# Math for CGI

## 3-5

### with Mark McClure

Summer, 2017

• Ph.D 1994 in Mathematics - Ohio State
• Professor at UNC-Asheville since 1997
• Third year working with FCR-STEM

## Scholarly interests

• Fractals and chaos
• Computational tools to
• Aide the research process
• Illustrate mathematics at all levels
• Teaching

## Why am I here?

• To teach you some math!

## Professional development

• As a participant in a professional development program, you're here to improve your teaching of mathematics.
• I'm told that many of the CGI techniques you're learning have been in practice for some time.
• Research shows they work.

## So..., why am I here?

• Research on CGI is ongoing.
• One new question: how does teacher content knowledge impact student learning?
• Much of the math we see will be applicable in your classroom fairly directly.
• Direct applicability is not necessarily the main objective, though.

## But..., why am I really here?

I happen to be the happy father of a proud 8 year old.

## What kind of math?

• Practical arithmetic
• with Algebra!

## Question

$4 \times 23=?$

## Using a little algebra

\begin{align} 4 \times 23 &= 4 \times (20+3) \\ &= (4\times20)+(4\times3) \\ &= 80+12=92 \end{align}

Looks like we used the

## Distributive Property

$$a\times(b+c) = a\times b + a\times c$$
$$3\times(2+4) = (3\times2)+(3\times4)$$
$$4 \times (20+3) = (4\times20)+(4\times3)$$

We also used

### substitution

If $a=b$, then $P(a)=P(b)$

$23=20+3$, so $4 \times 23 = 4 \times (20+3)$.

## A direct model

$3\times(2+4)$ $=$  $3\times2+3\times4$

## Can you think of another approach?

\begin{align} 4 \times 23 &= 4 \times (25-2) \\ &= (4\times25)-(4\times2) \\ &= 100-8=92 \end{align}

## A symbolic question

$4\times(2x+3)=?$

Note how similar this is to the last question.

In fact, if $x=10$, this is the last question.

## Using a little algebra

\begin{align} 4\times(2x+3) &= (4\times 2x) + (4\times 3) \\ &= 8x + 12 \end{align}

## Next few days

### The basic algebraic properties

• Substitution
• Associativity
• Commutativity
• Distributivity
• Arithmetic using these

### Algebraic properties and more

• How the rules arise from direct modeling
• Using the rules to do arithmetic
• Standard algorithms

### Relational thinking

• Uses of the equals sign
• Algebra

### More!

• Geometric/symbolic feedback
• Place value
• Something fun