Fun with Time

MILLION SECONDS,

BILLION SECONDS

Another idea that I often used to show the stunning power of mathematics was fairly simple. How long does it take a person to become one million seconds old? The answers I usually got from students were way off (that's what makes it interesting). We fairly soon could work out together that a person reaches one million seconds in a little over 11.5 days. So then I asked them to figure out how long it would take for a person to reach one billion seconds old. I might ask them if they thought they had already reached a billion seconds. Most thought they had since they had reached one million seconds in less that two weeks of life. Of course I wouldn't be using this problem if it came out the way most kids thought it would. Either later in that period or the next day, the students were stunned to find out that they would not become one billion seconds old until they were approximately 31.7 years old. Again most adults are amazed by that answer also.

This problem can be used in a variety of ways. One extension I had them do was to figure out on which day they would reach one billion seconds. That is not easy because of leap days and so forth, but also not that interesting to most of my students because it was so far away. For my 7^th and 8^th graders it was more interesting to figure out when they would be 5 000 days old. I had them assume that they had been born on noon of their birthday.

5 000 days is about 13.7 years so it was between their 13^th and 14^th birthday. I told them that if they came to me on their 5 000 day old birthday (and could prove to me that it was the right day) I would give them a little present. If they had already passed that day, they had to tell me what day it was and if they were correct then they got the present. The present was something small and inexpensive and the same for everyone that year. If the 5 000 day old birthday happened in the summer they could see me just before school got out or the first day of school the next year.

The creative teacher can find other uses for this situation, depending on the age and ability of their students.

Adults can figure out when they will reach or did reach one billion seconds. Those of us who are a little older can figure out when we will be two billion seconds old. Maybe you don't want to do that! Some of the students liked to use this information with their parents birthdays. Don't forget that problems like this can show the power and fun of doing math. Things like this help kids to think in a more mature and logical way.

60x60x24 =86 400 seconds in one day

1 000 000 / 86 400 = about 11.57 days

1 000 000 000 / 86400 = about 11 574 days

11 574 / 365.25 = about 31.7 years

Tissue Paper Problem

TISSUE PAPER PROBLEM

I often used this on the first day of school to let my students know that this class was going to be special. I used an ordinary piece of tissue paper (Kleenex). When you pull it out of the box, it is two-ply. That is there are two sheets together. I took those apart so I had only one sheet. I told the students that this one piece of tissue was so thin that 1024 of them stacked on top of each other, and then compressed down as far as possible, would be exactly one inch high. That is a way of telling them that the thickness or height of the one sheet was 1/1024 of an inch. I then took that one piece of tissue and folded in in half. I asked them how thick it was now compared to before I folded it. They could easily see that it was now twice as thick or, if you thought of the thickness as its height, it was now twice as high. I again folded it in half and asked them how thick or how high it was. They knew it was now four times as thick or high compared to before the folding. I continued folding and asking the same question. If you have ever tried this you know you can only fold it seven or eight times before it becomes too thick to continue. I did show them that it was ¼ of an inch high after 8 folds. I then asked them how thick or how high it would be if it could be folded in half a total of twenty times. I would say to them that even though we can't actually do that many folds that we could figure it out mathematically. I took some random answers from the students which were usually pretty far off. I then gave them another hint: after ten folds it would be one inch high. Of course that caused many of them to jump to the erroneous conclusion that after 20 folds it would be two inches high. I made sure that they knew that answer was wrong but I did not explain any further (that it would be two inches high after the 11^th fold). Their homework for the next day was to figure out the thickness or height of that one piece of tissue paper after it had been folded in half 20 times. When they came in the next day, I might have one or two kids who had figured it out. At any rate, they were all amazed that after 20 folds it would be 85 feet, four inches high. I find that most adults are amazed by that too.

When I first started using this activity, I was teaching on the third floor of a rather old building. The height of the building was about 48 feet. The building had a gym whose floor was below ground about 15 feet. It really stunned the students that a person could be down on their hands and knees on that gym floor and start folding that tissue paper and that after 20 folds it would be much higher than the top of the three story school building.

Most years I gave the students tissue paper to fold along with me. Sometimes that didn't work well because they didn't always hear the information I was giving them while they were folding. I would suggest that you try it with them folding their own tissue and if you don't like it, then just have them observe you.

(One ply of tissue paper is 1/1024 inch thick.)

0 folds – one thickness, 1 fold – two thicknesses,

2 folds – four thicknesses,3 folds- eight thicknesses,

4 folds – 16 thicknesses, 5 folds – 32 thicknesses,

6 folds – 64 thicknesses, 7 folds – 128 thicknesses,

8 folds- 256 thicknesses, 9 folds – 512 thicknesses,

10 folds – 1024 thicknesses, therefore one inch high,

11 folds – two inches high, 12 folds – four inches high,

13 folds – eight inches high, 14 folds – 16 inches high,

15 folds – 32 inches high, 16 folds – 64 inches high,

17 folds – 128 inches high, 18 folds – 256 inches high,

19 folds – 512 inches high, 20 folds – 1024 inches high.

1024 inches divided by 12 is 85 feet 4 inches.

Introduction

PRACTICAL and CREATIVE

IDEAS

FOR MATH TEACHERS

INTRODUCTION

Welcome to my blog! On here you will find a potpourri of interesting and novel ways to increase your effectiveness and your enjoyment of teaching. Everything I mention here has been used successfully in my own classroom but I know that doesn't guarantee that it will work for you. Some teaching ideas that worked very well for other people did not work well for me. Either way, reading and thinking about these things may cause you to come up with some creative ideas of your own that will be very effective for you.

I was a math teacher in the Spirit Lake, Iowa, school system from 1970-2003. My basic teaching assignment was 8^th grade Pre-algebra and Algebra 1 for 8^th graders. I also taught some 7^th grade math and some 9^th grade General Math. In many cases I found the textbook approach to certain topics to be ineffective as well as not much fun. Over the years I worked to develop better ways to teach those topics and ways of managing the classroom that solved some of the typical problems associated with math teaching or teaching in general. I also came up with several excellent projects that challenged my students and enabled them to use the math they were learning in some practical and creative ways.

Much of the content of this blog can also be used by elementary and high school math teachers and many ideas are applicable to other subjects besides math. You may want to tell your friends who teach other subjects, at any level, about some of the things you find here, especially the classroom management topics. Some of the projects I will mention are very good for inter-disciplinary work and you may find it enjoyable and valuable to work with some of your colleagues on those projects.

Please come back to this site often. I plan to post many ideas which could be very helpful to you. Tell you friends and colleagues about this site. Leave comments for me on any questions or problems you might like me to address. I look forward to sharing ideas with everyone in order to make teaching more effective and more enjoyable!