# Chapter 7 – Problem-Solving

Effective problem-solving requires open-minded approaches from teachers and students. This technique was demonstrated in another Egyptian problem, not found in the Rhind papyrus. Babylonians from 300 bce demonstrated a working knowledge of the Pythagorean theorem a^{2} + b^{2} = c^{2} for right triangles. Problem-solving is an essential ingredient in mathematical learning both inside and outside the classroom. Industry continues to request an increased emphasis on critical thinking and problem-solving skills for their employees.

## Exercises

### Exercise 7.1

- Solve this seemingly simplistic problem x
^{2}+ y^{2}= 169; area = xy = 60 square cubits, without the doubling technique. Did you resort to more sophisticated methods than were used by the Babylonian scribe in 350 BC?

### Exercise 7.2

- Write a definition of topology that would be appropriate for middle school students.
- Provide two different examples of topology that could be used in a secondary classroom to stimulate student curiosity about the subject.

### Exercise 7.3

- Identify two topics that would be of interest to middle school students and the mathematics related to those topics. Determine the grade level or subject in which each topic could be used to introduce the defined mathematical concept(s).
- Describe two topics that would be of interest to high school students and the mathematics related to those topics. Determine the grade level or subject in which each topic could be used to introduce the defined mathematical concept(s).
- Based on textbooks (use as many as possible from the same series) for Grades 6 – 12, present a written description of your opinion of the topics suggested for student analysis. What are the topics most often chosen? What topics would you use? Give examples.

### Exercise 7.4

- Describe at least one situation that does not have one right answer. Your discussion should include all the potentially correct responses. For example, what route do you travel to get to school? Many paths will lead to the same conclusion!
- Show the construction steps needed to create the two examples in Figure 6.5.

### Exercise 7.5

- Develop at least three additional questions a class could generate regarding the shapes in Figure 7.7 in the text.

### Exercise 7.6

- Create three higher-order thinking questions from the shapes in Figure 7.7 in the text. Discuss potential extension topics for each question.
- List at least one clue that could be used to guide students to an appropriate response for each of the questions you created in part 1 of this exercise. The clue should not be a dead give-away.
- Use another geometric shape to develop questions along the line of thinking developed with the example of five congruent squares in the text.

### Exercise 7.7

- Construct Figures 7.11 through 7.11 in the text using a dynamic geometry program. Determine the measure of the angles, move your construction, and describe the results.
- List the different topics that would be included in developing the idea that angle BCA is a right angle as shown in Figures 6.10 through 6.12.
- Find or create a problem that can be solved in at least three different ways. State the problem and show at least three alternate solutions.

### Exercise 7.8

- Describe how nine blocks, where eight are known to weigh the same and one is light, require two weighings to identify the lightweight. Can this be done in more than one way?

### Exercise 7.9

- Explain, in writing, the steps required to show that 27 blocks would require only three weighings to identify the lightweight block, given that you have a double-pan balance capable of holding as many blocks as you desire on each side and that all the look-alike blocks weigh the same except for the one that is light.
- Given that you have a double-pan balance capable of holding as many blocks as you desire on each side, determine the number of weighings it would take to identify one block that is different in weight, not knowing if the block is light or heavy. Establish a generalization for the problem.
- Given that you have a double-pan balance capable of holding as many blocks as you desire on each side and you have 9,999 blocks where all blocks are the same weight except one is different in weight, determine the number of weighings it would take to identify the one block that is different in weight.

### Exercise 7.10

- The Tower monk story makes a wonderful mathematics problem. Use a set of six disks or the following interactive website at the National Library of Virtual Manipulatives for Interactive Mathematics <URL LINK> (http://www.mathsisfun.com/games/towerofhanoi.html) to solve the puzzle in 63 moves. Then try to do the puzzle in 63 or fewer seconds. How did you do?
- Search the Internet and provide the web address for a different interactive site that allows you to do the Tower of Hanoi puzzle.
- Conduct some investigative research to determine a possible origination of the Tower of Hanoi puzzle. Produce a concise essay that describes the origination and a description of the puzzle for secondary students.

## Problem Solving Challenges

- Observe the five-column array of numbers. In what column will 1,000 appear and, more importantly, why?

Hint: Think about multiples of 4.

## Answer

Answer/solution: Column 2.

Notice that all multiples of eight are in column 2. 1,000 is a multiple of eight, therefore it will be in column 2.

- A school has a hall with 1,000 lockers, all of which are closed. A thousand students start down the hall. The first student opens every locker. The second student closes all lockers that are multiples of two. The third student changes (closes an open locker or opens a closed one) all multiples of three. The fourth student changes all multiples of four. And so on. After all students have entered the school, how many lockers are open and which ones? This is a good problem to solve more than one way.

Hint: Try a smaller problem with only 20 lockers. What do you notice?

## Answer

Answer/solution: 31 lockers will be open.

All perfect squares less than 1,000 will be open: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961.

Try a smaller problem. Imagine only 10 lockers and 10 students. After the first student passes, all locker doors are open. Use O for open and C for closed.

- 1 2 3 4 5 6 7 8 9 10
- O O O O O O O O O O
- The second student closes all lockers that are multiples of two, therefore, lockers 2, 4, 6, 8, and 10 are closed with 1, 3, 5, 7, and 9 remaining open.
- 1 2 3 4 5 6 7 8 9 10
- O C O C O C O C O C
- The third student closes/opens all multiples of three.
- 1 2 3 4 5 6 7 8 9 10
- O C C C O O O C C C
- The fourth student closes/opens all multiples of four.
- 1 2 3 4 5 6 7 8 9 10
- O C C O O O O O C C
- The fifth student closes/opens all multiples of five.
- 1 2 3 4 5 6 7 8 9 10
- O C C O C O O O C O
- The sixth student closes/opens all multiples of six.
- 1 2 3 4 5 6 7 8 9 10
- O C C O C C O O C O
- The seventh student closes/opens all multiples of seven.
- 1 2 3 4 5 6 7 8 9 10
- O C C O C C C O C O
- The eighth student closes/opens all multiples of eight.
- 1 2 3 4 5 6 7 8 9 10
- O C C O C C C C C O
- The ninth student closes/opens all multiples of nine.
- 1 2 3 4 5 6 7 8 9 10
- O C C O C C C C O O
- The tenth student closes/opens all multiples of ten.
- 1 2 3 4 5 6 7 8 9 10
- O C C O C C C C O C
- Which lockers are open?

## Additional Learning Activities

There is no Additional Learning Activities for this chapter.