MAT 156
Percents
Terminating Decimal
A decimal that doesn't go on forever.
Decimal Fraction
Whole number over a power of 10.
Can use 100 grid to change fraction to percent.
Per 100
Integers
Arithmetic with integers
Partition model
Repeated subtraction model
Repeated addition
Take away model
use the discrete set model of addition
Ordering integers
numbers to the right of -3 are greater than >
numbers to the left of -3 are less than<
Absolute Value
Look for the magnitude of the number
Number line approach
helps visualize
Opposite of an integer
Chip method
number line approach
a,-a are both the same distance from zero
a and -a
Fractions
Proportional Reasoning
Relative Thinking
Reconizing quantities you have and how they change
Unitizing
Ratio Sense
Understand rational numbers
Proportion
Cross Multiplication
Shortcut to multiply both sides by LCD
A statement that 2 ratios are equal
Relative Reasoning
Absolute Reasoning
A quantitative relationship showing number of times one contains or is contained within another value.
Compares two quantities regardless of whether the units are same.
may look like fraction
Repeated Subtraction (Division of Fractions)
Need to know the size of the group and find the number of groups
Parition(Long Division)
Multiplication
Add/Subtracting
Mixed fraction form
Advantages: no conversion, strengthens conception of place value, more consistent with add/sub on all types of numbers.
Disadvantages: regrouping of fractions
Improper method
Advantages: no regrouping of fraction needed, looks like part-whole context, process is similar to multiplication/division of fractions.
Disadvantages: work with larger numbers, more opportunity for mistakes, multiply/divide within problem.
Simplified
Improper/Mixed
Egyptian Fractions
Write as a sum of unit fractions
all must be unique unit fractions
Pattern Blocks
Help Student visualize fractions when working with shapes.
Interpretations
Ratio
Copies of a Unit Fraction
Is the accompaniement to part-whole. An example is 2/3 is two copies of the unit fraction 1/3.
Part-Whole
2/3 Represents 2 parts of a whole that was divided into 3 parts.
Divisibility
Algorithms
Column
Scaffolding
Partial-Quotients
Array
Tests
10
9
8
6
5
4
3
2
Equivalant Statements
Assuming bIa then,
b is a factor of a.
a is a multiple of b.
b is a divisor of a.
a is divisible by b.
Division is inverse operation of multiplication
Four fact Families
Has two multiplication and two division facts that go together
Multiplication Algorithiums
Partial Product
Multiplication Models/context
Cartesian Product Context (Can use tree diagram)
Area Model
Repeated addition Continuous (number Line)
Multiplcation as repeated addition of whole numbers (discrete)
Subtraction is inverse operation of Addition
Four Fact Families
An example is:
3 +7=10
7+ 3=10
10 - 3=7
10 - 7=3
Has two addition and two subtraction facts that go together
Subtraction Models/Context
Missing addend
Comparison
compare relative sizes of 2 quantities, determine how much larger or smaller one quantity is compared to the other.
Take-away
Have initial quantity and remove a specified amount
Addition Algorithiums
Left to Right Addition
Any column first
Low Stress
Lattice Method
Ways of recording numbers
Hindu-Arabic (usage of digits/numerals, and place values)
Base Blocks
Roman(Symbols in specific order)
Babylonians
Mayans (base 60)
Egyptian Symbols
Tally System
One-to-one correspondence
For each element of A there is an element of set B to match it with no extra elements and no repeated use of an element.
Sequences
Recurrence Relationship
current term is dependent on previous term
Geometric (common ratio)
Rule: an=a1 x r exponent n-1
Arithmetic (common difference)
Rule: an=a1+d(n-1)
Decimals
After the decimal point the place value has th at the end.
When you move the decimal point over on the divisor you have to do the same for the dividend.
Don't have to line up decimals
Doesn't matter which number is on top.
Line up decimal points
Must subtract numbers in order given.
addition
Can only add zero's to the left of the decimal.
Line up decimal points to line up place value.
Modualar Clocks
Division
Inverse multiplication
Multiplication
Arithmetic Properties
Closure Property- if we multiply any 2 values on the clock, do we get another value on the clock.
Commutative- a x b(mod 12)= b x a(mod 12)
Identity- a x ?(mod 12)=a
Inverse- a x b(mod 12)= 1
repeated addition
Subtraction
subtract the number by moving counter clockwise
Addition
Arithmetic Proerties
Closure Property- if we add any two values on the clock do we get another value on the clock.
Commutative- a+b(mod 12) = b+a(mod 12)
Identity- a+?(mod 12)=a
Inverse- a+b(mod 12)=0
move clock wise
Factoring
Relatively Prime
Two numbers don't share any common factors
Least Common Multiple
Greatest Common Factor
Use Cuisenaire Rods to visualize
Composite Numbers
Prime Numbers
Number Theory
Odd numbers
if is one more than even #, 2n+1
Even Numbers
The # is a multiple of 2, 2n
Division Models/Context
Measurement (repeated subtraction)
Need to know: Quantity starting with, and the size of each group.
Will Find: the number of groups
Partition (equal Sharing)
Need to know: Quantity starting with, and the number of groups.
Will find: the size of each group
Properties of Multiplication
Distributive property of multiplication over addition/subtraction on whole numbers (w)
Zero property of multiplication on whole numbers(w)
Identity proerty of multiplicatin on whole numbers(w)
Associative property of multiplication on whole numbers(w)
Commutative Property of multiplication on whole numbers(w)
Closure property of multiplication on whole numbers (w)
Subtraction Algorithiums
European Method
Scratch Method
Properties of Subtraction?
Identity property of subtraction on whole numbers(w)
Associative property of subtraction on whole numbers(w)
Commutative property of subtraction on whole numbers(w)
Closure property of subtraction on whole numbers(w)
No
Addition Tables
the numbers are diagonal and appear in bands
use the commutative property
Properties of Addition
Identity property of addition of whole numbers (w)
Associative property of whole numbers(w)
Commutative property of addition on whole numbers (w)
Closure property of addition on whole numbers(W)
Closure Property of addition
Models/Context
Continuous(number line) Model- Measured quantities
Characterized by combining two continuous quantities. Ex: time, distance, area, volume
Discrete (set) Model- Counted quantities
Characterized by combining two sets of discrete objects. Ex: markers, fruits, animals
Sets
Problem solving tool: Venn Diagram
Set difference
Set union
Set intersection
Proper Subset
Set complement
Subset
Equivalent Sets
Equal sets
Palya's Problem Solving Steps
4. Looking back
3. Impliment your plan
2. Devise a Plan
Some examples are to look for patterns, draw a diagram or symbols, act problem out, break it down, solve a simpler problem, or work backwards.
1. Understand the Problem
