Did you know that numbers can be polite? In math, a *polite number* is any number we can write as the sum of two or more consecutive positive whole numbers.

(*Consecutive* means numbers that come one right after another in the counting sequence.)

For example, five is a polite number, because we can write it as the sum of two consecutive numbers:

5 = 2 + 3

Nine is a doubly polite number, because we can write it two ways:

9 = 4 + 5

9 = 2 + 3 + 4

And fifteen is an amazingly polite number. We can write fifteen as the sum of consecutive numbers in three ways:

15 = 7 + 8

15 = 4 + 5 + 6

15 = 1 + 2 + 3 + 4 + 5

How many other polite numbers can you find?

### What Do You Notice?

Are all numbers polite?

Or can you find an *impolite* number?

Can you make a collection of polite and impolite numbers? Find as many as you can.

How many different ways can you write each polite number as a sum of consecutive numbers?

What do you notice about your collection of polite and impolite numbers?

Can you think of a way to organize your collection so you can look for patterns?

### What Do You Wonder?

Make a conjecture about polite or impolite numbers. A *conjecture* is a statement that you think might be true.

For example, you might make a conjecture that “All odd numbers are…” — How would you finish that sentence?

Make another conjecture.

And another.

Can you make at least five conjectures about polite and impolite numbers?

What is your favorite conjecture? Does thinking about it make you wonder about numbers?

Can you think of any way to test your conjectures, to know whether they will always be true or not?

### Real Life Math Is Social

This is how mathematics works. Mathematicians play with numbers, shapes, or ideas and explore how those relate to other ideas.

After collecting a set of interesting things, they think about ways to organize them, so they can look for patterns and connections. They make conjectures and try to imagine ways to test them.

And mathematicians compare their ideas with each other. In real life, math is a very social game.

So play with polite and impolite numbers. Compare your conjectures with a friend.

Share your ideas in the comments section below.

And check out the list of student conjectures at the Ramblings of a Math Mom blog.

CREDITS: Numbers photo (top) by James Cridland via Flickr (CC BY 2.0). I first saw this activity at Dave Marain’s Math Notations blog, and it’s also available as a cute printable Nrich poster. For a detailed analysis, check out Wai Yan Pong’s “Sums of Consecutive Integers” article.

Want to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.

https://brilliant.org/wiki/conjectures/

The article by Pong is a great example of a conjecture and proof- the results of the “social game”.

I like how he cites other mathematicians.

I read Dan Marain’s blog which you referenced above and see this is a very worthwhile exploration. I just got a little confused, at first. Did you mean whole numbers greater than zero or positive integers?

It just seemed like”positive whole numbers” was redundant until I realized you might mean whole numbers greater than zero which is the same as saying positive integers. Whole numbers include zero. Natural numbers are whole numbers greater than zero.

If you want to connect it to Number Theory, it is the study of integers.

Anyway, I was just thinking about it.☺

I meant to say that “positive whole numbers” seems redundant. All whole numbers,except zero are positive.

Ok. That’s all. I hope you get some conjectures.🙂

Different people use different definitions for “whole” and “natural” numbers. And I write for the general public, which tends toward math phobia, so I can’t assume my readers know these distinctions, anyway.

So by saying both “positive” and “whole” I can communicate to people who aren’t used to working number theory puzzles that they don’t have to worry about uncomfortable numbers — no negatives, and no fractions.

Hi Denise,

I didn’t know there was any disagreement about those definitions. Your writing is very friendly and you have links if anyone wants to go more in depth. And you have pictures, too.

So, why not add some humor?

Now, I have to share a link to Better Explained” (I just started reading his articles) because he has a funny “conversation about negative numbers of cows”:

https://betterexplained.com/articles/a-quirky-introduction-to-number-systems/

Thank you 😺

All I know about the definitions is that one of the AoPS books said they were reversed in some parts of the world. And that a lot of my friends wouldn’t know the term “natural numbers” — how can numbers be organic and better for your health?

Thanks for the Better Explained link. I’ll check it out. I do love his work!

I found 2 resources which expand on conjectures. One is at Brilliant.com; the other is a middle school curriculum for number theory.

I’d like to know if your general audience would find them useful.

https://brilliant.org/wiki/conjectures/

https://drive.google.com/file/d/1PNbzz9oXf9GbyGw2mCNXpgnoZAUIycxK/view?usp=drivesdk

I also wonder how much math anxiety occurs in a classroom vs. in tutoring and small settings. It might not be the math itself causing the anxiety.

I’m not sure where the Google Drive link is supposed to go, but it tells me I don’t have the correct permission.

Conjectures can be fun to play with. For example, I like Math Pickle’s version of the Collatz Conjecture.

Hi,

Sometimes you have to match the developmental stage of the students with something written in a more formal way. You have polite numbers tagged as high school math. I think high schoolers can be at any level of thinking and we need to offer challenges beyond observations ad open-ended questions, for those who want to get deeper into mathematics and or just play – see what happens when you work with others to prove what you observe. This article tells you how you might think seeing a pattern is enough, but when you go to prove it, you see your guess is wrong and real proofs come into the picture.

Since conjectures seem to be categorized as upper level math, I thought it might be appropriate to share an article link from Quanta magazine.

https://www.quantamagazine.org/where-proof-evidence-and-imagination-intersect-in-math-20190314/

😺

https://betterexplained.com/articles/how-to-develop-a-mindset-for-math/

Sorry, I gave you a different one above.

This is the one with the negative cows🐄🐮

https://betterexplained.com/articles/how-to-develop-a-mindset-for-math/

I love the negative cows!

Me, too!