Subtracting Fractions and Mixed Numbers
I. Subtracting Fractions With the Same Denominator
To subtract fractions with the same denominator:
a. Subtract the numerators.
b. Place the difference over the common denominator.
Least Common Multiple
Objective: Find the Least Common Multiple (LCM) of two or more numbers
The Least Common Multiple (LCM) of two or more numbers is the smallest number which is a
multiple of all the numbers being considered.
Multiplying and Dividing Fractions
1. The product of two or more fractions is the product of the numerators over the product of
2. Prime factorization will allow us to reduce fractions before multiplying so that the final
Introduction to Fractions
Figure A (Use for 15)
1. How many parts are there in this circle?
2. How many parts of the circle are shaded?
3. What fractional part of the circle is shaded?
b. It tells the number of parts that_.
c. The denominator o
Adding Fractions and Mixed Numbers
I. Adding Fractions with the Same Denominator
2 3 5
6 6 6
REMEMBER the denominator tells how many parts are in one rectangle. It tells us
something about the size of the parts. The numerator tells h
Multiplying Fractions and Mixed Numbers
of 6 is 3. When you get
of 6 objects, you make 2 (the denominator)
equal groups of the objects and you take 1 (the numerator) of those groups.
You know that
2 equal objects
one of the groups
Greatest Common Factor
Objective: Find the Greatest Common Factor (GCF) of two or more numbers
1. The Greatest Common Factor (GCF) of two or more numbers is the biggest number which is a
factor of all of the numbers being considered.
Dividing Fractions and Mixed Numbers
6 3 is asking, How many 3s are in 6?
one group of 3
a second group of 3
We see that there are 2 groups of 3 in 6.
6 asks "How many 's are in 6?"
Each item halved
We see that there are 12 one-half size o
Adding and Subtracting Rational Numbers
A rational number is a number that can be written as the ratio of two integers. A rational number
can also be written as a decimal.
In your class, you studied addition and subtraction of fractions. You also learned
Objective: Find the prime factorization of a natural number
Finding the prime factorization of a number means rewriting the number as a multiplication
that uses only prime numbers as factors.
Prime numbers are nu
Factoring Trinomials in the Form of ax2 + bx + c
This kind of trinomial differs from the previous kind we have factored because the
coefficient of x is no longer "1".
EXAMPLES of trinomials in the form of ax 2 + bx + c :
6a2 + 7a 24
a = 6 b = 7 c = 24
In this lab assignment we will look at two special kinds of factoring. One kind is the difference of two
squares, which is the product of the sum and difference of two terms.
a5 b5 = (a + b)(a b)
x5 4 = (x + 2)(x 2)
The other k
Factoring Trinomials in the Form of ax2 + bx + c by Grouping
The method of factoring by grouping can be used as an alternative to factoring trinomials by
trial and error. Let us recall the procedure for multiplying binomials using FOIL:
( x + 2
Factoring by Grouping
Factors are expressions joined by multiplication. In the expression 3x, the factors are 3 and x. In
the expression 4(x + 2), the factors are 4 and (x + 2). The factor (x + 2) is called a binomial
factor since it has two terms. In the
factor out the Greatest Common Factor (GCF)
factor by grouping (see example below)
x2 + bx + c .
ax2 + bx + c .
P2 2PQ + Q 2
x 2 y 2 = (x+ y)(x- y)
x2 + y2
Find factors of c t
LEAST COMMON MULTIPLES
Multiples of 4 are the_of 4 and the numbers
1, 2, 3, 4, 5 . (NOTICE 0 x 4 is not considered.)
Find the first TEN multiples of 4._
What is the smallest multiple of 4?_
Explain why there is not a largest mu
FACTORING TRINOMIALS IN THE FORM OF x2 + bx + c
In this standard form of a trinomial, where the coefficient of x is 1, x is a variable and b and c stand for
+ bx + c
x2 + 5 x + 6
b = 5,
y2 3 y + 2
b = 3,
b = 3,
c = 40
+ 3 a 40
GREATEST COMMON FACTORS
Factors of 12 are also the _ of 12.
Name all of the factors of 12_
Name all of the factors of 18_
What factors are in both lists?_
(These are the common factors of 12 and 18.)
What is the smallest
PRIME NUMBERS AND PRIME FACTORIZATION
Divisors and Factors of a Number
Previously, you learned the names of the parts of a multiplication
6 2 = 12
6 and 2 are the_
12 is the_
You learned the names of the parts of a division problem fr
Factoring means that we will be starting with a product and deciding
what was multiplied together in order to get that product.
polynomials, we will first want to see whether there are any common factors.
Let us look at some exa
Y = mx + b Word Problems
1. Suppose that the water level of a river is 34 feet and that it is receding at a
rate of 0.5 foot per day. Write an equation for the water level, L, after d days. In
how many days will the water level be 26 feet?
2. Seths father
We'll study this one in 4.2.
For any function to act as a
"model" for physical
phenomena, we must place
limits on the domain of the
First we did a little review on the
chalkboard. I'm not sure exactly
what the "numbers" w