|Module 2 - Computer Algebra|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|Lesson 2.3: Pattern Recognition|
In this lesson you will use a computer algebra system to create a mathematical laboratory where experiments are conducted, patterns are discovered, and conjectures are made. This type of inductive learning fosters a sense of ownership and interest in the concept discovered. Discovering patterns and making conjectures can motivate learning about proof.
Factoring the Difference of Two Squares
This example illustrates inductive, or discovery, learning. The concept to be developed is factoring the difference of two squares. In later modules, lessons that discover calculus concepts are developed.
The procedure we will use to discover patterns is
Explore Factoring x2 4
To factor x2 4,
The factors of x2 4 are x 2 and x + 2.
Explore Factoring x2 9
You may want to factor other examples of the difference of two squares to determine a pattern relating the original expression and its factors. After a few examples, you will probably notice a pattern. Can you predict the result of the next two commands before entering them? Write your predictions down before entering the commands into the calculator.
Explore by Factoring Other Examples
Describe the Pattern Verbally
2.3.1 When you see the pattern, describe it verbally and express it with an algebraic expression. Click here for the answer.
Predict the Pattern of the Factors of The Difference of Two Squares
2.3.2 Predict the factors of x2 a2 . Click here for the answer.
Test the Prediction
2.3.3 Check your prediction.
Extend and Check
2.3.4 Predict the factors of x2 5. Click here for the answer.
2.3.5 Factor x2 5 by using the factor( command. Click here for the answer.
2.3.6 Predict the factors of x2 a . Click here for the answer.
2.3.7 Try cfactor(x^2+4,x).
Click here for the answer.
You will now explore expanding expressions like (x + 1)n, which are called binomial expansions. The pattern of the powers of the variable in the expansion is
We want to find the values of the coefficients. In the following exploration you should look for a pattern in these coefficients and how they relate to the power of the expansion, n.
The Expand( Command
To explore binomial expansions, you will use the expand( command, which is in the Algebra menu of the Home screen.
Explore the Expansion of (x + 1)2
To expand the expression (x + 1)2,
When n = 2, the coefficients of the expansion are 1, 2, 1.
Explore the Expansion of (x + 1)3
When n = 3, the coefficients are 1, 3, 3, 1.
Explore the Expansion of (x + 1)4 and the Expansion of (x + 1)5
When n = 4, the coefficients are 1, 4, 6, 4, 1.
When n = 5, the coefficients are 1, 5, 10, 10, 5, 1.
Describe the Pattern
The coefficients found by expanding (x + 1)n, when n = 2,3,4,5 are shown in the respective rows of the triangle below.
2.3.8 Describe the pattern shown in the triangle above. Click here for the answer.
You can expand the triangle, both below and above the existing rows, and use the new rows to determine the coefficients of other expansions of (x + 1)n .
Predict (x + 1)6 and Confirm
2.3.9 Use the next line below the triangle shown above to predict (x + 1)6 and then check your answer with the expand( command. Click here for the answer.
We will not extend and generalize this topic at this time.
Using the TI-89, you are able to see the results of many binomial expansions more quickly than would be possible with only paper and pencil. This facilitates inductive learning. The Explore, Describe, Predict, Confirm, Extend, and Generalize procedure is a more active form of learning than reading or listening. Hopefully, when you use this procedure, you have more of a sense of discovering the theorem, rather than just being told the result.
2.3.10 Expand the product (x + 1) (x2 x + 1) by using the expand( command.
Click here for the answer.
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