Weeks 1,2,3,4,5,6,7,8

Cartesian Product Context: characterized by finding all possible pairings between all possible pairings between 2 or more sets of objects.

Investigating Quantity

This is the last topic of the course! Hooray!!

Addition of integers

Subtraction of integers

Multiplication of integers

Decimals

Percents

Properties of Modular Arithmatic(Multiplication)

Commutative Property of multiplication:axb(mod 5)=bxa(mod 5)

Closure Property of Multiplication

Commutative Property of Multiplication

Identity Property of Multiplication

Inverse Property of Multiplication

Properties of Modular Arithmatic(Addition)

Commutative Property of Addition:a+b(mod 5)=b+a(mod 5)

Closure Property of Addition

Communative Property of Addition

Identity Property of Addition

Proportional Reasoning

Relative thinking: identifying relationships between two quantities and comparing them.

Utilizing

Ratio Sense

Rational Numbers

Recognition of quantities and how they change

Relative thinking.

Proportion

a/b = c/d

Where two ratios are equal.

Ratio

Example of ratio: 2:3

Comparing two quantities regardless of whether units are the same.

Division of fractions

When using the partition model it's most important to know the # of groups as well as the size.

Repeated Subtraction

Partition

Multiplication of fractions

An area model is best used as a table.

Area Model

Fractions

Cuisennaire rods were an interesting manipulative to use to comprehend fractions.

Interpretations of a fraction

Part-Whole interpretation

Copies

Division

Separating groups

Ratios

Comparing 2 separate things

Rational Numbers

Any number that can be expressed as the quotient of two integers. a/b

Use of Cuisennaire Rods

Problem Solving

Don't forget to read over the problem over a second time so that you understand what you're reading.

Understand the Problem

Devise A Plan

Implement you Plan

Sequences

Geometric Sequences are sequences of numbers with a common ratio; that is, if you form the ratio of any two consecutive terms in the sequence the ratio is the same.

Arithmetic Sequence

Geometric Sequence

Recurrence Relationship Sequence

Sets

Equivalent Sets and Equal sets are not always the same.

Venn Diagram

Universal Set

Conjoined Set

Complements

Equivalent Sets

Equal Sets

Number Systems

The Mayan system is probably the most difficult number system to use.

Tally System

Egyptian System

Myan System

Babylonian System

Roman System

Hindu-Arabic System

Models

The Discrete Model is characterized by combining sets of two discrete objects.

Discrete Model

Counted Quantities

Continuous Model

Measured Quantities

Investigating Quantities

Cubes!

Bases

Cubes

a^3

Flats

a^2

Longs

a^1

Units

1Unit

Addition

Identity Property of Division was probably the most difficult to understand.

Closure Property of Addition

If aeX and beX then a+beX

Communative Property of Whole Numbers

If aeW and beW then a+b=b+a

Associative Property of Whole Numbers

If aeW, beW and ceW, then (a+b)+c=a+(b+c)=(a+c)+b

Identity Property of Division of Whole Numbers

If aeW then a+o=a=o+a

Division Models/Contexts

Divison Models/Contexts

Partition: Equal Sharing

Characterizing by distributing a given quantity among a specified number of groups(partition) and determining the size(amount) in each group(partition).

Know: Quantity of #'s starting with, and # of groups

Find: Size of each group

Measurement

Characterized by using a given quantity to create groups(partitions) of a specified size(amount) and determining the # of groups that are formed.

Know: Quantity, and size of each group

Find: # of groups

Multiplication Models/Contexts

Distributive Property: aeW, beW, ceW, then a(b+c) = ab+ac

Associative Property (grouping)

(axb)xc = ax(bxc)

Communicative Property

If aeW and beW, then ab = ba

Closure Property

If aeW and beW, then abeW

Identity Property

aeW, 1xa=a, 1 is the identity element

Zero Property

aeW, oxa=o

Distributive Property

aeW, beW, ceW, then a(b+c) = ab+ac

Subtraction Models/Contexts

Subtraction is characterized by starting with some initial quantity and removing a specified amount.

Closure Property (doesn't always work)

If aeW and beW, then (a-b)eW

Communative Property (doesn't always work)

If aeW and beW then a-b = b-a

Identity Property

If aeW then a-o = o-a

Number Theory

Prime numbers are only divisible by one and themselves.

Odd Numbers

An even number + 1

Even Numbers

Multiples of 2

Prime Numbers

Only divisible by 1 and itself

Composite Numbers

Has a positive factor other than 1 and itself

Prime Factorization

Prime factorization is a factorization containing only prime numbers.

Fundamental Theorem of Arithmetic

Means you can't find more than one prime factorization for a number.