**Please consider completing a brief anonymous research survey. Thank you!**

# 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
- 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
- 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?

**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

**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
(even numbers) = 0.5*d*

**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