Hypotenuse and Hippos

I have been a Sandra Boynton fan for years.  Our now-college-sophomore daughter can still recite "Moo, Bah, La la la" by heart with all the sounds, and singing and rhythms we used, every time.  I was not a math teacher when "Moo, Bah, La la la" first crossed our paths, but I am now, so when I saw Boynton's Geometry Lesson cartoon, I was smitten.  She had me at hippopotenuse.

Whimsical, Mathematical, Perfect.

Proportions at Play - Painting with tape

 The high school art students at our school created silhouettes as an art project.  They installed them after school on a Friday night creating a transformation for our open house the next day.  I was invited to participate, and created "Jumping Math Man" for lack of a better name.. maybe Mathuvian Man.  I used a reference photo, grabbed the tape, and "painted", thinking about proportions, angles, and dropped mental vertical lines for scale and spacing.  It was so much fun!!!  We used two widths of painters tape, so that it will come off with little impact to the school's walls.  Making art is like solving math problems... lots of thinking, chances to be creative, and to connect ideas from different corners of your brain.  This is the stuff that gets me jazzed up! 

Factor Flamboyantly! Paris and Sea Creatures!

 My seventh graders are learning about prime numbers, divisibility rules, factorization, Greatest Common Factor, and Least Common Multiple.  I love this content.  They get a real sense of the feel and weight of numbers and some tools to use for less familiar numbers.  And, we get to talk about monomials... but another day.  Back to the factoring.  We had worked on finding all the factors of numbers, and finding the prime factorization of numbers, and had applied our newly acquired divisibility rules throughout.  Then came....

 I was thinking of a number that the students could really dig into to with their factoring tools.  Why not 504?  It has promise.. So, we started making a factor rainbow to find all the factor pairs of 504.  We all started off leaving plenty of room, anticipating quite a few factors. We enjoyed the rainbow, watching it fill, enjoyed all the opportunities to divide.  Then, the rainbow was full!  We weren't finished!  What to do!?!

 

 

One of the students dropped a vertical line down through the rainbow and we continued, a "rainbow annex" of sorts.  There couldn't be that many more factors; we were closing in on the half way mark... so we made the "annex" optimistically a bit narrower.  (oops.)

 Two pair later, 4 simple factors and we had exceeded the "annex".  Someone said it looked like a factor umbrella at an outside table.  So, naturally, we called it the "Factor Cafe Table in Paris Where We Sit Outside and Eat Croissants While Drinking Cafe au Lait" (wouldn't you?)  Some of us spontaneously began speaking with French accents while we continued exploring.  Were we finished?

Of course not.  However, 12 and 42 were getting pretty close to one another in factoring terms, so we dropped another vertical, made "Annex le Deux", (that's Annex 2 to the language challenged among you).  More factoring, more dividing.  We were in uncharted territory here.. our divisibility rules had been left in the dust, all used up.  We were flying without a net, a drift in a sea of possible factors... we. were. terrified. and intrepid.

They found fourteen, and eighteen and their factor partners.  We filled Annex le Deux.  But then... using some exceptional logic, one student opined, "Hey, 3 and 7 are factors.  Doesn't that work like 6 with 2 and 3?  Is 21 a factor?"  They tried it, and there it was.. we needed Annex le Troisieme (You guessed it, Annex 3).
We finished.  It was long, it was exhausting, it was beautiful.  It was...  a factor Man o' War!!

Whew.  Exhausted, but beaming, we all agreed that 504 is our new, absolute favorite number... for now.



 

Sign of the Times: An Emotional Response

As we started back into the year, I assigned each of my Algebra I Honors students a topic to review with the class.  They chose from associative and commutative properties, inverses, identity, distributive, and the rules for conducting operations on real numbers.  One of my new students presented the rules for assigning the sign when multiplying and dividing numbers.  
She showed us this method for remembering the sign, and attributed it to "the best math teacher ever!" (not me, her 8th grade teacher from her district middle school. BTW, I sent the teacher an e-mail that night letting her know her praises were being sung in my room earlier in the day.) I made the graphic above based on my student's drawing and explanation.

Although I'm not a huge fan of gimmicks for remembering, nor do I use the word "hate" as a rule, I love this graphic, and this method.  I have some students who struggle to retain content.  I showed this graphic and now they own it.  For good.  "You love to hate.  You're a hater.  That's negative. Don't be a hater."

