(c) 2021 George Gadanidis

“I used the ideas in this workshop many years ago, when I taught HS Calculus. About 10 years ago, working in grade 3 classrooms, we also used them as a low floor, high ceiling way of teaching the

contentof area representations of fractions in a rich mathcontext.”George Gadanidis

#### MINDSET

Infinity in a parody of *Moonshadow* (Cat Stevens)

#### ABOUT THE INFINITY + LIMIT WORKSHOP

- For
**students**(enrichment; Covid catch-up),**teachers**(professional learning; online teaching resource) &**parents**(math therapy) **Self-serve workshops**: pick and choose activities to complete- Or, for
**digital certificate**: complete and submit tasks (get PDF) to receive a digital certificate- In collaboration with the STEAM3D Research Lab and Convergence.tech

MENU | TOPICS COVERED |

A. Infinity in a square B. Infinity in a walk C. Mathematician interview D. Infinity performances E. With code | — area representations of fractions — infinity & limit — how mathematicians think — history of mathematics — artistic/story representations — coding extensions |

#### A.1. INFINITY IN YOUR HAND

Is it possible to hold infinity in the palm of your hand, as William Blake’s poem suggests?

“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,Auguries of Innocence

What do you think?

#### A.2. **SHADING INFINITE FRACTIONS**

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

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

We’ll use 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

Now you need a pair of scissors.

- use 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? your house? your city?

#### A.3. **At the AI Academy**

At the AI Academy, robots learn to think mathematically by doing the same activity as above.

#### A.4. **Wonder**

The activity above was designed to help you experience the wonder of *holding infinity in your hand*!

Anthropologist Ellen Dissanayake (author of *Homo Aestheticus: where art comes from and why*) says that humans naturally enjoy experiencing, creating and sharing wonders.

What do you think? It this true for you?

#### A.5. STORYTELLING

Brian Boyd, distinguished professor of literature at the University of Aukland (author of *On the origin of stories: evolution, cognition, and fiction*) says that humans are storytellers.

He adds that creating a good story involves solving the artistic puzzle of offering surprise, insight and a sense of wonder.

#### A.6. AT THE MOVIES

The writer and filmmaker Jon Boorstin (author of *The Hollywood eye: what makes movies work*) explains that movies offer us the pleasure of surprise, where we can flex our imagination and see the world in new light.

When we watch a movie, we typically guess ahead. Boorstin explains that audiences pay for movies not to have their predictions confirmed, “but moment to moment they want to be wrong […] to be surprised” (p. 50).

Is this true for you? Do you prefer predictable movie plots or do you like to be surprised?

#### A.7. SHARE A MATH STORY

Share what you learned about infinity and fractions in a square with a friend or family member.

Share in a way so they experience math surprise and insight.

Ask them:

- What did you learn?
- What did you feel?
- What else do you want to know?

What did *you* learn from this sharing experience?

#### B.1. WALKING OUT THE DOOR

Here’s a puzzle: It is possible to walk out an open door?

Imagine walking to an open door in 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?

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

#### B.2. INFINITY IN HISTORY

The ancient Greek philosopher Zeno (495-430 BC) maintained that it was impossible for someone to walk to the end of a path because to do so they would have to travel an infinite number of distances:

- half of the path distance (1/2 of the path)
- half of the remaining 1/2 path distance (1/4 of the path)
- half of the remaining 1/4 path distance (1/8 of the path)
- and so on …

This creates a paradox:

- On the one hand, we know that we can walk to the end of the path, and even beyond.
- On the other hand, when we think of the infinite number of fractions we walk through, we are confused.

#### B.3. HARE & TORTOISE

Here is one more of Zeno’s paradoxes:

If a hare gives a tortoise a head start in a race, the hare will never be able to catch the tortoise.

Think of it this way:

- by the time the hare reaches where the tortoise started, the tortoise will move a little farther
- by the time the hare reaches to where the tortoise moved to, the tortoise will move a little farther
- by the time the hare catches up to where the tortoise moved to, the tortoise will move a little farther
- and so on …

Such infinity paradoxes puzzled mathematicians and philosophers of the past.

#### B.4. SHARE A MATH STORY

Share what you learned about infinity and walking to the door with a friend or family member.

