History

A. New Math -- Mid 60s - mid 70s

1. Based on Set theory

2. Showed the logical development of math

3. Designed by university professors to prepare students for upper division math classes starting in Kindergarten

4. Not well received by elementary teachers

B. Back to Basics -- Mid 70s - mid 80s

1. Concentrated on skills in order to get good results in achievement tests

2. Students were not able to get beyond basics

C. Reform Movement Mid 80s -

1. Emphasis on critical thinking and problem solving

2. Decreased emphasis on skills

D. The Math Wars Mid 90s -

I. Counting

A. Methods of counting

1. Tally

2. Roman numerals

3. Other systems

a. Egyptian

b. Babylonian

c. Hebrew and Greek

d. Mayan

e. Chinese

4. Hindu - Arabic decimal place holder system

II. Whole number arithmetic

A. Addition

1. Repeated counting

2. Single digit facts

3. Multiple digit techniques

B. Subtraction

1. Two ways of introducing subtraction

a. Take away

b. Un adding

2. Single digit facts

3. Multiple digit techniques

4. Alternative algorithms

a. Regroup before subtracting

b. Austrian Subtraction

c. Subtract from the base

d. Subtract from 999 and add

C. Multiplication

1. Repeated addition

2. Single digit facts

3. Multiple digit techniques

D. Division

1. Unmultiplying or repeated subtraction

2. Multiple digit techniques

a. quotient

b. remainder

E. Powers

1. Repeated multiplication

2. Table of powers

F. Different bases

1. Counting in different bases

a. Converting from base ten

b. Converting to base ten

2. Arithmetic in different bases

a. Addition

i. Table of single digit facts

ii. Multiple digit techniques

4. Subtraction

i. Table of single digit facts

ii. Multiple digit problems

5. Multiplication

i. Table of single digit facts

ii. Multiple digit techniques

6. Division

G. Bucket problems

III. Fractions

A. Expressing fractions

1. Proper and improper fractions

2. Improper fractions and mixed numbers

a. Divide the bottom into the top getting a quotient and a remainder to change from improper fractions to mixed numbers

i. The quotient is the whole number

ii. The remainder is the top of the fraction

b. Reverse the procedure to change back

3. Reducing fractions

a. Greatest common divisors

i. Allows you to reduce in one step

ii. Euclidean algorithm

b. Prime numbers

i. The Fundamental Theorem of Arithmetic

ii. Use products of primes to reduce fractions

B. Arithmetic with fractions

1. Addition and subtraction

a. Common denominators

i. Least common multiples of the denominators

ii. Break them up into products of primes and fill in the missing factors.

b. Mixed numbers

i. Carrying

ii. Borrowing

2. Multiplication and Division

a. Multiply the tops and bottoms, canceling if possible, to multiply

b. Invert and multiply to divide

c. Change mixed numbers to improper fractions if you want to multiply or divide.

C. Ordering of fractions

1. Find common denominators

2. Cross multiply

3. Find common numerators

4. Finding fractions between two fractions

a. Take their average: half their sum

b. Add the tops and add the bottoms

IV. Decimals

A. Definition

1. A decimal is a fraction whose denominator is a power of ten

2. Changing decimals to fractions

B. Arithmetic with decimals

1. Addition and subtraction

a. Line up the decimal points

b. Add or subtract using base ten arithmetic

2. Multiplication

a. Multiply using base ten arithmetic

b. The total number of places behind the decimal point in the numbers being multiplied is the number of places behind the decimal point in the answer

3. Division

a. Reverse your steps in multiplying

i. Move the decimal point to the end of the divisor and move it the same number of place in the dividend before bringing it up into the quotient

ii. Divide using base ten arithmetic

b. Changing fractions into decimals

i. Divide the bottom into the top and keep going if it doesn't come out even

ii. Repeated decimals

C. Irrational numbers

1. Decimals that go on forever without repeating

2. Square roots for example

V. Statistics

A. Presentation of data

1. Rank the scores

2. Frequency histograms

3. Stem and leaf plots

4. Box and whisker plots

B. Statistics

1. Max

2. Min

3. Mean

4. Median

5. Mode

6. Upper quartile

7. Lower quartile

8. Variance

9. Standard deviation

VI. Set theory

A. Primitive notion: a set is a collection of objects

B. Notation

1. Roster method

2. Set builder notation

C. Operations

1. Union

2. Intersection

D. Subsets

1. Definition of equality

2. Empty set

a. The empty set is a subset of any set

b. There is only one empty set

3. Pascal's triangle

E. Complements

1. The need for a universal set

2. DeMorgan's Laws

VII. Probability

A. Probabilistic model

B. Events

C. Counting

1. The multiplication principle

2. Permutations

3. Combinations

D. Expected value

E. Odds