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Twenty-seven is one of my favorite numbers, so I was excited to find that “my” edition of the carnival would be number 27.

Some fun facts via Wikipedia (click the link for even more):

- 27 is a perfect cube: 3³ = 3 × 3 × 3 = 27.
- 27 can also be written as
^{2}3 using the notation of tetration, which means that it is 3 taken to the power of itself, 2 times:

*a*exponentiated by itself,*n*times.

- 27 is the twenty-eighth (and twenty-ninth) digit in π. (3.141592653589793238462643383
**27**9…).*If you start counting with 0 it is considered one of few Self-Locating strings in pi.*

- 27 is the first composite number not evenly divisible by any of its digits.

I don’t know what I was thinking, agreeing to do a blog carnival in the middle of June! So please accept my apologies for the tardiness of this post, and the lack of creativity in presenting the posts!

We’ll start the carnival with a little humor as Patrick Vennebush presents Cats and Dogs math jokes at Math Jokes 4 Mathy Folks.

I always love the way Denise at Let’s Play Math! makes word problems fun by her choices of literary themes, as well as providing excellent explanations of to approach them. Hobbit Math: Elementary Problem Solving 5th Grade is a fantastic example of her wonderful style.

Imagine discussing the Fundamental Theorem of Calculus to a very gifted eight-year-old. Sue VanHattum provides a peek into her work with Artemis in Sneaking Up On the Fundamental Theorem of Calculus at Math Mama Writes….

What happens to the American flag if Puerto Rico becomes the 51st state? Read A mathematician figures out the best way to jam an extra star onto the American flag by Chris Wilson at Slate Magazine to find out! There’s also a fun widget provided so you can see what the flag might look like with an arbitrary number of stars, using a variety of different types of pattern.

Check out Caroline Mukisa’s brief interview with a UK-trained Maths tutor in A Maths Tutor Reveals All! at Maths Insider.

David Ginsburg tackles a huge pet peeve of mine — the lack of “sanity checking” of answers among math students — in Estimation Before Computation at Coach G’s Teaching Tips. (I posted a related rant — my Calculator Rant here a few years ago.)

Teachers stuck with silly word problems provided in the officially sanctioned textbooks will appreciate the ideas in One word problem – many word problems at CTK Insights. (This expands on a related post from Dan Meyer a few months back on his blog dy/dan.)

Tracy Beach presents New Teacher Downloadable: Give Parents Ideas to Avoid the “Summer Slide” posted at Math Learning, Fun & Education Blog : Dreambox Learning. (Note that the handout also contains an advertisement encouraging parents to purchase a subscription to their website. I don’t know enough about their subscription service to offer an opinion as to quality or value.)

C heck out Guillermo P. Bautista Jr.’s presentation of Rational and Irrational Numbers at Mathematics and Multimedia for clear explanations and diagrams illustrating that the real numbers are divided into rationals and irrationals.

Shana Donohue presents two zero-based posts at her appropriately-named blog The ZeroSum Ruler: To the Zero! [power] and Dividing by Zero Blows up the Universe!. The middle school students I work with seem to think that the rules about zero are mostly arbitrary. (I kind of agree with them on 0! — 1 is the value that makes all the combinatorial formulas work out cleanly, but I’ve never heard an actual good argument for it otherwise.) But these posts show why and how the rules make perfect sense when it comes to division by zero and raising numbers to the zeroth power! The “to the zero” lesson also extends gracefully into explaining how negative exponents work, and showing that they just keep following the same pattern.

Pat Ballew stumps his pre-calc students by asking the simple question, “Given two points, write the equation of a line containing the two points,” in Given Two Points???? at Pat’sBlog. Why are these bright honors students stumped? Because the points given are in three dimensions rather than two!

Jason Dyer offers his take on Dan Meyer’s “Be Less Helpful” TEDx speech in The varieties of Be Less Helpful at The Number Warrior.

At the intersection of Math and Computer Science, John Cook talks about what goes wrong when computations with large numbers must be carried out on computers with limited precision in What’s so hard about finding a hypotenuse? posted at The Endeavour.

And finally I decided to share some of my student’s responses to the prompt “Write the biggest number you can in this box” — an idea I got from Dr. Mike’s Math Games For Kids (check it out!)

For more Math Blog Carnival fun, check out the 66th Carnival of Mathematics posted at Wild About Math.

I missed tagging an incoming carnival post, so this one got accidentally overlooked when I put it together. So now you get an extra bonus post:

John Golden presents a great game to practice ordering decimals in Decimal Point Pickle at Math Hombre. It’s too late for this school year, but I’ll have to bookmark it for next year. Thanks John and sorry for forgetting your submission in my original post!

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The rules, from Dr. Mike were:

- You may write anything at all in the box on the entry form labeled “Write your number here”. I will try to interpret what you write as a number.
- If I cannot interpret what you write as a
**real, finite**number, your entry will be disqualified (in particular,

you can’t write ‘infinity’) - The winner will be the student whose number is bigger than those of all others in the contest
- Your number should fit entirely inside the box, which is 4 inches wide and 2 inches high.

As I mentioned above, we also limited the kids to 5 minutes.

The kids all had a lot of fun, and I think we got a lot more interesting entries than Dr. Mike did. I’ve scanned a few representative and interesting ones here. Please excuse the hard-to-read scans — most of them were written in pencil.

The most popular entry among the 5- and 6-year-olds was:

Slightly older kids (7- to 9-year-olds) tended to come up with entries like these:

This 9-year-old knew about powers (though not their notation) and came up with a really big number….