Interesting idea.  Thoughts?

Commas Have Names... use them.

 Large numbers can be hard for students to read.  They stumble with the place value names, and can be intimidated by the large number of columns. I reassure them that the numbers are easy to read if you know the names of the commas.  Each comma has its own name based on position. The name doesn't change, so you only need to know how to read numbers into their hundreds if you know the names of the commas.  

The first two commas you'll see in a number are named "thousand"  and "million", followed by "billion" and "trillion", etc... 

 
This approach always makes sense to my students.  They don't see large numbers that often, and feel in command when they can easily read:
  
forty-two "million"  three hundred forty-two "thousand" seven hundred eighty-three

Give it a try.  See what how it goes.

  

Light Bulb Moment #1 - What does completion look like?

School starts in two days.  This week, I readied my classroom, and checked off item after item on my "to do" list.  We had two days of professional development with time for meetings and more meetings.  During one of the meetings, our Special Ed liaison shared the IEPs of the students we will be teaching this year.  She gave us a concise summary of strategies recommended for each student on the list.  One of the strategies listed quite a few times was "Show what completion looks like".  
May seem very obvious, and it's something I do on occasion, but seeing the words in writing several times on the list of new students struck me. 
One of my checklist items this week is always to update the syllabus for each class I teach. In it, I describe the homework expectations, format and my grading policy.  That's when the light bulb ignited!  Each year,  seems I have a number of students who continue to negotiate what their homework should look like, and what level of completion is satisfactory.  Why, in the past, haven't I provided an example of what I expect?  So, this year I am, which is a good start on showing what completion looks like for the students for whom it is specified and for all of them.   My homework will be up on our Plus Portal system and assigned as the second day's homework.  Students will need to log onto Plus Portal, download the assignment, find the homework in the book, and finish the assigned problem in the correct format.
Seems obvious, but  sometimes light has to dawn.  I think it will make a big difference in the results I see for homework in the class.

What do you think?

Paper Bag to Book Cover, surface area and a tutorial

Book covers.  I like our math textbooks. They are a few years old, heavy, packed with info, and I insist that they remain covered. In fact,  I usually announce "text book cover quiz" several times each year whenever I notice the book covers becoming especially tattered or non-existent.  It's simple.  If your book is covered on the pre-announced day, you get 10/10 as a quiz grade.  My students love the idea, and see it as a great way to log an easy A as a quiz grade.  I had one very creative freshman this past year who completely forgot that I was checking book covers until he was at his locker before class.... he covered his book in his gym shorts.  "Hey, technically, it's covered, right?"  I reminded him how glad I was that he had chosen the debate team as his extra-curricular interest.  

I'm pretty tired of the stretchy fabric book covers.  They don't last.  You can't pick up the books without dropping them as they come from the lockers. Enough cracked spines.  This year, I'm having a paper bag book cover day in the first week, and I'll enfold it (pun intended) in a lesson about surface area of rectangular prisms.  I found the above picture of a drawing I did a couple of years ago showing how to cover the books.  It's a little basic.  Tonight I did some digging and found this great tutorial from the Childmade blog on how to accomplish the task.  

 So - hearty, personalized book covers,  recycling, surface area, decorating.  Sounds like a plan!

Thanks for looking. Please share your comments.  

                                                          

Problem-Solving Skills, Movement, and a great book

I have spent a lot of time this summer thinking about how to create a classroom environment that emphasizes and rewards problem-solving resilience, creative thinking, enthusiasm and risk-taking over the black and white of right and wrong. 
Here's a great article from Edutopia that discusses the growth mindset, and refers you to Carol Dweck's research. 
Don't misunderstand me, I teach math, and calculation skills, accuracy, correctness are all important.  However, the process by which you get there is even more important.  When it's all right and wrong, those who are accustomed to being wrong will find themselves back in familiar territory. Those who are used to being right, won't risk being wrong.  I want these students excited about their ideas, and eager to participate because their ideas are celebrated. 
My classroom environment leans this way because it is my nature.  However, this year, I am actively compiling strategies that support the thinking over the answer for part of each class.

