Saturday, October 31, 2015 Some Fun and Quick Class Tools to Hook Students

This is a crosspost from the technology blog I write with ProfeSeiden, called Tech4Scots. is probably one of the coolest things out there. It's really unique and students enjoy it. You can use it just for fun or as a learning tool. In a nutshell, you can create free games, fun activities, and diagrams in literally seconds. Here are some examples.

Breaking news generator:
I made this in about 30 seconds.

or use a pre-made picture from Anchorman:

Star Wars Movie Text:
Use this to scroll directions, for example, or to introduce something exciting (or otherwise not exciting, but you want to add some pizzazz...)

Fake Tweet:
To generate interest in a topic...haha I made this one up!

Use "Fakebook" for character development, to give a series of historical events, to give debates and relationships between people...this would make a great assignment. Here is an example from Classtools.

This is really cool. You have to see it to believe it. There are six sides to a box, so there are six key events that students can write side can be a video, one side can be a here to see it in action.

Fake Texts: 
This is just for fun...

Random Name Picker:
This would be great for our department, as currently, we play "nose goes."
click here to watch it spin--when a name is picked, you will hear applause. You can also remove the student so that they are not picked next time.

Fruit Machine:
This is a slot can use as a random name generator, but I think it's great for a vocabulary word list. Put in words and spin, and when a word comes up, students need to define it                                              

Countdown Timer:
This is nice to display, and you can select a soundtrack as well:

Need a timeline tool for students? There are a number to choose from:

Arcade Generator:
Need one? No problem. There are some pre-made ones as well:
Fishbone or Burger Diagram:
Because, why not? Who wouldn't want this?
 There's more, too. Just go to to check it out. There is a paid version as well for no ads.

Make something fun? Let us know in the reply section.

Tuesday, October 27, 2015

The Four-Color Theorem and The Pumpkin Time-Bomb

"CRAYONS!? We are COLORING? Wait, can I snapchat this????"

Yes, yes you may.
Here are the directions I gave to my Problem-Solving class the other day:

Take any map and color it so that no two adjacent states are the same color. Keep in mind that you want to use as few colors as possible so that coloring the map is not too expensive. It turns out that any map can be colored in at most 4 colors! And that this was the first proof proven using a computer. 

Here is the map I gave them, without telling them that they needed 4 may want to provide extras.

Here is the quick video I showed them after they colored, that talks about the four-color theorem.

After coloring the map, we drew the graph at the right, where each vertex represents a country and an edge connecting two vertices means those countries are adjacent. Then we colored the vertices and found the "chromatic number," that is, the least number of colors we can use. We needed at least three, as you can see by the red triangle drawn between Bolivia, Paraguay, and Argentina. The chromatic number for this map was 4. (And remember, for any map, this is the most it will ever be.)

Then we went on to scheduling committees, and I showed how you could make each vertex a club and each edge would represent any club that had a member in common, so a conflict in time. The chromatic number, in this case, would be the least number of club meetings required. We also answered some interesting problems that related to the Handshake Problem.

Here are my notes from class that day, that I took from my book. 
All in all, a fun class with a theorem that they have never heard before. And I am sure that they will never forget.
Today, we did the awesome Pumpkin-time-bomb-activity from Mr. Orr. Very fun predictions, and we were only 4 off from what we guessed! I highly recommend that you read his post, check out the data, and also watch the video from Jimmy Fallon.

The explosion took us by surprise so we didn't get a picture, but one student ended up with pumpkin all in her hair! I was amazed at what the rubberband clump looked like at the end! We didn't do anything to it. That's what it looked like after the explosion!

Monday, October 19, 2015

The Radioactive Cube Investigation

This is a problem that was presented originally by my department chair 21 years ago. I have used it often ever since, but not since I began blogging. It is a great problem at any level in honors or Pre-Calculus. Though some can get it rather quickly, there are definitely some who need it to be broken down. Often, the answer I get without much thinking is "27 cubic units." It is much less. Hopefully, before you read on, you will try the problem. It is great in that it uses a lot of Geometry and nothing that students are likely to be working on at the moment.

I am lucky to have a new course with only seven students. However, four of these students got the answer to this problem rather quickly. The remaining three had the weekend to think about it, but got nowhere. Insert Unifix Cubes. I've had these for a million years.
Students who needed to taped them together like this:

And saw that there were 7 cubes in all that were radioactive...the first cube itself, and the other 6 that radiated from the 6 faces of said cube. So this was 7 cubic units so far.

Then, they cut paper and taped it around the cubes. Students saw that this created 3 4-quarter cylinders, so this added 3 cylinders with radius and height both equal to 1, so π cubic units. These quarter-cylinders came from the edges of the inside cube that radiated radioactivity as a radius of 1 between any two cube faces.

