# 10,000 step challenge

This year I asked for a FitBit for my birthday.  (For those of you that don’t know a FitBit is a pedometer, counting your steps, flights of stairs, daily active minutes, and approximate number of calories burned).  I was excited and curious to clip on my FitBit and see just how far I was walking every day!

But, after a few, short days I was a little confused.  I thought that about 2,000 walking steps = 1 mile, but I was getting FitBit read-outs on my phone that looked like this:

So, if you’ve been reading this blog for any amount of time, you can probably guess what I did next . . .Yep, I Googled the length of a walking step and discovered that this website (which seems legit to me) estimates that the average length of a person’s walking step is about 2.5 ft, which means that in order for a person with average walking steps to walk 5 miles, they’d have to take 10,560 steps . . . not 10,000.

Then, I started wondering how long my steps were (on average of course); compared to the published average of 2.5 feet/step.  I used my FitBit output for 3 different days

and discovered that, despite my relatively short legs my walking stride length was pretty average!

Then, I started thinking about a project I used to have some of my students work one, which is now an activity on the NCTM Illuminations Site, called Walking to Class.

This summer, make a walking strides chart of your day (or a trip to and from the park, pool, etc.) but instead of measuring distance in steps, change the units from steps to miles using the average 2.5 foot length, or you could dust off a pedometer and calculate the length of your actual stride!

# 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!

# Screen time

It’s been a while since I’ve written about a mathematics problem from NCTM and I kinda of miss writing about them (You can read the other ones here and here)!  Last week they posted a question that only got snarky answers on Facebook, so I decided this would be the perfect one to write about 🙂

Here’s the post:

From the snarky answers it seems to have drawn criticism because of the number of movies Ryan supposedly has, but I just have to say from my own experience that my children could certainly given Ryan a run for his money in the movie collection department!

Anyway, Ryan’s movie collection is apparently quite extensive.  Also, he only likes movies that are fairly short (1 hour and 30 minutes in total).  If Ryan were to watch these movies back to back to back to back to (you get it, all 70 without stopping); how many days would it take him to watch said movies?

Can you tell right off the bat that this is a conversion situation?

My game plan:

Convert hours and minutes to the same unit of measure; then use this unit of measure to calculate the number of days of the marathon movie watching!

Except that as I was typing this, I came up with another game plan . . . each pair of movies equals 3 hours.  Then, I don’t have to worry about converting minutes to hours or hours to minutes.  AND, 70 is an even number of movies, meaning I can make pairs of movies without leaving any movies out.  AND, 3 hours is a really nice value to have when dealing with days, because there are 24 hours in a day and 3 divides 24 evenly.  So, I can watch 8 pairs of movies a day (or 16 movies a day).

New Game Plan:

16 goes in to 70 little more than 4 times (actually it goes into 70, 4.375 times)

That means Ryan will need to watch 8 pairs of movies for 4 days.  At the end of 4 days he will have watched 64 movies, leaving him the final 6 movies (or 3 pairs of movies, or 9 hours of movies) for the 5th day.

So my final answer is 4 days and 9 hours of movies to watch all 70 movies!

Just to double check, you could do the converting I described at the beginning of the post . . .OR you could describe a different method to do the calculations if the mood strikes you!  If you do this another way, I’d love to hear about it in the comments below!

I also feel the need to address the posts on Facebook about Ryan’s movie situation.  Most of the comments were along the lines of “only a mathematics website, textbook, teacher, etc., etc. would ask such a silly question.  This question isn’t realistic.”  I must say I have to respectfully disagree.  The first thing I thought about when I read this question was the Brita water filter commercial showing plastic water bottles stretched across the Earth’s surface.

If this commercial had been written as a word problem in a mathematics class, it probably would have sounded a little something like this:

The makers of Brita water filters claim that 1 filter is able to filter the equivalent of 300, 16.9 oz. bottles of water.  Suppose Ryan uses 1 Brita water filter per month for one year.  If Ryan had used plastic bottles, instead of water filters how many times could the number of plastic bottles used wrap around the equator of the Earth?

I’m willing to admit that this type of problem situation isn’t one that we encounter in our every day lives; however these types of conversion situations come up whether they be for impact, or marketing, or something else.  Isn’t nice to know that you . . . reader of the It’s Just Math Blog have a way to evaluate such statements for their accuracy?

And finally, with regard to Ryan and his movie watching I’m willing to bet that the following problem statement may have been a little more well-received.  Why?  You might ask?  Because most of us know someone who’s done this 🙂  If this problem statement had been written about me it would have been about watching Law and Order SVU marathons 🙂

Suppose Ryan is a huge fan of James Bond movies.  In fact, this weekend he plans to watch every James Bond movie ever made back-to-back.  Will he have enough time to watch all of the movies if he starts Saturday morning at midnight and finishes Monday morning in time for school at 8am?

# NCTM Circle Problem

If you’ve been here before, you probably know I really like the NCTM problems they post on their Twitter and Facebook pages.  Well they posted one last week that I just couldn’t seem to stop thinking about (doesn’t that usually mean you’ve got a good question on your hands!?!)

