## Tuesday, July 17, 2007

### Zeller's algorithm

Monday's child is fair of face,
Tuesday's child is full of grace,
Wednesday's child is full of woe,
Thursday's child has far to go.
Friday's child is loving and giving,
Saturday's child works hard for a living,
But the child born on the Sabbath Day,
Is fair and wise and good in every way.

So which day of the week were you born on? You can find out in a while if you don't know and more importantly if you want to.... (that's my rubbish attempt to sound enthusiastic. :( )

This week has been a busy one, but apart from today I'm just going to summarise my observations of the lessons at the end of the week (rather than everyday!). Truth be told I've been pretty tired and so my brain doesn't want me type. BTW repetition is not good for the brain, as says Dr Kawashima (my brain is 30 years old now- well he did tell me that sleep was important!). I think it's more of a point that you shouldn't always restrict yourself to repeating the same activity, without moving on when you can.

I was meant to plan part of a lesson for the year seven set 4 class and I had been considering asking them the same question that I just asked a minute ago- in a more enthusiastic manner of course. That lesson wasn't meant to be today, so I was going to discuss my ideas with the maths teacher during a free period. We didn't get a proper chance to discuss it, but we tried to see how we could make the lesson student friendly and more wordy. (It had too much going on).

I'll discuss the rest of the day later, but alas today I was once again attacked by the English department. :/ I had to help cover an English lesson for about 10 minutes. Thankfully the class had students who are now familiar with seeing my scary face and some know me from Maths lessons. (Phew!) I didn't want to do 'nothing' and I didn't want to do English, so I asked them the same question as well (about the day they're born on)! A few were interested and so after some persuasion we worked through the algorithm and most of them managed to work out what day of the week they were born on. (apart from those who didn't take part and glared at me for even suggesting doing some 'maths' in an English lesson!). Before I had to leave a few students groaned that this wasn't English and so I did an ickle 'syllogism' with them (did I mention that I liked them!). Thankfully I didn't stay in that lesson for long. Next lesson it was the year sevens. Set 4.

Before I describe the disastrous events that followed I will post how Zeller's algorithm works, through an example (my friends birthday). Zellers algorithm is a series of steps through which you can work out the day you were born on. It doesn't have to be for the day you were born, but it can work out the day of any other date as well. It's going to be in 'baby' language if that's OK. (Some notation before we start: [x] = the largest integer not exceeding x or 'integral part'.)

My friends DOB: 1/07/1988 (Write your DOB here!)

Date= D= 1
Month = M= 7
Year= Y= 1988

We take the year (1988) and split it into two components- 19 and 88. Call the 19 bit C and the 88 bit Y'.

C= 19
Y'=88

Then calculate:

$\displaystyle A= \left[ 2.6M-5.39 \right] + \left[\frac{Y'}{4} \right]+ \left[\frac{C}{4}\right]=12+22+4=38$

Now work out the sum:

$\\Z=D+Y'-2C+A\\ Z= 1 +88-38 +38\\ Z=89$

Divide this by seven and take the integer part,

$\displaystyle\left[\frac{Z}{7}\right]=\left[\frac{89}{7} \right]= 12$

Finally we calculate the remainder which turns out to be: 89 - (12 x 7)= 5, and so my friend was born on a Friday. *Quickly goes to check* Yep- that seems to be the case!! (If it's 1 you're born on a Monday, 2 a Tuesday.... ). Here's a more comprehensive version of the algorithm, and you might notice that dividing by 7, taking the integer part and then working out the remainder is the same as calculating 89 mod7.

I couldn't find the LaTeX code for the integer brackets properly (not that I looked hard enough!), but year sevens don't know about this notation. We did this in college but all I had from my notes was the calculations with no words. I also didn't have anything called Z or A, but to make things look 'nicer' I introduced them. However, that wouldn't look nice for the year sevens. Fair enough, I was going to write up a sheet with blanks and words describing what goes in each box, and instructions on what to do next. (I don't have my D1 book anymore, but if I remember correctly the algorithm was a set of instructions and just that!)

Originally I was meant to do this bit on Thursday's lesson, if indeed I did do it. But in today's lesson the teacher learnt that it was the last maths lesson with this class. A quiz was going to be prepared for the last ever lesson but that couldn't happen anymore. The teacher contemplated on whether to continue with angles and after some discussion I was at the front of the class all of a sudden, asking the same question again. ;(

I'm annoyed at myself for not preparing properly and even more so because I didn't react to the situation in a positive way. I just ... melted. Well you could still see my head (slightly), but inside I was shouting, 'Nooooooooooooooo' in a dramatic way as my life went zooming past. Yes- everything is still dramatic! Most students didn't have calculators and I confused many of them because I had decided not to call A 'A', and not to call Z 'Z'. Looking back I should have written sentences on the board for them to follow, but I didn't think. Some students actually did follow everything and waited impatiently as the other's struggled but the challenge is getting the students who struggle to overcome their difficulties. Apart from doing this on one occasion I messed up big time. The teacher helped a lot but I think when the teacher's present I become nervous all of a sudden. It's a weird sensation where normally my mind goes blank and I look expectantly at the teacher hoping that I don't have to say anything. (Which is why another teacher probably observed me on another occasion).

