Math for CGI

3-5

with Mark McClure

Summer, 2017

A little about me

  • 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!

Don't freak out!!

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

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

Distribute

Can you think of another approach?

How about:

\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