So, here's a book I picked up at the NCTM conference in April.  The first section has a set of great, quick activities titled "What Doesn't Belong?".  There are four choices for each and multiple interpretations.  Students are encouraged to record their observations.  I'm planning to hang the letters ABCD in my room, and will ask students to stand by the letter representing the choice they want to explain.  Thinking, moving, sharing... I'm all over it.

 

Example of "What Doesn't Belong"

 School begins soon.  My brain is buzzing...  thanks for reading.  Please share your strategies for encouraging creative thinking.

Divisibility Rules... with a surprise.

 Okay.  Divisibility rules.  Students have been taught them before they come to my seventh grade class.  Maybe they retained some of them, maybe all of them.  But, we review.  Owning the divisibility rules is a HUGE help in pre-Algebra. Percentages, proportions, fraction operations, ratios, similarity, congruence... just to start.  
We use lots of vocabulary around this concept; factor, divisible by, multiple of, reduce, simplify, is in proportion to, equivalent ratios, etc.

This year, my seventh graders made posters for the divisibility rules.  I am a great believer in reuse and recycle.  We used old maps.  The number itself had to be at least half the size of the entire poster, and each poster had the rule, an example, and a "be careful".  We hung the posters around the room, where they were a great resource for the rest of the year.  Also, I chose one student to be the "go to person" for each rule.  Each student was very comfortable with their role, and was welcome to stand up to read the poster as a reference as long as needed, but eventually they all knew every rule cold.  
Quick review?                                                       

2 - one's digit is even
3 - if the sum of the digits is a multiple of 3
4 - if last 2 digits is divisible by 4 (as in 712)
5 - you know this one (ends in 5 or 0)
6 - 2 and 3 are factors
7 - there is no rule
8 - last 3 digits is divisible by 8 (as in 91,824)
9 - sum of the digits is a multiple of 9 

So, what's the surprise?   All this time, I thought there was no rule for seven.  You just divide.  Truth is, I never minded, and would toss in a few big multiples of 7 just so that my students could get the practice dividing.  But, turns out there is a rule for divisible by 7.  Whether you choose to use it or not is your choice...  I will teach it this year to give the students the option.
Here's the way it works.  Let's think about 5292.  

Double the last digit and subtract it from the remaining 3.  529 - 4 is 525.  Now double the last digit and subtract it from the remaining 2.  52 - 10 is 42.  42 is a multiple of 7 so 5292 is also a multiple of 7. 

Thanks for reading.  Were you surprised by the 7 rule?  I was.

Let me know.

                            

Vitruvian Man and First Week Activity

PG-Clothed Vitruvian Man for Middle School

One of my goals in my middle-school math classes is to cross pollinate with lots of common knowledge. The younger grades in our K-12 charter school use the Core Knowledge Sequence of books edited by E.D. Hirsh Jr, which provides a basis for common language our multi-cultural students can share.  For example, if I say "We're in the red", students may have no context for the expression.  

I like to use "The Vitruvian Man" as a first week activity for my 8th grade students.  I provide some background information about Leonardo Da Vinci.  (Most of the students know him as a Ninja Turtle.) I was so glad to see the exhibition of Da Vinci's drawings at the Museum of Fine Arts earlier this summer. 

While looking at the Vitruvian Man, we talk about the proportions of the human figure, and the hypothesis that our height and wingspan are close to equivalent.  Note:  use rich vocabulary and use a small pre-lesson to introduce.  Assign students to add the vocab to the Word Wall.  The students then measure each others wingspans and heights, and they each record the results for the whole room in the table provided.  It has 5 columns; name, height I, wingspan I, height II, wingspan II. We use the columns II later in the year to re-measure and look at growth throughout the room... great lesson on percent of change!  

(Measure across the back!!!  Girls will quickly realize why.)  I have prepped by creating two measurement stations in the room with meter sticks (or yard sticks) taped to the wall, or door jam.  When I use yard sticks, I tape the stick one yard from the floor. 

We use this data periodically throughout the year, but finish this activity by finding mean, median, any mode and range.  The collected data is a great basis for later lessons on scatter plots, percent of change, data representation in box and whisker plots and other graphs.  Have fun with it!

BTW, when you pull the data out later, you'll need to know the idiom "Good things come in small packages" because there will always be one student who hasn't grown even a smidge.

Thanks for checking out my new Math blog.  I look forward to your comments.