Tips on Teaching Math
           MAT156

5.1 Adding and Subtracting Integers

Addition:

Chip Model:
Black= positive
Red= Negative

Charged Field Model:
(+) and (-) charges

Number Line Model:
ALWAYS START AT ZERO

Pattern Model:
4+3=7
4+2=6
4+1=5
4+-4=0
4+-5=1
4+-6=-2

Absolute Value:
The distance between the number and zero
The absolute value of both 4 and -4 is 4
ALWAYS POSITIVE OR ZERO!

Properties of Integer Addition:
a.) Closure
b.) Commutative
c.) Associative
d.) Identity

Subtraction:

Same models as those used for addition

Properties:
CANNOT do commutative or associative

5.2 Multiplying and Dividing Integers

Multiplication:

Same models as those used for both addition and subtraction of integers

Properties:
Closure
Commutative
Associative
Multiplicative
Distributive
Zero
Additive Inverse: (2x3) is -(2x3) thus (2x3) +(-2)(3)=0

Division:

The quotient of 2 negative integers is positive
The quotient of a negative and positive integer is a negative

Order of Operations:

PEMDAS

Order of Integers:

x+3<-2
x+3+-3<-2+-3
x<-5 (-6,-7,-8,...)

-3^4= -81
HOWEVER
(-3)^4= (-3)(-3)(-3)(-3)= 81

5.3 Divisibility

Divisibility Rules: A number is divisible by...

2 if the last digit is even

3 if the sum of the digits is divisible by 3

4 if the last two digits are divisible by 4

5 if the last digit is 0 or 5

6 if it is divisible by 2 and 3

8 if the last 3 digits are divisible by 8

9 if the sum of the digits is divisible by 9

10 if the last digit is 0

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

5.4 Prime and Composite Numbers

Prime:

Definition: Numbers in which there are only 2 factors or positive divisors

Prime Factorization:

Composite numbers can be expressed as products of 2 or more whole numbers greater than 1

Definition: a factorization containing only prime numbers

Factor Tree

Ladder Model:
2 l12
2 l 6
3 l 3
1

Number of Divisors:
How many positive divisors does __ have?

Sieve of Eratosthenes: method of identifying prime numbers

Composite:

Definition: Numbers in which there are more than 2 factors or positive divisors

5.5 GCF and LCM

Greatest Common Factor:

Definition: The greatest number that divides into both a and b

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.

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

Ladder Method

Least Common Multiple:

Definition: The least natural number that is simultaneously a multiple of a and multiple of b

Use same methods as used for GCF

6.1 The Set of Rational Numbers

a/b, a is the numerator and b is the denominator

Uses of Rational Numbers:

1. Division problem or solution to a multiplication problem
2. Partition, or part, of a whole
3. Ratio
4. Probability

Proper Fraction: Numerator is smaller than the denominator
Improper Fraction: Numerator is larger than the Denominator

Equivalent or Equal Fractions:

Represent the same number on the number line

The value of the fraction does not change if its numerator and denominator are multiplied by the same nonzero integer

Let a/b be any fraction and n a nonzero integer. Then a/b= an/bn

Can be found from dividing n/n into a fraction such as 12/42= 2/7x6/6=2/7

Simplifying Fractions:

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

Equality of Fractions:

Both fractions to the same simplest forms

Rewrite both fractions with the same LCM

Rewrite both fractions with a common denominator

Ordering Rational Numbers:

Rewrite fractions with the same positive denominator

Denseness of Rational Numbers:

1/2 and 2/3 = 3/6 and 4/6

6.2 Addition, Subtraction, and Estimation with Rational Numbers

Addition:

Area Model

Number-line Model

Addition of Rational Numbers with Like Denominators:

a/b + c/b= (a+c)/b

Addition of Rational Numbers with Like Denominators:

a/b + c/d = (ad)+ (cb)/bd

Rational Numbers Properties:

For any rational number a/b, there exists a unique rational number -(a/b) called the additive inverse of a/b

