Math for CGI

K-2

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

$26+4=?$

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)

Slip

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

Flip

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