Combinations and Pascal's Triangle

 

The first 12 rows of Pascal's Triangle perhaps the most famous triangular array of numbers in mathematics!

Assignment Exercises Combinations Pascal's Triangle

Readings Bello & Britton, Sections 9.3-9.4 Higher Level Thinking Skills Slide Show by Dr. Lori Campbell http://www.wscc.cc.tn.us/socialsci/ LCampbell/chapter8/sld001.htm

Web Resources These web resources have links to combinations and information about Pascal's Triangle.

Assignment

1) Post to the discussion forum:

After looking at the reading for the week by Dr. Campbell, visit the Math Forum's Web Unit on Pascal's Triangle with grade level standards lessons and links (http://forum.swarthmore.edu/workshops/usi/pascal/), review the unit for your grade level, and post a critique of that unit from the point of view of developing higher level thinking skills.

Click here to go to the discussion forum (if you are not logged onto Blackboard, log in to bb.wpunj.edu and click on Communication and choose the Discussion Board.

2) Complete the following exercises.

 

Exercises

1) pp. 622-623, Ex. 2, 8, 16, 22, 24, 29, 32, 35

2) pp. 630-632, Ex. 1, 2, 4, 9, 15, 18, 19, 24

3) Amanda won the big prize on a TV quiz show - 3 brand new cars. She can chose from Nissan, Saab, Mitsubishi, and Suburu in any combination, OR she can select all three of the same kind, OR she can select 2 of the same kind. How many different choices does she have?

4) p. 623, Ex. 39, 40

4) Find 112, 113, 114, and 115, What do you observe?

5) Fractal patterns using Pascal's Triangle are a new way of looking at different number multiple patterns. a) Visit Coloring Multiples in Pascal's Triangle http://204.219.224.5/interactivate/activities/pascal1/index.html and click on "Auto-Color!" after you have set your own number to 2 b) Do the same at the Coloring Remainders in Pascal's Triangle http://204.219.224.5/interactivate/activities/pascal2/index.html c) Is there any difference in the pattern? ? d) Visit both sites and use the number 3. Explain the difference in the color patterns.

*Choose one problem from 6, 7, 8, or 9 depending on your curriculum! Problem 6 is for a geometry classroom, problem 7 is for a general classroom, and problem 8 is for an algebra classroom.

6) In Pascal's Triangle, the third diagonal gives the sequence of what is called the tetrahedral numbers. Why?

7) In Pascal's triangle, take any number n in the interior of the triangle, and consider its 6 neighbors. Calculate a*b*c and d*e*f. Do this for several choices of n. What is the pattern? Can you prove it using the combination formulas?

8) In algebra, the Binomial Theorem states how to raise a binomial , say (x + y), to an integral power. How is the Binomial theorem related to Pascal's triangle?

Web Resources

Ask Dr. Math about Pascal's triangle
http://forum.swarthmore.edu/dr.math/problems/pascal.html

with a link to an interactive Pascal's triangle which generates as many rows of Pascal's Triangle as you wish
http://forum.swarthmore.edu/~ken/pascal.cgi

Ask Dr. Math about Permutations and Combinations
http://forum.swarthmore.edu/dr.math/faq/faq.comb.perm.html

Exploring Fractals website - by Dr. MaryAnn Conners with the ways to generate fractals
http://www.math.umass.edu/~mconnors/fractal/fractal.html

one page of which relates Pascal's triangle to the Sierpinksi triangle
http://www.math.umass.edu:80/~mconnors/fractal/generate/pascal.html

The HEARTS Lesson - a lesson on "how many paths" for grade 4 with the standards connection
http://score.kings.k12.ca.us/lessons/hearts.html

Introduction to Pascal's Triangle and Counting - complete lesson plans for using Coloring Multiples and Coloring Remainders in the exercises of this week's homework.
http://204.219.224.5/interactivate/discussions/pascal.html

Javascript Calculator - calculates the number of combinations of C(n, k)
http://www.io.com/~ritter/JAVASCRP/PERMCOMB.HTM#Combinations

NASA's Learning Technologies Project - Using Excel with Combinations
http://www.lerc.nasa.gov/Other_Groups/K-12/Basic_Tutorials/Excel/excel.htm  

Pascal's Fractals at Ivars Peterson's Mathland - a good article on "Fascinating patterns can arise out of arrays of numbers defined by simple rules."
http://www.maa.org/mathland/mathland_2_10.html

Pascal's Triangle in the Mathematics by Kryss Katsiavriades web site - emphasizes Pascal's Triangle and the Binomial Theorem
http://www.krysstal.com/binomial.html

Pascal's Triangle and the Combinatorial Coefficients - relates the factorials, permutations, and combinations
http://www.uwm.edu/~ericskey/TANOTES/Algebra/node20.html

Pascal's Triangle Interface - interactive applet which generates rows up to100 and mods up to 16
http://www.cecm.sfu.ca/organics/papers/granville/support/pascalform.html

Spreadsheets, Pascal's Triangle and Sierpinski Gaskets -where you can download an Excel spreadsheet to generate and modify the colors
http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/BrombacherAarnout/spreadsheets/SpreadsheetsnPascal.html

StudyWeb's site on Pascal's triangle- lots of links to lesson plans and activities, designated by grade level
http://www.studyweb.com/links/774.html

Two printable worksheets for Pascal's Triangle:

Pascal's triangle worksheet for primary grades
http://teachingideas.co.uk/maths/nopattern/pascalwsht.htm

pdf file for an activity sheet
http://www.harcourtschool.com/teacher_resources/math/grade_08/g8_honeycomb.html

  

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