REFLECTIONS ON NUMBER

GOALS

1. Find the factors and multiples of a number (including prime factors).

2. Write composite numbers as products of prime numbers.

3. Find a product or quotient of two- and three-digit numbers.

4. Understand that division by zero is undefined.

5. Understand how multiplication and division algorithms work.

6. Understand the relationship between operations and their inverse operations.

7. Develop number sense by operating on numbers in a convenient way.

8. Begin to understand the structure of the real number system, including rational and irrational #’s.

SECTION SUMMARIES

SECTION A: FACTORS AND DIVISORS

1. Pascal’s Triangle can be used to find the total number of codes..

2. Factors of a number can be represented by different rectangular patterns and by a graph. Factor graphs do not have the same number of points for all numbers. Prime numbers will only have two points. Perfect squares will have an odd number of points. Composite numbers will have more than two points. One is not a prime number.

3. Prime numbers have exactly two different factors; composite #’s have more than two factors.

SECTION B: USING PRIME FACTORS

1. Upside-down trees (factor trees) can be used to factor composite numbers.

2. The end numbers of the trees are prime numbers; this is called the number’s prime factorization.

3. Every number can be written as a product of prime numbers (prime factorization) and each number has its own unique set of prime number factors. This is called the Fundamental Theorem of Arithmetic.

SECTION C: INVESTIGATING ALGORITHMS

1. An algorithm is a predetermined set of rules used to perform computations. (+,-,*,/)

2. Dividing by zero is undefined – it has no meaning

3. Multiplying by zero always gives the result of zero. (Zero Property)

4. Multiplying sometimes makes numbers smaller. (if multiply by a number between 0 and 1)

SECTION D: SQUARE AND NOT SO SQUARE NUMBERS

1. Squaring a number and taking the square root of a number are inverse operations. (undo each other)

2. Only perfect square numbers will give you an exact square root, and thus be rational numbers – all others will be irrational as they can’t be written as fractions since they are approximations.

SECTION E: FINAL REFLECTIONS

1. You found that when some whole numbers are subtracted from others, the results are negative numbers, so you must use integers.

2. When some whole numbers are divided, we get fractions; thus, to divide whole numbers, you must use rational numbers.

3. Numbers that cannot be written as whole numbers or fractions are called irrational numbers.