MAT 156 Math Instruction

Polya's 4 Steps to Solve a Problem

4 Steps

Understand the Problem

Devise a Plan

Carry Out the Plan

Look Back

Examples of Problem Solving Plans

equations

working backwards (input/output problem)

make a table/diagram

identify a subgoal

guess & check

indirect/direct resasoning

Sum of Sequential Numbers

Normal Sequence

S = 1+2+3+4+5...+100

1+100=101, 2+99 =101, 3+98=101...

So there are 100 sums that equal 101

2S = 100 x 101

S = (100 x 101)/2

S = 5,050

Even

S = 2+4+6+8+...+100

Odd

S = 1+3+5+7...+101

Main topic

Numeration Systems

Roman Numerals

= 1

= 5

= 10

= 50

= 100

= 500

= 1,000

Arabic

Tally Marks

Bases

Chinese

One-to-One Correspondences

Outfit example:

2 pants = blue, brown; 3 shirts = yellow, red, green

Make a table or tree

Blue - blue/yellow. blue/red, blue/green

Brown - brown/yellpw, brown/red, brown/green

2 pants & 2 shirts = 6 possible outfits

Simple Vocabulary

Addition

addends

sum

subtraction

difference/remainder

subtrend

minuend

multiplication

factors

product

division

dividend

diviisor

quotient

function

f(x)=x+2; (1,3)(2,4)(3,5)

Every domain (input, x) has exactly one range (output, y)

Order of Operations

Parenthesis

Exponents

Multiplication

Division

Addition

Subtraction

Properties

Associative

Commutative

Distributive

Identity

of One - addition/mulitiplication

of Zero - addition/ multiplication

Closure

Prime & Composite Numbers

Rainbow

Factor Trees

Color Map

Subtopic

Least Common Multiple & Greatest Common Denominator

Intersection of Sets

20: 1, 2, 4, 5, 20

32: 1, 2, 4, 8, 32

40: 1, 2, 4, 5, 8, 10

Number Line

Ladder Method

Subtopic

Prime Factorization

Subtopic

Adding & Subtracting Integers

Chip Model

Number Line

Equations

5 + (-6) = -1

Understanding Addition and Subtraction

Subtraction of Whole Numbers

Manipulatives, then Paper & Pencil

counting back/on

fact families

take away models

counting blocks

doubles

tens

number line

Number Sequences

Arithmetic

addition/subtraction

an = a1+(n-1)d

Fibonacci

previous term + current term = next term in the sequence

1, 2, 3, 5, 8, 13...

Geometric

multiplication/division

an = a1r^n-1

Finding the nth Term

Venn Diagrams

Inverse Operations

Multiplication is the inverse of division

Addition is the inverse of subtraction

Understanding Multiplication & Division

Mental Calculation

32 x 6 = (30x6 )+ (2x6) = 180 + 12 = 92

Repeated Addition

3 x 4 = 3+3+3+3

Long Division

Divide

Multiply

Subtract

Check if less

Bring down

Circle

Functions

Ordered Pairs

Equation

Graph

Evaluate a Function

f(x) = x+2

f(1) = (1) + 2

f(2) = (2) + 2

f(3) = (3) + 2

Make a Table

Graph

Subtopic

Ordered Pairs

Arrow Diagram

Subtopic

Main topic

Subtopic

Main topic

Subtopic

Fractions

Fraction Diagrams

2-D Parts of a Whole

3-D Example

Number Line

-3...-2...-1...0...1...2...3

Subtopic

Rounding Fractions

4 6/7 + 5 2/9 =

5 + 5 =

= 10

Decimals to Fractions, etc.

Equivalent Fractions

3/4 = 6/8 = 12/16

Adding & Subtracting Fractions

Fraction Bars

1/2 + 1/2 = 1

Subtopic

Vocabulary

Whole

Part

Fraction

Numerator

Denominator