Elementary Mathematics

collection of properties and symbols that represent numbers systematically

Egyptian Numeration System

grouping system based on powers of 10

includes additive property

Babylonian Numeration System

place- value system

numbers greater than 59 represented by groupings of 60

Mayan Numeration System

3 symbols (including 0)

groupings of 20 vertically

Roman Numeration System

additive property

VII=5+1+1=7

subtractive property

IV=5-1=4

multiplicative property

bar placed over numeral to multiply by 1,000

Hindu-Arabic Numeration System

numerals constructed by 10 digits

0,1,2,3,4,5,6,7,8,9

place value based on powers of 10

Base Ten System: 0,1,2,3,4,5,6,7,8,9

Divisibility

An even whole number has a remainder of 0 when divided by 2.

An odd whole number has a remainder of 1 when divided by 2.

Theorems

If d is a factor of a, d is a factor of any multiples of a.

Divisibility Test for 2

A whole number is divisible by 2 only if the units digit is even.

Divisibility Test for 5

A whole number is divisible by 5 only if the units digit is 0 or 5.

Divisibility Test for 10

A whole number is divisible by 10 only is the units digit is 0.

Divisibility Test for 4

A whole number is divisible by 4 only if the last 2 digits represent a whole number divisible by 4.

Divisibility Test for 8

A whole is divisible by 8 only if the last 3 digits represent a whole number divisible by 8.

Divisibility Test for 3

A whole number is divisible by 3 only if the sum of the digits is divisible by 3.

Divisibility Test for 9

A whole number is divisible by 9 only if the sum of the digits is divisible by 9.

Divisibility Test for 11

A whole number is divisible by 11 only if the sum of the digits in places that are even powers of 10 minus the sum of digits in places that are odd powers of 10 is divisible by 11.

Divisibility Test for 6

A whole number is divisible by 6 only if the number is divisible by both 2 and 3.

Prime and Composite Numbers

A prime number is any whole number number with exactly two distinct whole number divisors.

A composite number is any whole number greater than 1 that has a whole number factor other than 1 and itself.

1 is neither prime nor composite.

Prime Factorization

A factorization containing only prime numbers.

Factor Tree

Theorems

Each composite number can be written as a product of primes in one, and only one, way except for the order of prime factors in the product.

Greatest Common Divisor

greatest whole number that divides both a and b

Methods

Colored Rods

Intersection of Sets

Prime Factorization

Euclidean Algorithm

Least Common Multiple

least non-zero whole number that simultaneously is a multiple of a and b

Methods

Number Line

Colored Rods

Intersection of Sets

Prime Factorization

Euclidean Algoritm

Theorem

GCD (a, b) x LCM (a, b)= ab

Rational Numbers

Representations for Rational Numbers

Bar Model

Number-Line Model

Set Model

Theorems

Fundamental Law of Fractions

If a/b is a fraction and n is a non-zero number then a/b= an/bn

Denseness Property of Rational Numbers

If there are two different rational numbers a/b and c/d there is another rational number between the two

Equality of Fractions

1. Simplify

2. Rewrite both fractions with the same least common denominator

3. Rewrite both fractions with a common denominator

Addition

Theorems

Additive Inverse Property of Rational Numbers

a/b+(-a/b)=0=(-a/b)+a/b

Addition Property of Equality

If a/b=c/d, then a/b+e/f=c/d+e/f

Subtraction

Estimation

round fractions to a convenient fraction

1/2,1/3,1/4,2/3, or 1

Multiplication

Theorems

Multiplicative Identity Property of Rational Numbers

1 (a/b)=a/b=(a/b) 1

Multiplicative Inverse Property of Rational Numbers

(a/b)(b/a)=1=(b/a)(a/b)

Distributive Properties of Multiplication Over Addition and Subtraction for Rational Numbers

a/b(c/d+e/f)=(a/b)(c/d)+(a/b)(e/f) and a/b(c/d+e/f)=(a/b)(c/d)-(a/b)(e/f)

Multiplication Property of Equality for Rational Numbers

(a/b)(e/f)=(c/d)(e/f)

Multiplication Properties of Inequalities for Rational Numbers

If a/b>c/d and e/f>0, then (a/b)(e/f)>(c/d)(e/f)

If a/b>c/d and e/f<0, then (a/b)(e/f)<(c/d)(e/f)

Multiplication Property of Zero for Rational Numbers

(a/b) 0=0=0 (a/b)

Repeated Addition

Division

Theorems

Algorithm for Division of Fractions

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

Properties of Exponents

Proportional Reasoning

Ratio

A comparison of two quantities

a/b, a:b

Proportion

two given ratios are equal

Theorem

a/b=c/d is a proportion if ad=bc

Constant of Proportionality

if y=kx (or k=y/x) then y is proportional to x and k is the constant of proportionality between y and x