They then realized that they were missing the triangular pieces, shown above. This was from the corner of the middle cube, and it created eight 1/8 spheres, so 1 full sphere with radius 1. Or, 4/3π. 

So the total answer is 7+3π+4/3π, or approximately 20.6 cubic units of radiation.

Some visually talented students drew it like this:

The 7 cubes

The cylinders and spheres built in

Some final results

Students were very proud of themselves either way!

Wednesday, October 14, 2015

The Art (Benjamin) of Problem Solving

When I saw him in the crowd at Twitter Math Camp '15, I knew I recognized him from somewhere. Then it hit was Art Benjamin, famous mathemagician and speaker whose DVDs I had shown in my classes some time ago. I was completely starstruck, and my colleague had to calm me down a bit...but then...then he was actually going to do a show for us? WHAT?? It was absolutely amazing. Art Benjamin is not only brilliant, he has an energy about him that catching to others. His love of math is pure and he engages his audiences by multiplying large numbers in his head (he converts some of these numbers to words to help him!)

Today in my Problem Solving Class, I showed this amazing and highly recommended Ted Talk video. Here, Mr. Benjamin squares 2 digit, 3 digit, 4 digit, and yes, even 5 digit numbers in his head, much to the surprise of his audience. He even can tell audience members the day they were born if they tell him their birthdays.
My students were in awe. They were perplexed. How did he do it? Well, we googled and found this video next, where he explains the secrets of how to square a two-digit number in your head.

And they could do it! Well, some could do it in their was hard! 

Here is his method.

Take any two digit number and square it:
For example, 23x23

First, find a number more or less than the number that is easier to this case, we will use 20. Twenty is 3 less than 23, but you also have to go up 3 more than 23, like this:

(3 up)
(3 down)

Now, multiply the bottom and the top numbers: 20 x 26...easy to multiply 2x26 and add the 0, and get 520. Then, take the number you added or subtracted (3, in this case), and square it (9) and add it to the 520, and you get 529. 

Well, of course we had to try this out. We went to the Algebra 2 class next door and showed our newly learned talent during their brain break. They were impressed, and we taught them how to do it. Then a tour came in. I think the student on the tour was a bit scared when I excitedly asked her to pick a number any number, and we would square it for her. I just could not contain myself. 

So why does this work? Well, it's all about FOILing, if you don't mind that word. Essentially, what we did to square 23 was to say:
23x23 = (23 + 3)(23 - 3) + 3x3

Art Benjamin may explain it better, so watch the videos!

The other day I saw on twitter that he wrote a new book The Magic of Math: Searching for X and Figuring out whY, so I had to buy it. It's fantastic, and I can't wait to share more gems with my students.

But this is still not the best news. The best news is that I found his website, and I saw that he does shows. Lots of them. And he is doing one near my school. And I contacted him. And I contacted my head of upper school. And GUESS WHAT!! Art Benjamin is coming to do a show at my school in March! I am so excited. When I told my students that, they couldn't believe it. They can't wait, and neither can I. The president of my math club came into our class after we showed his videos, and we told him that he was coming to the school. His eyebrows went to the back of his head. He had watched a series of his videos before. He was so impressed that Art Benjamin will be at our school. So am I. I am so happy my school is hosting him, and I can't wait for his love of math to rub off on our students.

Saturday, October 10, 2015

Teach Students to Think Critically and to Practice Out-of-the-Box Questions Often

