A number of blog postings have arisen over the past week discussing fascinating facts about the number 2010. Check out the posts at MathNotations and 360.  I read about this gem:

2010 = 1+2-(3-4-5)*6*7*8-9

on a Mathletics Facebook post. And Theasmet has taken that to a whole new level!

With my middle schoolers, I figured it would be too challenging to ask them to find an expression such as the one above, so I used it as an order of operations review.  I provided 4 variants on the expression above (each changing a single symbol) and asked them to compute the answers, carefully, using order of operations.  I gave them the hint that one of the problems had 2010 as a solution.  It was a fun way to use that expression, and showed us the parts of order of operations that still need a little shoring up.

For homework, I asked them to come up with some fascinating facts about 2010.  I asked some leading questions to get them rolling, namely:

• How many factors does 2010 have?
• How can 2010 be expressed as a sum of squares?  Of cubes? Of primes? Of other interesting kinds of numbers?
• Can 2010 be expressed as a sum of consecutive integers? Of consecutive odd or even integers?

Here are some of the things they came up with. (More to come…)

• 2010 has 16 factors
• The sum of its digits is 3 (and prime)
• it has 4 distinct prime factors
• The product of its digits is 0
• 2010 = 40^2 + 20^2 + 3^2 + 1^2
• 2010 = 44^2 + 7^2 + 5^2
• 2010 = 13^3 + -4^3 + -4^3 + -4^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
• 2010 = 127 + 128 + … + 141
• 2010 = 669 + 670 + 671
• 2010 = 668 + 670 + 672
• 2010 cannot be expressed as the sum of any two consecutive integers
• The closest two integers that have a sum of 2010 are 1004 and 1006
• The closest two primes that have a sum of 2010 are 1013 and 997
• The largest number of consecutive integers that have a sum of 2010 is 15 (not true — need to get her to think about the cases with negative integers included)
• The only square factor of 2010 is 1
• You can sum to 2010 using addition and only (multiple copies of) the first two cube numbers.
1. January 9, 2010 10:06 am

Hi, Mathmom, and welcome back!

I like what you did with the order of operations exercise, and your students came up with some great facts. I would like to try a similar exercise with Math Club, but my students are too young this year. I think it will be as much as they can handle just to find the prime factors.

2. January 9, 2010 3:08 pm

Hi Denise,

I actually gave them the prime factors, though finding them is something that they know how to do. And we reviewed how you can calculate how many factors a number has, if you know the prime factorization.

How old are your Math Club kids? I did “fascinating facts” about 97 and 153 a couple of years ago with a slightly younger group. I think if you prime the pump with age/ability-appropriate suggestions, almost any age can do this activity. I’ve done variations on this with kids as young as 8 or 9yo.

3. January 9, 2010 4:39 pm

My kids range from 3rd to 5th grade, and most of them are absolute newbies. They’ve only just learned what prime numbers are, so I was planning to spend part of this month’s lesson on defining and finding prime factors. They also tend to panic with big numbers, so 2010 will be a nice challenge for them: big enough to look scary, yet easy to factor.

4. January 9, 2010 11:41 pm

Denise, you might try something like this with your Math Club kids — I used it with a group about at that level, and it worked out really well. It turned out to be one of my favorite things I’ve done with that age group.

February 20, 2010 7:44 am

“The largest number of consecutive integers that have a sum of 2010 is 15 (not true — need to get her to think about the cases with negative integers included)”
I actually am going to compete in the chapter mathcounts competition this morning, and doing these exercises was a good warmup- the longest string is -2009 to 2010, which when summed will give 2010. That is, 4020 numbers.

6. February 20, 2010 10:33 am

Eliot, good job! I hope Mathcounts goes/went well for you and you’ll be prepping for State next. :)

April 18, 2010 6:17 am

1+2-(3-4-5)*6*7*8-9

= -1017

8. June 12, 2010 10:36 am

2010 = 1+2-(3-4-5)*6*7*8-9 ==> That managed to get my attention!

So simple and elegant … and it leaves you with a taste of … why didn’t I think of that!

Q: How did Ted manage to get -1017?

BTW … just came across your page of Math Goodness.

Enjoy!