INFINITY

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Infinity is big! But you can hold it in your hand!
Hold infinity in your hand! Click To Tweet


“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

1. INFINITY IN YOUR HAND

In brief:

The fraction one-half
  • start with some squares, as shown below [get the PDF]
  • shade the first square to represent the fraction 1/2 (as shown at right)
  • shade the second square to represent half of 1/2, or 1/4
  • repeat for fractions 1/8, 1/16 and 1/32
  • use scissors to cut out the shaded parts
  • imagine doing this forever, shading and cutting out
  • then join all the parts to form a new shape
  • how big would the new shape be?

At the AI Academy

In the video below, 3 bots investigate the above activity.

Wonder

Humans naturally enjoy experiencing, creating and sharing wonders. This activity helps you experience the wonder that you can hold infinity in your hand!

In Grade 3

This activity was first designed when teachers in three Grade 3 classrooms asked, “do you have any interesting ideas for teaching area representations of fractions?”

What did you do in math today?

Infinity. It’s real. Tangible.

After completing the activity above, and some of the activities below, the Grade 3 students created skits to practice sharing at home what they learned in math class. Parents were surprised to see that it’s possible to hold infinity in your hand. See some of the students’ skit ideas in the music video below, in 4. Walk out the door.

If you can’t share an experience or tell a story that illustrates the wonder of mathematics, how well have you understood the math concepts you are studying?

infinity painting

Infinity poster you may download.

After the Grade 3 students finished their activities, a painting was created by artist Ann Langeman, depicting key experiences. The painting was donated to their school. Here is a PDF poster of the painting.

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.

In Grade 12 Calculus

The same activity has been used in Grade 12 Calculus, to help students engage with and develop a conceptual understanding of infinity and limit. For example, the sum of the infinite series of fractions 1/2 + 1/4 + 1/8 + 1/16 + … has a limit of 1 .

In teacher education

This activity has also been used in mathematics teacher education settings, to help teachers see that it is possible to use wonderful mathematics contexts (infinity and limit) when teaching more specific mathematics topics (area representations of fractions).

A low floor and a high ceiling

Notice that the above activity is designed to be accessible with minimal prerequisite knowledge (low floor) while also offers connections to more complex ideas (high ceiling). Such a design allows for differentiated instruction and student engagement.

Occasional experiences of wonder

Working in research classrooms in Canada and Brazil, our goal is to engage students with such activities occasionally, say once per unit of study.

We believe that occasional, well-designed aesthetic mathematics experiences “that are immersive, infused with meaning, and felt as coherent and complete” (Parrish, 2009, p.511), and the associated experience of complex, surprising, emotionally engaging, and viscerally pleasing mathematics, can serve as “a process of enculturation” (Brown, Collins and Duguid, 1989, p. 33) with lasting impact on students’ (and teachers’) dispositions, living fruitfully in future experiences (Dewey, 1938), by raising expectation and anticipation of what mathematics can offer.

Gadanidis, Borba, Hughes & Lacerda (2016). Designing aesthetic experiences for young mathematicians: A model for mathematics education reform. International Journal for Research in Mathematics Education, 6(2), 225-244

2. INFINITY IN PIECES

In brief:

Some fractions are squares. Some are not!
  • notice that some of the fractions are squares (1/4, 1/16 …) and some are not (1/2, 1/8 …)
  • how much of the whole square do the square fractions fill?
  • how much of the whole square do the other fractions fill?

At the AI Academy

In the video below, 3 bots engage with this activity.

Mathematician Graham Denham (Western)

The idea of seeing two different patterns of fractions in the square came from Dr. Denham.

See Dr. Denham think through this idea in the video below.

3. INFINITY ART

In brief:

Math art!
  • notice that the patterns in the square look like art
  • find new ways of representing the fractions in the square, to create new infinity patterns and new math art

At the AI Academy

In the video below, 3 bots investigate this math + art activity.

4. WALK OUT THE DOOR

Is it possible to walk out the door?

Try this:

  • 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 door.

Stuck? Try it another way:

  • don’t think about the fractions
  • just walk out the door
  • then stop and look back
  • use your imagination to see the infinite number of fractions you walked to get to the door

Paradox

Why is it that when you think about the fractions in your walk to the door, it is difficult to imagine reaching the door, but easy to do if you don’t think about the fractions?

Zeno of Elea (495-430 BC)

Zeno was a Greek philosopher who posed infinity paradoxes.

Can a fast hare catch up to a slow tortoise with a head start?

In one such paradox, he suggested that if a hare gave a tortoise a head start in a race, the hare would never be able to catch up to the tortoise.

By the time the hare reached the spot of the head start, the tortoise would have moved a bit farther. Then, by the time the hare reached the spot where the tortoise moved to, the tortoise would have moved a bit farther again.

This process never ends and the tortoise is always a little bit ahead of the hare. What do you think?

Infinity is history

Mathematicians struggled with such paradoxes for a long time. Infinity, especially where a limit is involved, was very difficult to grasp.

In a song

Preparing to share with family and friends, a group of grade 3 students wrote a skit of what they might say and do if their parents asked them to take out the garbage.

The skit was then turned into lyrics.

The song performed at a math concert below by Indigenous recording artist Tracy Bone and Bob Hallett of Great Big Sea. Funded by the Fields Institute for Research in Mathematical Sciences and the Social Sciences and Humanities Research Council.

5. DOES 0.99999… EVER END?

Think about this:

  • 0.99999… is the sum of 0.9 + 0.09 + 0.009 + 0.009 + …
  • 0.99999… is the sum of 9/10 + 9/100 + 9/1000 + …
  • the more of its parts that you look at, the closer you get to 1
  • will 0.99999 … ever reach 1?

Try to walk to the door:

  • walk nine tenths of the way to the door
  • then walk nine tenths of the remaining distance to the door
  • then walk nine tenths of the remaining distance to the door
  • keep doing this forever
  • will you ever get to the door?
  • will you ever walk out the door?

Need to think about this some more?

  • 0.99999… = 0.33333… + 0.33333… + 0.33333…
  • 0.99999… = 1/3 + 1/3 + 1/3
  • 0.99999… = 1

Mathematician Graham Denham (Western)

See Dr. Denham think about the value of 0.99999…

6. MODELLING INFINITY + LIMIT WITH CODE

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

Sum of 1/2 + 1/4 + 1/8 + …
Plotting the fractions 1/2, 1/4, 1/8 … (in blue) and their sum (in green)