It's easier to stay "in the box." My cat, Tiki, thinks so at least. But I say, let's help students step out-of-the-box and take some risks. Let's teach them to learn some grit and begin to think critically and not just through rote memorization. Students will rise to the occasion when you expect them to. 
I have caused a little bit of dissent in our department over the years...insisting on putting a few out-of-the-box questions on tests and quizzes. Some colleagues have firmly stood with me, saying a test should not just be a "worksheet," but others have been worried that their students would not be able to solve the problem(s) and would therefore do badly on the test. 
Most of these colleagues that originally were skeptical of putting problems on a test that were not directly taught in class have come around to the idea of giving them problems that are not in the textbook. I used to say that this type of question "separates the A students from the B students," but that's not the reason I do it anymore. It's to try to get ALL students to think critically and not just regurgitate what I have given to them. It's more about teaching students to problem solve rather than to teach them how to memorize isolated facts and then just spit them back out. I have found that at first, students are nervous about these kinds of problems, but the payoff is great..they learn the power of grit, and they feel fantastic when they get these problems right. Therefore, my review sheets are often harder than the test, because I put several questions on them that cover the same topics, but are asked differently or require several ideas to be combined into one.
Today I came across the article Why our Smartest American Students Fail Math. It is a fantastic read, and I strongly recommend that you read it. Basically, Carol Lloyd, exec editor at, writes that our top students who go to college for a STEM (Science, Technology, Engineering, or Math) tend to drop the major because they get terrible grades in these courses. They are not learning problem solving or perseverance in their high school courses and, as Richard Rusczyk, former Math Olympiad winner and founder of the awesome website stated, “These were kids who had never gotten anything but 95s and 100s on their tests and suddenly they were struggling and were getting 62s on tests and they decided they weren’t any good [at math].” Lloyd writes, "Indeed, traditional math curriculum is to teach discrete algorithms, a set of rules that elicit a correct answer, like how to do long division, say, or how to use the Pythagorean theorem. Then students 'learn' the material by doing a large quantity of similar problems." She continues that Rusczyk says the result is that "students are rarely asked to solve a problem they are not thoroughly familiar with. Instead, they come to think of math as a series of rules to be memorized. The trouble is kids don’t necessarily learn how to attack a new or different kind of equation."
This is what I have been saying, too. We cannot just give students a worksheet, have them memorize all of the topics, and then test on the same material. What does that teach them? I think it teaches them to remember things just for a test and not how to critically think in high school. Students need to fight through problems and talk through them with others. They need to hear all the different approaches others have to solving the problem. I spoke about grit here, and my problem solving students are learning about grit daily.  I gave the following problem to them on a problem set:
I was shocked at how many students did not know how to do this. There are three Calculus students, three Pre-Calculus students and one Pre-Calculus regular student in the class, and they did not even know how to start it. Finally someone said, "Oh, it's like a composition of functions problem?" Then they got it. The next week, I gave them this problem on their problem set:
This time, they got it right away. But it is important to give different questions to ALL students, not just those in a Problem Solving course. They begin to recognize the type of problem that it represents, not the exact one on a worksheet.
My Mu Alpha Theta students, which I wrote about here, are exposed to new and different problems each week at our club meetings. I feel like the exposure to these problems is definitely making them better thinkers. Spending an hour on math that is not following the curriculum is SO COOL, and I would love to hear from others of you who are either an advisor to a math club or are just thinking about it...and if you are thinking about it but aren't sure, JUST DO IT! Not only is it great for the kids, but it is also great for you--it truly makes my "cup runneth over."
Attached are two review sheets I gave my Algebra 2 Honors classes that had some problems on it that were out of the ordinary, but were related to what we were learning.

These are from I know the font is pretty old on the first worksheet. It's because I am using the oldest tests from the '90s in class, as we are using the most recent ones during Mu Alpha Theta practice. In math club, I have students work in pods on these worksheets, and it's fun to see them do this when they get the question right:

Here are some great places to get problems:
Do you know of any others?

Thursday, October 8, 2015

Fold and Cut Theorem and Walking Through an Index Card
Yesterday, I threw my lesson plan out the window. It doesn't (can't) often happen in my Algebra 2 Honors or Pre-Calculus classes, as we have a lot to cover in those classes. But it does happen in my Honors Problem Solving Seminar, and those normally tend to be the best lessons.

I try to show students great videos, and this on The Fold and Cut Theoremone popped in my inbox. I didn't watch it, but planned to. Then, one of my Mu Alpha Theta members forwarded it to me, raving about it. So the next morning, we watched it together. It turns out that you can take a piece of paper and fold it several times and may ANY shape with just one cut, as long as the shape is made of straight lines. What???
Check out this video to watch Katie Steckles cut out the entire alphabet and more with one snip of the scissors.

What was very cool for me is that I LOVE to watch these videos with my students without previewing much so I can discover along with them. As soon as she started mentioning the 5 pointed star, I stopped the video and told students that I played the role of Betsy Ross in 5th grade and found out how she impressed George Washington by cutting a 5 pointed star in one snip. I continued the video, and that was exactly the history lesson that was of course, we had to try it!

I was shocked at how "folding challenged" my students are. I am, too, so this is why I was so shocked. I pulled up this diagram from and we had a lot of fun cutting stars (and recutting when they did something wrong.)
This reminded me of a different activity where you can cut an index card in such a way that you can literally walk through it. I gave my students some time to think about how to do this on their own. They were so funny and cute when they thought they had it, but then they realized it was actually open on each end. I think it's important to let them try it on their own. One student actually had seen this before and demonstrated the right way to the rest of the class (remember I told you that this class, as brilliant as they are--so smart!--are self-claimed craft-challenged), and most finally got was very cool. Below is the video.

