It Figures! Great Blue Journal Issue 5 May 2000

It Figures
Contents

-Awesome Ninesome
-Divisors and Patterns in Them
-1998-1999 Football Attendance
-What Comes After Infinity?
-K'Nex Towers
-Prime Numbers
-Scale: What Do We Use It For?
-Dog and Cat Babies
-How Many Kids Were Hot Lunch?

Introduction

My question is will the Awesome Foursome work with nines? I found my question by looking in old issues of Great Blue.

Procedure

This is what I did first. I made a number chart. The chart had the numbers 1 to 100. I was trying to figure out some answers for each number. My question started getting a little harder as it went on.

But then some luck came along. Then I started getting some answers. I was still using only + - x ÷. Then on Wednesday, February 16, 2000 my teacher, Mr. Jenks, told me about the square root of 9, which really equals 3. He also told me about the fraction 9/9, which really equals 1. That helped me a lot because I did not know about they were even possible.

On Tuesday, March 7, 2000 Mr. Jenks told us there was a due date. I went and worked so hard that I felt like giving up. But I said, "No!"

I made a new chart for my answers. On Tuesday, March 14, 2000 Mr. Jenks said check over my answers. Most of them were wrong! But I did find some new ones that were right. Then came the deadline.

Results

I found 1, 2, 4, 6, 8, 9, 11, 12, 14, 18, 19, 24, 27, 30, 31, 33, 36, 39, 42, 45, 54, 57, 63, 75, 77, 81, 87, 93 and 99.

Interpreting Results

If I had more time I think I could find all of the answers to 100.

Acknowledgements

I would really like to thank myself, of course. I would like to thank my friend Andrew Matje for helping me find some of the answers. I would also like to thank my teacher, Dave Jenks.

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Divisors and Patterns in Them
by Eliza
Randall School

Introduction

As soon as I learned about some of the rules for numbers dividing evenly into other numbers, I became interested. When we needed It Figures! articles I wanted to write one on divisibility.

Procedure

A math professor, Jon Kane, comes in every Friday to talk to the Algebra group about different areas in math. One day he introduced divisors; I took lots of notes. Then, when we could choose a math topic to work on for a long time, I choose divisibility. I wrote down the rules for dividing the numbers 1 through 7. As it turns out, I was wrong on dividing 4 and 7. A week later Jon came in and helped me figure out the right rules for 4 and 7 and all the rules for the rest of the numbers all the way up to 15. By myself I wrote down at least 3 numbers that were divisible by each number up to 15.

Results

Never before this did I know there were rules to help you take shortcuts in seeing if a number was divisible by a certain number, like 6,345 by 15, and knowing that it works. So here are all the rules for dividing by numbers up to 15:

1- Any number is divisible by 1.

2- Any even number is divisible by 2.

3- Add up all the digits and if that number is divisible by 3, then the whole number is. For example, 393 is divisible by 3 because 3+9+3=15 which is a multiple of 3.

4- If the last two numbers are divisible by 4 then the whole number is. Like, the number 3,444 which is divisible by 4 because 44 is a multiple of 4.

5- Any number ending in a 5 or 0 is divisible by 5.

6- 6 goes in if both 3 and 2 go in, so the number 222 is divisible by 6. It won't work if just 3 or 2 goes in.

7- Take the last digit, double it and subtract it from the remaining number. For example with the number 147, you take the 7, double it and you get 14, then subtract it from the remaining number which is 14 and 14-14=0 which is a multiple of 7.

8- To see if a number is divisible by 8, just take the last 3 digits and see if 8 goes into them, and if it does 8 will go into the whole number.

9- This number works the same as three, just add up all the digits and see if it is a multiple of 9. The number 981 works.

10- Any multiple of 10 (10, 20, 40, 50 ...).

11-Take the last digit and subtract it from the left-over number. For example the number 154 works because when you subtract 4 from 15 you get 11 which is obviously a multiple of 11.

12-12 will go in if both 4 and 3 go in, like the number 3744 is divisible by 12.

13- For this number multiply the last digit and add it to the rest. So the number 221 works because 1 times 4 equals 4 and 22+4=26 which is a multiple of 13.

14- You know 14 works if 7 and 2 go in, so 742 is divisible by 14.

15- 15 will work if 3 and 5 go in, so the number 135 works.

