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How Old Are You — In Days?

The question “How old are you?” or “How old is he (or she)?” actually has more than one type of answer, depending on the individual concerned. For instance, you, the readers of this webpage, will probably reply with statements like “I’m 14 years old”, or “I’ll be 25 next month”. (Scroll to the bottom of this article for a humorous treatment of this concept.)

But if the parents of a newborn baby or a very young child are asked the same question, their answers are would probably be
something like “He’s just 5 days old.”, or “She just turned 3 weeks old today.”, or “Last week he was 10 months old”. You see, we tend to use units of time that are appropriate for the situation.

This brings up an interesting question, that I bet you never thought of before: “How old are you today IN DAYS?” In reality it is nothing more than a basic math problem to calculate. After all, it
involves only a little simple arithmetic, perhaps a calculator, knowledge of a calendar, and genuine problem solving — all with a personal touch.


As a schoolteacher, I feel this activity is guaranteed to capture the interest of most students. When I present it (usually to students at the 4th-5th grade levels, with their teacher accompanying them), I begin by asking a child selected at random: “How old are you?” Usually the answer is: “I’m 9 (or 10, etc.) years old.” To which I reply: “Oh, nice, today is your birthday.” There follows a lot of giggles. “No, it’s not!” I then say: “Well, you’re not really 9 after all. You are 9 years and some days more, right?” Of course, they can agree to that. So we then dive into the main aspect of the activity: to determine one’s age IN DAYS, instead of the usual years.

I use an example to demonstrate, often my own son’s age. [Those are photographs of him, his mother and myself at the top of this page, if you haven’t figured that out. ūüėČ ] Since he was born on Dec. 22, 1983, I usually do the activity around March or April. [The reason will
soon be apparent.]

Let’s assume a presentation date of March 14, 1997. The following is written on the board and discussed/explained step-by-step. [The words inside the brackets are presented merely here for explanation, not written.]

Kevin Trotter: Dec. 22, 1983
1. 365 x 13 = 4745 [age at last birthday party (1996 – 1983) times the number of days in a basic year; this is where the calculator helps with younger students]
2. LYD(96, 92, 88, 84)      = 4 [LYD = leap year days; the years in which a Feb. 29 occurred; here counting backwards by 4s is fun, and instructive.]
3. Dec = 9

Jan = 31

Feb = 28

Mar = 14

—-

82

[no. of days left in the birth month, after the party, so to speak]

[no. of days lived in January]

[no. of days lived in February]

[no. of days lived SO FAR in March]

[no. of days lived SINCE his last birthday party]

4. 4745 + 4 + 82 = 4831 days

Then we finish things off with a “summary statement”: “Kevin
is 4831 days old today.”


Teaching Advice

I often then repeat the procedure for the class’s own teacher’s age. That really perks up the kids’ interest! Then they find their own ages, or mine as well. Later they are encouraged to do likewisefor their parents and other family members. A wealth of practice is thus generated that is quite meaningful to any student.

I’ve found that it’s better to use, as an initial example, a date of birth (i.e. month/day) 4-5 months prior to the lesson date. It’s so that in Step 3 there is a medium length list of post-birth-month entries — not too long, not too short. In practice, if it’s too long, it worries the kids (it’s a LOOONG problem!); if it’s too short, they get “the wrong idea” (they generalize that this is a quick list). However, they must learn to accept variations and surprises as they come up. Some Step 3 lists are long; some are short. “It all just depends.”

One of the most interesting aspects of this involves whether or not the kids know the number of days in the months of the year. This causes a lot of errors and discussion. It provides a good
opportunity to introduce the famous nursery rhyme: Thirty days hath
September…

Thirty days hath September,

April, June, and November;

February has twenty-eight alone,

All the rest have thirty-one,

Excepting leap-year, that’s the time

When February’s days are twenty-nine.

The fun thing about this activity is that one’s age sounds so BIG! And it keeps changing by the day.


Footnote:

Additional challenging questions can be posed, depending on the mathematical maturity of the students, that use this basic concept. For example: On what date were you exactly 1000 days old? (That number could be changed to 2000, 3000, etc. to quickly provide more problems at no great cost of preparation time.)

When I do this with older students, at the middle school level, I make it into a little quiz, giving two invented dates, one that would be appropriate for a parent’s birthdate and another for their child’s birthdate. After finding the respective ages, in days, of course, I ask them to solve this problem: “The child’s age is what percent of the parent’s age?”

