Pascal’s Triangle for Middle School
streets2.gsp
Starting out with a card trick! Why does this work?
Generating Pascal’s Triangle
Problem: Starting at the top of the grid of streets, how many different paths are there from the starting point to any intersection? You must follow these conditions: All paths must follow streets; you are only permitted to move downward, left, and right.
Open Geometer’s Sketch Pad and open the file streets2.gsp located on the desktop. Color the segments to make a path to each intersection and write down how many paths there are to each intersection. To check your answers, click the buttons on the bottom of the file. While you are determining the paths, you also may want to write the number of paths on your blank triangle.
To generate more of the triangle, click the “Remaining and More” button or go to http://mathforum.org/dr.cgi/pascal.cgi?rows=10
Types of Sequences
1. Arithmetic
2. Geometric
3. Polygonal
Points on a Circle
Problem – Given a specified number of points on a circle, find the number of segments, triangles, quadrilaterals, pentagons, hexagons, and heptagons. Then, find this sequence in Pascal’s Triangle.
Use the circular geoboard found at http://nlvm.usu.edu/en/nav/vlibrary.html grades 6-8 to complete the worksheet.
Points on a CircleImage / Points / Segments / Triangles / Quadrilaterals / Pentagons / Hexagons / Heptagons
/ 1
/ 2
/ 3
4
5
6
7
Finding Sequences
Find the following patterns in Pascal’s Triangle and describe where they are located within the triangle.
What they look like / Where you found them1. Points on a Circle
2. Natural Numbers
3. Prime Numbers
4. Powers of 2
5. Magic 11s
To verify that you found the natural numbers and prime numbers, go to http://nlvm.usu.edu/en/nav/frames_asid_181_g_4_t_5.html?open=activities.
Hints for Powers of 2:
Row # / Sum of Row / = / Value / = / 2 row#Hints for Magic 11s
Row # / Expression / = / Value / Viewed in TriangleThe Hockey Stick Pattern
Given the following diagram, write a description the relationship between the numbers on the straight part where you would hold a hockey stick versus the number where the puck would hit the stick.
Description of Relationship
Making Patterns
Multiples
Multiples of 2One more than a multiple of 2
Two more than a multiple of 2
Multiples of 3
One more than a multiple of 3
Two more than a multiple of 3
Three more than a multiple of 3
Multiples of 4
One more than a multiple of 4
Two more than a multiple of 4
Three more than a multiple of 4
Four more than a multiple of 4
Using your charts, color the multiples of 2, 3, and 4 in the triangle. This means you will have 3 different triangles. To color the triangle, go to
http://www.shodor.org/interactivate/activities/ColoringMultiples/ for students who need more guidance
on multiples or
http://nlvm.usu.edu/en/nav/frames_asid_181_g_4_t_5.html?open=activities for students who are
confident in multiples.
To extend into the next activity, use a different color for each “more than” category of number.
Remainders
Fill in the following charts. Notice any patterns that occur.
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9Q / R / Q / R / Q / R / Q / R / Q / R / Q / R / Q / R / Q / R / Q / R
20 / 21 / 28 / 35 / 36 / 56 / 70 / 84 / 126
Q / R / Q / R / Q / R / Q / R / Q / R / Q / R / Q / R / Q / R / Q / R
Choose a divisor and color the remainders in Pascal’s triangle different colors. We have completed all values up to and in row 8. To color your triangle, go to
http://www.shodor.org/interactivate/activities/ColoringRemainder/ for students who need more guidance
on remainders or
http://nlvm.usu.edu/en/nav/frames_asid_181_g_4_t_5.html?open=activities for students who are
confident in remainders.
Pascal Petals
Go to http://nlvm.usu.edu/en/nav/frames_asid_181_g_4_t_5.html?open=activities. Shade 15 the color red. The seven shaded cells make a flower with a center cell and the 6 surround petal cells. Color the cells numbered 5 20, 21 green. Color the remaining petals numbered 6, 10, 35 blue.