Share in a way so they experience math surprise and insight.

Ask them:

- What did you learn?
- What did you feel?
- What else do you want to know?

What did *you* learn from this sharing experience?

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

#### C.1. FRACTIONS 1/2, 1/4, 1/8 …

- How can we represent the fractions 1/2, 1/4, 1/8, 1/16, 1/32 and 1/64 as shaded areas?
- How do they relate to one another?
- What if we extend the pattern?

#### C.2. FRACTIONS IN A SQUARE

- ‘How much of a square will the fractions 1/2, 1/4, 1/8, 1/16, 1/32 and 1/64 fill?’
- ‘It would nice to visualize these in the same place. Let’s stuff all of these fractions in the same box. In how many ways can I actually do this?’

#### C.3. 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.’

#### C.4. GEOMETRIC SERIES

- 1/2 = 1 – 1/2 = 1/2

1/2 + 1/4 = 1 – 1/4 = 3/4

1/2 + 1/4 + 1/8 = 1 – 1/8 = 7/8 - ‘You can get as close to 1 as you want. So mathematically we’d say that this represents a limit that converges to the number 1. It converges from below.’
- ‘It’s a famous problem. It’s called a geometric series.’

#### C.5. 1/3 + 1/9 + 1/27 + … = 1/2

- ‘We can do the same game with one-third.’
- 1/3 + 1/9 + 1/27 + 1/81 + … = 1/2
- ‘I like that picture a lot because as I look into it I see this fairly important basic mathematical truth inside the picture.’

#### c.6. TWO SERIES IN A SQUARE

- 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + …

= (1/2 + 1/8 + 1/32 + …) + (1/4 + 1/16 + 1/64 + …)

= 2/3 + 1/3

= 1 - ‘That’s kind of fun. I suppose then if you tried to re-arrange these in other ways you could probably also find other patterns and see some other ways in which the infinite sum fits in the box.’

##### At the AI Academy: Two series in a square

#### C.7. INFINITY ACROSS THE GRADES

- ‘I see that some kids have been doing this in the classroom … someone even dissected the square into triangular regions … that’s very beautiful as well … it’s nice to have a project where you can feel free to play and put things together in more than one possible way.’
- ‘In university level mathematics it’s not so far off. This particular topic is pretty interesting because … this process of taking a limit … is one of the core ideas in Calculus.’

#### C.8. 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.

#### C.9. SHARE A MATH STORY

Share what you learned from the interview with Graham Denham with a friend or family member.

Share in a way so they experience math surprise and insight.

Ask them:

- What did you learn?
- What did you feel?
- What else do you want to know?

What did *you* learn from this sharing experience?

#### D.1. in grade 2

Grade 2 students in Brazil engaged with similar investigations of fractions and infinity, and prepared the song below, called “Infinito”, which they performed for their school community.

#### D.2. **In Grade 3**

In Ontario, Canada, grade 3 students wrote skits of how they might share their learning at home. The skits were turned into lyrics for a song, which was performed by Indigenous recording artist Tracy Bone and Bob Hallett of *Great Big Sea* at a math concert.

#### D.3. **In A PAINTING**

The painting below, hanging in the main hall of St. Matthew the Evangelist Catholic School, in Whitby, Ontario, shows the math/infinity art created by students.

#### D.4. **At the AI Academy**

Three bots investigate infinity + art.

#### D.5. in a summary of ideas

#### D.6. CREATE A MATH PERFORMANCE

Create a math performance about infinity, such as:

- a poem
- a song
- math art
- a short story

#### D.7. SHARE

Share your performance with a friend or family member.

Ask them:

- What did you learn?
- What did you feel?
- What else do you want to know?

What did *you* learn from this sharing experience?

**E. WITH CODE**

#### E.1. MODELLING INFINITY WITH CODE

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

#### E.2. MODEL OTHER FRACTION PATTERNS

Edit the code to model the sums of each of these series:

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

Investigate series of other fraction patterns.

#### E.3. SHARE

Share what you learned about infinity and coding with a friend or family member.

Share in a way so they experience math surprise and insight.

Ask them:

- What did you learn?
- What did you feel?
- What else do you want to know?

What did *you* learn from this sharing experience?