spreadsheet.

6th Grade Version:

Info required...

Questions to explore...

What is the unit price for each size?

What is the percent change in size, price, unit price?

What is the least amount of popcorn from the large container (in cups) you would need to eat so that you don't get ripped off? (This is not as interesting a question as the 8th grade version because you can't usually tell how many cups of popcorn are left in a bucket.)

8th Grade (or beyond) Version:

Info required...

Volume of a truncated cone:

Source: http://keisan.casio.com/exec/system/1223372110

You will notice that there is a little bit of popcorn above the rim of each bucket. There is also a small gap on the bottom of each bucket. I assumed that the added and subtracted volumes of this popcorn would more or less cancel each other out. I could be wrong about this!!!

Questions to explore...

What is the capacity of each size?

What is the unit price for each size?

What is the percent change in size, price, unit price?

How many inches of popcorn would be left in the large bucket if you eat just as much as the small bucket?

What is the least amount of popcorn from the large container you would need to eat so that you don't get ripped off? In other words, how many inches of popcorn can I leave at the bottom of the bucket?

The answer....

I'm not leaving my full solution here because I'm curious to see how others might solve it. Basically, I used a spreadsheet to test different heights of popcorn eaten to determine where the unit price of the large matches the unit price of the small. If you think about it, this is further complicated because as you eat popcorn, the height AND top radius changes. You will have to come up with a formula that calculates the top radius based on the height.

I determined that you get ripped off if you leave more than two inches of popcorn at the bottom of the bucket.

My Grudge with "Grudge"

I'm flying home from Twitter Math Camp near Los Angeles, and after successfully figuring out how to steal the airplane's wifi, I decided to write another post. This is what I do. I go to a conference, get inspired to contribute to the MTBoS community, and write a blog post. You must understand that once I get home, all motivation to do such a thing will be lost. That's what Netflix would like me to believe anyway.

There is one contribution I've made to the online community that has received a lot of good feedback from students and other teachers. This is a game called Grudge. I gave a survey to my students at the end of this year and asked them what were their favorite things were from my class. Grudge was near the top of the list. ("Mr. Kraft" was at the very top of the list, of course.) 

There is no question in my mind that it is a review game that engages almost all of my students almost all of the time. I also feel that I present it in such a way that students seriously consider their answers and are eager to understand their mistakes. But there is a problem with the game. On occasion, students will team up on other students, and while it is not always expressed, I do believe that feelings can be hurt. As Matt Vaudrey once expressed in a tweet, it hurts the class culture. It promotes competition instead of collaboration.

I've learned that any activity I use in my class should not only be engaging and promote academic growth, but should also encourage students to be respectful to one another.

What the hell is mean absolute deviation?

When I first started looking at the Common Core standards for sixth grade a couple of years ago, admittedly, there was one standard I had to do a double-take on:

6.SP.B.5.C: Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

And, like many of my colleagues, I thought, "What the hell is mean absolute deviation?" My horror was confirmed when I googled it and saw how complicated it would likely be for my students.

Looking in some textbooks and online resources, I was continually left wondering why my students would even care about mean absolute deviation. I mean, you do all of these steps, you get a number, and then what? What does mean absolute deviation tell you?

I figured that the only way my students are going to have any access to this would be to compare different data sets, make a quick judgement about which one has more variability (which can be very subjective) and find some way of quantifying that variability. On top of that, I wanted my students to create their own data where the goal would be to have the least amount of variability.

I then remembered the "Best Triangle" activity I did with Dan Meyer. In this activity, Dan asked four teachers to draw their best equilateral triangles. (Notice that Andrew and I have points in our nostrils.)

Rather than having the students evaluate the teachers' triangles, I had them create their own. I started the lesson off by asking the students to draw, what was in their mind, the perfect triangle. Immediately, there were several hands that shot up from students who wanted some clarification, but I told them to just do what they thought was best. After a quick walk-around and throwing some random triangles up on the document camera, it seemed that almost everyone was trying to draw an equilateral triangle. A few students argued that a right triangle could be considered a perfect triangle and I admitted that my instructions were very vague and their interpretations were justified.

We then brainstormed all the things we should look for in the perfect equilateral triangle. Students agreed that we needed three equal sides and three equal angles. They then made a second attempt on the whiteboards to draw perfect equilateral triangles. I asked everyone to make a quick judgement about which triangles they thought were the best, but soon ran out of time for the day. After the students left, I quickly took pictures of their triangles and took measurements in millimeters. (Admittedly, this is something I would have preferred having the students do on their own, but my class time is unbelievably short...37 minutes.)

The next day, I told my students that I took those measurements and found a way to rank all of the triangles from all of my classes. Next, I showed them the five triangles which represent the minimum (best), first quartile, second quartile, third quartile, and maximum (worst) of the data (in order below). This was a nice way to show a sample of the triangles as my students had just finished learning about box-and-whisker plots.

When I first showed them these triangles, I asked them to figure out which triangle represented the maximum and the third quartile. The other three triangles were not easily identified, however, we noticed that if you reorient the triangles so that one of the other two sides was on the bottom, the inferior triangles no longer looked equilateral (leaning to the left or right).

I explained that ranking these five different triangles didn't provide too much difficulty, but I was confused how to rank triangles that looked very similar. I gave the three following triangles as an example and had students vote on which one they believed looked the best:

In each class, there was a lot of disagreement about which triangle was the best, and more often than not, the majority picked the wrong one. I then provided the side lengths of each triangle (above in millimeters) and asked the students, "how can we use these measurements to rank these three triangles?"

After a few unproductive guesses, someone usually asked to find the differences between the measurements, which lead to someone else asking to find the sum of those differences or the range. They notice that the ranges for each triangle are all 20 mm. Someone usually calls me out for doing this intentionally...which I did.

Next, somebody will ask about the mean of the numbers. I act dumb, as I do with every suggestion, and we explore that possibility. We find the means, and it would seem that we have again hit a dead end.

I have say that at this point, some classes were completely stuck, and some kept going with it. For those that were stuck, I told them that to me, the mean (157 mm for the first triangle) represented the side length that the triangle drawer had intended for each side, but sometimes he or she fell a little short of that goal (149 mm), or overshot it (169 mm). I then asked them to compare each drawn side to "the perfect side length". We found the differences of each length and the mean, and soon after, someone suggested finding the sum of those differences.

At this point, most of my classes were satisfied that we found a method of comparing the triangles. We just had to look at the sum of the differences from the mean. The best triangle was the triangle that had the lowest sum. A couple of classes even went one step further to find the mean of those differences. In reality, there was nothing wrong with either of those methods. However, the second method WAS THE MEAN ABSOLUTE DEVIATION!!! When I first started planning this lesson, never did I think my students would intuitively come up with this concept.

This was the first time I've taught this lesson and I realize that there was a lot more I could have done with it. Given more time, I could have had students work in groups to come up with their own methods for determining the best triangle (similar to Dan's lesson plan) and we could have compared the methods later.

Side note: Dan says that "the best solution is to use the fact that an equilateral triangle is the triangle that encloses the most area for a given perimeter". Sixth graders are not at a point yet where they can find the area of a triangle just given the side lengths, so some other solution was necessary. Technically, my method is flawed because it favors smaller triangles. If you double or triple the size of a triangle, it doubles or triples the mean absolute deviation. This is noticeable in the data as smaller triangles were preferred. A better method would have been to compute the percent differences from the mean, but this would have greatly complicated an idea I was just trying to introduce for the first time.

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.