Nothing quite boggles children—and adults—like the concept of infinity.

Photo By: Mommy, You’re Going To Be My Wife One Day, Right? / Lotus Carroll / CC BY 2.0

What is the biggest number? How big is infinity? What’s bigger than infinity? In order to explore the mathematical concept of infinity, parents and teachers can apply a number of practical tools.

1. The Infinity Game

To introduce the concept to students in grades 6-12, teachers can play an integer-naming game with the students. Players should take turns naming integers, where the winner is the player who names the largest integer. Quick-learning students will soon discover that the strategy to winning this game is simply going last (a billion plus one, for example).

After the game, the teacher should discuss with the class the role of infinity in the game. In the game, the fact that the player going last can always win is a consequence of the ‘infinitude’ of the sets of numbers. A teacher should use this game to get students thinking about the concept of infinity: something to which there is effectively no cap or limit.

2. A Concrete Approach

Digi-Block blocks can also be used as a concrete model that helps children visualize infinity. Due to the nesting property of the blocks, many children can quickly conceptualize larger and larger blocks. Below is a picture of the blocks. First, ten singles fit into a holder to create a block-of-10. Next, ten blocks-of-10 create a block-of-100. Then ten blocks-of-100 nest to create a block-of-1000.

While Digi-Block only manufactures blocks up to 1000, your students can imagine what they could create if they had ten blocks-of-1000. What if we had ten more blocks of that block?! Take some time to make up names for these bigger blocks. A few of my favorite are ginormahumongous or ‘HUGE! Digi’. If your students are familiar with creating and joining equal groups (aka muliplication), this might be a good time to reflect on what we are actually doing when we create the next larger block (i.e., multiplying by ten).

Now that you have discussed infinitely large whole numbers, ask your students to think about how many times we can unpack the blocks. What if you could unpack the single blocks?

3. Infinity in Literature

Want to explore the concept through a literary lesson? Try The Cat in Numberland, by Ivar Ekeland. It’s a story of the Hotel Infinity being observed by a cat.

The Cat in Numberland

At Mr. and Mrs. Hilbert’s Hotel Infinity, the resident cat is puzzled. The hotel is fully booked — the rooms are full of Numbers, both Odds and Evens — yet guests continue to arrive...

4. Group Activity - Examples of the Infinite

Have the students make lists of quantities that may or may not be infinite and share them with their classmates. Some examples that encourage students thinking about infinity include the number of planets in the universe, the number of humans who have ever lived on Earth, the number of words in the English language and so on.

Bring up the following questions, which frequently double as puzzlers and philosophical queries:

• What are examples of finite vs. infinite things?  Grains of sand on a beach?
• Can one infinity be larger than another? How?

5. Presentation - The history of the Infinite

Divide students into small groups and have them further research the concept of infinity and its role throughout the history of mathematics. Have each group choose a particular topic and prepare a short presentation from their research on their use of infinity. After each presentation, have a whole-class discussion about the topic.

The Common Core Mathematics Standards for grades 6-12 include the following standards relating to infinity:

• Apply and extend previous understanding of numbers to the system of rational numbers
• Know that there are numbers that are not rational, and approximate them by rational numbers
• Use properties of rational and irrational numbers

The Common Core Mathematics Standards Practices include:

• Reason abstractly and quantitatively
• Construct viable arguments and critique the reasoning of others
• Look for and express regularity in repeated reasoning