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4. INFINITY

To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour.

— William Blake

MAKING 10 MENU	MATH + CODING MENU
4. Infinity — In the classroom 4A. Infinity in your hand 4B. Infinity in a walk 4C. Walk fractions with Scratch 4D. Sum of fractions with Python 4E. Modelling infinity with Python 4F. What is natural density? 4G. Natural density with Scratch 4H. Mathematician interview	Home page 1. Squares and spirals — introducing variables [gr. 3-9] 2. Making 10 — random coordinates line up [gr. 1-10] 3. Inequalities — in 1D, 2D & 3D [gr. 1-10] 4. Infinity — it’s big but fits in your hand [gr. 3-12] 5. Binary choice — probability, Pascal’s triangle & algebra [gr. 1-12]

IN THE CLASSROOM

Ontario Curriculum

Gr. 3-8
C1. – identify, describe, extend, create, and make predictions about a variety of patterns, including those found in real-life contexts
C3.1 – solve problems and create computational representations of mathematical situations
- C1.2 – create and translate repeating, growing & shrinking patterns using various representations
Gr. 4/5
- B1.4 – use drawings and models to represent fractions
- C.3.2 – read and alter existing code; sequential, concurrent, and repeating & nested events
Gr. 9
- B1.1 – research a number concept to tell a story
- B1.3 – use patterns and number relationships to explain density, infinity, and limit as they relate to number sets
- C2 – apply coding skills to represent mathematical concepts and relationships dynamically, and to solve problems, in algebra and across the other strands
Gr. 12
- infinity & limit in Calculus

Implementation

Gr. 3-9
- Using 8×8 square grids, student shade to represent the fractions 1/2, 1/4, 1/8, 1/16 & 1/32.
- They use scissors to cut out the shaded parts and combine them to form a new shape.
- They are surprised to discover that all the fraction (to infinity) pieces fit in a single square, and that they can hold infinity in their hands.

Gr. 7-9
- Using Scratch code that models the relative density of number sets, they edit the code to solve density puzzles.
Gr. 12
- Students investigate infinity and limit in the context of Calculus

4A. INFINITY IN YOUR HAND

Let’s shade squares to represent an infinite number of fractions.

For example, the image below shows how to represent the fraction 1/2.

And the image below shows how to represent the fraction 1/4.

Q1. Now you do this, using identical squares, like the ones shown below.

Sketch these squares on a piece of paper
Or, print this PDF handout of the squares

Shade the first square to represent the fraction 1/2
Shade the second square to represent half of 1/2, or 1/4
Repeat for fractions 1/8, 1/16 and 1/32

Q2. Now you need a pair of scissors.

Use the scissors to cut out the shaded parts (as shown on the right)
Then join all the shaded parts to form a new shape
Imagine doing this forever, shading, cutting out, joining
How big would the new shape be?
Would it fit in your room? In your house? In your town or city?

The animation below summarizes the above activity.

4B. Infinity in a walk

Q3. Imagine walking to an open door this way:

Walk half way to the door
Then, walk half of the remaining distance to the door
Then, walk half of the remaining distance to the door
Keep doing this forever
Will you ever get to the door?
Will you ever walk past the door?

The fractions you walk on the way to the end of a path

Q4. Stuck? Try it another way:

Don’t think about the fractions
Just walk to the door
Then stop and look back
Use your imagination to see the infinite number of fractions you walked to get to the door

4c. WALK FRACTIONS WITH SCRATCH

Q5. The code below walks fractions 1/2, 1/4, 18 and so on. You can run the code at https://scratch.mit.edu/projects/1056011059/editor

Alter the code to find sums of these fractions of 300:
- 1/2, 1/8, 1/32, 128 and so on
- 1/4, 1/16, 1/64, 1/256 and so on
- 9/10 + 9/100 + 9/1000 + …
What did you learn?