Theorems

a/b=c/d if b/a=d/c

a/b=c/d if a/c=b/d

Scale Drawings

Bar Models

Addition of Whole Numbers

binary operation because two numbers are added

two disjoint finite sets

shown using set model and number line

Basic Addition Facts

counting on

doubles

making ten

Theorems

Closure Property of Addition of Whole Numbers

sum of two whole numbers is a unique whole number

Commutative Property of Addition of Whole Numbers

a+b=b+a

Associative Property of Addition of Whole Numbers

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

Identity Property of Addition of Whole Numbers

a+0=a

Mental Computation

adding from left

breaking up and bridging

trading off

using compatible numbers

sums easy to calculate mentally

making compatible numbers

Estimation

front-end with adjustment

grouping to nice numbers

clustering

rounding

using the range

Subtraction of Whole Numbers

operations that undo each other are inverse operations

subtraction is inverse of addition

subtraction models

take-away model

number line

missing addend model

comparison model

base ten blocks

Mental Computation

breaking up and bridging

trading off

drop the zeros

Multiplication of Whole Numbers

Multiplication Models

Repeated Addition

8+8+8=24

Rectangular Array Model

objects arranged with the same number of objects in each row and column

Area Model

4-by-5 grid

Cartesion-Product Model

the number of ways objects in sets can be combined

tree diagram or table

Theorems

Closure Property of Multiplication of Whole Numbers

a x b is a unique whole number

Commutative Property of Multiplication of Whole Numbers

a x b = b x a

Associative Property of Multiplication of Whole Numbers

(a x b) x c= a x (b x c)

Identity Property of Multiplication of Whole Numbers

a x 1= a =1 x a

Multiplication Property of 0 for Whole Numbers

a x 0=0=0 x a

Distributive Property of Multiplication Over Addition of for Whole Numbers

a(b+c)=ab+ac

Distributive Property of Multiplication Over Subtraction for Whole Numbers

a(b-c)=ab-ac

Alternative Methods

Partial Products Algorithm

Lattice Multiplication

Mental Computation

Front-End Multiplying

64 x 5= 60 x 5=300 and 4 x 5=20= 300+20=320

Using Compatible Numbers

Thinking Money

Estimation

Front-End Multiplying

524 x 8= 500 x 8= 4000 and 20 x 8= 160 = 4000+160=4160

Compatible Numbers

524 x 8= 500 x 8=4000 and 25 x 8=200 = 4200

Division of Whole Numbers

Division Models

Repeated Subtraction

10-2=8,8-2=6,6-2=4,4-2=2,2-2=0 (five groups of 2 in 10)

Set Model

Missing Factor Model

3c=18, 3 x 6=18, c=6

Division by 0 and 1

n/0= undefined, 0/n= 0, 0/0= undefined, n/1= 1

division by 0 is undefined

Mental Computation

Breaking Up the Dividend

Using Compatible Numbers

Estimation

Using Compatible Numbers

Addition and Subtraction

Negative Integers

-1, -2, -3, -4...

Positive Integers

1, 2, 3, 4...

Absolute Value

distance between the point corresponding to an integer and 0

Representation of Integers

Chip Model

Number Line Model

Charged Field

Temperature Cube Model

Theorems

Closure Property of Addition of Integers

a+b is a unique integer

Commutative Property of Addition of Integers

a+b=b+a

Associative Property of Addition of Integers

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

Identity Property of Addition of Integers

0+a=a=a+0

Additive Inverse Property of Integers

a+-a=0=-a+a

-(-a)=a

Addition Property of Equality for Integers

if a=b, a+c=b+c

-a+-b=-(a+b)

Multiplication

Integer Multiplication Models

Chip Model

Number Line

Pattern Model

Theorems

Closure Property

ab is a unique integer

Commutative Property

ab=ba

Associative Property

(ab)c=a(bc)

Identity Property

1(a)=a=a(1)

Distributive Property of Multiplication Over Addition for Integers

a(b+c)=ab+ac

Zero Property

a(0)=0=0(a)

(-1)a=-a

(-a)b=-(ab)

(-a)(-b)=ab

Distributive Property of Multiplication Over Subtraction of Integers

a(b-c)-ab-ac

(b-c)a=ba-ca

Difference of Squares

(a+b)(a-b)=a^2-b^2

Division

Division Models

Chip Model

Number Line

Ordering Integers

-5<-3

-3>-5

Theorems

if x<y then x+n<y+n

if x<y then -x>-y

if x<y and n>0 then nx<ny

if x<y and n<0 then nx>ny

Decimals

Terminating Decimals

numbers that can be written with a finite number of digits to the right of the decimal point

Theorem

a/b is simplest form can be written as a termination decimal if prime factorization of the denominator contains no primes other than 2 or 5

Ordering Terminating Decimals

1. Line up the numbers by place value

2. Start at the left and find the first place where the values are different

Compare the digits, the digit with the greater face value represents the greater of the numbers

Multiplication

Scientific Notation

Addition

Subtraction

Division

Mental Computation

Breaking and Bridging

Using Compatible Numbers

Making Compatible Numbers

Balancing with Decimals in Subtraction

Balancing with Decimals in Division

Repeating Decimals

Percents

n%=n/100

Percent Bar

Proportions

Mental Math

using fraction equivalents

using a known percent

Computing Interest

amount (a), principal (p), interest (i)

A=P+I=P+Prt=P(1+rt)

Real Numbers

Irrational Numbers

infinite number of non-zero digits to the right of the decimal point

cannot be a repeating block of digits

any number that can be represented by a decimal