54 votes
Accepted

The concept of infinity for a 5 year old

This does not directly concern the $\infty+1=\infty$ issue and I am not certain that I understand what you mean by his previous understanding of mathematics, but I wanted to give the following ...
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  • 1,012
26 votes

The concept of infinity for a 5 year old

First of all, regardless of age, people need to understand that "infinity" is not a number, and not a placeholder for a number, but an attribute of them (i.e. the fact that you can increase numbers ...
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  • 709
19 votes

The concept of infinity for a 5 year old

On a piece of paper, he started with writing 10, then 100, then 1000, .... and he stopped after writing 40 zeros with 1. Then he came to me and said, "I understand infinity now; infinity is a number ...
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  • 771
17 votes

The concept of infinity for a 5 year old

I'm not sure why the two basic things adults seem to say about infinity are "infinity is not a number" and "∞+1=∞", both of which are at best misleading. (Infinity doesn't name a number, but it does ...
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16 votes

The concept of infinity for a 5 year old

Speaking as someone who was that kid, you might be able to explain $\infty + 1 = \infty$ via the Hilbert hotel. Imagine a hotel that has an infinite number of rooms, one for every number. Imagine the ...
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  • 261
14 votes

Nice examples of limits to infinity in real life

If you are willing to take some time to explain the model and do some simulation, I really like to show students a logistic growth model (in discrete time). The basic setup is something like the ...
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11 votes
Accepted

Motivating example for sequences, sums and limits in high school

This application is known as "gross-up" in accounting. You run the finances for a small business. The boss would like to give an employee a \$100 bonus for their hard work. However, the ...
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10 votes

Reasoning outcomes of simplifications involving infinity

"How to describe these situations to our students" ... At any level below calculus, tell the students: "You will cover this later." This is a special case of: Do not try to tell ...
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  • 6,377
9 votes

The concept of infinity for a 5 year old

My son, also 6 yo, regularly talks about millions and billions and infinity. Obviously, large numbers have some attraction to children of this age. I try to explain that infinity is not a number. ...
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9 votes

Reasoning outcomes of simplifications involving infinity

What resolved these questions for me was when a teacher explained that you cannot add or multiply infinitely many things together: all infinite sums are actually limits. It can help to think of ...
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  • 2,109
7 votes

Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?

This has certainly been tried before. See for example, H. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. On-line Edition. This has also been published in print by Dover. ...
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7 votes

Nice examples of limits to infinity in real life

Deeba and Rushkady Go to Town: A Fanciful Real-Life Story A festival was just starting in town, and Deeba and Rushkady walked toward the square headed to the Infinite Pancake Eating Contest. The ...
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6 votes

Can we explain to undergraduates how points make a line?

This is what I came up with thinking about your question. I would start by exploring what it means to say that a line is 'made up of' points, because I think that is a really important thing that ...
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  • 5,568
6 votes

The concept of infinity for a 5 year old

I am going to answer your question by suggesting a couple of books which might be fun to read with your son: The Phantom Tollbooth by Norton Juster. The book is a rather surreal adventure trip ...
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6 votes

Motivating example for sequences, sums and limits in high school

I'm not sure that starting with an applied motivation (derivation, word problem) is the best way to introduce this topic. Look at how your experiment failed. This is because "word problems are ...
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6 votes

Reasoning outcomes of simplifications involving infinity

"When you keep on multiplying 1's such as, 1×1×1×... can you say here we have product of infinite number of 1's or is this limit of product of n number of 1's as n tends to infinity?" In ...
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5 votes
Accepted

Intuition explanation about Lebesgue measure zero of the rational numbers

Let me build on the idea of Steven Gubkin in his comments. One way to visualize this scenario is to use Ford circles. The standard picture is to plot a circle tangent to the $x$-axis at $\frac{p}{q}$ ...
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  • 7,703
5 votes

The concept of infinity for a 5 year old

There is a well-known Christian hymn, Amazing Grace, whose last lyric captures the idea of (countable) infinity quite well, and may be more effective to a five year old because it includes a context ...
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5 votes

Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?

Speaking as a former student, though an engineering one . . . It was hard enough learning to integrate tricky expressions and solve differential equations, without having to learn a new number system ...
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  • 551
3 votes

The concept of infinity for a 5 year old

My children both learned about infinity at around four to five years old (now 5 and 7). For both of them it was fairly straightforward; it came about with my eldest when he was talking to other kids ...
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  • 161
3 votes

The concept of infinity for a 5 year old

I would start by saying something along the following lines... "You're asking some very grown-up questions for someone that's only 5. Are you ready to do some really, really, grown-up thinking about ...
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  • 1,214
3 votes
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Cardinal vs. ordinal: When learned? When needed?

Is the distinction between cardinal numbers and ordinal numbers taught as part of mathematics (as opposed to part of learning the language distinction between "one" and "first") in pre-college or ...
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3 votes

The concept of infinity for a 5 year old

I've no idea whether this would work, but would relating it to forever hekp? Infinity is like forever but for nunbers. Doing something for a week and then forever is the same as just doing it forever. ...
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  • 551
3 votes

The concept of infinity for a 5 year old

I suppose one problem is that your son looks at $\infty$ the same way he looks at $10$. But infinity is not a natural or real number, even though it has a symbol and can be used in "equations" like $\...
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3 votes

Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?

It's not really that relevant since the bulk of a normal calculus course (e.g. AP BC, Thomas Finney, Stewart) just does a small amount of epsilon-delta (so student is exposed to it) and then moves to "...
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  • 39
3 votes

Can we explain to undergraduates how points make a line?

I agree with the premise, lines are (in some way) made out of points, and points have no length. If, restricting ourselves to a straight line, we can consider these as subsets of the real numbers ...
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  • 333
3 votes

How can I familiarize elementary school students with infinities larger than $\aleph_0$?

Los Alamos National Laboratory wrote a nice lesson plan in the mid-90s around an embellished version of the Hilbert Hotel story, including the alignment to the 1989 NCTM standards: Hotel Infinity (...
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