Sometimes there are unpredictable diversions in my lessons that are really hard to resist. They don’t really fit into any Specific Outcomes in our curriculum. I have an excuse for those. The Alberta Program of Studies front matter has a couple of goals that I like to quote. “Students will gain understanding and appreciation of the contributions of mathematics as a science, philosophy and art” and “Students will exhibit curiosity”. And eight year olds are curious about all sort of things when it comes to mathematics.

This lesson happened some time ago. The pictures and notes from it has been sitting in my folder for a while. Sometimes I come back to look at them because they amaze me. I’ve been sharing bit and pieces with my colleagues and on twitter, but I wanted it all in one place, and this post will be the place.

**Can Number Line Be a Number Circle?**

We have just had a few clothesline lessons on integers which extended our number line to the left of zero. Students were gathering on the math carpet, and I overheard a few boys arguing about infinity on the opposite sides of the number line. Will it be the same, or will the other one be negative? If it is the same, can’t you just connect infinity with infinity, and then make a number line a number circle? More students were joining the conversation. I grabbed an anchor chart and started putting down some questions that were happening.

Based on the idea that numbers are creatures that live on the number line, we tried to figure out if infinity is a number.

Someone said, “I saw a video about an infinite hotel, but I don’t know how it works, can we do this puzzle?” My original lesson plan would have to wait until tomorrow.

**Infinite Hotel**

Here is the link to the paradox in question.

So, I told my students a story about a hotel manager, and a lonely traveler arriving at night, and asked what room number will he get if the hotel is full but infinite. At first, everyone jumped with “I get it!”.

S: “There is infinity of rooms, so he will just go to the next room!”

Me: “But what number is that next room?”

Ss: “Infinity.” “Infinity is not a number!” “Maybe it is!”

Me: “And what room number is right before infinity room? How will the guest find it?”

Ss: “Is it an actual normal number?”

Me: “It is a normal number that you can count, so you can give the guest a key.”

Students got together with their table groups to discuss their ideas.

“There are rooms with fraction numbers. So he can go into 1 ½ room.”

“He can quickly build another infinity hotel”.

“He can give the guest his own room.”

“There was a secret room 403 that was left empty.”

Me: “Nope, it was really full”.

S: “People can share a room.”

Me: “Nope, hotel’s motto is *We Have a Room Just For YOU.”*

**Infinity And Us**

By the afternoon, I was curious to know how my students would describe infinity. I tried to come up with an art project, but nothing really came to mind. So the proper art lesson did not happen. Instead, we read the beautiful book *Infinity And Me *by Kate Hosford, and then I asked students to write how they imagine infinity. Then draw it on a square of black paper. Not a full blown art lesson, but it worked with the writing.

*“I imagine infinity as millions of dust in the air.”*

*“I think infinity is like a rainbow because it disappears and comes back.”*

*“Infinity means to me is freedom.”*

*“I imagine infinity as my friendship and love for my family and friends.”*

*“I imagine infinity as a bunch of rocks cause you can always break rocks smaller and smaller and smaller and smaller and… a lot of rocks.”*

*“Infinity mean to me a color wheel with many colors and the wheel spins around and around forever.”*

*“When I think of infinity I thing of earth because it is a circle and circles are endless,”*

*“I imagine infinity would look like a ball rolling around a cup for forever.”*

*“I picture infinity as a dragon going around and around looking for something.”*

*“I think of infinity as the space because if it ends what is next?”*

*“I imagine infinity like someone walking through time without even noticing he is.”*

It will be interesting to look at a different kind of infinity when we get to fractions. I wonder what images will come to my students’ minds when infinity will appear between the familiar numbers, when it will be close, not at a far distance, not vast, but unimaginably small. Sometimes I wonder if diversions like this are justified from the point of view of learning intentions, assessment and outcomes. I think, sometimes it’s worth to stop and look at mathematical landscape and just admire the beauty.