Math for CGI

K-2

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

$26+4=?$

Using a little algebra

\begin{align} 26+4 &= \color{blue}{(20+6)}+4 \\ &= 20+\color{blue}{(6+4)} \\ &= 20+10=30 \end{align}

Looks like we used the

Associative Property

$$(a+b)+c = a+(b+c)$$
$$(1+2)+3 = 1+(2+3)$$
$$(20+6)+4 = 20+(6+4)$$

We also used

substitution

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

$26=20+6$, so $26+4=(20+6)+4$.

Direct modeling

(20+6) + 4 = 20 + (6+4)

Another question

$26+17=?$

Using a little algebra

\begin{align} 26+17 &= (20+6) + (10+7) \\ &= 20+6 + 10+7 \\ &= 20+10 + 6+7 \\ &= (20+10) + (6+7) \\ &= 30 + 13 = 43 \end{align}

A direct model

Looks like we threw in the

Commutative Property

$$a+b = b+a$$
$$2+3 = 3+2$$

A symbolic question

$(2x+6)+(x+7)=?$

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} 2x+6+x+7 &= 2x+x+6+7 \\ &= 3x + 13 \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