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.
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.
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 into gender-specific groups (all boys or all girls). Each group needs to have 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. When you use sources, include all in-text citations and references in APA format.
Note: Please submit all graphics and equations in *.pdf (Portable Document Format) files.
Note: When bulleted points are present in the task prompt, the level of detail or support called for in the rubric refers to those bulleted points.
Note: For definitions of terms commonly used in the rubric, see the Rubric Terms web link included in the Evaluation Procedures section.
Note: When using sources to support ideas and elements in a paper or project, the submission MUST include APA formatted in-text citations with a corresponding reference list for any direct quotes or paraphrasing. It is not necessary to list sources that were consulted if they have not been quoted or paraphrased in the text of the paper or project.
Note: No more than a combined total of 30% of a submission can be directly quoted or closely paraphrased from sources, even if cited correctly. For tips on using APA style, please refer to the APA Handout web link included in the APA Guidelines section.