Patterns In Pascal's Triangle
Abstract
This lesson is designed to show students that patterns exist in the Pascal's Triangle, and to
reinforce student's ability to identify patterns.
This is a good follow up lesson to Finding Patterns In Fractals.
Objectives
Upon completion of this lesson, students will:
- have been introduced to Pascal's Triangle and its patterns.
- have practiced identifying and determining patterns in Pascal's Triangle.
Standards
The activities and discussions in this lesson address the following
NCTM Standards:
Number and Operations
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
- work flexibly with fractions, decimals, and percents to solve problems
- understand and use ratios and proportions to represent quantitative relationships
- use factors, multiples, prime factorization, and relatively prime numbers to solve problems
Algebra
Understand patterns, relations, and functions
- represent, analyze, and generalize a variety of patterns with tables, graphs, words,
and, when possible, symbolic rules
- relate and compare different forms of representation for a relationship
Use mathematical models to represent and understand quantitative realtionships
- model and solve contextualized problems using various representations, such as
graphs, tables, and equations
Links to other standards.
Student Prerequisites
- Arithmetic: Students must be able to:
- perform integer and fractional arithmetic
- Technological Students must be able to:
- perform basic mouse manipulations such as point, click and drag
- use a browser such as Netscape for experimenting with the activities
Teacher Preparation
Students will need:
- Access to a browser
- pencil and paper
- Copies of supplemental materials for the activities:
Key Terms
This lesson introduces students to the following terms through the included discussions:
Lesson Outline
- Focus and Review
Remind students what has been learned in previous lessons that will be pertinent to this lesson
and/or have them begin to think about the words and ideas of this lesson:
- Ask the students to recall what a
multiple
is and to think of an example. Have a student share his example with the class.
- Have the students also consider
Pascal's Triangle. If your class has not studied
it previously, ask, "Has anyone ever heard of Pascal's Triangle?" If your class has studied it, have one student remind the class what Pascal's Triangle looks like.
- Ask students, "Did you know that multiples make a pattern in Pascal's Triangle?"
- Objectives
Let the students know what it is they will be doing and learning today. Say something like this:
- Today, class, we will be talking about the patterns that multiples create in Pascal's Triangle.
- We are going to use the computers to learn about these patterns, but please do not turn your
computers on or go to this page until I ask you to. I want to show you a little about Pascal's
Triangle and its patterns first.
- Teacher Input
In this part of the lesson you will explain to the students how to do the assignment.
You should model or demonstrate it for the students, especially if they are not familiar with how
to use our computer applets.
- Ask a student to describe the pattern that she sees after all the multiples have been found.
Ask the students what types of shapes are made by the multiples within the Pascal's Triangle.
- Pass out the Worksheet to
Accompany "Finding Patterns In Pascal's Triangle." Have the students draw the pattern that the
class determined as a group for multiples of 4 in Pascal's Triangle.
- Guided Practice
Try another example, letting the students direct your moves. Or, you may simply ask,
"Can anyone describe the steps you will take for this assignment?"
- If your class seems to understand the process for doing this assignment, simply ask,
"Can anyone tell me what I need to do to complete this worksheet?" or ask, "How do I run
this applet?"
- If your class seems to be having a little trouble with this
process, do another example together, but let the students direct your actions:
- This time, choose a number such as 8 to try the example with. Let
the students call out multiples of 8 that they see in the triangle.
- The multiples of 8 include: 8, 56, 120, 792, and 3432. You might want to ask students to
compare this pattern to the one that was formed by the multiples of 4. Be sure to point out
that all of the multiples of 8 are also multiples of 4 and yet the patterns are very
different (since the multiples of 4 are not necessarily multiples of 8).
- Independent Practice
- Allow the students to work on their own to complete the rest of the worksheet. Monitor the room
for questions and to be sure that the students are on the correct web site.
- Students may need help with finding the multiple of the harder numbers, such as 7. Encourage
the students to devise their own methods for determining the multiples. Suggest that the students
attempt to use their knowledge of the patterns they already discovered to aid in finding the
harder patterns!
- Closure
It is important to verify that all of the students made progress toward understanding the
concepts presented in this lesson. You may do this in one of several ways:
- Take up the individual or group worksheets to evaluate for completion.
- Bring the class together and have different groups or individuals share their result for a
particular number with the rest of the class. Allow students who did not get to finish that
number to sketch the result so that they will not lack some of the information needed for full
understanding.
- Have the students write a short paragraph explaining the type of
patterns that they saw including the similarities between the different pictures, and the type of
shapes that reoccurred in the pictures.
Alternate Outlines
This lesson can be rearranged in several ways.
- The students may wish to tackle the worksheet in groups.
- You may wish to assign different groups with particular numbers to ensure that every
option is attempted for the class discussion later.
Suggested Follow-Ups or Extensions
- As an extension, you may have students predict the entended pattern for a particular
number when the Pascal's Triangle is made larger. The class could work together to extend
the triangle by hand (on a bulletin board, perhaps) and see if the predictions were correct.
Again, 4 may be a good number to use for this extension.
- You may wish to do a similar lesson to discuss patterns formed by
Coloring Remainders In Pascal's Triangle.
This activity may prove to be a little more challening for students, may require more supervision,
and may best be done as a class discussion and demonstration.
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