Math 156

Properties

Addition

Closure propeties of addition

aEx and bEx then a=b E

Closure propety of adding whole numbers

if aEw and bEw then a+b E w

Communitive property of addition of whole numbers

If aEw and bEw then a+b = b+a

Associative property of whole numbers

aEw bEw and cEw then (a+b)c = a(b+c) or (a+c)b

Identity property of addition of whole numbers

aEw then a+0=a=0+a

Subtraction

Closure property of subtraction of whole numbers

aEw and bEw then (a-b) E w

Communitive property of subtraction of whole numbers

aEw and bEw then a-b=b-a

Identity property of subtraction of whole numbers

aEw then a-0=a= 0-a

Multiplication

Closure property of multiplication for whole numbers

aEw and bEw then abEw

Communitive property of multiplication of whole numbers (changes order)

aew and bEw then ab=ba

Associative propoerty of multiplication of whole numbers (changes grouping)

(a*b)*c =a* (b*c)

Identity property of multiplication of whole numbers

aEw then 1(a)= a b/c 1 is the identity element

Zero property of multiplication

aEw then 0(a)= 0

Distributive property of multiplication over addition and subtraction

aEw, bEw, and cEw -> a(b+c) -> ab+ac

Division

Sequence/ Patterns

Is an ordered list of numbers or events that may be referred to as elements of a sequence, members of the sequence or terms of the sequence.

Arithmetic Sequence

Is a number sequence with a common difference.

Method: a(n)=a1+d(n-)

Geometric Sequence

Is a number sequence with a common ratio.

Helpful: # on the right/ # on the left.

Method: a1(r) to the n-1 power

Reoccurence Sequence

A sequence in which the current term is dependent on the previous term.

Method: b, sub unknown= 2(b, sub last term solved)+ (b, sub the term before last solved term.)

Main topic

Ratios and proportions

proportional (part of a whole) between two different numbers or quantities.

Clocks

Has different module that determine what numbers go around it. They usually want to know what one number is congruent to another number by using a specific Mod. clock.

Ex 12=0 we see if twelve is congruent to zero, which it is when using a mod 12 clock.

Understanding Integers

can be done visually with chips or on a number line

The opposite of the number a is the number that must be added to a to produce an additive identity. a + ??= 0.This is referred to as the opposite of a written as -a.

Arithmetic with integers

Division

We know the number of groups. We need to find the size of groups.

Repeated subtraction

We know the size of groups. We need to find the number of groups.

Properties of Modular Arithmetic

I'm thinking these are pretty simple enough to declare what property a problem is by using these definitions and examples.

Closure prop. of Addition

Closure prop. of Multiplication

When we add/mult. any two values on the clock, and we get another answer that is on the clock.

EX: 0+4 mod5 = 4

Communitive prop. of Addition

Communitive prop. of Mulitplication

Gives a mirror image across the mult/or addition chart when looking at the diagonal.

EX: a+b mod5 = b=a mod5

Identity of Addition

Identity prop. of Multiplication

When we add/mult. two numbers and ending with the same number you first started with.

EX: a = ? mod5= a

Inverse prop. of Addition

Inverse prop. of Multiplication

When we add/mult. two numbers and end with the number zero (the inverse.)

EX: a + b mod5 = 0

Modles/ Context

Addition

Discrete (set): countities

Characterized by combining two sets of discrete objects

Examples: markers, animals, kids, chairs, fruit

Individually separate and distinct objects

Continuous (number line): measured quantities

Characterized by combining of two continuous quantities

Example: Time, distance, area, volume

Subtraction

Take Away

Characterized by starting with some initial quantity and removing or (taking away) specified amount

Comparison

Characterized by comparison of the relitive sizez of two quantities and determining either how much longer or how much smaller one quantity is compared to the others.

Missing Addend

Characterized by the need to determine what quantity must be added to a specified number to reach some targeted amount.

Multiplication

Repeated addition (Discrete)

Ex: 3 rows of seats with two students in each. How many students altogether?

Repeated addition (continuous)

Is characterized by repeatedly adding a quantity of continuos quantities. A specified number of times.

Area Modle

characterized by a product of two numbers representing the size of a rectangular region, such that the product represents the number of unit sized squares within the rectangular region

Cartesian Product

characterized by finding all possible pairings between 2 or more sets of objects.

Division

Partition (equal product)

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

Measurement (repeated subtraction)

Characterized by using given quantity to create groups or (partitions) of specified size (amount) and determining the number of group (partitions) that are formed

Tradition Algorithm (seperate sheet)

I like to call this "doing it the old fashion way"

12/2=6 and 12/6=2 (four fact families)

Number Theory

Even

If the number is a multpiple of 2.

Odd

If the number is one more than an even number or one less than an even number.

