“When exactly in my whole life am I ever going to have to simplify:”

Bernie Frankpitt

What is algebra?

What is algebra?

• “Algebra is the Language of Mathematics”

What is Algebra?

• “Algebra is the ‘Language of Mathematics’”– Yuck!– Languages of Mathematics are:

Greek, Sanskrit, Arabic (Ancient)Latin (To 19th Century)French, German, Russian (Present)English (Most Papers are published in English)Maori (Language for the Future?)

What is Algebra?

• “Algebra is the part of mathematics that deals with the abstraction of structure”

• You need to know something about the structure before you think of abstraction

• Pre-algebra is about identifying structure in concrete contexts

• Warning!I’m kind of opinionated about this

For Example

• Numeracy (arithmetic) is based on– Commutative Property– Associative Property– Distributive Property Structural Properties– Negatives of Fields– Reciprocals– Identities (0 and 1)

The key to understanding algebra is…

The key to understanding algebra is…

Strong concrete examples of the algebraic structure that you are trying to abstract

So ….

“When exactly in my whole life am I ever going to have to simplify:”

When using prime factorizations to calculate factorizations, lowest common denominators and largest common divisors.

... One Answer

Teaching Sequence

1. Factor Trees and Prime Factorizations2. Divisors 3. Simplifying Fractions4. Multiples 5. Adding and Subtracting Fractions

These are all ‘Hard Problems’

Prime Factorisations

• Calculate prime factorisations for all the numbers from one to 200

• Use this as a table for conversions between positive integers and prime factorizations

• Sort Prime Factorizations in “Alphabetical order” for a reverse lookup table

Divisors

Is 21 a divisor of 798?

and

Answer = _________

Simplifying Fractions

• Simplify

• Answer =

Multiples

• What is the lowest common multiple of 105 and 708?

• Answer =

Adding and Subtracting Fractions

Where’s the algebraic structure here?

We’re digging into:• Multiplicative structure of rings• Polynomial rings• Primes• Rational Functions

The prime factorization tables are more of a teaching aid than a useful mathematical idea.