Nspired Connections from Outside the Box

Goals of Today's Presentation

Highlight unexpected connections among early grades, secondary, and post-secondary mathematics

Illustrate how these connections are strengthened through the use of hands-on materials and technology

Highlight ways in which technology affords users opportunities to find deep connections that otherwise might be missed

Up Next: Completing Squares

A typical completing the square

example with algebra tiles

Surely, you are . . .

Joking!

Where do we

go next?

Are there other ways

to complete a square?

What if we allow ourselves

the luxury of adding x-squared

tiles in order to complete

the square?

How can

we square a

circle?

How can we square a line?

In conclusion

The BIG idea

When we limit ourselves to conventional

ways of thinking about conventional tasks,

we risk missing important new ideas and

connections.

We miss opportunities to share new discoveries, genuinely cool mathematics . . . original mathematics . . . beautiful mathematics, with our students.

Graph of Square Completers

Graph of Negative Square

Completers

False Conjecture

Negative Square Completers

with Generating Function

Did you find all of these? Are we missing any?

While factored forms associated with these shapes are

algebraically equivalent, geometrically speaking the shapes

are distinct.

Making Squares: Solve with a neighbor

Using at least one square tile and as many other pieces as you want, how many different squares can you make?

A First Example:Making Rectangles

2

Making Rectangles: You have a collection of one x-square tile, four x-tiles, and five unit tiles. Using the x-square tile and any of the other tiles, construct as many rectangles as you can.

These squares will play an important role in the remainder of our talk.

Ladies and Gentlemen, consider the graph of the family of square completers for a given p(x).

Steve Phelps

Nspired Connections from Outside the Box

Michael Todd Edwards

What if we allow ourselves the luxury of adding whatever tiles we need in order to make a square?

Let's explore this "live" with TI-Nspire CAS by graphing the family of square completers.

At a recent teaching conference, a presenter shared his use of virtual tiles on an interactive whiteboard.

While discussing pedagogical advantages and limitations of such an approach, the presenter made an

off-handed comment that was rather intriguing to us.

"Tiles aren't helpful when completing the square with trinomials containing odd linear coefficients (such as x^2+5x+1) because odd numbers of x tiles can't be split into two piles with the same number of pieces."

Steve wasn't so sure about this (and we're glad he wasn't)!

x+8

The notion that mathematical ideas are connected should permeate the school mathematics experience at all levels. As students progress through their school mathematics experience, their ability to see the same mathematical structure in seemingly different settings should increase (NCTM, 2000, p. 64).

As a matter of fact . . .

We should really listen to Billy more often. Rich mathematics is overlooked when our gaze is fixed "within the box."

2x

15

x

Where the "conventional"

completing the square

model fails

(Billy)

3x+15

Generate a list of expressions that could complete a square

An odd number of x tiles?

????

Student: "How can we make a square from this?"

Countless Teachers: "You can't, Billy."

Student: "Really?"

with additional x-square

tiles whilst thinking

What completer is suggested here?

Completing a square

outside the box.

A graph of a family of square completers,

each member containing at least one square tile.

How can we

complete a

cube?

with quadratic family members.

A family of cube completers

with cubic family members.

A family of cubic completers

Inside the classroom,

Sly as a fox,

Mathematics rocks,

Outside the box.

Going outside,

Outside the box,