# Chapter 10 – Foundations of Mathematics

Knowing where to begin this chapter and what to include is difficult. There are so many topics to be covered. What is the best order? What background information will students possess? Should topics be integrated into this discussion? You are expected to blend topics as you teach by relating one to the other. Perhaps this discussion should follow an integrated format. Still, if that is done, continuity will be next to impossible. This comment brings to mind the story of an individual wanting to borrow a chainsaw. The request was made, and the response was “No.” Rather stunned, the individual followed with “Why not?” “My dog is sick” was the reply. “What does your dog being sick have to do with my borrowing your chainsaw?” was the next question. The response was “Nothing, but I don’t want to lend it to you, and one excuse is as good as another.” The moral of that story is that no matter what order is selected, others could serve equally well.DOWNLOAD

## Exercises

### Exercise 10.1

- Reflect on your experiences with courses taken prior to Algebra I. Describe how the NCTM suggested changes would alter the mathematics education you received in Grades 5 – 8.
- After reading the introduction to
*A Vision for School Mathematics*(NCTM, 2000, p. 3), select one (and only one) sentence that stands out to you. Why did you select that sentence?

### Exercise 10.2

- Have the Common Core Standards for Mathematics been implemented in your state or country? If so, in which grades?
- Rank the Standards for Mathematical Practice in order of importance to you. Should these standards be in a priority order for teachers?
- Interview a current math teacher. Ask about the teacher’s perception of the Common Core Standards for the content he/she is teaching.

### Exercise 10.3

- Select a topic, like “What percent of 8 is 5?” and research how to teach it and all the prerequisite experiences the student should have prior to broaching this subject. Describe the skills you would expect the student to have prior to the study of the topic and the method you would use to introduce it.
- Find a real-world application of the topic you select for part 1 of this exercise. Describe the application and how you would relate it to students.

### Exercise 10.4

- Do The Big 20 listed here. Calculators are not permitted. Do not study the problems prior to attempting to do this assignment. Your objective is to do it in two minutes or less and miss none.
- Get a friend to try The Big 20. Describe to them what they will be working as they do The Big 20 and then time them as they do it. Discuss their emotions as they did The Big 20. What did you learn that could be used as you teach mathematics?

### Exercise 10.5

- Describe how egg cartons would be used to show $$\frac{4}{5}+\frac{2}{3}$$.
- Use egg cartons to describe $$\frac{4}{5}-\frac{2}{3}$$.
- Use egg cartons to demonstrate that $$\frac{1}{2}=\frac{2}{4}=\frac{4}{8}=\frac{8}{16}$$. Explain the result.

### Exercise 10.6

- Assume fractions with denominators of 3 and 5 are to be added. Describe a process to determine the unit rod that would permit expression of thirds and fifths at the same time, which could be used with students.
- Do each of the following problems using Cuisenaire rods.
- Show $$1\frac{2}{3}=\frac{5}{3}$$
- Show $$\frac{2}{3}=\frac{8}{12}$$
- $$\frac{2}{3}+\frac{3}{4}=$$
- $$\frac{13}{12}-\frac{2}{3}=$$
- $$\frac{2}{3}\xf7\frac{3}{4}=$$
- $$\frac{3}{4}\xf7\frac{2}{3}=$$

Note: Parts e and f are very difficult to rationalize and yet, if you work through them, your understanding of division of fractions will increase significantly.

### Exercise 10.7

- Look at a minimum of three secondary texts that deal with the teaching of addition of fractions. Compare the sequence of presentation of problem types with Figure 10.11 in the text. Is the text sequence laid out well? Is consideration given to avoiding issues like dividing out common factors, conversion of improper fractions to mixed numbers, and so on? Would you alter the sequence? Why or why not? Is the sequence adequate as presented? Why or why not?
- List the readiness skills necessary for a student to understand the explanation for doing a problem like $$\frac{3}{4}-\frac{2}{7}$$.
- Find a secondary text that uses manipulatives to introduce division of fractions. Describe the presentation. Do you think students would learn from that explanation (assuming the teacher followed the pattern and supplemented it as needed)? Why or why not?

### Exercise 10.8

- The last few paragraphs contained points for rationalizing the use or nonuse of a calculator in the curriculum. Present a defense for both sides of the issue. Where do you stand, and why?