… though I think not as big as the second of the numbers listed by this eight-year-old (who I think believed that he had written out a googolplex at the top, though in actual fact it is only a googol):

The googol/googolplex theme was popular among the 11-year-olds, though interestingly they all spell it like the search engine

The 12-year-olds remembered about factorials!

…and the 13-year-olds combined factorials and powers:

But I’m pretty sure the largest number of all was this entry by the lone 14-year-old in the group (it was his 14th birthday the day of the exercise):

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Using the following integers and an unlimited number of +, -, ÷, ×, and parentheses, create an expression that is equal to 2010. (You must use all the numbers in the set exactly the number of times they are listed.)

- {2, 2, 2, 2, 3, 3, 5, 5, 17}
- {1, 3, 3, 3, 7, 7, 7, 7}
- {1, 1, 1, 1, 3, 3, 3, 3, 3, 5, 5, 5}

If you like puzzles like this, also check out the 2010 game over at Let’s Play Math!

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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!

edited: See also MAA Number A Day

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:

Things to think about:

- 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.

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Despite being inspired by my elementary-school-aged son, this post is not about elementary or middle school math, in my usual style. (For elementary Pi Day ideas, check out last year’s Pi Day post.) But stick with me, if you will, and maybe you will learn something fascinating about irrational numbers.

Let’s step back a moment. What does it mean for a number to be irrational? My son knows that the decimal expansion of pi goes on forever and never repeats. An irrational number is defined to be a number that cannot be expressed as a *ratio* or fraction (and a rational number is a number that can be expressed as a fraction). All rational numbers have decimal expansions that are either terminating (such as 1/2) or repeating (such as 1/3). Maria over at Homeschool Math has an excellent introduction to irrational numbers and proof that is irrational — the former should be accessible to middle schoolers, the latter high schoolers and beyond. (Pi, as it turns out, is not only irrational but also transcendental, which means that it is not a solution to any polynomial with rational coefficients.)

Alexander Bogomolny at Cut the Knot offers an easy-to-understand proof that it is possible to raise an irrational number to an irrational power and obtain a rational result! Many are familiar with Euler’s famous equation or equivalently . Somehow one can raise an irrational number to an imaginary irrational power (!!!) and get an integer! But what does it *mean *to raise a number to an irrational power??

Most of us have an intuition about what it means to raise a whole number to a whole number power. means you multiply 3 by itself 7 times: .

What does it mean to take a number to a rational power? , but why is this the case? One intuition is to think that because it’s taken to the 1/2 power, you need 2 of them multiplied together just to get up to the original base, in this case, 3. Or if you’re comfortable with the rules of exponents, you’ll know that . So if times itself equals 3, it must be equal to Similarly

But is there any intuition that can help us undertand what it means to take a number to an irrational power? What could it mean to multiply by itself times? The best explanation I can find is over at Ask Dr. Math, where Doctor Rob explains that we find the value of a number to an irrational exponent by raising the number to successively more precise rational approximations of the irrational exponent, and then using calculus to find where that sequence converges. (Although, that does not explicitly cover the case where the base is also irrational, the same general approach applies.)

Finally, Thomas Christie offers a nice explanation of Euler’s Equation. (This link has been broken for the past few days, but I hope it will be back!) In case Christie’s page doesn’t come back, you might wish to look here or here.

I hope you had fun exploring irrational numbers and powers with me today! Now, go eat some pie. Happy Pi Day!

**Edited to add:** The Math Less Traveled has a nice series of posts up going through the proof of the irrationality of pi, starting here.

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Since we won’t be in school for Pi Day (3/14), which falls on a Saturday this year, you might want to mention this one to your classes, if you read about it in time.

Happy Square Root Day!

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I decided that first we should find all the prime numbers less than 100 using a Sieve of Erastothenes. Having the prime numbers in front of them is useful when trying to figure out how many factors a number has, since they might notice that 74 is 2 x 37 but then not immediately know if 37 can be factored further or not. So, we spent the first 20 or so minutes of class crossing out multiples of all the primes up to and including 7 on a 100 chart. We also figured out why we only had to go up to 7.

We counted 25 primes less than 100. The kids noted that none of those numbers have exactly four factors, since primes have exactly two factors. (Yay!) One girl also pointed out that no perfect squares could have exactly four factors, because squares always have an odd number of factors. (Double Yay!!) I asked the kids to each guess how many numbers we would find under 100 with exactly four factors. Their guesses ranged from 14 to 25. I also asked them to guess whether they would find more numbers less than 50 or greater than 50 with exactly four factors. Everyone (myself included) thought we would find more such numbers below 50. (Perhaps you would like to make your own guesses before reading further — I will post the actual number and distribution of the numbers at the end of this post. )

I had to clarify Dave’s “higher order” question: “These numbers fall into 2 categories. Describe these categories.” The first “guess” I got as to the categories were “odd and even”. Well, yes, some of them are indeed odd, and some of them are even. So I clarified that I was looking for descriptions of two categories of numbers, *all of which* have exactly four factors, and that al numbers with exactly four factors must fal into one of the two categories. I hinted that the categories have something to do with the prime factorizations of the numbers.

Most of the kids got most of the way through finding the numbers during class. I assigned finding the rest of the numbers (after comparing my results with a few fast finishers, we told everyone how many numbers they were looking for) and coming up with the two categories.

We have in the past discussed how to find the number of factors a number has, based on its prime factorization. (I can’t find a good page to link for that, so perhaps I’ll write it up myself at some point.) I wonder if any of them will think about it in those terms. I’ll report back to let you know how they did with that part of the challenge.

Related: Dr. Math page about finding numbers with a given number of factors

Answers: There are 32 numbers less than 100 with exactly four factors. And more than half of them are greater than 50!

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Thanks for the kind words, folks!

I’ll have more content posted soon, I promise!

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