4D. SUM OF FRACTIONS WITH PYTHON

Q6. The code below finds the sum of the first 10 fractions 1/2, 1/4, 1/8 …

Enter the code at https://cscircles.cemc.uwaterloo.ca/console
Run the code.
Edit the code to model the sums of each of these fractions:
- 1/4 + 1/16 + 1/64 + …
- 1/2 + 1/8 + 1/32 + …
- 9/10 + 9/100 + 9/1000 + …
What did you learn?

4E. MODELLING INFINITY WITH PYTHON

Follow this link to investigate infinity + limit by editing snippets of code: https://colab.research.google.com/drive/1D7Z7Uxsw-qgwG5odFxzWZ6oDbL3SH4u7?usp=sharing

Fractions 1/2, 1/4, 1/8 (blue) and their sums (green)

Q7. Edit the code to model the sums of each of these fractions:

1/4 + 1/16 + 1/64 + …
1/2 + 1/8 + 1/32 + …
9/10 + 9/100 + 9/1000 + …

Investigate sums of other fraction patterns.

4F. What is natural density?

Natural density of even numbers

Natural numbers = {1, 2, 3, 4, 5, 6, 7, 8 …}
Even numbers = {2, 4, 6, 8 …}
The natural density of the even numbers = the chance of randomly selecting a natural number that is even = 0.5
We may also write this as d(even numbers) = 0.5

Q8. Does d(even numbers) = 0.5 makes sense?

Every other natural number is even.
Natural numbers = {1, 2, 3, 4, 5, 6, 7, 8 …}
Therefore, the natural density of the set of even numbers is 1/2 or 0.5.
How might you also explain this using a dot density diagram, like the one shown on the right?

Natural density of other multiples

Q9. What is the natural density of the each of the following subsets of multiples?

{100, 200, 300, 400, 500 …}
{3, 6, 9, 12, 15, 18 …}
{5, 10, 15, 20, 25, 30 …}

Q10. More

The natural density of the number 1 = 0.
- How might this make sense?
The natural density of square numbers (1, 4, 9, 16 …) = 0.
- How might this make sense?

4G. Natural density with Scratch

Natural density of {1} using Scratch

The code below uses experimental probability to approximate the natural density of even numbers. The code is available at https://scratch.mit.edu/projects/550227059/editor/

Q11. Alter code to estimate the natural density of {1}.

Execute the code to see the output.
Alter the code to randomly pick numbers from 1 to 1000.
What do you notice?

NATURAL DENSITY OF SQUARE NUMBERS USING SCRATCH

The Scratch code below models calculating the density of square numbers in the interval 1-10 of natural numbers. The Scratch code is available at https://scratch.mit.edu/projects/565845359/editor

Q12. Change the interval size and notice the effect.

4H. Mathematician interview

We interviewed mathematician Graham Denham (Western University) and he completed, extended and discussed some of the infinity activities above.

1/2 + 1/4 + 1/8 + … = 1

How can the sum of a never-ending sequence of fractions have a finite sum?
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64
= 63/64
= 1 – 1/64
“It looks like there’s some suspicious pattern going on here.”
“It turns out that you can make some real mathematical sense out of saying that an infinite sum is actually equal to 1.”

Animation of 2 series in a square

0.9999… = 1?

“I have to confess something that bothered me when I was in about grade 5.”
“How is it possible that 0.99999… = 1?”

Such infinity paradoxes puzzled mathematicians and philosophers of the past.

MAKING 10 MENU	MATH + CODING MENU
4. Infinity — In the classroom 4A. Infinity in your hand 4B. Infinity in a walk 4C. Walk fractions with Scratch 4D. Sum of fractions with Python 4E. Modelling infinity with Python 4F. What is natural density? 4G. Natural density with Scratch 4H. Mathematician interview	Home page 1. Squares and spirals — introducing variables [gr. 3-9] 2. Making 10 — random coordinates line up [gr. 1-10] 3. Inequalities — in 1D, 2D & 3D [gr. 1-10] 4. Infinity — it’s big but fits in your hand [gr. 3-12] 5. Binary choice — probability, Pascal’s triangle & algebra [gr. 1-12]

by George Gadanidis, PhD