In case you missed it, here was the question:

What’s the value of the question mark?

The first time I saw the picture I immediately noticed that 4 and 8 were on the circle and that they were only one wedge apart.  I really, really wanted the question mark to be worth 16, but then that would mean that the 13 should have been a 32 and that the 23 should have been a 64 and that the 92 should have been a 128 . . . clearly I was wrong.  (P.S. can you tell what I was going in order to fill in the wedges of the circle?).

Then, I noticed that there were a few prime numbers on the circle (remember how to check to see if a number is prime or composite (via Math Warriors)?).

Here’s my tip . . . anytime you’re looking for a pattern in a situation prime numbers should make you a little suspicious.  You should be suspicious because they don’t mix well with other numbers, when considering “typical” pattern generators such as multiplication, division, or power properties.

Given my prime number predicament I felt like I had two choices . . . I could look for a pattern generated by adding or subtracting the numbers or I could look at the prime factors of the numbers in the circle and see what happened.  I wasn’t really compelled to try the whole add/subtract thing because from looking at the entries, that didn’t seem very likely.

Instead rewrote the numbers on the circle as a product of their prime factors and I came up with something that looked like this:

Based on my little diagram, I was able to fill in the missing value.

If you happened to come across this question via Twitter or Facebook last week, you know there were two common answers floating around.  The first common answer was 120.  The second was 79.

1. How did people arrive at both of these answers?

2. Do you think that one is “right” and one is “wrong”?  If so, why?  Take a side and convince me the other answer is incorrect.

Another point that was brought up online was that if you started at different points on the circle, you could get a different question mark value.

3. What do you think about that argument?

# Finding Prime Numbers

This post first appeared at kkdegner@blog.com on 7/22/13.

Last week I came across this little doozy on my Facebook page, which was posted by The Belin Blank Center, which it looks like they must have reposted from the National Council of Teachers of Mathematics

I love the question for so many reasons!  First, I love it because its not complicated.  If you know what a prime number is and you know how to find square roots, you’re set!  Trial and error might be the weapon of choice for many people answering this question.  And while there’s nothing particularly wrong with trial and error, it can take a while and, well, its not really that interesting.  But, we could use just a little, tiny bit of strategy to try to cut down on the amount of numbers we have to test in order to answer the question.

To address this question we have to check 3 things 1)is the number prime? 2) is the square root less than 20? 3)is there a larger prime number whose square root is also less than 20?  Let’s start by setting an upper bound to eliminate the 2nd and 3rd questions.  We know that 20 squared is 400, meaning all numbers less than 400 have a square root less than 20.  So now we know we’re looking for prime numbers which are less than 400.  If you’re feeling very adventurous, you could just list all of the numbers between 1 and 400.  The largest prime in that list of numbers would be the answer to the question.  But, do you really want to check 399 numbers too see if they’re prime? No?  Me neither.  (We don’t have to check 1, we know its not prime or composite).

Good news!  We can use our knowledge of numbers and their factors to whittle this list down pretty quickly!  First, get rid of all the even numbers (except for 2–its prime), because we know they’re divisible by 2.  Additionally, get rid of all of the numbers ending in 5 (except for 5–prime!), because they’re divisible by 5.  Now we’re down to, what 161 numbers by my count?  That still seems like a lot of numbers to test, no?  There are still other numbers we could systematically check off of our list of possibilities, for example the multiples of 11 and 3 are pretty easy to spot (prime and prime, respectively).  Taking the grand total of possibilities down to 105 numbers.

From here, we could do a few things.  Starting with the largest number left on the list (the largest number I have left is 397), we could check to see if the number is prime.  If it is, then great!  We’ve found the answer. If you don’t like that method we could continue along the path we’ve been on.  That is, now cross out all of the multiples of 7, 13, 17, and 19.  See what’s left standing, check to make sure that its prime and just for fun, calculate an approximation of the square root.  Happy Number Crunching!

Wait, you didn’t think I’d actually give you the answer, did you?  What would be the fun in that?  I’ll tell you what I will do though, I’ll tell you about a little tool/algorithm/rule that is very helpful in answering this question (and its one I’ve been using in this post).  Its called the Sieve of Eratosthenes and it says this:

List all of the numbers from 1 – n (in our case 400).  To find all of the prime numbers, start with the smallest prime (2) and cross out every second number in your list.  Now, go to the next number in the list that you haven’t crossed out (3).  Cross out every 3rd number (notice, some of the numbers like 6 and 12 in your list you will have now crossed out twice).  Continue this process through all of the numbers 1 – square root of (n).  The numbers remaining are all of the prime numbers 1 – n.

Now you can answer that nice little problem!

What’s that you say?  You don’t understand why 1’s not a prime number?  Don’t worry, you wouldn’t be the first to ask.  Have you ever heard of Math Warriors?  Felicia does a great job of explaining this question in this episode.  (If you haven’t seen Math Warriors yet, you might as well just start at the beginning by going here.)

P.S. I included lots of great math resources in this post.  Follow them on Facebook and Twitter!