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Week 1 Math Task

The Adult Numeracy Network’s Professional Development Principles state that sound professional development should begin with teachers as mathematics learners and thinkers. Following that principle, in addition to the discussion questions, a Math Task related to the reading will be offered each week. These are tasks that can certainly be shared with students, but as we go through the book study together, approach the task as a mathematics learner and thinker so you can have your own experience with the excitement of discovering mathematical ideas. Each task was chosen to be accessible and challenging regardless of the math level that is familiar to you. As you approach these tasks, please try to let go of ideas of math that you can and can’t do and just work with the activity as it is given. Share your experiences and thoughts about the math task in the group. Have fun!

Math Task:  Make several copies of Pascal’s Triangle, p 222. Use crayons, markers, colored pencils, or paint as you look for patterns. Just dive in! All of your conjectures may not show patterns. Share your work on the discussion board.

Comments

MarkTrushkowsky's picture

I found an applet on Geogebra that has the first 12 rows of Pascal's Triangle and lets you color in different patterns. Check it out - https://www.geogebra.org/m/vgskJ9pM

Mark

 

 

Amy Vickers's picture

Thanks Mark!  I tried it with my nephew.  We found primes, factors of 12, multiples of 5, 7, and 9, and the number 8.

I posted the image as a pdf under the documents tab in this group.

 

ecappleton's picture

Hi all,

A group of teachers in NYC met on Monday to talk about the book and do some related activities. We used part of the YouCubed Pascal activity and found the following patterns. Some people found a number of other interesting things in the triangle, but these were the ones we shared as a group.

 

Looking at the differences between numbers (rate of change) in diagonals...

pascal-triangular

Look at the rate of change of the rate of change...

Looking at row totals...

Highlighting the odd numbers...

 

 

 

Pam Meader's picture

Hi Eric and Mark,

This morning I tried Mark's link and was looking at the odd numbers like your post Eric. I want to now extend this to see what happens and if my hypothesis is correct. I think the large white middle will form another inverted triangle. I also notice at the very top and the two yellow triangles, one on the left bottom and one on the right bottom are similar. I am curious how that plays out. So much fun!

 

 

ecappleton's picture

I had a similar hypothesis to Pam and currently working on a question Amy shared on Twitter: What proportion of the numbers in Pascal’s Triangle are even?

Our exploration of highlighting odd numbers led me to seeing new patterns as the triangle continues. It's amazing how this simple pattern of adding two numbers produces such a complex arrangement of numbers.

 

Sarah Lonberg-Lew's picture

Based on the similarity of the picture with the odds or evens colored in to Sierpinski's Triangle (http://mathworld.wolfram.com/SierpinskiSieve.html), I'm guessing that the fraction of the numbers in the triangle that are even gets closer and closer to 100% the more rows you include in your drawing. 

Can you see why the pattern occurs? It has to do with what happens when you add different combinations of odd and even numbers (odd + odd = even etc.).

Here's doodle I did a while ago showing the pattern when numbers are shaded based on the remainders they give when divided by 3. There are interesting similarities and differences between this and just shading the even or odd numbers.

 

 

 

Marcy Cordova's picture

When I taught high school we explored Pascal's triangle, but then we looked at the game Plinko from The Price is Right. Here's a website that explains one way to do the exercise: http://www.mathdemos.org/mathdemos/plinko/bigboardplinko.html I did it slightly differently: I gave each student a paper version of a Plinko board. I had each student pick a slot where they would drop the chip. Then they filled in their boards using Pascal's triangle. They compared their boards, and had to write an explanation of where they would drop the Plinko chips and why. Then we used this website: http://mimage.opentext.com/alt_content/binary/ot/newmedia/ot_plinko/index.html to play Plinko on my Smartboard (they had to use the strategy they wrote about). We compared what they expected to happen with what really happened. Finally we watched the video called "The Worst Plinko Luck. Ever." at https://www.youtube.com/watch?v=0Ou-bhlgneM

This lesson was a fun way to relate Pascal's triangle to "real life". I imagine with the new game show The Wall, there would be even more interest now. 

Kerry Cook's picture

 

Thanks for the link to Geogebra - it's really helpful!   I've enjoyed looking for the  patterns that you have all been sharing, and those led me to see other interesting connections as well.  For example, the diagonal that begins with 1, 7, 28 contains numbers that are all multiples of 7.  When I looked at the " differences of the differences" they are all multiples of 7 as well.  Cool!      Here is my work:  (by the way I tried posting a picture but it doesn't seem to let me while using my iPad)  

       First level of sequence:  7  28  84  210  462  

              First    differences: 21  56  126  252  

           Second differences:      35  70  126  

                third differences:   35  56  

                   Fourth difference:21

I'm teaching sequences and series to my College Algebra students now, so I think I will bring this task to them for some practice in finding patterns.  Thank you!

 

 

 

sadkins2009's picture

I'm not sure where this one is headed....I think I need to do more rows on the sequence. But I started exploring the prime factorizations of the numbers in the sequence to see if there was a pattern. Then I started coloring each side of the hexagon with a different color representing the prime number (2 = pink, 3 = blue, 5 = green, 7 = purple)

2017-04-24 14.01.12.jpg

Dropbox is a free service that lets you bring your photos, docs, and videos anywhere and share them easily. Never email yourself a file again!

 

SharonShoe's picture

I'm glad that we are still welcome.  I keep busy.

I enjoyed coloring Pascal's Triangle.  I would have enjoyed a math class like this!

Here is a link:  https://drive.google.com/file/d/0B3nenlHFGhmmeUhNQXp5emQ5RWM/view?usp=sharing