{"id":52,"date":"2026-01-25T01:14:56","date_gmt":"2026-01-25T01:14:56","guid":{"rendered":"https:\/\/learnx.ca\/um\/?page_id=52"},"modified":"2026-02-16T20:04:13","modified_gmt":"2026-02-16T20:04:13","slug":"infinity","status":"publish","type":"page","link":"https:\/\/imaginethis.ca\/u\/infinity\/","title":{"rendered":"Infinity in your hand"},"content":{"rendered":"\n

Infinity in your hand?<\/h2>\n\n\n\n

Can you hold infinity in your hand?<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

Most students answer: \u201cNo,\u201d or, \u201cYou could try\u2014but it would spill over.\u201d<\/p>\n\n\n\n

But now and then, a different kind of thinker emerges.
They write infinity<\/em> on a small scrap of paper and place it on their open palm.<\/p>\n\n\n\n

\u201cLook,\u201d they say with a soft triumph. \u201cInfinity\u2014in my hand.\u201d<\/p>\n\n\n\n

Let\u2019s see where this question may lead us next.<\/p>\n\n\n\n

\n\n\n\n

Calculus<\/strong><\/h2>\n\n\n\n

When I taught Calculus, I liked to start with a puzzle. Before we talked about limits or infinity, I\u2019d ask my students to imagine this: Start walking toward the door. Go halfway. Then go halfway again. Then halfway again. Keep halving forever.<\/em> So\u2026 do you ever reach the door, or are you stuck in the classroom for eternity?<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Students rarely agreed, which made conversations wonderful. Some insisted you\u2019d eventually get there. Others were convinced you\u2019d be stuck in an endless \u201calmost\u2011but\u2011not\u2011quite\u201d loop. What do you think?<\/p>\n\n\n\n

Infinity has been confusing\u2014and entertaining\u2014mathematicians for thousands of years. Zeno certainly had fun with it. In one of his famous paradoxes, he gives a turtle a head start in a race against a rabbit. According to Zeno, the rabbit can never<\/em> catch up.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Every time the rabbit reaches the spot where the turtle was<\/em>, the turtle has already shuffled a little farther ahead. This keeps happening forever. Just like your steps toward the door, the gap shrinks and shrinks but never quite disappears\u2026 or does it?<\/p>\n\n\n\n

\n\n\n\n

Grade 3<\/strong><\/h2>\n\n\n\n

While co\u2011teaching in Grade 3 classrooms, the teachers tossed me a challenge: \u201cDo you have any fun ideas for teaching fractions with area models?\u201d<\/em> My brain immediately whispered, Hmm\u2026 what if we use the fractions from our \u2018walk to the door\u2019 adventure?<\/em><\/p>\n\n\n\n

So off we went, shading area representations of 1\/2, 1\/4, 1\/8, and so on.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Out came the scissors. Students snipped out the shaded pieces\u2014little islands of fractions floating on desks.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Then came the twist. We asked them to rearrange all those fractional bits to build a brand\u2011new shape. When they finally pieced everything together, their eyes widened: every single shaded piece fit perfectly into one square!<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Once they got the hang of it, the creativity exploded. Students shaded their fractions in colourful patterns and proudly turned them into math art.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The best part? They marched home to share their creations and announcing with their parents, \u201cLook! I can hold infinity in my hand!\u201d Not a bad day\u2019s work for Grade 3 mathematicians.<\/p>\n\n\n\n

\n\n\n\n

Through the eyes of a mathematician<\/strong><\/h2>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n

I invited Dr. Graham Denham from Western University to take this little fraction adventure for a spin.<\/p>\n\n\n\n

He spotted something the rest of us had completely missed: some of the shaded bits were perfect little squares\u2026 and some were definitely not<\/em>. Naturally, he couldn\u2019t resist rearranging everything, until the two types of shapes stood out clearly.<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

Then he paused and asked: \u201cHmm\u2026 what fraction of the whole square is made up of the square\u2011shaped pieces?\u201d<\/strong><\/p>\n\n\n\n

A wonderful new puzzle was born.<\/p>\n\n\n\n

What do you<\/em> think the answer might be?<\/p>\n\n\n\n

Take a peek at this video clip to watch Dr. Denham puzzle it out in real time.<\/p>\n\n\n\n