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LOGO Contents Math words quiz Factors-prime factors Multiples-LCM Patterns and sequences SETS!!!

LOGO Math words quiz 10 minutes!!!

LOGO Factors  Factors of a number are the whole numbers that multiply together to give the original number  E.g. The factors of 12 are?  12 is the the original number  So which numbers can multiply together to give 12?  1×12, 2×6, 3×4  That is, 1,2,3,4,6,12 are factors of 12.  We use F(12) as a short way of writing factors of 12.  F(12)={1,2,3,4,6,12}  Factor pairs of 12 are (1,12), (2,6), (3,4)  Among these 6 factors 2 and 3 are prime factors. [Prime factors of a number are factors of the number that are also prime number.]

LOGO Writing numbers as the product of prime factors  Prime factors  2,3,5,7,11,13…  12=4×3,but 4 is not prime number, we break 4 down further 12=2×2×3 that we have written 12 as the product of prime factors

LOGO Step 1 Step 2 Step 3 Try to divide the given number by the first prime number -- 2 Continue until 2 will no longer divide into it Try the next prime number, 3, then 5, 7 and so on, until final answer is 1 Steps

LOGO Examples  Write 60 as a product of prime factors.  Write 3465 as a product of prime factors.  So 3465=3×3×5×7×11

LOGO Several rules  The number is ended by 0,2,4,6,8 can be divided by 2.  The sum of all digits of the number can be divided by 3, that is, the number can be divided by =18 18/3=6 so 3465 can be divided by 3  So does 5, 7, 11 and 13

LOGO Multiples  Definition: the multiples of number are the products of that numbers and 1,2,3,4,5…(Natural number)  E.g. The multiples of 3 are???  3, 6, 9, 12, 15…  The first five multiples of 3: M(3)={3,6,9,12,15}

LOGO LCM-Lowest common multiple  最小公倍数  The smallest number that is a multiple of two or more numbers  12, 24, 36 are multiples of 3 and 4. BUT, 12 is the smallest one, that is, 12 is the LCM of 3 and 4.

LOGO Two ways to find LCM  ONE: ① List the multiples of each numbers of each numbers ② and then pick out the lowest number that appears in every one of the lists. (applicable for small numbers)  ANOTHER: Expressing each number as a product of prime factors

LOGO Sets  Any collection of objects – have sth in common, some connection with each other.  { } braces, comma  The object in a set we called element of the set ∈  5 ways to express sets

LOGO 5 ways 1.Listed set {1,2,3,4.5} 2.Described set {first five natural numbers} 3.Set builder notation to describe sets mathematically {x:x ≦ 10 and x is an even number} 4.Represented by a name or a letter {red, blue, yellow} {Thomas, Joise} 5.Venn diagram

LOGO Special sets  Finite sets and infinite sets  Universal set rectangular Venn diagram It can change from problem to problem  { } empty set

LOGO Relationships between sets  Equal sets: same cardinality and same elements “=“  Equivalent sets: same number of elements  Subsets: A is the subset of set B if all of the elements of A are elements of B A ⊂ B （子集） B ⊃ A (superset 扩散集） In our book ， different from Chinese book How many subsets? Include {} and equal set Use permutation and combination to prove.

LOGO Continue  Complement set A’ contains all of the elements of the universal set not in A. set A and its complement A’ are disjoint- A ∩ A’=empty set  Power set: All subsets of a given set A  If a set has n elements it will have 2^n subsets

LOGO Intersection and union of sets  A ∪ B : the union of sets A and B.  A ∩ B : the intersection of sets A and B. The elements common to set A and B.

LOGO Laws  1. A ∩ A = A  2. A ∩ B = B ∩ A (commutative law)  3. A ∩ B ∩ C = A ∩ (B ∩ C) (associative law)  4. A ∩ φ = φ ∩ A = φ  5. A ∪ (A ∩ B) = A  6. A ∩ (A ∪ B) = A  7. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (distributive law)  8. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (distributive law)

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