Tag Archives: Proofs

How Tall is That Tree?

One of my most favorite lessons to teach is featured in this month’s issue of NCTM Middle School Mathematics (February, 2014).  If you’re a member of NCTM, click here to check it out and download the handy classroom printables.  If you’re not a member, click here to join NCTM; then click here to read the lesson and download the handy printables (or, you could make friends with someone who is already a member of NCTM . . . like maybe your department head and talk them in to letting you read the lesson!).  However you get your hands on it, I hope you enjoy!

Here’s a little excerpt from the full text:

“The American Forests organization documents the largest trees int he 50 states and the District of Columbia.  Each state, responsible for locating its largest trees to add to the national database, has its own method for measuring and locating tall trees.  Some states rely on amateur tree hunters for nominations.” (MMS pg. 386)

Turn your students into amateur tree hunters in this lesson!

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Unbelievable Math Problem

During my first year of teaching one of my Algebra I students printed out this chain email and brought it to me to solve:

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I don’t remember the exact conversation, but I do remember that it was something along the lines of

“My mom and sister and I spent the weekend trying to do this and we bet you can’t figure it out!”

Just in case you can’t read the problem in the photo, I’ll re-type it for you (but I have to admit I added the missing capitalization and punctuation . . . I just couldn’t leave alone!):

Wow, this is spooky.

UNBELIEVABLE MATH PROBLEM

Here is a math trick so unbelievable that it will stump you.  (Or at least it stumped me and I have a degree in math!)  Personally I would like to know who came up with this and where they had the time to figure this out.  I still don’t understand it!

1. Grab a calculator (you won’t be able to do this one in your head).

2. Key in the first three digits of your phone number (NOT the area code).

3. Multiply by 80.

4. Add 1.

5. Multiply by 250.

6. Add the last 4 digits of your phone number.

7. Add the last 4 digits of your phone number again.

8. Subtract 250.

9. Divide the number by 2.

Do you recognize the number?

*Disclaimer about this email . . . I was very motivated to explain to the student how this worked, because I’m not a fan of the whole “math is incredibly complicated and hard to figure out,” movement that seems to sweep across quite a few mathematics classrooms.  (We’ve talked about that in my “Calculating Tips is Calculus” post).

Anyway, let’s get to the bottom of this so-called “math trick.”  If you haven’t already go ahead and follow the steps in the email.  If you’re doing this on a calculator, as the email suggests be sure to press “enter” or “=” at the end of each step.

So, what happened?  Do you recognize the number?  It’s your phone number, right?

Hmmm, I wonder if this works with everyone’s phone number?  Try your mom’s phone number, or your gramma’s phone number, or your best friend’s phone number . . . does the trick work with their numbers too?

Well that’s tricky, isn’t it? (or is it?!?)

Let’s start at the beginning of the math trick email.  You start by entering the first 3 digits of your phone number.  Now, we know this trick already works for your phone number, and the other phone numbers you tested, but let’s see if we can figure out why/if it works for all phone numbers.  Instead of a specific number, I’m going to say the first 3 digits of my phone number are:

#$%

This is important, do you know why I’m doing this (this being symbols instead of numbers)?

No?  Let me tell you; If I pick the numbers 123 as the first three digits of my phone number, I’m not explaining why this math trick works for all phone numbers.  I’m only showing that this math trick works for phone numbers that have 123 as their first 3 digits.  If I wanted to assign the first 3 digits actual numbers, I’d have to test this math trick for all possible combinations of the first three numbers!  That a total of 729 different first three numbers . . . that’s a lot of “math trick” tests (and it doesn’t even include the different combinations of first 3 digits, plus last 4 digits . . . that’d be a total of 4,782,969 different combinations to check this math trick for).  So while it seems funny to use symbols, since the symbols can represent any first 3 digits it ends up saving me a lot of time in the long run.

So, step 1: Grab a calculator (I won’t be able to use a calculator if I’m going to replace the digits with symbols, so I’ll grab a piece of paper instead).

Step 2: Instead of keying in the first 3 digits of my phone number, I’m going to use the symbols.

Step 3: Multiply by 80: 80(#$%)

Step 4: Add 1: 80(#$%) + 1

Step 5: Multiply by 250: 250[80(#$%)+1)]–OK at this point the expression is getting a little ugly, so let’s go ahead and use the distributive property to simplify this mess!  When doing that, I get:

20,000(#$%)+250

Step 6: Add the last 4 digits of your phone number: I’m going to use ^&*@ as the last 4 digits.  This means now I have:

20,000(#$%)+250+^&*@–I know, I know its getting a little weird just stick with me!

Step 7: Add the last 4 digits of your phone number again: OK at this point I’m adding another round of ^&*@, giving me 2^, 2&, 2*, and 2@ so, I’m writing this expression as:

20,000(#$%)+250+2(^&*@)

Step 8: Subtract 250: Thank goodness, that 250 was getting to be annoying!  Now you have

20,000(#$%) + 2(^&*@).

Step 9: Divide by 2:  Well, what do you know?!?  The coefficient of #$% and ^&*@ are both divisible by 2!  That means I’m left with

10,000(#$%) + ^&*@

That’s a phone number!  Look, you take the first three digits of a phone number and multiply by 10,000.  Remember, multiplying anything by a power of 10 just moves the decimal point, so 10,000(#$%) = #$%0000 and what do you know; when I take #$%0000+^&*@, I get #$%^&*@–which is my phone number!

But, I have a challenge for you: Change the math trick so that it works if the person includes their area code.  When you come up with one, let me know!  I’d love to see it!

Fibonacci Number Sequence and Prime Numbers

Please don’t let this title turn you away from the post.  Its a really great post, I promise . . . I just couldn’t come up with something cute and catchy today!

Anyway, I’ve been very vocal about the fact that NCTM asks great questions on their Facebook and Twitter pages, but today I have another Tweet that I really enjoyed from Maths Jam:

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So, this is a great question, right?  You probably know what makes a number a perfect square (just in case you don’t, look here), but you might not know about the Fibonacci Number Sequence.  I sort of wish that we’d talked about it before today, because there are many great and interesting things about this sequence of numbers, but we’ll just have to talk in more detail about those great and interesting things later.  For the purposes of this question you just need to know that the Fibonacci number sequence is a sequence of numbers which is generated by adding the previous two numbers together.  So terms 1 – 5 of the Fibonacci Number sequence are:

1, 1, 2, 3, 5, . . .

It seems that this particular Tweet asserts that the number sequence is actually

0, 1, 1, 2, 3, 5, . . . which I’d never seen before, until I did a little investigating via Wolfram Mathworld, but it isn’t really central to the answering of this question.

So, just to recap; we’re looking for numbers in this sequence that are also perfect squares.  In that case, I guess we should start by listing the Fibonacci Numbers, starting with the “0” term:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, . . .

Here’s the thing about answering this question, the number sequence generates an infinite number of terms meaning that, yes–I would venture a guess that at some point there will be another perfect square number in this sequence of numbers, but I’m not sure what it is.

This person calculated and factored the first 300 Fibonacci Numbers; and from this list it looks like 0, 1, and 144 are the only Fibonacci Numbers that are also perfect squares.

There are proofs, involving Lucas Numbers that also show that the only Fibonacci Numbers that are also perfect squares are 0,1,144.  That means that with the use of mathematics we are able to prove things about numbers that we know exist, but that no one has discovered yet.  Wow.  Just wow.