While collecting my data I noticed a lot of interesting patterns. The most amazing one was when I was dividing by 11. When I would divide by 11 the middle number would disappear and the answer would be the outer two numbers. Like with the number 275, when I divide it by 11 the answer was 25, or with the number 154 when it was divided by 11 the answer was 14.

I think with this project I found the answer to my question. I found all the rules for dividing numbers up to 15 and I found at least one pattern while dividing.

New Directions

If I could do this project again I would definitely try to have more time so I could work on even more numbers and hopefully find more patterns.

Acknowledgements

I would like to thank Jon for helping me. I would also like to thank Mr.Wagler and my mom and dad for their support.

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Introduction

In my project I was wondering how many people attend NFL football games and if there is a difference in the number in a losing season and a winning season. If there is, how much of a difference? I chose this project because I like football and I'm interested in math.

Procedure

First, I looked up the team standings for 5 teams on the internet. I picked the five teams (Denver Broncos, Indianapolis Colts, San Francisco 49ers, St. Louis Rams and the Atlanta Falcons) because they all had a winning year and a losing year in 1998 and 1999. Next, I had to look up attendance figures for all eight home games for the winning season and eight home games for the losing season to get the yearly totals.

Results

My results are shown on the graph below.

For the Denver Broncos there is a 150,000 person difference and the winning season is higher. For the other four teams the losing year had a tiny bit more attendance than the winning year.

Interpreting Results

All the teams but one have a little higher attendance in the losing year, so it seems like losing or winning didn't really make a difference. Overall, comparing the winning and losing year totals, I found out that the difference was only 128,000 more fans out of 2,000,000 that went during the winning season. I don't think that is a big difference. But for the Broncos it seemed like it made a big difference and a lot more people went during the winning year. Maybe they had a new coach or a new stadium or something. I was surprised when only one team had a higher attendance in the winning year.

New Directions

I might have looked to see if attendance at the game might have been because of bad weather or holidays.

Acknowledgments

Thanks Mom and Dad and Mr. Wiesner for helping me finish my project.

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Introduction

I have always been interested in infinity and what happens after it. Last year I did my "It Figures" project based on infinity but I never got it written up and I really wanted to.

This year I had already made an article for Great Blue but Mr. Wagler said that the Great Blue Journal didn't have many "It Figures!" articles, and he said that if anybody wanted to, they could have a second article as long as it was an "It Figures!" one. Four other people and I decided to write an extra article. I love writing Great Blue articles, and I liked my question from last year, so I decided to keep it.

Procedure

The first thing that I did was get a book called Infinity and the Mind. I thought I still had my materials from last year. My problem was that I found out that I didn't keep the materials, so I had to start over. Once I got the book, I went on-line. I tried tons of searches like "Infinity" and "Aleph-one" and got many results. The problem was that there are many groups with infinity in their name and there is a kind of new car called Infiniti. There was the same problem with the "Aleph-one" search, because there is a music group called that, and there are many places called "Aleph-one." I printed all of those results and even talked a bit with a mathematician. From all of those things I got the right information and put it in the article that you are now reading. Here is what I found out from my work.

What is Infinity?

Almost everyone knows about infinity (also known as omega or W) with the sideways eight sign. Of course, that's not nearly all. In fact, there is nothing called "infinity." There is only "actual infinity" and "potential infinity." Potential infinity is the infinity idea that says that infinity never stops. Actual infinity is the infinity that has an ending point. It's a lot easier to find potential infinity, and it's easier to understand than actual infinity, but for this article I'm mostly addressing actual infinity.

What is Aleph-one?

Aleph-one is the number that comes after infinity. It is in a group of numbers called the "Alephs" (also spelled "alefs"). They are written in many different ways like, w (omega) + 1, and many others. The most common way is just "aleph 1."

What Does Come After Infinity?

At the beginning of my research (and why I used it as a search), I thought that it was aleph-1. That was wrong. The short real answer is "C", also known as the continuum. The problem is whether there is anything in between "C" and aleph-one. People have proved that there is and that there is not. The same goes for "C" and aleph-two, aleph-three, etc. So, from infinity on, you can't really prove anything, only decide what you want.

New Directions

If I were to do this project again, I would go to the library to get more than one book. I would also ask more than one person and work a lot longer.

Acknowledgements

I would like to thank Jon Kane, the mathematician, for giving me tons of information, Mr. Wagler, for always being there and most of all, my parents, for being the best of all.

