# Chapter 11 – Algebra I

Investigate some historical foundations of algebra

## Exercises

### Exercise 11.1

- You are teaching algebra in your school. What will you do if another colleague teaching algebra in your school disagrees with your teaching philosophy to use technology for a particular concept or skill? Should students be impacted by different teaching philosophies in school? Should all students use the same tools in the same classes?
- As a professional mathematics educator, do you feel that these basic assumptions by NCTM are viable? Why or why not? Are these assumptions necessary for full development of algebraic skills during the middle grades?
- Make two position statements:
- Position 1 summarizes your recommendations for the use of technology to teach algebra in the middle school.
- Position 2 presents your recommendations for the use of technology to teach algebra in the high school.

Compare and contrast your two positions.

### Exercise 11.2

- Have the Common Core Standards been adopted in your state or region?
- Is there a significant difference between the objectives for Algebra I for Mississippi and Massachusetts and the Common Core? Does this surprise you? Justify your response.
- Is universal agreement on the content to be covered in first-year algebra important? Why or why not?
- What topics should form the basic framework of a first-year algebra course?
- Locate a copy of your state’s curriculum frameworks for Algebra I. Are they similar to Mississippi and Massachusetts? Why or why not?

### Exercise 11.3

- Describe or create an activity that could be used to show (
*a*+*b*)^{2}= (*a*+*b*)(*a*+*b*). - Describe or create an activity that could be used to show (
*a*–*b*)^{ 2}= (*a*–*b*)(*a*+*b*).

### Exercise 11.4

- Develop an alternative to * that is located on the keyboard. How could this be universally implemented?
- Develop a new symbol for multiplication that is not currently on the keyboard. How will this key be universally accepted in the mathematics and computer community?
- Do you think questions about determining a key to represent multiplication on the keyboard merit time in a mathematics classroom? Why or why not?
- Do the differences in writing multiplication involving variables have a negative impact on the performance of students? Why or why not?

### Exercise 11.5

- Create a paper set of manipulatives including units,
*X*s, and X^{2}s. Do an example like 5(2*X*^{2}+ 3X + 4) using the manipulatives. Describe your thoughts and reactions to the manipulation in light of thinking about how this could help students understand the operation.

### Exercise 11.6

- Create a paper set of manipulatives including units,
*X*s, and*X*^{2}s. Do an example like (*X*+ 4)(2*X*+ 3) using the manipulatives. Describe your thoughts and reactions to the manipulation in terms of how this could help students understand the operation. - Connect a product like (3
*X*+ 4)(2*X*+ 1) to a partial product approach used to multiply (34)(21), which would look like:

34

×21

4 from 1 × 4

30 from 1 × 30

80 from 20 × 4

600 from 20 × 30

714

Describe your thoughts and reactions to the manipulation, focusing on how this could help students understand the operation.

### Exercise 11.6

- Are there any hints that help tell how many points of intersection two linear equations will have? If so, what are they and how could you describe them to students? If not, what do we do to help students?

### Exercise 11.7

- There are other applications of the principle used in the discussion of the speed trap in this chapter. Find at least one more and describe how it could be used to answer the typical student question, “When am I ever going to use this?” The example should be from the world as viewed by teenagers, not from your adult perspective.
- Like adults, children are often motivated by money. Find a real-world algebraic example that could be used during an algebra lesson that involves money.

### Exercise 11.8

- Given

1 + 2 = 3

4 + 5 + 6 = 7 + 8

9 + 10 + 11 + 12 = 13 + 14 + 15

16 + 17 +18 + 19 + 20 = 21 + 22 + 23 + 24

what is the first element of the one-hundredth row? How many terms are in the row? What is the last element of that row? Describe the Nth row.

### Exercise 11.9

- What is wrong with determining the sum of the first
*N*even counting numbers to be $$\frac{-{N}^{2}+N}{2}$$ in the text? Would the approach taken to arrive at this conclusion appear logical to a student? Why or why not? - Present another example of a “proof” that leads to an incorrect conclusion like 2 = 1.

### Exercise 11.10

- In the text, 4 was found to be a solution to the problem “Suppose 9 and 5 are two of the three digits in a three-digit number. You are to find a third digit so that an addition equation can be formed so both addends and the sum are permutations of the same three-digit number comprised of 9, 5, and the third digit you find.” Is 4 the only answer?
- Pick any prime number greater than 3. Square it. Add 15. Divide by 12. What is the remainder? Why? Will this always occur?

### Exercise 11.11

- Provide a list of five online Internet-based algebra tutorials for students. How would you rate each of these applications?
- Develop a lesson that would make use of an algebraic tutorial outside the classroom setting.
- Is it possible that mathematics tutorials could replace the importance of classroom teachers? Justify your response.

## Problem Solving Challenges

**Chapter 11: Problem-solving Challenges**

- Evaluate $$\frac{1001!-1000!}{999!}$$

Hint: Try a smaller problem with values such as 4, 3, and 2.

## Answer

Answer/solution: 1,000,000

$$\begin{array}{l}\frac{1001!-1000!}{999!}=\frac{999!(1001\times 1000-1000)}{999!}\\ =1001\times 1000-1000\\ =1000000\end{array}$$The key is to factor out 999! in the numerator.

- If three hens lay four eggs in five days, how many days will it take a dozen hens to lay four dozen eggs? Please round your answers to the nearest egg!

Hint: How many eggs would a dozen chickens lay in 5 days?

## Answer

Answer/solution: 15 days

A dozen hens would lay 16 eggs in five days. Therefore, since 48 is 3 times as many as 16 eggs, it would take 3 times as many days or 15 days to lay 48 eggs.

## Additional Learning Activities

There is no Additional Learning Activities for this chapter.