Finally, we did go back to the lesson, which was to work on a Speed Math challenge from Here is a copy in case you would like to try it! 

Sunday, October 4, 2015

Figuring out Figurative Numbers--a mini unit

Last week, my Honors Problem Solving Seminar students wrote a collaborative guest blog about their unit on Figurate (or figurative) numbers here.

Here are the handouts. They are mostly from a book I wrote 16 years ago called Investigating Discrete Math. Unfortunately, I did not publish the book. Fortunately, I use it a lot as a resource. 

Most of the students had already seen The Locker Problem, which I blogged about here. This is the first example of a figurate number investigation, though they may not know it at the time.

I gave them the handout below and had them work in pairs, investigating the patterns in oblong and triangular numbers. Interestingly, they can get the oblong number explicit formula and the triangular number recursive formula quickly, but they have a hard time with the oblong recursive and the triangular number explicit. 
What I like about the worksheet is it leads right into the sums of  counting and even numbers, and that the nth triangular number represents the sum of the first n counting numbers, and that the nth oblong number represents the sum of the first n even numbers.

Many of my students needed a hint on finding the explicit formula for triangular numbers. I drew a diagonal through the middle of an oblong and soon they discovered triangular numbers were half the oblong numbers. So oblong number formula is n(n+1) and triangular is that divided by 2.

Since they remembered the locker problem, we reviewed square numbers and showed that the nth square number represents the sum of the first n odd numbers.

I also talked about the story of Gauss whose teacher was trying to keep him out of trouble as a very young boy.  She told him to add the numbers from 1 to 100, thinking it would take him quite a while to do so.  As the story goes, Gauss had the answer within moments.  How did he do it?  Problems and 2 demonstrate two quick ways to do it:

1.         Shown below is the list of numbers he was told to add.

            Notice it you add together the first and last number, you get 101, as shown above.

            a.  Add together the second and the second to the last number.
            b.  If you continue adding (i.e., 3 and 98), you continue getting the same number of 101.  
                 How many pairs of numbers sum to 101?

            c.  Find the answer to the problem using the information above.

            d.  Generalize this for adding together the numbers from 1 to n.

2.  Another interesting way to find the sum of the numbers from 1 to 100 is to list the numbers from 1 to 100 and then underneath it, list the numbers from 100 to 1, as shown below.  Let both sums equal X.

          X =   1 + 2 + 3 + 4 +…+ 97+98+99+100
          X = 100+99+98+97+…+  4  +3 + 2 + 1   

Now add each of the pairs of numbers that are underneath each other; for example, add together the 1 and 100, the 2 and 99, etc.  How many pairs of the same number do you get?  Note that this answer equals 2X.  Why?  Use this new method to solve for X, the sum of the numbers from 1 to 100.

Once we had all of that "figured" out, I gave them the Handshake Problem. Many knew it, but could not remember the formula. I try to encourage students not to remember the formula, but to get it from scratch.

It was pretty cool when they eventually realized that the sequence for the number of handshakes with n people: 0,1,3,6,10,15,... (i.e., for 1 person there are 0 handshakes, for 2 people there is 1 handshake, etc) is one off of the triangular numbers: 1,3,6,10,15, .... Thus, they developed the formula n/2(n-1) by substituting n - 1 into n for the triangular numbers formula. Or, they came up with the formula by realizing that for n people, each person shakes hands with n-1 people, but you need to divide that by 2 since when I shake your hand and you shake my hand, that counts only once not twice.

Next, I gave them the Pentagonal Numbers Investigation.

This was much harder, and again, I encourage you to look at the blog the students wrote here to see their thinking. But one student figured out that pentagonal numbers (by looking at the 3rd figure) is just the sum of the two nth triangular numbers + one (n-1)st triangular number - 1. Pretty cool!!

Finally, I gave them the Pizza Problem.

Students discovered the sequence to be 2,4,7,11,... which most recognized as the triangular numbers + 1, which was pretty cool. I did not do the Pyramid Numbers Investigation, but next year I will, as it is a nice intro to the Method of Finite Differences, which demonstrates how to solve explicit polynomial formulas without looking at patterns.

Here is the Pyramid Number Investigation:

And the Method of Finite Differences plus homework handout. Keep in mind that this was made 16 years ago and so the typeset is a little "old!"

Students really enjoyed that this was out of the normal "math" that they see in their other math classes. We also went to and did some regression there as well. There are many ways to solve for the formulas!!

The highlight of this was twofold. One was the collaborative blog they wrote. But the second was when they were the teachers for the Algebra 1 classes, and taught them the classic handshake problem with a twist: The Fist Bump Problem, a la Sam Shah. It was amazing. I will blog about that soon.