Fraction Flags - Creative Thinking

My students have graph paper journals I gave them this year.  Recently I asked them to design a 300 square flag, where they made a pattern using 1/3 of the flag, 1/5 of the flag, and 1/6 of the flag and the rest of the flag as patterns. I suggested they lay out their flag patterns in something more sophisticated than rectangles.  We got some interesting results.  My goal is to let students be creative while thinking, and for them to be successful.  Allowing space for thinking, and for open-ended questions leaves room for success for all my students, even those who find math a struggle.

Three Fourths How? Good thinking!!!!

This year I gave each of my students a composition book filled with graph paper.  They are kept in bins in the classroom, and at least weekly, we use them for some interesting critical thinking problems.  The first I gave them was to depict the amount 3/4 in as many ways as they could... using words, numbers, pictures, music, design, whatever they could come up with.  I told them I expected at least five ideas per student.  We got great stuff!!!  3/4 Time in music, 3 quarters taped to the page, 3/4 pizza, 3/4 as equivalent fractions... We shared in groups, and then with the whole class.  I then challenged the students to bring in another creative way to show 3/4 the next day as extra credit on the first test.
Here are some of the interesting ideas I received.

"Three Fourth Awakens" with 3/4 Darth Vader's face

 

"3/4 a Bearded Dragon's Shed Skin"

"3/4 the way there"

"3/4 the Cranes are Green"

"3/4 the Gloves are Pink"

Card Houses, Patterns, and Algebraic Expressions

 My pre-algebra and algebra classes are learning about algebraic expressions, also called variable expressions.  It is important for them to be able to observe patterns, to identify the constant and changing parts of those patterns and to write variable expressions that represent the patterns. Yesterday we built card houses with that intent.  I demonstrated and off they went.  With each layer, they had to measure how high the top was from the floor. They quickly found they didn't have to measure from the floor each time, because the height of the table didn't change. The house of cards were approximately 9cm tall per story, so each additional story added 9cm to the total.  We built a table of the data, and decided what would happen with a card house any number of stories high. We made the table below, then the students told me how far from the floor a card house of 10 stories would be, 15 stories would be.  They've got this... finding patterns, writing expressions, making predictions.  And, they had a blast.

Happy Pi Day - Dominos, Measurement, and Differentiation

   

So today is Pi Day, as in 3.14.  My math students think about pie, and an infinitely long decimal, and sometimes stories of a classmate long ago who could reel off the first 25 digits, or the first 100 digits.  However, I want the students to come away with an understanding that Pi is actually the ratio of every (yes, every) circle's circumference to its diagonal. Pi is an irrational number, meaning it cannot be written as a fraction with integer numerator and non-zero integer denominator.  We didn't have a chance to organize a pie-a-palooza for today, so I brought in round cookies (okay, a stretch to think of them as mini-pies, but they certainly are desserts and they are mostly round). 
  

 

 I intended to have my students watch my favorite Pi Day video  (you've got to check this out!!), especially because after its mesmerizing 3 minute 14 seconds (hey, wait a minute....) it explains the math in a very accessible way.  Then they would measure the cookies, measuring circumference with a string, and diameter and finding the ratio, and going from there.  Thanks to my colleague for sharing her idea, which was a table for measuring lots of round objects.  I borrowed it, wrote "cookie" in the first row, stacked a variety of round objects on the table,  and added some differentiation for my different classes as follows:  

My seventh graders watched the video and measured, calculating ratios.  We will discuss their results tomorrow.
One eighth grade class watched the video, measured, then built a scatter plot of their results with the data and drew a line of fit. I will show them how to find the line's equation tomorrow.
The other eighth grade class did all of the above and generated equations for the line of fit, which we wrote on the board.  Tomorrow we will compare equations, thinking the slope of each should be near 3.14.  Hmmmm. 

So, I spent Pi Day without eating a bite of pie, which is fine with me.  My students came away from Pi Day having eaten a well-measured cookie, understanding that Pi is irrational, and that it is the ratio of circumference to diameter of each and every circle.  Everyone was engaged! It was a very good day.



                                           

"Hall Math" or "How to Accomplish Review, Collaboration, Movement and Confidence with a Few Index Cards"

My high school algebra class is working with linear equations.  There is a lot of vocabulary.  Much of the vocabulary sounds vaguely similar so it is easy for the students to feel "mushy" about the whole unit.  In my book, "mushy" is not good, and leads to "mushier" as we move forward into inequalities, linear systems, and eventually quadratic equations.  Ideally, students will have facility with equation solving tools, with which they can manipulate linear equations into various forms.  And, they can see that certain forms of linear equations lend themselves to different applications.  We aren't there yet, but closer.  As I write this, I realize I need to design a lesson where they read problems and identify which form of the linear equation would be more applicable.  

