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As Graphs: Horizontal- inputs Vertical- outputs
As Tables and Ordered Pairs: 0 1 1 3 2 5 3 7 4 9 (0,1), (1,3), (2,5), (3,7), (4,9)
One input cannot have more than one output. X cannot be the same as another x, but y can be the same as another y
Function? (1,2), (1,3), (2,3), (3,4) NO, 1 INPUT is used twice.
As Arrow Diagrams: Used to examine whether a correspondence represents a function. Domain Range 0 1 1 4 2 7 3 10 Yes, a function 1 2 2 4 3 No, since element 1 is paired with 2&4
As Equations: f(0)= 0+3=3 f(1)= 1+3=4 f(3)= 3+3=6
As Machines: x f(x) 0 3 1 4 3 6 4 7 6 9
As Rules: 1 3 0 0 4 12 Rule= nx3
Always multiply your quotient times the divisor to get the dividend to check your answer.
Instruction for "long division": Divide Multiply Subtract Check Break Down
"Does McDonalds Sell Cheese Burgers?" DMSCB
Multiplication with Two-Digit Factors: 2 3 x 1 4 (10+4) ________ 9 2 (4X23) + 2 3 0 (10X23) ___________ 3 2 2
Single Digit times two digit: 1 2 10+2 x 4 ---> x 4 _____ ______ 4 8 40 + 8
n divided by 0 is undefined 0 divided by n is 0 0 divided by 0 is undefined
Division is the inverse of multiplication
a=bq+r with 0< or equal to r
Repeated subtraction model: 18 divided by 6 could be shown as 18-6=12-6=6-6=0
KEY VOCAB: (a/b)=c a is the dividend b is the divisor c is the quotient
Missing- Factor Model: 3xc=18. Using multiplication we know that 3x6=18, therefor c=6
Set (Partition) Model: Set up a model of the total number of items in the problem then partition them into sets. ex.) 18 cookies divided by 3 would be 3 sets of 6 cookies
Properties of Multiplication:
Closure: axb= whole number Commutative: axb=bxa ex.) x+5=5+x Associative: (ab)c= a(bc) ex.) 9(xy)=(9x)y Distributive: a(b+c)=ab+ac or a(b-c)=ab+-ac ex.) 8(x-2)= 8x-16 Multiplication Identity of One: bx1=1xb Multiplication by 0: 0xb=bx0=0
Cartesian-Product Model: Use of a tree diagram to solve multiplication problems.
Be aware of how multiplication is modeled: AxB, A(B), AB where A and B are the factors and AxB is the product
Repeated Addition Model: 3+3+3+3=12
Use base-ten blocks to show adding and taking away AFTER a manipulative
Models:
Number line
Comparison Model: Susan has 3 blocks Timmy has 8 blocks
Missing Addend: 3+__=8 --> Put in 3 blocks plus__=8 --> Number line --> Fact families --> Cashiers- Movie costs $8, you paid $10 which means 8+2=10
Take-Away: You have 8 blocks, take away 3
Subtraction is the inverse of addition
Counting Back: 9+7= 9 is one less than 10 which equals (10+7)-1=16
Making 10 then adding leftovers: 8+5= (8+2)+3=13
Doubles: 3+3=6 3+4= by 3+3=6 plus one more =7
Counting On: 4+2=4,5,6
Infinite if anything other than finite
Finite if its cardinality is 0 or a natural number
Ex.) A= {p,q,r,s} B= {a,b,c} C= {x,y,z} D= {b,a,c} A=C False A~C False A=B False B~D True C cannot equal D True
Do NOT confuse "equal" with "equivalent"
Ex.) A= (p,q,r,s) B= (a,b,c) C= (x,y,z) D= (b,a,c) Set A and B are not equivalent and not equal. Set B and C are equivalent, but not equal.
Two sets A and B are equivalent (A~B) if and only if there exists a one-to-one correspondence.
If elements of sets P and S can be paired so there is one element of P for each of S and one element of S for each of P then P and S are in one-to-one correspondence.
Anything to the zero power is one
Ex.) 10 base 6= 100,000
Read as 1,2,3 base 10
No base noted means the number is base 10
Number of power= how many zeros used
1,1,2,3,5,8,13,21,34,55,89
Sum of first two numbers equal third number
an= 3x2^(10-1) an= 3x2^9 an= 3x512 10th term= 1536
Ex.) 3,6,12,24... 10th term
a (little n)= a (little 1) x r ^ (n-1)
Uses multiplication of the ratio
an=1+(20-1)2 an=1+(19x2) an= 1+38 an=39
Ex.) 1,3,5,7,... 20th term
a (little n)=an=a (little one)+ (n-1)d
Must have common number pattern!