Fractions

Multiple Interpretations of a fraction

I prefer the term part whole vocabulary and find it a better fit when refering to fractions rather than saying copies and such.

Part-While

2 parts of a whole that was divided into 3 parts

Division

The fraction bar becomes the alternative tool to indiccate division

Copies of a unit fraction (accompaniement to part-while)

2/3 is two copies of the [unit fraction 1/3]

Ratio

Comparing two seperate things ratio of girls to boys is 2 girls/ 3 boys

Rational Numbers

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

Operations with Decimals

Addition, subtraction, line up decimals

Multiplication and division solve ignoring decimal till very end then count how many place values over.

how you would say 42.31 is "Forty-two and thirty-one hundreths"

Percents

I like using the is over of and percent over 100 method when working with percent problems, That could be just becaus thats what I am used to but I think I'm going to start givin the other way a try more.

Ex: What is 12% of 350 people?, 1% of 350 people is 3.5 peoplpe, then you take 3.5 and multi by 12.

Three types of fractions are:

Improper and mixed numbers

7/8 + 1/2

Simplified

6/8 = 3/4

Egyptian

Thank you for showing us the youtube video in class. I feel that it was worth spenging our time on and gave us a better idea of how their fractions are.

All fractions must be unti fractions 1/n. All unit fractions must be unique. 3/2doesn't = 1/2+1/2+1/2

Division

Before going onto the division part of fractions we discussed the advantages of adding and subbtracting mixed numbers by two different ways. I think I have a better understanding of them both and prefer to keep it as a mixed number instead of the improper fraction back into a mixed number.

Partition (long division)

Know the number of groups/partition

Find the size of partition

Repeated subtraction (Divion of fractions)

Know size of group

Find number of groups

Sets

This part would've been easier I think if I had learned this before just now.

A group of numbers, variable, geometric figures that are using set braces.

Compliment of A

A with a line over the top

Union A intersects Union B

A u B

A Union B

A u B

Examples:

B C A

B = A

B ~ A

B - A

Ways to Record Numbers

Tally

single strokes or tally marks to represent numbers

Egyptian

uses grouping system to represent certain numbers

Babylonian

place value system

Mayan

primarily on 20 with vertical groups

Hindu- Arabic

Base ten system

Roman

uses additive subtractive and multiplicative properties

Problem solving

Understand the Problem

Devise a plan

Pattern

Guess, Check, Revise

Diagram, Picture

Table

Algebraic Symbols

Act it out

Break it down

Work backwards

Impliment the plan

Cusenaire Rods

I found that using theCusenaire Rods helpful and a good visual aide when learning new concepts especilly when working fractions I think it would help too.

Used as a mulipulative for factoring and multiples.

Greatest Common Factor

The biggest number that can go into both two numbers

Least Common Multiple

I like the way you showed us how to find LCM's. Ex: (6,8) 2*3*2*2* = 2*2*2*3=24

The smallest numbers you can multiple the original number by to get the remaining total the same.

Set Relationships

a~b is when two sets are equal. Meaning they are equivalent to each other

XnY=X is when elements in X must be in Y too.

XnY= XuY If only elements in X and Y and the only elements that are in common and X’s and Y’s elements intersecting are the same, then X and Y are equivalent to each other

one to one correspondence: for each element there is an element of set B to match it with no extra elements and no repeated use of an element.

Methods

I think we should be taught additive methods at an earlier age.

Lattice

adding using two four digit numbers

Low stress

list of single digit numbers being added together

Scratch

series of addition only using two digit numbers

Any Column first

large numbers with the same amount of digits being added together

Left to Right

the one that emphasizes place value when you start on the left, add numbers in the first column together then write it down with the place value represented.

Multiplication

Partial product

lattice method

Divisibility

Whole number a is divisible by whole number b if, and only if there exisists a third whole number c.

b is a factor of a, b is a divisor of a, a is a multiple of a, and a is divisble by b

Cuisenaire Rods

Factoring

GCF

Greatest Commmon Factor

The biggest number that goes into all of the comparing numbers.

Multiples

LCM

Least Common Multiple

Smallest multiple that is divisble by all of the comparing numbers.

Ratio, proportions, and reasonings

Ratios

May look like a fraction

Comparing two quantities regardless of whether the units are the same. If units happen to be different this is referred to as a ratio.

Quantitive Relationship

SHowing the number of times one contains or is contained witrhin another value.

Proportion

Are when two ratios are equal

ex: a/b = c/d

Analogy a is to b as c is to d.

Ex: a:b as c:d

Proportional Reasoning

Absolute reasoning and relative reasoning

Rational Numbers

Quantities and knowing how they change

Relative thinking

Ratio Sense

knowing how to use them

Utilizing

Recognizing and comaparing units