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Introduction

We are crazy about K'Nex; we also like building towers. So we decided to do a project on K'Nex towers. Our question is how much weight can a K'Nex tower hold? We decided to see how much weight a K'Nex tower could hold because we always wondered how buildings are constructed so they will stand up.

What are K'Nex? K'Nex are kind of like Legos but they are put together differently. There are two kinds of K'Nex, rods and connectors. The connectors attach to each end of the rods, and then you can start building a tower.

Hypothesis and Procedure

We wanted to test three different building designs: triangles, squares, and a combination of both. We built one tower using squares and tested it, then put K'Nex rods into the squares to make triangles and tested it, and then alternated squares and triangles on the third trial and tested the strength. We thought that the lower the towers get, the more weight they could hold. We tested each tower at different heights: 36 inches, 30 inches, 24 inches, 18 inches, 12 inches and 6 inches. We used dictionaries to test the strength of the towers. We piled on dictionaries until the towers broke or crashed.

Results

Our hypothesis was right. The lower the towers got, the more weight they could hold. But, the weird thing was that the 6-inch high triangle structure and the combination structure held one less dictionary than the 12-inch triangle and combination.

Acknowledgements

We would like to acknowledge our parents for encouraging us and helping us type up our project. We would also like to acknowledge our teacher, Mrs. Bostrom, for defining what an experiment is and also allowing us to build towers out of her K'Nex.

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Introduction

What are some prime numbers? How many are from 1-1000? What are some of the ways to find if a group of numbers is prime or not? These are some of the problems I have been working on. I find this valuable because prime numbers are used almost every day for factoring. Also, it's kind of interesting to think, when your spending two or three dollars, 'Hey, that's prime.'

If you don't know, a prime number is a number that is only divisible by one and itself. The majority of numbers are not prime numbers.

Procedure

I started out by finding all the prime numbers from 1-100 (I changed that to 1-1000 when a parent said I would get more data that way). To do that I used a method that I made up, which is like the following theorem: cross out all the numbers that are divisible by whole numbers that are less than or equal to the square root of the largest number (for example, the square root of 1000 is 33).

You're probably wondering how I got the 1000 numbers written down. I did that by using a formula on the computer, on a spreadsheet program. Then I looked for patterns and formulas.

Results

From 1-1000 there are 167 prime numbers. Of those numbers, 25 are from 1-100, 20 are from 100-200, 16 are from 200-300, 16 are from 300-400, 16 are from 400-500, 14 are from 500-600, 16 are from 600-700, 14 are from 700-800, 16 are from 800-900, and 14 are from 900-1000.

I also found out some rules that help you eliminate numbers faster: " two represents all the even numbers" and "5 represents all the numbers that end with five." I found many patterns that only lasted for a little while, and not lasting forever like a real pattern. There have been many formulas and hypotheses tried during the history of the world. Unfortunately almost none of them work.

Some mathematicians program their computers to look for more prime numbers while they're not using them (for example, while they're sleeping). Every six months a prime number is usually found that has never been found before.

If you want to know all the prime numbers from 1-1000 you can read this section, otherwise don't. These prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 471, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 826, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 889, 907, 911, 919, 929, 937, 941, 947, 953, 957, 953, 967, 971, 977, 983, 991, 997.

New Directions

If I had more time or if I did this again, I would read articles by other mathematicians, test theorems, and try my own formulas.

Acknowledgements

I would like to thank Jon and Janet, two parent volunteers for math, and Mr. Wagler.

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Introduction

How do we use scale to represent existing or imaginary buildings? Just to remind you what scale is, it is what architects or anyone else uses to make a model or a floor plan by reducing large things to a smaller size. There are many good reasons for this question. The first is that you can use scale lots of times and it's much easier if you know how to use scale well. Another is that I'm doing a project with an architect where we use scale. That tempted me to do this project; another thing that tempted me was that my class did something called Box City where we made models of our own houses.

Procedure

To answer my question I first made a drawing of my house from Box City. Then I went home and measured my house again. Maybe it was because there wasn't any snow or maybe it was for a different reason, but it turned out that the measurements I took were different than the ones I had for Box City! This is an example of what can happen if you do not understand scale. Another way I've learned about how scales are used is by making a lot of drawings to scale. I usually use 1/4" = 1' (one quarter inch equals one foot).