Subtraction:

a/b - c/d = e/f

a/b - c/d = (ad-bc)/ bd

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

6.3 Multiplication and Division of Rational Numbers

a/b x c/d = (a x c) / (b x d)

Properties of Multiplication of Rational Numbers:

Distributive Prop of Multiplication Over Addition
a/b ((c/d) + (e/f)) = ((a/b) x (c/d)) + ((a/b) x (e/f))

Multiplicative Identity:
1 x (a/b) = a/b = (a/b) x 1

Multiplicative Inverse:
(a/b) x (b/a) = 1 = (b/a) x (a/b)

Multiplication Property of Equality:
(a/b) x (e/f) = (c/d) x (e/f)

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 Zero:
(a/b) x 0 = 0 = 0 x (a/b)

Multiplication with Mixed Numbers:

Use Improper Fractions

Use Distributive Property

7.1 and 7.2 Introduction and Operations on Decimals

Convert to decimals: 25/10= (2 x 10+5)/10 = (2x10)/10 + 5/10 = 2 + 5/10 = 2.5

Terminating Decimals: Decimals that can be written with only a finite number of places to the right of the decimal point.

Ordering Terminating Decimals: Change decimal by adding place value such as 0.36 and 0.9 change to 0.36 and 0.90 to ease in ordering decimals.

Algorithm for addition and subtraction of terminating decimals:

2.16
1.73
_____
3.89

Algorithm for multiplying decimals:

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

Scientific Notation:

93,000,000= 9.3 x 10^7

0.000078= 7.8 x 10 ^(-5)

Dividing Decimals:

1.2032/ 0.32 becomes 120.32/ 32

7.3 Nonterminating Decimals

Ways to convert some rational numbers to decimals:

7/8 = 7/ (2^3) = (7x(5^3))/ ((2^3)(5^3))= 875/1000= 0.875

OR numerator divided by denominator point zero, zero, zero
7/8 = 8.000/7

1.1 Problem Solving

4 Step Problem Solving Process:
     1. Understand the Problem
     2. Devise a Plan
     3. Carry Out the Plan
     4. Looking Back (CHECK)

Gauss's Approach to Find the Sum

Ex.) 1+2+3+4...+999

S=1      2      3      4     
S=999   998  997   996
___________________
2S= 1000

999 Sums of 1000= 999,000/2
          999x1000

= 499,500

Skipping Numbers

Ex.) 1+3+5+7...+103

103+1=2n
104=2n
104/2=52
52x104=5408
5408/2=

     2704

1.2 Patterns

Arithmetic Sequence

Must have common number pattern!

a (little n)=an=a (little one)+ (n-1)d

Ex.) 1,3,5,7,... 20th term

an=1+(20-1)2
an=1+(19x2)
an= 1+38
an=39

Geometric Sequence

Uses multiplication of the ratio

a (little n)= a (little 1) x r ^ (n-1)

Ex.) 3,6,12,24... 10th term

an= 3x2^(10-1)
an= 3x2^9
an= 3x512
10th term= 1536

Fibonacci Sequence

Sum of first two numbers equal third number

1,1,2,3,5,8,13,21,34,55,89

2.1 Numeration System

10 block breakdown:
Unit =1
Long=10
Flat=100

Roman Numeral System

I=1

V=5

X=10

L=50

C=100

D=500

M=1000

IV=4 (5-1)

IX=9 (10-9)

XL=40 (50-10

Ex.) MDCCLXXIII=
1000+700+70+3

Base 10

Number of power= how many zeros used

No base noted means the number is base 10

Read as 1,2,3 base 10

Ex.) 10 base 6= 100,000

Anything to the zero power is one

Base 5

5 zero power=1

5 first power=5

5 second power=25

5 third power=125

5 fourth power=625

5 fifth power=3125, etc.

