Math 67 Unit 2: Fun with Numbers
OVERVIEW
In this unit, students gain a deeper understanding of concepts and applications of number theory.
In Grade 5 Mathematics, students studied classification of counting numbers into subsets with distinguishing characteristics such as odd and even numbers and prime and composite numbers.
Students also developed a strong foundation for understanding and applying multiples and factors. They will extend this concept to include greatest common factors and least common multiples which in turn will build a deep understanding of the Fundamental Theorem of
Arithmetic. This unit is a building block to students’ deeper understanding of rational numbers which students will study extensively in unit 3 of the 6 th
grade mathematics framework.
Instruction should include the representation of these numbers and their relationships to other concepts, such as multiplication and division using diagrams, charts, tables, multiple number lines, and explanations. Number theory is a topic that begs for students to reason, discuss, make sense of and justify their thinking. This can be accomplished by students playing games that are based on number theory, working and debating with their peers and in sharing ideas through a teacher-facilitated whole class discussion. Students also should be provided with opportunities for revisions. In order to demonstrate mastery of the learning in this unit, students will explain the Fundamental Theorem of Arithmetic to a friend who has been absent for the unit and solve a puzzle involving factors, multiples and prime numbers.
By the conclusion of this unit, students should be able to demonstrate the following competencies:

Students should be able to find all the factors of a number by constructing or drawing arrays, thus “proving” the presence of 1 as a factor of all numbers.

Students should be able to list all the factors of any given number and discuss how they know that the number is prime, composite or neither (the number 1 is neither prime nor composite).

Students should be able to determine the greatest common factor of two or more numbers and offer situations in which it would be useful to know common and greatest common factors of two or more numbers.
ENDURING UNDERSTANDINGS:

Factors and multiples are related in ways that are similar to the way that multiplication and division are related.

All natural numbers greater than one are either prime or can be written as a unique product of prime factors.

The number 1 (one) is always a factor of any number.
ESSENTIAL QUESTIONS:

When or why would it be useful to know the factors of a number?

When or why would it be useful to know the multiples of a number?

What features does a number have if the number is prime?

What role does the number 1 have when you are finding factors of any number?

How can I use the array model to represent factors of a number?
Burch/Chen VMMS Math 67 Unit 2 Fun with Numbers Resources GA DOE Math Frameworks

Math 67 Unit 2: Fun with Numbers

How applicable are multiples in everyday life?

How applicable are factors in everyday life?

How can I use models to represent multiples of a number?
Mathematics standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical ideas.
:
M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems. a.
Apply factors and multiples. b.
Decompose numbers into their prime factorization (Fundamental Theorem of
Arithmetic). c.
Determine the greatest common factor (GCF) and the least common multiple (LCM) for a set of numbers.
M6P1. Students will solve problems (using appropriate technology). a.
Build new mathematical knowledge through problem solving. b.
Solve problems that arise in mathematics and in other contexts. c.
Apply and adapt a variety of appropriate strategies to solve problems. d.
Monitor and reflect on the process of mathematical problem solving.
M6P2. Students will reason and evaluate mathematical arguments. c.
Develop and evaluate mathematical arguments and proofs. d.
Select and use various types of reasoning and methods of proof.
M6P3. Students will communicate mathematically. a.
Organize and consolidate their mathematical thinking through communication. b.
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c.
Use the language of mathematics to express mathematical ideas precisely.
M6P4. Students will make connections among mathematical ideas and to other disciplines. a.
Recognize and use connections among mathematical ideas. b.
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. c.
Recognize and apply mathematics in contexts outside of mathematics.
M6P5. Students will represent mathematics in multiple ways. a.
Create and use representations to organize, record, and communicate mathematical ideas.
Burch/Chen VMMS Math 67 Unit 2 Fun with Numbers Resources GA DOE Math Frameworks

Math 67 Unit 2: Fun with Numbers b.
Select, apply, and translate among mathematical representations to solve problems. c.
Use representations to model and interpret physical, social, and mathematical phenomena.
SELECTED TERMS AND SYMBOLS:
The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

Arrays: rectangular arrangements that have equal numbers in the rows and columns.

Decompose: The process of factoring terms and numbers in an expression.

Exponent: The number of times a number or expression (called base) is used as a factor of repeated multiplication. Also called the power.

Factor: When two or more integers are multiplied, each number is a factor of the product.
"To factor" means to write the number or term as a product of its factors.

Fundamental Theorem of Arithmetic: Every integer, N > 1, is either prime or can be uniquely written as a product of primes.
 GCF: Greatest Common Factor: The largest factor that two or more numbers have in common.

Identity property of multiplication: A number that can be multiplied by any second number without changing the second number. The Identity for multiplication is “1”.
 LCM: Least Common Multiple: The smallest multiple (other than zero) that two or more numbers have in common.

Multiple: A number that is a product of a given whole number and another whole number.

Prime factorization: The expression of a composite number as a product of prime numbers.
 Prime number: A positive number that is divisible only by itself and the number one.

Square number: A number that is the product of a whole number and itself. This is also known as a “Perfect Square”.

Composite number: A composite number is a number that has factors in addition to one and itself.
Burch/Chen VMMS Math 67 Unit 2 Fun with Numbers Resources GA DOE Math Frameworks

Math 67 Unit 2: Fun with Numbers
The collection of the tasks we will complete with this unit represents the level of depth, rigor and complexity expected of all sixth grade students and will include seventh grade standards for the accelerated learner. These tasks or tasks of similar depth and rigor should be used to demonstrate evidence of learning.
Burch/Chen VMMS Math 67 Unit 2 Fun with Numbers Resources GA DOE Math Frameworks