### Exercise 10.9

- Create an addition problem involving three addends with units, tenths, and hundredths in each of them so that each column requires regrouping. Show through a series of steps using F, L, and S how the regrouping from each place would be accomplished, being careful to adequately show the trades as they are made, doing one exchange at a time. Relate each step to an abstract representation of the same problem.
- Create a decimal subtraction problem that requires regrouping in both the tenths and hundredths places. Show through a series of steps using F, L, and S how the regrouping from each place would be accomplished, being careful to adequately show the trades as they are made, doing one exchange at a time. Relate each step to an abstract representation of the same problem.
- Develop a position paper on whether or not there is value to spending so much time in the curriculum teaching students to deal manually with decimals in light of the existence of inexpensive technology that deals with them so easily.

### Exercise 10.10

- Investigate perfect numbers. In the process, answer questions like the ones that follow. You should not limit your research to answering the listed questions. What are the next two perfect numbers after 28? How many perfect numbers have been found to date? Has it been established that all perfect numbers have been found?
- The following are listed as abundant numbers: 12, 18, 24, and 36. Are all abundant numbers multiples of 6? Is there a multiple of 6 that is not an abundant number? Are there abundant numbers that are not multiples of 6? List examples and explain your position.
- Prime numbers are deficient numbers. Is there a pattern for nonprime deficient numbers? List examples and explain your position.

### Exercise 10.11

- It is possible to extend the sieve of Eratosthenes so there will be 100 consecutive composites. Where does that occur?
- Excluding the first row, will there be another row in a 10-column sieve of Eratosthenes that will have four primes in it?
- Create a 6-column sieve as opposed to a 10-column sieve like that attributed to Eratosthenes. Define the generalizations students would be expected to make using this new creation. Describe the benefits and disadvantages to presenting this 6-column sieve to a class of secondary students.

### Exercise 10.12

- Write a summary of at least three instances where art could be inserted into the mathematics curriculum. For each of the examples, describe the mathematical application.
- Create a design that will tessellate the plane. Your design should not be a regular polygon.
- Many mathematicians choose between art or music and mathematics as their career. Describe the background of one such individual.

### Exercise 10.13

- Consider Figure 10.20 in the text to be a body of water containing two islands. How many colors would be needed to color Figure 10.20 in the text with the restriction that no two edges share the same color?
- Research four-color maps. Has the idea ever been clearly proven?

### Exercise 9.14

- List at least three areas of geometry where you see possibilities for students to become confused. In each instance explain the source of confusion and describe how you would clarify it for students.
- Use textbooks or objectives to determine the major geometric concepts covered in a mathematics course prior to Algebra I. Discuss what topics, if any, should be eliminated from, or added to, the list, including a rationalization for each entry.
- Many geometric topics are introduced in prior grades. Still, they are revisited in successive work, often with no elaborations or extensions. How can we, as professionals who should teach a concept right the first time, justify this repeated visiting of an idea with no alterations to our approach?

### Exercise 10.15

- Should the curriculum of courses prior to Algebra I be changed to reflect items discussed in this text, NCTM publications, and the Common Core Standards? Why or why not?
- Defend or take issue with the statement, “General mathematics is foundational work for future mathematical study. There is no need to connect general mathematics curriculum with the real world of students.”

## Problem Solving Challenges

1. A camel merchant willed his 17 camels to hYou have a digital clock that shows only hours and minutes. How many different readings between 11:00 a.m. and 5:00 p.m. (of the same day) contain at least two 2s in the time?

**Hint: **Try a smaller problem (less hours)

## Answer

Answer/solution: 34.

There is one such reading between 11:00 and 12:00, 1:00 and 2:00, 3:00 and 4:00, and 4:00 and 5:00 each of which being 22 minutes past the hour. There are 15 such readings between each of 12:00 and 1:00 and 2:00 and 3:00. 12:2X accounts for 10 of them, while 12:02, 12:12, 12:32, 12:42, and 12:52 account for five. 2:2X accounts for 10 of them, while 2:02, 2:12, 2:32, 2:42, and 2:52 account for five as well. The total reading is 1 + 1 + 1 + 1 + 10 + 5 + 10 + 5 = 34.

- Start with a square piece of paper. Draw the largest circle possible inside the square, cut it out and discard the trimmings. Draw the largest square possible inside the circle, cut the square out and discard the trimmings. What fraction of the original square piece of paper has been cut off and thrown away?

Hint: Try the problem with scissors and paper? What do you notice?

## Answer

Answer/solution: Half the area.

Try it yourself. The following diagram shows the results. B is the midpoint of AC, and D is the midpoint of CE. Since BG is congruent to CD and DG is congruent to BC, triangle BCD is congruent to BGD (they also share side BD) by SSS. Therefore the cut-away portion (triangle BCD) of square BCDG is half of the square. This is the same for each portion of the original square. See Figure 9.2 below.

## Additional Learning Activities

There is no Additional Learning Activities for this chapter.