Ex.) 1 0 3 0 base 5= 1x5 to the 3rd power (1x125)+0x5 to the
2nd power+3x5 to the 1st power (3x5)+0x5 to the zero
power...125+0+15+0=140

2.2 and 2.3 Sets

A set is any  group or collection of objects

The objects that belong to the sets are called members or elements

Order is NOT important

Each element is only listed once

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.

Equivalent Sets:

Two sets A and B are equivalent (A~B) if and only if there exists a one-to-one correspondence.

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.

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

Finite vs. Infinite

Finite if its cardinality is 0 or a natural number

Infinite if anything other than finite

3.1 Addition and Subtraction of Whole Numbers

Set Model:
Set A n(A)= {a,b,c,d}
Set B n(B)= {e,f,g}
n(A)+ n(B)= {a,b,c,d,e,f,g}= 4+3+7= n(AuB)

KEY TERMS:
numbers a+b are the addends
a+b is the sum

Number Line (Measurement) Model:
ALWAYS START AT ZERO

Addition Properties:
Closure: a+b= whole number

Commutative: a+b=b+a

Associative: (a+b)+c= a+(b+c)

Identity: a+0=a

Basic Addition Facts:

Counting On:
4+2=4,5,6

Doubles:
3+3=6
3+4= by 3+3=6 plus one more =7

Making 10 then adding leftovers:
8+5= (8+2)+3=13

Counting Back:
9+7= 9 is one less than 10 which equals (10+7)-1=16

Subtraction:

Subtraction is the inverse of addition

Models:

Take-Away:
You have 8 blocks, take away 3

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

Comparison Model:
Susan has 3 blocks
Timmy has 8 blocks

Number line

3.2 Algorithms for Whole-Number Addition and Subtraction

ALWAYS START WITH A MANIPULATIVE!!

Subtraction:

Use base-ten blocks to show adding and taking away AFTER a manipulative

3.3 Multiplication and Division of Whole Numbers

Multiplication:

Repeated Addition Model:
3+3+3+3=12

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

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

Division:

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

Missing- Factor Model:
3xc=18. Using multiplication we know that 3x6=18, therefor c=6

KEY VOCAB:
(a/b)=c
a is the dividend
b is the divisor
c is the quotient

Repeated subtraction model:
18 divided by 6 could be shown as 18-6=12-6=6-6=0

Division Algorithm:

a=bq+r with 0< or equal to r<b
q: quotient
r: remainder

Inverse Operations:

Division is the inverse of multiplication

Division by 0 or 1:

n divided by 0 is undefined
0 divided by n is 0
0 divided by 0 is undefined

Order of Operations:
PEMDAS
Parenthesis
Exponents
Multiplication or Division (LEFT TO RIGHT)
Addition or Subtraction (LEFT TO RIGHT)

3.4 Algorithms for Whole-Number Multiplication and Division

Multiplication Algorithms:

Single Digit times two digit:
1 2 10+2
x 4 ---> x 4
_____ ______
4 8 40 + 8

Multiplication with Two-Digit Factors:
2 3
x 1 4 (10+4)
________
9 2 (4X23)
+ 2 3 0 (10X23)
___________
3 2 2

Division of Algorithms:

Instruction for "long division":
Divide
Multiply
Subtract
Check
Break Down

"Does McDonalds Sell Cheese Burgers?"
DMSCB

Always multiply your quotient times the divisor to get the dividend to check your answer.

4.3 Functions

A function is a relationship that assigns exactly ONE output for each input.

Ways to represent functions:

As Rules:
1 3
0 0
4 12

Rule= nx3

As Machines:
x f(x)
0 3
1 4
3 6
4 7
6 9

As Equations:
f(0)= 0+3=3
f(1)= 1+3=4
f(3)= 3+3=6

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 Tables and Ordered Pairs:
0 1
1 3
2 5
3 7
4 9
(0,1), (1,3), (2,5), (3,7), (4,9)

Function?
(1,2), (1,3), (2,3), (3,4)
NO, 1 INPUT is used twice.

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

As Graphs:
Horizontal- inputs
Vertical- outputs

Relations:
Every function is a relation, but not every relation is a function.