Results

First, I'll describe the process for creating a scale drawing of an existing building. First you measure the building. Then you choose a scale to represent the building. A commonly used scale is 1/4" equals 1'. For example, if the house was 36' long, this would be a good scale because four goes evenly into thirty-six.

But if the house was 25' long, I would not use that scale because four does not go into 25'. I'd probably use 1/5" equals 1'. You could use any scale for any length, but it is easier if you use one that goes in evenly. Now that you understand that, I'll move on to the question: How do we use scale to represent existing or imaginary buildings?

Scale represents things by just taking something you want to see and shrinking it down proportionally so you can see it well. Scale drawings are good because you can get a picture of what your house is going to look like even before it is built. If the front porch is bigger than the rest of the house, then you know there are problems. So really, we just use scale to represent buildings so that it is easier to communicate with other people about them without making a full size drawing or looking at the building.

My Hypothesis

My hypothesis was that people use scale because it is easier and I even thought that before I understood scale. I was right.

New Directions

I don't think there is really anything else to do unless you find a different way to calculate scale.

Acknowledgements

I would like to thank my teacher, Mr. Wagler, for helping me come up with the idea and helping me to get started.

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Introduction

I was riding in the car thinking about Great Blue and I thought of my idea. I wanted to learn how many cat and dog babies are born each winter. I was going to ask veterinarians my question. I was going to take a survey about domestic cats and dogs, and wild cats and dogs.

Procedure

I called three animal specialists (Dr. Chuck Schobert, Dr. Ann Plata and Dean Anderson) and asked my questions. They gave me some web sites and articles. I asked these questions: Do you have an idea where I can find information on how many wild or domestic dog and cat babies are born each winter in Wisconsin (in 1997, 1998, & 1999)? Do you recommend certain web sites or books? Do you know anybody that can help me?

A friend and I asked the school kids survey questions about cats.

My teacher called the Humane Society and they told him how many cats and dogs were taken in in the months January, May, August, November & December of 1998 & 1999.

My dad (Norm) and I went on the Internet and found a formula to figure out numbers of domestic cats and dogs per household in a city:

My dad and I made a table of my surveys of the Lapham kids:

The problems I had were not knowing what to do next, not having enough time to work at home and at school, and I needed more information on the topic.

The things I liked were getting to work on my own project, finding out new things, working alone, and trying new things.

Results

The question that I asked is impossible to answer because nobody keeps data on when their dogs or cats have their puppies and kittens! When I calculated the surveys that I did with a friend, Rebecca Toetz' class was the wierdest compared to the national averages. I didn't find anything in the library. This is the formula and its results from the website my dad and I found that told us how to estimate the number of pet-owning households in a city (see TABLE 2).

Domestic dogs and cats can have babies whenever they want because they temperature in the house is always the right temperature whether it is hot or cold outside. There is a huge overpopulation problem with domestic cats and dogs. class, so I could add up to find out how many were in school.

The Humane Society and many veterinarians recommend stopping cats and dogs from mating and having babies.

Useful Web Sites:

www.avma.org - the American Veterinary Medical Association (pet care section).

www.hsus.org - the US Humane Society (pet ownership statistics & overpopulation statistics).

Acknowledgements

I want to thank Mr. Swift , Dean Anderson, Ann Plata, Chuck Shobert and my dad, Norm Stockwell.

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Introduction

I got my idea from my second grade pen-pal, Matt, because he did it in last year's Great Blue.

Procedure

I got my information from Ken and Monica, who works in the kitchen. She has a menu for the whole month and she writes how many people are hot lunches. I used the calculator to help me solve how many people had cold lunch and I wrote it down. I added all of Monica's numbers for how many people were hot lunch. Then I subtracted that number from the number of kids in school to find out how many ate cold lunch. Janet, our secretary, wrote down how many kids were in each Other things I learned:

Results

The most popular lunch was heart shaped chicken nuggets on Valentine's Day. 163 kids ate those. After that it was Pizza Hut. Twice 156 kids ate Pizza Hut.

In the first week of January 697 kids ate hot lunch and 408 ate cold lunch. Now I'm going to put the information on a chart.

Interpreting Results

Hot lunch was always more popular than cold lunch.

Acknowledgements

I want to thank Monica, Janet and Mr. Swift.

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