have been introduced to Pascal's Triangle and its patterns

have practiced identifying and determining patterns in Pascal's Triangle .

Standards Addressed:

Grade 10

Estimation and Computation

The student accurately solves problems (including real-world situations).

Numeration

The student demonstrates conceptual understanding of real numbers.

Grade 3

Numeration

The student demonstrates conceptual understanding of whole numbers up to one thousand.

Grade 4

Numeration

The student demonstrates conceptual understanding of whole numbers to ten thousands.

Grade 5

Numeration

The student demonstrates conceptual understanding of whole numbers to millions.

Grade 9

Estimation and Computation

The student accurately solves problems (including real-world situations).

Numeration

The student demonstrates conceptual understanding of real numbers.

Algebra

Arithmetic with Polynomials and Rational Expressions

Use polynomial identities to solve problems

Fifth Grade

Operations and Algebraic Thinking

Analyze patterns and relationships.

Fourth Grade

Operations and Algebraic Thinking

Generate and analyze patterns.

Sixth Grade

The Number System

Compute fluently with multi-digit numbers and find common factors and multiples.

Grades 3-5

Algebra

Understand patterns, relations, and functions

Use mathematical models to represent and understand quantitative relationships

Grades 6-8

Algebra

Understand patterns, relations, and functions

Grades 9-12

Algebra

Understand patterns, relations, and functions

Grade 3

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 5: The learner will recognize, determine, and represent patterns and simple mathematical relationships.

Grade 5

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 5: The learner will demonstrate an understanding of patterns, relationships, and elementary algebraic representation.

Technical Mathematics I

Number and Operations

Competency Goal 1: The learner will apply various strategies to solve problems.

3rd Grade

Algebra

The student will demonstrate through the mathematical processes an understanding of numeric patterns, symbols as representations of unknown quantity, and situations showing increase over time.

7th Grade

Algebra

The student will demonstrate through the mathematical processes an understanding of proportional relationships.

Intermediate Algebra

Algebra

The student will demonstrate through the mathematical processes an understanding of sequences and series.

3rd Grade

Algebra

Content Standard 2.0 The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

4th Grade

Algebra

The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

5th Grade

Algebra

The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

6th Grade

Algebra

Content Standard 2.0 The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

7th Grade

Algebra

The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

8th Grade

Algebra

The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

Algebra I

Algebra

Students will describe, extend, analyze, and create a wide variety of patterns and functions using appropriate materials and representations in real world problem solving.

Algebra II

Algebra

Students will describe, extend, analyze, and create a wide variety of patterns and functions using appropriate materials and representations in real-world problem solving, and will demonstrate an understanding of the behavior of a variety of functions and their graphs.

Geometry

Algebra

Students will recognize, extend, create, and analyze a variety of geometric, spatial, and numerical patterns; solve real-world problems related to algebra and geometry; and use properties of various geometric figures to analyze and solve problems.

Grade 4

Patterns, Relationships, and Algebraic Thinking

7. The student uses organizational structures to
analyze and describe patterns and relationships. The student is expected to describe the
relationship between two sets of related data such as ordered pairs in a table.

3rd Grade

Patterns, Functions, and Algebra

3.24 The student will recognize and describe a variety of patterns formed using concrete objects, numbers, tables, and pictures, and extend the pattern, using the same or different forms (concrete objects, numbers, tables, and pictures).

4th Grade

Patterns, Functions, and Algebra

4.21 The student will recognize, create, and extend numerical and geometric patterns, using
concrete materials, number lines, symbols, tables, and words.

4.21

5th Grade

Patterns, Functions, and Algebra

5.20 The student will analyze the structure of numerical and geometric patterns (how they change or grow) and express the relationship, using words, tables, graphs, or a mathematical sentence. Concrete materials and calculators will be used.

7th Grade

Patterns, Functions, and Algebra

7.19 The student will represent, analyze, and generalize a variety of patterns, including arithmetic sequences and geometric sequences, with tables, graphs, rules, and words in order to investigate and describe functional relationships.

The science that studies the numbers of different combinations, which are groupings of numbers. Combinatorics is often part of the study of probability and statistics

fractal

Term coined by Benoit Mandelbrot in 1975, referring to objects built using recursion, where some aspect of the limiting object is infinite and another is finite, and where at any iteration, some piece of the object is a scaled down version of the previous iteration

multiples

The product of multiplying a number by a whole number. For example, multiples of 5 are 10, 15, 20, or any number that can be evenly divided by 5

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 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.

Check to be sure that the students understand how to make Pascal's Triangle by having them
create a portion on paper, or by drawing one on the board or overhead projector as they tell
you what to write.

Open your browser (but don't let the students open theirs yet) to
Coloring Multiples in Pascal's Triangle in order to demonstrate this activity to the students. Ask students if the triangle that they
created looks like the one displayed on the screen.

You must now explain the applet to the students. This can best be done by setting your own
number: 4 is a good number to choose when explaining this to students

Ask students to name multiples of 4 that they see in the triangle. They will probably name
numbers such as 4, 8, 12, 20, 28, and 36. Click on these numbers to highlight them as the
students call them out

You may have to give hints to help students determine the larger multiples of: 56, 84, 120,
220, 252, 364, 792, 924, 1716, and 3432. Encourage the students to look for the pattern and
make an educated guess about the larger multiples of 4.

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.

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
recurred in the pictures

Alternate Outline

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-Up

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.