This Algebra I class is particularly fidgety after lunch, so I proposed "hall math" to them.  The eight questions above were taped in the hall, within view of my room.  I asked them how they should comport themselves in the hall, and we reviewed.  Then I asked how many times I should have to prompt them to keep the noise down before I called them all back in.  They suggested five! (not surprised)  I told them that they would get one warning before I rounded them up, but I was sure they could handle it, and this was a trial run to see if we could have more hall math in the future... And, I asked them to pair up as follows:  If you are comfortable with the material, find someone less comfortable, and vice versa.  I gave them each clipboards with an eight-graph sheet attached and off they went.  They were fantastic!  

Last year, I attended the National Council of Teachers of Math conference which was held in Boston.   May I say, what a treat to be among hundreds of people who are enthused about mathematics?  I had sort of forgotten that many people continue to believe that Math is a mystery, or it became a mystery to them at some point in school and it moved forward without them.  That's a topic for another day. But, one of the sessions I went to was about incorporating some social conscience into math class.  It's easy to start; rather than having Roxanne contemplate which deal is better at the health club, you can formulate problems that involve giving, or that highlight some local or global concerns.

Here's an example of slope-intercept form in action:
 
Alex has set a goal of donating 76 books to the Boys and Girls club. He started by using part of his birthday money to buy 10 books.  Each week he uses some of his allowance to choose 4 books from the sale rack at the local bookstore. 

a. Write an equation representing Alex's progress toward his goal as a function of how many weeks have passed. 


b. Graph the equation.

c. How many books will Alex have left to donate after 6 weeks have passed?


Thanks for reading along... I'm excited about hall math... "Get thee to a laminator".


 

"Forever January" and some Roman history

In my classroom, one of my homeroom students writes the date up on the whiteboard each morning.  Same place, every day.  This past week, my last block of the day was working, and someone remarked, "We've been in January for a long time!"  None of us took notice, because this is middle school and the occasional non-sequitur is de rigueur.  Then we noticed the student looking curiously up at the date.  Gone was January 20th.  It was now mid-day January 80th!  (Middle-school mischief)   Interesting.  So, I prompted the students for a little extra credit.  Seems we had landed in an alternate universe where it is always January. If we are in the perpetual January world, where our February 1st is Januaryland's January 32nd, then what day on our calendar corresponds to January 80, 2016?  

Would love to see your comments with the date you think is the answer, and a clue about how you got there. (Don't use a calendar.  It's more fun.)

So, then I did a little research about January itself and made up the following problem for my other classes the next day.  You can click on Numa P's name to learn more about the 2nd king of Rome below.

 

Numa Pompilius reigned 715–673 BC was the legendary second king of Rome, who reigned from 715-673 BC, succeeding Romulus. Many of Rome's most important religious and political institutions are attributed to him. In fact, King Pompilius gave January and February their names.  January was named after Janus, the god of beginnings and endings.

There is a folktale that King Pompilius liked the name January so much, he decided not to have any other month names.  The whole year would be called January. So, what we think of as February 1^st would have been called January 32.  If the folktale came true and King Pompilius had his way, think about January 80^th, 2016.

a. What month and day would that be on our calendars?

b. What day of the week is it? 

  

 

Happy New Year - 2016 Fabulous Factoring and a Balrog

During my Christmas break, I began thinking about a good Do Now for the students on their first day back.  What could I do with the number 2016?  Hmmm.  Well, excitingly, it has SO MANY FACTORS!  That would be a great way to get their brains cooking again.  Back in October, my 7th graders explored 504.  We blew out the factor rainbow with a factor "man-oh-war", and proclaimed 504 our favorite number.  It was exhaustingly exciting.  But. Today, 2016 bumped 504 from its pedestal.  This was such a good review of divisibility rules, and dividing, and number patterns, and association.  Once we got into the two digit numbers, the students conjectured whether, for example, 28 would be a factor of 2016.  After all, 2, 4, 7, and 14 were all factors so didn't it make sense?  And they were right.  We're a little stumped why 27 is not a factor of 2016 because 3 and 9 are there.  Someone suggested maybe the larger factor had to be even, but there sat 21 in all its odd glory belying the hypothesis.  We think it might have to do with the fact that 9 is a power of 3 but we're not sure yet, and want to be able to make predictions. 
When you divide 2016 by 4 you get our old friend 504.  That was a huge help when we figured out that if 504 is a factor of 2016, then ALL its factors are also factors of 2016.  Truthfully, took a little convincing on my part, which I demonstrated by showing 36's factors and later on asking for 72's factors.  But, we were rusty from the break.
My 8th graders were worn out by this activity, as they were over a year from factor rainbows, etc.  I told them the sad news that next January, I wouldn't be able to entertain the future 8th graders with a  factoring adventure starring 2017 because, well, 2017 is a factoring dud. "What?!  That's not fair!  Why did we have to have the bad luck to have Jan 2016 in our 8th grade year!!?"  (Did I tell you I love middle school?) I did promise  to inflict 2016 on next year's 8th graders in December just before the break with a "So Long 2016" factoring event. That was some comfort.
So, my 7th graders, who love to factor, loved 2016.  It's not a factor umbrella.  It's not a factor Man-o-War.  It's almost akin to Tolkien's Balrog just before he wraps his sinewy glowing tendrils around Gandalf's leg and drags him into the depths of Moria, but I haven't told them that.  Yet.