# Chapter 8 – Discovery

Discovery learning is a method of indirect instruction. The teacher structures a learning environment that allows students to develop conclusions. Normally, when doing a direct instruction lesson, students know the teacher will eventually “tell the big secret, formula, or conclusion.” Students quickly learn there is no need to work at discovering. Teachers need to provide learning time, appropriate tools, and prompts and expect students to arrive at conclusions. This means teachers should not “tell” the answer.DOWNLOAD

## Exercises

### Exercise 8.1

- Summarize and react to at least three major points raised in one of these publications: Everybody Counts, Curriculum and Evaluation Standards for School Mathematics, Professional Standards for Teaching Mathematics, Assessment Standards, Standards 2000, or Reshaping the Schools. Are the suggestions practical? Do you think they could have been instituted in your high school? Why or why not?
- Complete an annotated bibliography on all documents mentioned in part 1 of this exercise.

### Exercise 8.2

- Use dynamic geometry software to construct a quadrilateral. Experiment by moving the vertices to show that the sum of the exterior angles will always be 360°.
- If the quadrilateral from part 1 is concave, what is the impact on the sum of the exterior angles? How do you rationalize this with students?
- Construct a regular 12-gon (dodecagon) using dynamic geometry software. Show that the sum of the measures of the exterior angles of this 12-gon is 360°. Describe the advantages and disadvantages of increasing the number of sides of the
*n*-gon. What is the impact of this construction on determining for students that a circle is 360° in full rotation?

### Exercise 8.3

- Develop or find a number trick using at least four instructions (remember to give appropriate bibliographical credit). Have someone do the trick and record their reaction.
- Will complex numbers work for your trick? What conditions must be placed on your trick?
- Should calculators be allowed for students when a number trick is presented? Justify your reasoning. Ask two practicing teachers of mathematics the same question. Did their response surprise you?
- Create a number trick where the solution is the same for all participants regardless of the value each participant selects as a starting value. Try this with a group of students. What percentage of students obtained the correct solution? Did this surprise you? Why or why not?

### Exercise 8.4

- Are there sets of values where commutativity of subtraction does exist?
- Does commutativity for multiplication exist in all number systems? Don’t forget to consider the reals, complex, and quaternions.

### Exercise 8.5

- Research howlers. Provide at least two more examples similar to.
- Generalize your discussion of howlers from part 1 of this exercise.

### Exercise 8.6

- Another example similar to the development just done for “reduce” can be made for the expression “Give me a number larger than 1.” What would be encountered as this question is investigated? How could the pitfalls be avoided, or extended?

### Exercise 8.7

- Can twiddle be an arithmetic mean? (
*B = D*in the original definition). - Can twiddle be a harmonic mean? (
*A = C*in the original definition).

## Problem Solving Challenges

1. A camel merchant willed his 17 camels to his three sons. In the merchant’s will, the camels were to be divided among them as follows:

- The eldest son was to receive half of the camels.
- The middle son was to receive a third of the camels.
- The youngest son was to receive a ninth of the camels.

The executor of the merchant’s estate was perplexed. Finally, he devised a method for dividing the camels without having to slaughter any of the animals. How many camels did each son receive? Explain his solution. Hint: What would be the result if you had 18 camels?

## Answer

Answer/solution: The eldest son received 9 camels, the middle son received 6, and the youngest received 2. The executor added one camel to the total giving 18 camels, which can be divided by 2, 3, and 9. Once the will was executed, the camel that had been added was still the property of the executor and not a part of the will.

2. Suppose the NCAA decided to have a single elimination tournament involving all Division 1A teams at the end of the basketball season. If there are 303 Division 1A teams, how many games will be played before a national champion is declared?

Hint: Try a smaller problem with 6 teams. 10 teams.

## Answer

Answer/solution: 302 games to declare a winner. Since this is a single elimination tournament, each loss sends some team home. Out of the 303 teams, there is only one national champion, which means there must be 302 losses or 302 games to declare the winner. You could say the following:

302 teams means 151 games (one team has a bye)

152 teams means 76 games (bye team back in)

76 teams means 38 games

38 teams means 19 games

18 teams means 9 games (one team has a bye)

10 teams means 5 games (bye team back in)

4 teams means 2 games (one team has a bye)

2 teams means 1 game (one team has a bye)

2 teams means 1 game (bye team back in)

TOTAL 302

## Additional Learning Activities

There is no Additional Learning Activities for this chapter.