*Written by Sara at Classically Homeschooling*.

Eventually kids run into the concept of limits during their math classes. In traditional classes, it’s common to use graphs to explain this idea. But when you homeschool, you can make learning a bit more hands-on, which is how I prefer to teach my children.

This fun hands-on math activity is a great (and simple) way to explore this concept with middle school students!

## How to Teach Limits: Hands-on Middle School Math

Sometime around 450 BCE, Zeno of Elea, one of the ancient Greek Philosophers, gave a puzzle to his students.

*A tortoise came up to Achilles and proposed a race.*

*You know who Achilles is, right? The hero of the Illiad (think Trojan War). The guy who died of an arrow in his ankle. The mighty hero himself!*

*Well we’ve met the tortoise in Aesop’s Fables. He’s a pretty smart guy.*

*So the tortoise comes up to Achilles and proposes a race. Achilles fell down laughing. A slow, crawling, reptile is challenging ME to a race?*

Why yes. However I need a head start. 10 meters should be enough.

*Achilles laughs harder, even with a head start he’d quickly catch the tortoise!*

*Then the tortoise begins to explain.*

No matter how fast you run, you will only catch up to where I WAS, not where I am now. After you’ve run the 10 meters, I will have moved farther ahead.

No matter how fast you run, you will only run to where I have been. Not where I will be. I will have always added a small distance before you reach where I am. So you can never catch up to me.

*Achilles sadly conceded the race.*

**So let’s take a look at this concept on the kitchen table.**

Pull out your Achilles and Tortoise. We used two different Lego minifigures.

Set the Tortoise ahead of Achilles. Place a piece of tape or use a piece of posterboard to mark the spot where the tortoise is standing.

Now move “Achilles” to the designated spot, but move the “tortoise” slightly ahead.

Do this again and again and again. No matter how fast Achilles runs, he always ends up where the Tortoise WAS, not where he is.

In the pure world of mathematics, Achilles will always run the where the Tortoise WAS, not where the Tortoise IS. Achilles will get close to the Tortoise. He may be atoms away from the Tortoise, but he will never catch Achilles.

The tortoise is the **limit** for Achilles in the race.

A limit is the value that a function or sequence approaches as the input, or index, approaches some value.

Achilles can approach the tortoise but never quite reaches the tortoise. The tortoise is the Achilles limit!

Isn’t the pure world of mathematics fun!

Let’s take another look at limits through a joke my math professor once told us.

An engineer, physicist, and mathematician were placed on one side of the room. A line was drawn on the other side. They were told they could only go halfway each time.

So the engineer whipped out a calculator and measuring tape. After a few measurements and calculation, he quickly crossed the room.

The physicist pulled out a slide rule and a yard stick. After a few measurements and scribbled calculations, she quickly crossed the room as well.

The mathematician…. well the mathematician has yet to arrive!

So again, we pulled out the LEGO figure to take a look at what’s going on.

First you place one figure at one end of the table. Now measure the table and go halfway.

Measure the table and go halfway again!

As you can tell, if you only go halfway each time, you’ll never reach the end of the table.

The measurement of the table is the **limit **of the infinite series 1/2(table) + 1/4(table) + 1/8(table) + 1/16(table) + 1/32(table) and so on. Once you reach infinity, supposedly you’ll have reached the end of the table.

But not until then!

Hence our poor mathematician is trapped trying to get to the other side of the table.

Try again moving the mathematician 1/4 or 1/8 of the way each time. Does it change the limit or does the limit remain the length of the table?

Limits is a fascinating and fun concept to explore with kids!

Check out these other math activities for homeschoolers!

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