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Key Words/Topic
and Assignments
2.1 Identity and
Zero Properties
Topic 2 Guided Notes
Equivalent Expressions
Information, Definitions, Solutions
New Terms
Identity Property of
Addition The sum of 0 and any number is that number. 0 + 4 = 4
Identity Property of
Multiplication The product of 1 and any number is that number. 4 x 1 = 4
Zero Property of
Multiplication The product of any number and zero is zero; 4 x 0 = 0
Review Terms
Equivalent Expressions that always have the same value.
Expressions
Today’s Concept Mathematical properties are always true. Today you will learn 3
properties.
The Identity Property of Addition states that
For any value a; a + 0 = a
so 3 + 0 = 3
The Identity Property of Multiplication states that
For any value a; a • 1 = a
so 6 • 1 = 6
The Zero Property of Multiplication
For any value a; a • 0 = 0
so 5 • 0 = 0
Group Work
Homework
Key Words/Topic
and Assignments
2.2
Information, Definitions, Solutions
The
Commutative
Properties
New Terms
Commutative
Property of Addition You can add numbers in any order. For any numbers a + b = b + a
(ex. 3 + 4 = 4 + 3)
Commutative You can multiply numbers in any order. For any numbers a•b = b•a
Property of (ex 3 • 4 = 4 • 3)
Multiplication
Addends Numbers that are added together to find a sum. (ex. In 3 + 4 = 7, 3
and 4 are addends.)
Factors Numbers that are multiplied to give a product. (ex. In 3 • 4 = 12, 3
and 4 are factors.)
Review Terms
Today’s Concept The Commutative Property of Addition states that the order in
which we add addends doesn’t matter.
a+b=b+a
so 3 + 4 = 4 + 3
The Commutative Property of Multiplication states that the order
in which we multiply factors doesn’t matter.
a•b=b•a
so 2 • 6 = 6 • 2
You can use the commutative properties of addition and
multiplication to add, or multiply, more than two numbers together.
For example 3 + 4 + 5 = 4 + 3 + 5
and 4 • 5 • 6 = 6 • 4 • 5
Group Work
Homework
Key Words/Topic
and Assignments
2.3
Information, Definitions, Solutions
The
Associative
Properties
New Terms
Associative Property The way in which addends are grouped doesn't change the sum. {ex.
of Addition 3 + (4 + 5) = (3 + 4) + 5}
Associative Property The way in which factors are grouped doesn't change the product.
of Multiplication {ex. 3(4 • 5) = (3 • 4)5}
Review Terms
Today’s Concept The Associative Property of Addition states that the way in which
addends are grouped doesn’t matter.
a + (b + c) = (a + b) + c
so 2 + (3 + 4) = (2 + 3) + 4
The Associative Property of Multiplication states that the way in
which factors are grouped doesn’t matter.
a(b • c) = (a • b) • c
so 2(6 • 3) = (2 • 6)3
Group Work
Homework
Key Words/Topic
and Assignments
2.4
Greatest
Common Factor
Information, Definitions, Solutions
New Terms
Common Factor A number which is a factor of two or more given numbers. (ex. 3 is
a common factor of 6 and 9.)
Greatest Common The GCF of two or more whole numbers is the greatest number that
Factor is a factor of all of the numbers. (ex. The GCF of 12 and 10 is 2.)
Prime Number A whole number greater than 1 with exactly two factors, 1 and the
number itself.
Composite Number A whole number greater than 1 with more than 2 factors.
Prime Factorization The prime factorization of a composite number is the expression of
the number as a product of its prime factors. (ex. 30 = 2•3•5)
Review Terms
Factors Numbers that are multiplied to give a product. (ex. In 3 • 4 = 12, 3
and 4 are factors.)
Today’s Concept When two, or more, numbers have the same factor, the factors are
called common factors. Common factors are very helpful in math.
The greatest (largest) common factor between the numbers is
called the greatest common factor (GCF).
There are different methods to find the GCF. We will look at three
methods.
Method 1: List all of the factors of a number using factor pairs.
Compare the factors of the numbers and find the greatest common
factor.
What is the GCF of 18 & 24?
1. List the factors of 12 and 24. Factor pairs can help us.
Factor pairs for 18: 1,18; 2,9; 3;6
Group Work
Homework
Factors for 18: 1, 2, 3, 6, 9, 18
Factor pairs for 24: 1,24; 2;12; 3;8; 4;6
Factors for 24: 1, 2, 3, 4, 6, 8, 12, 24
2.4
Greatest Pick out the largest common factor between the two numbers: 6 is
Common Factor the GCF.
Continued
Method 2: For the 2nd method, you need to know how to prime
factor numbers. In math we often have to “break” numbers down
into their simplest parts (factors). The simplest factors of all are
prime factors. There are a number of prime factorization
methods. One of the most common methods is the tree method.
