Name Key Words/Topic 2.1 Identity and Zero Properties Topic 2 Guided Notes Equivalent Expressions Identity Property of Addition Identity Property of Multiplication Zero Property of Multiplication The sum of 0 and any number is that number. 0 + 4 = 4 The product of 1 and any number is that number. 4 x 1 = 4 The product of any number and zero is zero; 4 x 0 = 0 Equivalent Expressions Expressions that always have the same value. 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
Key Words/Topic 2.2 The Commutative Properties Commutative Property of Addition Commutative Property of Multiplication Addends Factors You can add numbers in any order. For any numbers a + b = b + a (ex. 3 + 4 = 4 + 3) You can multiply numbers in any order. For any numbers a b = b a (ex 3 4 = 4 3) Numbers that are added together to find a sum. (ex. In 3 + 4 = 7, 3 and 4 are addends.) Numbers that are multiplied to give a product. (ex. In 3 4 = 12, 3 and 4 are factors.) 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
Key Words/Topic 2.3 The Associative Properties Associative Property of Addition Associative Property of Multiplication The way in which addends are grouped doesn't change the sum. {ex. 3 + (4 + 5) = (3 + 4) + 5} The way in which factors are grouped doesn't change the product. {ex. 3(4 5) = (3 4)5} 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
Key Words/Topic 2.4 Greatest Common Factor Common Factor Greatest Common Factor Prime Number Composite Number Prime Factorization Factors A number which is a factor of two or more given numbers. (ex. 3 is a common factor of 6 and 9.) The GCF of two or more whole numbers is the greatest number that is a factor of all of the numbers. (ex. The GCF of 12 and 10 is 2.) A whole number greater than 1 with exactly two factors, 1 and the number itself. A whole number greater than 1 with more than 2 factors. 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) Numbers that are multiplied to give a product. (ex. In 3 4 = 12, 3 and 4 are factors.) 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 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 Common Factor Continued Pick out the largest common factor between the two numbers: 6 is the GCF. Method 2: For the 2 nd 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 1 st 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 2.5 The Distributive Property Distributive Property Greatest Common Factor 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) The GCF of two or more whole numbers is the greatest number that is a factor of all of the numbers. (ex. The GCF of 12 and 10 is 2.) 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 We can use the distributive property to break up or join numbers and make some problems easier to solve. 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
2.5 The Distributive Property Continued take the algebraic expression 25x + 15. 5 is a common factor for both terms. If we factor out the 5 from both terms, we get 5(5x + 3). 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 2.6 Least Common Multiple Multiple Common Multiple Least Common Multiple The product of the number and a whole number. (ex. Multiples of 3 are 3, 6, 9, 12...) A multiple that two or more numbers share. (ex. 12 and 24 are common multiples of 4 and 6) The LCM of two or more numbers is the least (lowest value) multiple shared by all of the numbers. (ex. The LCM of 4 and 6 is 12.) 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. 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 Multiple Continued two numbers, but you must know the exception. 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