After this had been done one year, a student presented me with this poem that she wrote:

Mr. LeeI told Mr. Lee

to please tell me

what percent of his life

had he spent with his wife

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Kevin & Dad Kevin Kevin & Mom

The STAGES of AGES

Do you realize that the only time in our lives when we like to get¬†old is when we’re kids? If you’re less than 10 years old, you’re¬†so excited about aging that you think in fractions. How old are¬†you?…. “I’m four and a half” …. You’re never 36 and a¬†half …. you’re four and a half going on five!

That’s the key. You get into your teens, now they can’t hold you¬†back. You jump to the next number. How old are you? “I’m gonna¬†be 16.” You could be 12, but you’re gonna be 16.

And then the greatest day of your life happens …. you become¬†21. Even the words sound like a ceremony …. you¬†BECOME 21 … YES!!!

But then you turn 30 …. ooohhh what happened there? Makes you¬†sound like bad milk …. He TURNED, we had to throw him out.¬†There’s no fun now.

What’s wrong?? What changed?? You BECOME 21, you TURN 30,¬†then you’re PUSHING 40 ….. stay over there, it’s all slipping¬†away ……..

You BECOME 21, you TURN 30, you’re PUSHING 40, you¬†REACH 50 ….. and your dreams are gone.

Then you MAKE IT to 60 ….. you didn’t think you’d make it!!!!

So you BECOME 21, you TURN 30, you’re PUSHING 40,¬†you REACH 50, you MAKE IT to 60 …… then you build up so much speed you¬†HIT 70!

After that, it’s a day by day thing. After that, you HIT¬†Wednesday ….

You get into your 80’s, you HIT lunch. My grandmother won’t even¬†buy green bananas …. it’s an investment you know, and maybe a¬†bad one.

And it doesn’t end there …. into the 90’s you start going¬†backwards …. I was JUST 92 …

Then a strange thing happens. If you make it over 100, you become¬†a little kid again …. “I’m 100 and a half!!!!”

–found on the internet.

Happy & Dizzy Numbers

INTRODUCTION

Before we can explain what a happy number is, you have to learn a new idea, called “recurrent operations.” As the word “recur” means “to happen again”, a recurrent operation must mean a mathematical
procedure that is repeated. A very simple example would be the rule “add 5 to the result”. If we started with the number 0 and applied that recurrent operation rule, we would produce the sequence 0, 5, 10, 15, 20, … ; this list is the famous “multiples of five”.

Of course, there are all kinds of recurrent operation rules in mathematics. Another important rule is “multiply each result by 2″. If we used 1 as our first number, this sequence shows up: 1, 2, 4, 8, 16, 32, … ; this list is the also famous “powers of two”. So, you see it’s really not such a difficult idea now, is it?

However, in order to produce “happy numbers”, we will invent a rule that is just a little bit more complicated. (After all, you
didn’t expect this to be that easy, did you?) Our rule now will be given in two steps: (1) find the squares of the digits of the starting number; then (2) add those squares to get the result that will be used in the repeat part of your work.

Here is an example. Let’s start with 375. We write:

32 + 72 + 52 = 9 + 49 + 25 = 83

Now we repeat the R.O. procedure with 83. This gives us:

82 + 32 = 64 + 9 = 73

Of course, we continue with 73. This will produce 58.


HAPPY NUMBERS

But we can hear you saying: “When do I stop? What’s the point of all this?” That’s the beautiful part of the story. The answer is: when you see something strange happening. The strange thing that tells when a number is happy is simply this: the result of a 1 eventually occurs. Here is an example, starting with the number 23:

4 + 9 = 13; 1 + 9 = 10; and 1 + 0 = 1.

It’s that easy! When you reach a 1, the starting number is called happy. [But don’t ask why it’s happy, instead of sad; that’s just what the books say.]

Once you determine a number is happy, you can say all the intermediate results are also happy. The numbers 13 and 10 must also
be considered as happy, because they too produce a 1.

Can you find some more happy numbers? Yes. If you know a certain number is happy, it’s easy to find many more. How? One way
is to insert a zero or two. Look: above we saw that 23 was happy, right? This means that 203 is also happy; so is 230. A larger example is 2003. See? Now you can make many, many happy numbers, using an old one with as many zeros as you wish.

But that’s the easy way. You want something a bit more challenging, don’t you? Well, that’s your task now — find some more happy numbers without using the “zeros” technique. Okay?