How can we use the tree method to find out the prime factorization
of 72?
1. Put 72 at the top of the tree.
2. Use your divisibility rules to break 72 into factor pairs.
3. If one of the factors is a prime number, then continue to bring
the “branch” down to the “ground”.
4. Any numbers that are still composite numbers and not prime,
need to be broken down into factor pairs.
5. Once all of your numbers are prime, you are done!
6. Then rewrite the prime factors into one row (factor string).
The factor tree on the last page is not the only way to break 72 down
into its primes. How you break a number down may vary, but the
result should be the same.
2.4
Greatest
Common Factor
Continued
1. Factor the numbers into prime factorization.
2. Rewrite both factor strings on “top” of each other.
2*3*3
2*2*2*3
3. Circle any common factors in both strings, picking one number
from each string per pair.
2*3*3
2*2*2*3
4. Pick JUST ONE number from each circle and rewrite them
as a multiplication problem. Multiply the numbers together and
that is the GCF.
2*3=6 GCF
Method 3: Upside Down Birthday Cake or Cake Method
1. Write the factors inside the 1st layer of your upside down cake.
2. Ask yourself what common factor does 24 & 18 have? 2; write
the factor on the side of your cake layer. Divide both numbers
inside of your cake by the factor outside of your cake. Write the
quotients in your next layer.
3. Repeat the process until the numbers at the bottom don’t have
any common factors. Multiply the numbers on the outside of the
cake and you get 6!
Key Words/Topic
and Assignments
2.5
The
Distributive Property
Information, Definitions, Solutions
New Terms
Distributive Property Multiplying a number by a sum, or difference, gives the same result
as multiplying that number by each term in the sum, or difference,
and then adding or subtracting the corresponding products. a(b + c)
= a • b + a • c and a(b - c) = a • b – a • c so 36(14 + 85) = (36 • 14)
+ (36 • 85)
Review Terms
Greatest Common The GCF of two or more whole numbers is the greatest number that
Factor is a factor of all of the numbers. (ex. The GCF of 12 and 10 is 2.)
Today’s Concept When we distribute something it is like spreading it out, or giving it
to other numbers in the problem.
a(b + c) = a • b + a • c we “give” the a to the b and the c.
3(2 + 4) = 3 • 2 + 3 • 4
3 • 6 = 6 + 12
18=18
or
a(b - c) = a • b - a • c we “give” the a to the b and the c.
3(4 – 2) = 3 • 4 – 3 • 2
3 • 2 = 12 – 6
6=6
Group Work
We can use the distributive property to “break up” or “join”
numbers and make some problems easier to solve.
Homework
Break it up; 6 • 37 = 6 • 30 + 6 • 7
Join; 6 • 31 + 6 • 9 = 6 • 40
We can also use common factors to make equivalent expressions
using the distributive property. Take the expression 24 – 16. The
GCF for 24 and 16 is 4. We can set 24 – 16 = 4(6 – 4) by factoring
out the GCF.
The distributive property is essential to mastering algebra and we
can use the distributive property with algebraic expressions. Let’s
take the algebraic expression 25x + 15. 5 is a common factor for
2.5
The both terms. If we factor out the 5 from both terms, we get 5(5x + 3).
Distributive Property
Continued If we start with the algebraic expression 12(42x – 3), we can
distribute (or give) the 12 to each term in the parentheses and end up
with 504x – 36.
Key Words/Topic
and Assignments
2.6
Information, Definitions, Solutions
Least Common
Multiple
New Terms
Multiple The product of the number and a whole number. (ex. Multiples of 3
are 3, 6, 9, 12...)
Common Multiple A multiple that two or more numbers share. (ex. 12 and 24 are
common multiples of 4 and 6)
Least Common The LCM of two or more numbers is the least (lowest value)
Multiple multiple shared by all of the numbers. (ex. The LCM of 4 and 6 is
12.)
Review Terms
Today’s Concept Like finding the GCF, there are numerous methods for finding the
Least Common Multiple.
Method 1: List the common multiples of the numbers until you
find the smallest match.
Although 24 & 48 are both common multiples, we need to pick the
smallest value – 24 = LCM.
Method 2: Use factor trees to find the prime factorization.
Once you find the prime factorization, write the factor strings of the
numbers.
Group Work
Homework
Circle the greatest number of times a factor appears in the
different strings. Then multiply the circled factors together. The
product is the LCM.
Method 3: You can also use the CAKE method as long as you are
using it for just two numbers. It is possible to use it for more than
2.6
Least Common two numbers, but you must know the exception.
Multiple
Continued Once you reach the point where there are no common factors
between the two numbers, you form an “L” around the outside
numbers and the lowest level of the cake. Then multiply all of the
numbers inside the “L”. This gives you the LCM.
2x3x4=24