103+1=2n 104=2n 104/2=52 52x104=5408 5408/2= 2704
Ex.) 1+3+5+7...+103
= 499,500
999 Sums of 1000= 999,000/2 999x1000
S=1 2 3 4 S=999 998 997 996 ___________________ 2S= 1000
Ex.) 1+2+3+4...+999
OR numerator divided by denominator point zero, zero, zero 7/8 = 8.000/7
7/8 = 7/ (2^3) = (7x(5^3))/ ((2^3)(5^3))= 875/1000= 0.875
1.2032/ 0.32 becomes 120.32/ 32
0.000078= 7.8 x 10 ^(-5)
93,000,000= 9.3 x 10^7
4.62 x 2.4 = 462/100 x 24/10 = 462/(10^2) x 24/(10^1) = (462 x24)/ ((10^2) x (10^1)) = 11088/ (10^3) = 11.088
2.16 1.73 _____ 3.89
Use Distributive Property
Use Improper Fractions
Multiplication Property of Zero: (a/b) x 0 = 0 = 0 x (a/b)
Multiplication Property of Inequality: (a/b) > (c/d) and (e/f)>0, then ((a/b) x (e/f)) > ((c/d) x (e/f))
Multiplication Property of Equality: (a/b) x (e/f) = (c/d) x (e/f)
Multiplicative Inverse: (a/b) x (b/a) = 1 = (b/a) x (a/b)
Multiplicative Identity: 1 x (a/b) = a/b = (a/b) x 1
Distributive Prop of Multiplication Over Addition a/b ((c/d) + (e/f)) = ((a/b) x (c/d)) + ((a/b) x (e/f))
Greater Than and Less Than:
a/b < c/d if c/d - a/b > 0 c/d > a/b if and only if a/b < c/d
a/b - c/d = (ad-bc)/ bd
a/b - c/d = e/f
Rational Numbers Properties:
For any rational number a/b, there exists a unique rational number -(a/b) called the additive inverse of a/b
a/b + c/d = (ad)+ (cb)/bd
Addition of Rational Numbers with Like Denominators:
a/b + c/b= (a+c)/b
Number-line Model
Area Model
1/2 and 2/3 = 3/6 and 4/6
Rewrite fractions with the same positive denominator
Rewrite both fractions with a common denominator
Rewrite both fractions with the same LCM
Both fractions to the same simplest forms
A rational number a/b is in the simplest form if b>0 and GCD (a,b)=1; that is, if a and b have no common factor greater than 1 and b>0
Can be found from dividing n/n into a fraction such as 12/42= 2/7x6/6=2/7
Let a/b be any fraction and n a nonzero integer. Then a/b= an/bn
The value of the fraction does not change if its numerator and denominator are multiplied by the same nonzero integer
Represent the same number on the number line
1. Division problem or solution to a multiplication problem 2. Partition, or part, of a whole 3. Ratio 4. Probability
Use same methods as used for GCF
Definition: The least natural number that is simultaneously a multiple of a and multiple of b
Ladder Method
Prime Factorization Method: 180= 2x2x3x3x5= ((2^2) x3) 3x5 168= 2x2x2x3x7= ((2^2)x 3)2x7 Thus the common prime factorization is (2^2)x3=12
The Intersection of Sets Method: List all members of the set of positive divisors of both integers, then find the set of common divisors and pick the greatest element in that set.
Definition: The greatest number that divides into both a and b
Definition: Numbers in which there are more than 2 factors or positive divisors
Prime Factorization:
Sieve of Eratosthenes: method of identifying prime numbers
Number of Divisors: How many positive divisors does __ have?
Ladder Model: 2 l12 2 l 6 3 l 3 1
Factor Tree
Definition: a factorization containing only prime numbers
Composite numbers can be expressed as products of 2 or more whole numbers greater than 1
Definition: Numbers in which there are only 2 factors or positive divisors
11 if the sum of the digits in the places that are even powers of 10 minus the sum of the digits in the places that odd powers of 10 is divisible by 11
10 if the last digit is 0
9 if the sum of the digits is divisible by 9
8 if the last 3 digits are divisible by 8
6 if it is divisible by 2 and 3
5 if the last digit is 0 or 5
4 if the last two digits are divisible by 4
3 if the sum of the digits is divisible by 3
2 if the last digit is even
x+3<-2 x+3+-3<-2+-3 x<-5 (-6,-7,-8,...)
PEMDAS
The quotient of 2 negative integers is positive The quotient of a negative and positive integer is a negative
Properties: Closure Commutative Associative Multiplicative Distributive Zero Additive Inverse: (2x3) is -(2x3) thus (2x3) +(-2)(3)=0
Same models as those used for both addition and subtraction of integers
Properties: CANNOT do commutative or associative
Same models as those used for addition
Properties of Integer Addition: a.) Closure b.) Commutative c.) Associative d.) Identity
Absolute Value: The distance between the number and zero The absolute value of both 4 and -4 is 4 ALWAYS POSITIVE OR ZERO!
Pattern Model: 4+3=7 4+2=6 4+1=5 4+-4=0 4+-5=1 4+-6=-2
Number Line Model: ALWAYS START AT ZERO
Charged Field Model: (+) and (-) charges
Chip Model: Black= positive Red= Negative
