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**SUBDOMAIN 209.1 - FINITE MATHEMATICS**

**Competency 209.1.3: Number Theory** **-** *The
graduate represents numbers in different forms, recognizes
relationships among numbers and number systems, deduces the meanings of
operations, and demonstrates a conceptual understanding of numbers*.

**Competency 209.2.1: Knowledge of Numbers and Operations -**
The graduate demonstrates computational proficiency, including a
conceptual understanding of numbers, ways of representing numbers,
relationships among numbers and number systems, and meanings of
operations.

**Introduction: **

There are many interesting concepts associated with number theory in exploring real-world applications.

In this task you will solve and explain problems that focus on
rounding, truncation, primes, composites, and modular operations. Title
and organize your paper in outline form in **three** parts. Use appropriate APA style and referencing as needed.

**Task:**

A. Rounding and
Truncation: In a classroom, students will receive a letter grade based
on the percentage of points gained in the term out of the total points
possible. There are 334 points possible. To get an A in the class, the
student must have a percentage that, when properly rounded to a whole
number, is at least 90%.

1. Determine whether the
teacher will give Student 1 an A for the class if the student has
earned 299 points, justifying your answer.

2. Use your answer in
part A1 to explain whether Student 1 will receive an A for the class if
the teacher truncates the percentage to a whole number.

3. Explain the following (*suggested length of 1–2 pages*) as if you were teaching a middle school mathematics classroom (grades 5–9):

a. Why a taxpayer whose
income tax rate is 27.8% would hope that the rate could be truncated to a
whole number when calculating the amount of tax owed on the tax form

1. Why the government prefers and requires the taxpayer to round the tax rate

b. Use your mental math
process in calculating the above situation and how you would use
rounding and truncating in real-world scenarios

1. Provide **two** examples of *each* rounding and truncation (four total examples) to illustrate mental math skills.

B. Primes and
Composites: There are 20 boys and 24 girls in an Algebra I class. The
class is so large that the teacher wants to divide the students by
gender into cooperative groups composed of the same number of students.

1. Explain the process the teacher will use to determine how many students will be in *each* group using appropriate mathematical terms from number theory.

2. Determine the largest number of students that can be placed in a group, showing all work.

3. Given your answer in
B2, determine how many groups will be created from the Algebra I class,
providing support (i.e., showing all work or, if you use mental math,
explaining your answer).

4. Explain how to prove that there is an infinite number of primes.

C. Modular Operations:
You want to explain the concept of modular operations to a middle school
mathematics classroom, starting with a demonstration from clock
arithmetic. Your explanation should include the following support:

• Appropriate examples for modular addition using positive integers

• Appropriate examples for modular addition using negative integers

• Appropriate examples for modular multiplication using positive integers

• Appropriate examples for modular multiplication using negative integers

• A sentence or two about the use of modular operations in real-world scenarios

1. Explain how you would discuss the following (*suggested length of 1–2 pages*) in the classroom setting:

a. Modular arithmetic and its relation to time

b. What is meant by 10 mod 6

c. How to add or multiply in mod 7

D. If you use sources, include all in-text citations and references in APA format.