# Chapter 13 – Algebra II – Advanced Algebra and Modeling

Should we continue with the emphasis on symbolic manipulation? How do we integrate technology into the picture? What is the impact of a more interdisciplinary and connected view of mathematics? Will the complexities of the world that demand more flexible thinking, less manipulation of symbols, and more creative problem-solving be strong enough to impact what algebra is taught and how it is delivered? Algebraic modeling brings real-world examples into focus allowing students to connect the mathematics classroom with the world around us.

## Exercises

### Exercise 13.1

- Compare and contrast the expectations for students in high school between Standards 2000 and the Common Core Standards. Is there a significant different in the expectations? Justify your position.

### Exercise 13.2

- Establish a position on the question of a national curriculum for mathematics. Defend your position while considering questions like a national test and teaching to the test.
- Determine if there is a mandatory mathematics test each student must pass to graduate from high school in your state. What happens if the student passes all required mathematics classes in high school but cannot pass the test? What happens if the student passes the necessary test but does not pass all of the required math courses to graduate? Is there an appeals process for the student?

### Exercise 13.3

The questions in this section refer to points in the preceding paragraphs. For each of them, describe what you would use to influence your decisions and what those decisions would be.

- What will you do when the students have difficulty on a particular concept? If you slow down to adequately cover foggy material, what algebra concepts will you choose to omit at the end of the year?
- Will all algebra teachers omit the same concepts?

### Exercise 13.4

- Compare an Algebra I and Advanced Algebra text from the same publisher. List all topics that are repeated and to what extent.
- Compare three Advanced Algebra texts. How does the material differ from chapter to chapter? Did any of the texts omit a chapter that you would not have? How many review chapters does each text have? Is the review too much, too little, or just right? Why?

### Exercise 13.5

- Create an ellipse on paper. The classic method is to establish two points, A and B, as foci and a third point, C, not on the segment between A and B. The sum of the distances AC and BC can be made constant by using a string. Put a pencil at point C, use tacks to anchor points A and B, and move the pencil, tracing the ellipse.
- Create an ellipse using dynamic geometry software. Describe the steps necessary to create the ellipse.

### Exercise 13.6

- Explain what went wrong in the example immediately preceding this question where the result was R = –1. Describe how you would have a class with the appropriate background become aware of the results.
- Locate and read the book
*The King’s Chessboard*by David Birch (1993). For what age student is this book appropriate? What mathematical concepts can be taught using this book?

### Exercise 13.7

- Fold a standard sheet in half, then fold that in half again, and so on. What is the maximum number of folds you can make? Does the maximum number of folds change if you use a different size or kind of paper?
- Assume that the thickness of a standard sheet of paper is 0.004 inches. How high would the stack be if that sheet could be folded 50 times? How does this result compare with the text description of the sheet of paper being 0.003 inches thick? Should you have anticipated that result? Why or why not?
- Devise a lesson using approximations that could be used to convince students of the plausibility of the result you get when folding a 0.004-inch-thick piece of paper in half 50 successive times.
- Continue the algebraic expression of the Fibonacci sequence. Square the eighth term, and verify that the absolute value of the differences between the product of the square of the eighth term and the product of terms, 1, 2, 3, 4, 5, and 6 places removed in both directions, yields squares of the respective first six terms of the Fibonacci sequence.

### Exercise 13.8

- Extend Figure 13.10 to be at least 6 by 6. Include nonstandard responses in your extension. Play the game with two of your peers and describe their reactions.
- Develop a game that would enhance a second-year algebra student’s knowledge of an Algebra I concept.

## Problem Solving Challenges

- Find a three-digit number XYZ such that X! + Y! + Z! = XYZ. For this problem, X cannot equal 0 and X, Y, and Z must all be whole numbers less than 10.

Hint: One value is 1.receive? Explain his solution. Hint: What would be the result if you had 18 camels?

## Answer

Answer/solution: 145

1! = 1, 4! = 24, and 5! = 120. 1+ 24 + 120 = 145.

- In Pascal’s Triangle, how many odd numbers will there be in the 65th row?

Hint: Look for a pattern and consider rows that are powers of two.

## Answer

Answer/solution: 2

The number of odd numbers in each row forms the pattern 1, 2, 2, 4, 2, 4, 4, 8, 4, 8, 8, 16, … such that the rows of powers of two such as the second row has 2 odds, the 4th row has 4 odds, the 8th row has 8 odds, the 16th row has 16 odds, then the 32th row will have 32 odds, revealing that the 64th row would have 64 odds containing 64 numbers. Since the first and last numbers are one, there must be at least 2 odds in the 65th row. If the 64th has all odd numbers, then when you find the sum of any two numbers to determine the 65th row, the numbers will be even. Therefore, only the first and last numbers, the ones, will be odd.

## Additional Learning Activities

There is no Additional Learning Activities for this chapter.