What is the product of the green petals? ______
What is the product of the blue petals? ______
Why do you think the products are the same?
Write the prime factorizations of each number in the green petals ______
Write the prime factorizations of each number in the blue petals ______
Write an explanation of your discovery:
Do you think that if a different center cell were chosen that the results would be the same? ______
Choose any cell on Pascal’s triangle that is surrounded by 6 other cells. Shade this cell using red.
Starting with the petal above and to the left of the center cell, alternating petals are colored green. My
numbers are ____, _____, ______
Color the three remaining petals around the center blue. These numbers are ____, ____, ___
What is the product of the green petals? ______
What is the product of the blue petals? ______
Write the prime factorizations of each number in the green petals ______
Write the prime factorizations of each number in the blue petals ______
Compare the prime factorizations of the two sets of numbers. Write an explanation of your discovery.
Handshake Problem
Problem- A baseball team is practicing for the first time. In order to get to know each other, the players shake hands. Each player shakes hands with every other player exactly once. How many total handshakes are there? (A team is comprised of 9 players)
Use Excel to create a chart to figure out the sequence.
Find this sequence in Pascal’s Triangle.
Figurate Numbers
Task A: Using the GSP file Figurate Numbers.gsp and geoboard at http://nlvm.usu.edu/en/nav/vlibrary.html to construct the 5th triangular and square number, and the 4th hexagonal number.
Task B: Find the triangular, square, and hexagonal numbers in Pascal’s Triangle. Use the GSP file to
verify your findings.
Middle School Seating
Problem: In middle school, girls and boys don’t want to sit next to each other for fear of getting cuties. So, a teacher lets each girl sit next to at least one other girl, and each boy sit next to at least one other boy. Since there are more girls in the class, each row must begin with a girl. How many ways can we seat n children in a row?
Use Excel to create a chart to figure out the sequence.
What sequence is this?
Fibonacci Sequence
Find the Fibonacci Sequence in Pascal’s Triangle.
Use the GSP file Fibonacci Numbers to verify your answer.
If you still do not see it, consider rearranging the elements in the triangle such that
Column / 0 / 1 / 2 / 3 / 4 / 5 / …0 / 1
1 / 1 / 1
r / 2
o / 3
w / 4
5
…
Probability Patterns
Problem- It’s Friday night and the downtown pizza place is crowded and you and your group are trying
to order a large pizza. Downtown pizza has 8 toppings: anchovies, extra cheese, green peppers, mushrooms, olives, pepperoni, sausage, and tomatoes. How many different pizzas can be ordered if a pizza can be selected with any combination of toppings?
- How many different pizzas can you order with only one topping?
- How many different pizzas can you order each with seven toppings?
- Are the number of one-topping pizzas and the number of seven-topping pizzas related?
- How many different pizzas can you order with two toppings?
- How many different pizzas can you order with six toppings?
- Are the number of two-topping pizzas and the number of six-topping pizzas related?
- Can you find these numbers in Pascal's triangle?
- Can you use Pascal's triangle to help you find the number of pizzas that can be ordered with three, four, or five toppings?
- In all, how many different pizzas can be ordered?
For more information on anything you have seen here please visit:
1. Card trick - http://binomial.csuhayward.edu/CardTrick.html
2. Streets & Handshake – www.dpgraph.com/janine/mathpage/patterns.html
3. Sequences – http://ptri1.tripod.com
4. Remainders & Multiples http://www.shodor.org/interactivate/lessons/PatternsInPascal/, http://mathforum.org/workshops/usi/pascal/mid.color_pascal.html , http://nlvm.usu.edu/en/nav/frames_asid_181_g_4_t_5.html?open=activities
5. Pascal Petals - http://mathforum.org/workshops/usi/pascal/pascal_lessons.html
6. Pizza Toppings - http://mathforum.org/workshops/usi/pascal/pascal_lessons.html
7. Fibonacci Sequence - http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html#pascal
8. Geoboards - http://nlvm.usu.edu
9. Chinese Triangle - http://www.ualr.edu/lasmoller/pascalstriangle.html