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!

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 me...it 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 heads...it 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 square...in 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:

26

|

(3 up)

|

23

|

(3 down)

|

20

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.

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 http://www.greatschools.org/, 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 artofproblemsolving.com 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 famat.org. 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:

• Florida Mu Alpha Theta
• Art of Problem Solving AMC past contests
• Phillips Exeter problem sets
• NCSSM test bank
• Marywood University past contests
• Kings College London Challenges
• Corbett Math 5-a-day and Conundrums
• Brainbashers

Do you know of any others?

Fold and Cut Theorem and Walking Through an Index Card

http://www.origamiplayer.com/orimath/embed560.php?ori=betsyrossstarfroma4

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???

erikdemaine.org

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 taught...so 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 http://www.ushistory.org/betsy/flagstar.html 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 it...it 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 famat.org. Here is a copy in case you would like to try it! 

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 1 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 https://www.desmos.com/ 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.

~Lisa

Don't kill a kitten!! Please!!

It was Meg's puppy/kitten/unicorn post and Sarah's kitten blog that got me thinking about creating a list of common student mistakes, and when my Algebra 2 Honors students came in at the beginning of the year making the same ones over and over, I thought it was time to take the "unicorn" by the horn.

Here is my Don't Kill a Kitten handout for Chapter 1 in Algebra 2 Honors. Whenever a student begins to solve a quadratic equation by setting it to the constant and factoring out an x (the horror!), I have them go to their handout and star the mistake they are making. The hope is that it will really stick with them. So far it's working. I may make one for every chapter.



Which mistakes are your students making now?

Transformations of FUNctions: Headbanz and Space Race Fun

I totally stole this Headbanz idea from Mary Bourassa, and I took many of the equations from Meg Craig's Speed Dating handout. I had leftover stock card and ribbon from old projects and I used a box cutter to make the slices on either side of the stock card to put in the ribbon. (Note that we do not teach students transformations with both a horizontal stretch/shrink and slide until we get to trig functions.)

My rules: (I have never really played Headbanz, but here's what I did)

1. Once you have the headbanz on (ask someone if it's right side up), you may ask one person a question.

2. The person who answers CANNOT speak! (They can move their arms in the shape of parent functions, for example.)

3. That student now asks you a question in the same way.

4. Each of you moves on to a new person and continues in this fashion until you know the equation.

5. When you think you know the answer, come to me and I will tell you if you are right.

This was really fun to watch. They could ask, do I have a vertical stretch? Then next student, what is the vertical stretch, and that student could show with their fingers. They could not, for example, say, what is the number inside of the function? So they had to use math language like: reflection, horizontal stretch, shrink, translation left, and right.

It was very fun, and the students recommended that I keep the rules the way we did them. However, I definitely would do this DAY 2 instead of day 1 when I taught the lesson because they did not all have time to process the information enough on day 1.

Space Race from Quizlet

The day before I taught transformations, I taught parent functions. For homework, I assigned this Space Race from Quizlet. I told them to work on it for at least 10 minutes and email me a snapshot of their highest score. Kids love it, and they really learn the parent functions better than any other way I have used in the past. I tell them to study the phrases and equations as they are on the page you see when you open the link, because when they click on space race in the upper right hand corner, the graphs or names fly across the screen and they have to type it exactly as they see it...try it! There are high scores, and I give a prize for the student with the highest score in each class. Desmos needs Space Race, am I right??

Notes:

Here are my notes for parent functions (this includes piecewise functions) and then transformations of functions.




And my notes from Transformation of functions...I actually use the Chalkboard font, but here it looks much more playful...not sure why it changed!

Speed dating:

I am going to use this on Monday after we go over homework. (we have a 30 minute period) by lining up desks and having students sit across from each other. I will give them a few minutes a problem and then have one side shift to the left (with the last person coming around to the first desk.) They will first have to check their answer from the last "date" with the new "date," and then they can move on. I will try out the datexx timer I just bought. Did I see that from Julie? or Meg? I love twitter.

Homework and answers:

Who did I get this worksheet from??

Explain Everything Video:

As I said, I didn't think my students had enough time to process, so I made this video on Explain Everything, which I blogged about once here (my first blog ever.) I uploaded it to Google Classroom so they could watch it this weekend if they needed any help. It is NOT perfect...and I don't necessarily recommend that you use it...but I am showing it to show how easy it is to do--you can make a quick video to recap what you did and send it out to students. 

My rule for videos (as you can see)--make it once. Don't keep trying to fix what you did or what you said or it will take forever. Just make the correction and move on...unless REALLY wrong...I hope I did not make any mistakes that I didn't fix!

And that's it for transformations 2015. Any other transformation ideas?? Please send them along!