Elementary Mathematics

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Elementary Mathematics

Decimals, Percents, and Real Number

Real Numbers

any number that can be represented by a decimal

Irrational Numbers

cannot be a repeating block of digits

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

Percents

Computing Interest

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

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

Mental Math

using a known percent

using fraction equivalents

Proportions

Percent Bar

n%=n/100

Decimals

Repeating Decimals

Balancing with Decimals in Division

Balancing with Decimals in Subtraction

Making Compatible Numbers

Breaking and Bridging

Division

Scientific Notation

Terminating Decimals

Ordering Terminating Decimals

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

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

1. Line up the numbers by place value

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

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

Integers

if xny

if x0 then nx

if x-y

if x

Ordering Integers

-3>-5

-5<-3

Difference of Squares

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

Distributive Property of Multiplication Over Subtraction of Integers

(b-c)a=ba-ca

a(b-c)-ab-ac

(-a)(-b)=ab

(-a)b=-(ab)

(-1)a=-a

Zero Property

a(0)=0=0(a)

Distributive Property of Multiplication Over Addition for Integers

Identity Property

1(a)=a=a(1)

Associative Property

(ab)c=a(bc)

Commutative Property

ab=ba

Closure Property

ab is a unique integer

Integer Multiplication Models

Pattern Model

Addition and Subtraction

Additive Inverse Property of Integers

a+-a=0=-a+a

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

Addition Property of Equality for Integers

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

-(-a)=a

Identity Property of Addition of Integers

0+a=a=a+0

Associative Property of Addition of Integers

Commutative Property of Addition of Integers

Closure Property of Addition of Integers

a+b is a unique integer

Representation of Integers

Temperature Cube Model

Charged Field

Number Line Model

Chip Model

Absolute Value

distance between the point corresponding to an integer and 0

Positive Integers

1, 2, 3, 4...

Negative Integers

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

Whole Number Operations

Division of Whole Numbers

Breaking Up the Dividend

Division by 0 and 1

division by 0 is undefined

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

Division Models

Missing Factor Model

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

Repeated Subtraction

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

Multiplication of Whole Numbers

Compatible Numbers

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

Front-End Multiplying

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

Thinking Money

Using Compatible Numbers

Front-End Multiplying

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

Alternative Methods

Lattice Multiplication

Partial Products Algorithm

Distributive Property of Multiplication Over Subtraction for Whole Numbers

a(b-c)=ab-ac

Distributive Property of Multiplication Over Addition of for Whole Numbers

a(b+c)=ab+ac

Multiplication Property of 0 for Whole Numbers

a x 0=0=0 x a

Identity Property of Multiplication of Whole Numbers

a x 1= a =1 x a

Associative Property of Multiplication of Whole Numbers

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

Commutative Property of Multiplication of Whole Numbers

a x b = b x a

Closure Property of Multiplication of Whole Numbers

a x b is a unique whole number

Multiplication Models

Cartesion-Product Model

the number of ways objects in sets can be combined

tree diagram or table

Area Model

4-by-5 grid

Rectangular Array Model

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

8+8+8=24

Subtraction of Whole Numbers

drop the zeros

subtraction models

base ten blocks

comparison model

missing addend model

number line

take-away model

operations that undo each other are inverse operations

subtraction is inverse of addition

Addition of Whole Numbers

using the range

rounding

clustering

grouping to nice numbers

front-end with adjustment

Mental Computation

making compatible numbers

using compatible numbers

sums easy to calculate mentally

trading off

breaking up and bridging

adding from left

Identity Property of Addition of Whole Numbers

a+0=a

Associative Property of Addition of Whole Numbers

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

Commutative Property of Addition of Whole Numbers

a+b=b+a

Closure Property of Addition of Whole Numbers

sum of two whole numbers is a unique whole number

Basic Addition Facts

making ten

doubles

counting on

shown using set model and number line

two disjoint finite sets

binary operation because two numbers are added

Rational Numbers and Proportional Reasoning

Proportional Reasoning

Bar Models

Scale Drawings

Constant of Proportionality

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

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

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

Proportion

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

two given ratios are equal

Ratio

A comparison of two quantities

a/b, a:b

Division

Properties of Exponents

Algorithm for Division of Fractions

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

Multiplication

Repeated Addition

Theorems

Multiplication Property of Zero for Rational Numbers

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

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 Equality for Rational Numbers

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

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)

Multiplicative Inverse Property of Rational Numbers

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

Multiplicative Identity Property of Rational Numbers

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

Subtopic

Estimation

round fractions to a convenient fraction

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

Subtraction

Addition

Addition Property of Equality

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

Additive Inverse Property of Rational Numbers

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

Rational Numbers

Equality of Fractions

3. Rewrite both fractions with a common denominator

2. Rewrite both fractions with the same least common denominator

1. Simplify

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

Fundamental Law of Fractions

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

Representations for Rational Numbers

Set Model

Number-Line Model

Bar Model

Number Theory

Least Common Multiple

Theorem

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

Euclidean Algoritm

Number Line

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

Greatest Common Divisor

Methods

Euclidean Algorithm

Intersection of Sets

Colored Rods

greatest whole number that divides both a and b

Prime and Composite Numbers

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.

Prime Factorization

Factor Tree

A factorization containing only prime numbers.

1 is neither prime nor composite.

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

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

Divisibility

Theorems

Divisibility Test for 6

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

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 9

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

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 8

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

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 10

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

Divisibility Test for 5

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

Divisibility Test for 2

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

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

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

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

Numeration Systems

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

collection of properties and symbols that represent numbers systematically

Hindu-Arabic Numeration System

place value based on powers of 10

numerals constructed by 10 digits

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

Roman Numeration System

multiplicative property

bar placed over numeral to multiply by 1,000

subtractive property

IV=5-1=4

additive property

VII=5+1+1=7

Mayan Numeration System

groupings of 20 vertically

3 symbols (including 0)

Babylonian Numeration System

numbers greater than 59 represented by groupings of 60

place- value system

Egyptian Numeration System

includes additive property

grouping system based on powers of 10