1.a.4. EXTENSIONS

1. WHAT IF ZERO VANISHED?

We asked Western University applied mathematician Lindi Wahl what would happen if zero vanished. Here is her reply:

My sisters and I had some fun ... and we have a song for you.
We decided that the concept of zero is so basic that you could never really get rid of it. You could stop calling zero a number, but it would still be exactly what it's always been -- nothing.

I AM NOTHING ... download the mp3 file

You can take away my name
I am nothing all the same...
You can say that I'm not there
But I never have been ...  I am nothing.

I'm the absence, I'm the nil
I'm the void you cannot fill...
Call the numbers what you will,
I'm platonic. I'm ideal. I am nothing.

You can take away my sign (my oval)
I exist outside of time...
Erase me from your number line
But that empty space is mine. I am nothing.

 

I'm the absence, I'm the nil
I'm the void you cannot fill...
Call the numbers what you will,
I'm platonic.  I'm ideal.  I am nothing.

You can say I don't belong
In your cluttered counting song...
But when all the rest have gone
 5 4 3 2 1 ...  I am nothing.

I'm the absence, I'm the nil
I'm the void you cannot fill...
Call the numbers what you will,
I'm platonic.  I'm ideal.  I am nothing.

 

The ancient Babylonians (3,000 BC) used a sexadecimal (60 digit) number system. We use a decimal (10 digit) number system.

Interestingly, the ancient Babylonians did not have a symbol for zero.

They left a blank space in numbers where a zero would have appeared.

What if zero did vanish?

What would change in your life if we didn't have a zero?

 

 

2. ODDS, EVENS & NATURALS WITH PYTHON

Use the University of Waterloo Computer Science Circles website to write computer code using the computer programming language Python.

Natural Numbers

The code below prints the first 10 Natural numbers and their sum.

 

 

Even Numbers

The code below prints the first 10 even numbers and their sum.

 

 

Odd Numbers - version 1

The code below prints the first 10 odd numbers and their sum.

 

 

Odd Numbers - version 2

The code below prints the first 10 odd numbers and their sum using a different coding method.

 

 

3. ODDS, EVENS & NATURALS WITH SCRATCH

Scratch is an easy-to-learn computer coding language, developed at MIT.

You code in Scratch by dragging/dropping/combining blocks of code.

For example, the code for creating a bar graph of the first 10 Natural numbers looks like this.

Go to http://scratch.mit.edu/projects/39711816/#editor to see and use this code. Click on the green flag to run the code.

At the above link you will also find code for creating bar graphs of:

  • the first 10 even numbers
  • the first 10 odd numbers
  • the first 10 natural, even and odd numbers, side-by-side, as shown below

 

Use-edit-create

Go to http://scratch.mit.edu/projects/39711816/#editor

  • use the code to see what it does
  • edit the code to see what the effect is and to better understand how it works
  • create your own code, to model a different math relationship

 

 

4. WHO'S SCARING YOUNG CHILDREN AWAY FROM MATH?

Young children enter school mathematically curious, capable and enthusiatic - they have to learn to be otherwise (Papert, 1980).

Children become conditioned by the negative attitudes in our society towards math. What can we do about ?it

Here is a song based on comments made by teachers in an online math for teachers course.

When I was young
Math was hard
Overwhelming
Because of my fear for math

My parents disliked it
It’s scary, scary, scary
I always heard from everyone
It’s “the big bad wolf”

Now I’m the parent
That parent from the past
Scaring my children
With fears of math

Children love to copy cat
Copy, copy cat
Listening
ALWAYS listening!

But my mind has been refreshed
Math’s not scary after all
My feelings have changed
It’s something fun, delicious

I don’t want to be
The Big Bad Wolf
I’m excited about math
It’s “cool” to understand it

One negative comment
Creates a whole imagination’s worth
Of a child’s self-doubt
Self-doubt

Children love to copy cat
Copy, copy cat
Listening
ALWAYS listening!

 

 

5. INTERVIEW WITH A MATHEMATICIAN

See an interview with Dr. Ken Davidson on "The Sum of Odd Numbers", University of Waterloo, Canada.

 

What different methods does Dr. Davidson use to find the sum of odd numbers?

View the clip "As a mathematician". What does Dr. Davidson say interests him as a mathematician?

 

 

6. SONG FROM THE ALDERVILLE FIRST NATION

Middle school students from the Alderville First Nation Learning Centre developed digital stories of (1) of their community and (2) of mathematics.

Using these stories as a basis, the students worked with aboriginal performance artists Tracy Bone, JC Campbell and Dave Mowat to perform live performances for their community.

Notice what the students wrote about odd numbers in their second verse. What story can you tell about odd, even and natural numbers?

Numbers in my hands
That make L patterns
And fit like spoons
Numbers that make squares

See their performance of their song.

 

 

7. ODDS, EVENS & NATURALS ACROSS THE GRADES

The "where do odd number hide" activity was first done in grade 2, to address the content topic of "growing patterns."

It can also be used to address other content topics, across the grades.

For examples, here is a partial list of Ontario curriculum connections:

  • Grade 2— Patterning and algebra: identifying and describing repeating patterns and growing.
  • Grade 3— Patterning and algebra: creating and extending growing and shrinking patterns.
  • Grade 6— Patterning and algebra: describing pattern rules in words; calculating any term when given the term number.
  • Grade 7— Patterning and algebra: This concept fits into sections covering linear growing patterns and representing patterns algebraically.
  • Grade 8— Patterning and Algebra: This concept fits into representing a pattern or finding the term number in an algebraic pattern.
  • Grade 11— MCR3U Discrete functions. This concept fits into an introduction to arithmetic sequences and series.

What connections can you make across the grades?

 

 

8. WHAT DID YOU DO IN MATH TODAY?

If someone asked you "What did you do in math today?", what story might you share to capture their imagination and surprise them mathematically?

What have you learned about:

  • using coding to represent odd, even and natural numbers?

How did you feel while learning these things?

  • happy or sad?
  • surprised or bored?
  • why?