Part II: Dizzy Numbers

The term “dizzy numbers” was invented by me. It is based on an idea that should occur to anyone searching for happy numbers, because often they find themselves “going in circles”, literally, i.e. getting
dizzy. Here’s why:

Recall the number 375 from above? It produced the sequence 83, 73, 58,… But we stopped there in our explanation of the RO procedure. If we had continued, we would have had 89, 145, 42, 20, 4, 16, 37, and then back to 58! Hmm… now that’s strange, isn’t it? We’ve returned to where we were (58) just eight steps earlier; we’ve gone in a circle. We’ve produced an 8-term numerical cycle. Hence, we’re getting a little dizzy. (Get it?)

So we can now define more formally a dizzy number to be one that is either part of that cycle or produces a sequence that enters the cycle eventually (like 375 did).

Now, do you want to hear something really strange? All numbers that are not happy are dizzy! That’s right. No matter how big or small a number may be, if you use the sum-of-the-squares-of-the-digits RO procedure on it, you either reach a 1 or the 8-term cycle. Amazing,
isn’t it?

Now armed with this new knowledge, you are ready to classify any number as happy or dizzy.

Have fun!


For more activities about recurrent operations, go to Kaprekar or Ulam.


Update: (6/24/02)

For additional information about this interesting topic, go to Mathews: Happy Numbers.

Angle Activity

Background Comments

One of the more common lessons in measurement in middle school¬†math classes is that of measuring the number of degrees in an angle.¬†This, of course, implies the use of a fairly technical instrument, for¬†students of this age anyway: the protractor. All those strange numbers¬†going in two different directions is certainly more complicated than an¬†ordinary ruler.¬†But once that aspect is taken care of, there still remains the¬†matter of homework exercises for practice. One thing that has always¬†bothered me is that when a textbook shows several examples of angles¬†to be measured, a big problem emerges. The angles are often rather hard¬†to measure, from a physical standpoint. By this I mean they are often¬†too small to fit the protractor being used; or they are positioned too¬†close to the center of the opened book, hence it is difficult to make¬†the protractor lie flat, etc.¬†The obvious solution is to have the angles drawn on flat pages.¬†This implies one of two things: a separate workbook or lots of single-sheet photocopies for the teacher to prepare.¬†But wait! There’s another way. Why not have the student draw¬†his own angles on his own paper before measuring? It certainly would¬†give him added practice with the concept.¬†But wait again! How would one know if all his work was being¬†done at least reasonably well? With a large class doing such a home-work each with their own angles of all sizes, it would be a nightmare¬†to evaluate, even in a general way.¬†A solution to that dilemma would be: design a way to “force”¬†the students to make angles of a size that the teacher knows in advance¬†what the degree measure will be. This can be done. Watch.

The Lesson

About five years ago, I designed a lesson that involves three¬†unusual allies: graph paper, the protractor, and a scientific calculator.¬†Here is how it’s done.

1. First we will draw our angle on graph paper, using a set of coordinate axes.
2. Next we will measure it with our protractor.
3. Then we will use our calculator to check our measurement accuracy.

Example: Measure the angle formed by the x-axis and a ray that passes through the point (8, 6).

Solution: The angle has been drawn on a pair of coordinate axes, as shown below.

When we measure it, we see that it is approximately 37¬ļ.

But how can the student be sure that the answer is correct, or at least reasonably close? It is sometimes difficult to handle that instrument. So I told my students to get out their scientific calculators and do this key sequence:

1. Press 6 [√∑] 8 [=] (Result: 0.75)

2. Press [Inv], or [2nd], or [Shift] (depending on one’s model)

3. Press [tan].

The result in the display is 36.869898, which rounds to 37 (to the nearest whole number), or 36.9 (to the nearest tenth). Our work was close enough to be acceptable.

And that, in a nutshell, is all there is to it. Simple, direct,¬†and multi-faceted in approach. We used skills in plotting points in the¬†coordinate plane, measuring with protractors, and utilizing some unusual¬†keys (for middle school students anyway) on a calculator. By the way,¬†no mention is made about the trigonometry concepts that are implied in¬†the calculator checking process. My experience showed that the kids¬†could accept the procedure for what it was: a way to find the number¬†of degrees in the angle. They were instructed that in Step #1 that it¬†was always “divide the y-value of the ordered pair by the x-value“.

Homework Sample

Exercises: Draw the angle first; measure it with your protractor; and finally do a calculator check.

1) (7, 5) 2) (2, 9) 3) (10, 3)

4) (6, 11) 5) (14, 12) 6) (15, 15)

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