I guess it's because I feel under the light, but I was told to not pay any attention to the teacher being there, which is easier said than done! The lesson I'm afraid was awful (my bit anyway- the teacher recovered it later on). Some students were really eager to work out the day they were born on, but sadly they couldn't. The teacher told me not to worry about it too much, but I'm annoyed because it didn't go as I wanted it to. Maybe this was a stupid topic to pick after all. :( Thankfully this was the last lesson of the day, so I was able to come home and rub my wounds. Meh- I mustn't dwell on this for too long but I keep on thinking about what I should have done but didn't.

Now onto the 'beginning' of the day.

In the morning I was with a year 9 class. This is the final week of school (which signals my full time return to help Bob!), and the students were desperate for a free lesson (much to my dismay :D). The teacher asked the students who wanted a free lesson to put their hands up. All students hands went up apart from Lennie, who I mentioned last time, is my maths buddy! Another student did try and force Lennie's hand up, but Lennie was getting annoyed at the time wasting- there was maths to be done!

Since I was sat near Lennie's table and since Lennie was asking for maths, I decided to do some maths with him whilst the class discussed the free lesson. From the book, I opened the page on quadratics to see what to do. I then asked Lennie (and a curious friend), the answer for: $x(3x+6)$. They wrote: $3x+6x=9x$. They hadn't multiplied the x terms, and after asking them what 10 x 10 equals in terms on 10 (10^2), and a few other numbers, they realised it's meant to be x^2. A few other students 'mocked' Lennie, but I offhandedly mentioned to them that although they were not to quote me on anything, if you did a Maths A level you're likely to earn 9% more! (or something like that). Lennie seems to have his head screwed on straight and knows what he wants to achieve and what he must do so thankfully didn't care to much about what his classmates were saying.

By the time the class teacher had decided to do a lesson I was pleasantly surprised to find that it was to do with quadratic factorisation and solving quadratic equations. I didn't know this was year 9 material, and had only opened the page for Lennie since I thought he might find it useful in preparation for year 10. I think they had done something like this before but all of a sudden many students were confused when they had to solve the following: (x- 1)(x +1)=0. The teacher did explain this but a few didn't follow. (Why do I have a stinking suspicion that the teacher had heard me talking to Lennie about quadratic equations and then decided to do them?)

The problem I had was that I didn't know how much these students knew and what they knew. Had they encountered the quadratic graph before, and were they to know that when the graph cuts the x-axis we have a root? I asked questions to deduce what they knew but one student seemed to be having a bad day. (It was nice when the concept actually 'clicked' since I was with this group for a while.) They all did seem to be slightly more lethargic than normal and I also finally got the 'graffiti artist' student to do some questions. Woo hoo. (Although I spotted one student who had not even started the work!)

This lesson was quite good and I really enjoyed it. Now I'm jumping from lesson to lesson but let me tell you a little secret. I was going to go to SG's maths class today, and as I entered I was greeted with the site of graph paper! *alarm alarm* That was it- I quickly apologised to the teacher and said that I was with the other set today and made my get away to the other year 8 class. That wasn't much better, but better than using graph paper to do what not! I ended up sitting next to the worst table in the classroom, but I think they were playing up since I was sat there. (A lot of swearing and a lot of not nice words were said!) As soon as the lesson had been delivered and the students were told to get on with their work I was on the move, and quickly positioned myself on the opposite side of the classroom. The lesson was on transformations and a lot of the students did recklessly rotate a shape by 90 degrees and didn't check if it was correct, but they could do most of the work. Apart from the loud group the lesson went well.

The class teacher had thought I was doing my PGCSE (Post Graduate certificate of Education?)!!! So I've been mistaken for a year 10 student (15 years old), a college student (17/18), and now a post graduate (21+). No one seems to guess that I'm at university, and none of them believe that I'm seven! You could also look at my age mod 25, rather than it belonging to the group S_25 I suppose...

All the teachers seem to looking forward to the end of the week. I wonder why?!

(There's something I've forgotten but I can't remember what!! Duh!)

steve said...

I couldn't find the LaTeX code for the integer brackets properly

Do you mean the normal [ ] brackets on the keyboard? You can use them but, if you want to make them match the height of the expression they are bracketing, use \left[ ... \right] as in \left[\frac{89}{7}\right]

steve said...

It occurred to me that you actually wanted this:
\left\lfloor\frac{89}{7}\right\rfloor

beans said...

I hadn't realised that '[' was on the keyboard. :o I'd gone to the AoPS reference page and saw the word floor and remembered that floor meant to take the integer bit (I hope!).

So does \lfoor 2.65 \rfloor mean the same thing as [2.65]? (I'm thinking it does, but one can never be too sure.)

Thanks for the codes. :) If the above statement is true(!) then both of them do nicely. (I prefer using '[' though).

steve said...

Yes, \lfoor 2.65 \rfloor = [2.65]. [..] was always used in the past but there was some confusion caused by different computer languages having different meanings for the integer part of negative numbers.

I prefer the traditional use of [x] = the largest integer not exceeding x or 'integral part' (see the classic Hardy & Wright which I regard as the authority on the subject). But this term is often called 'integer part' which then disagrees with computer usage. Thus [-3.5]=-4 but Int(-3.5)=-3 and of course -4 is no longer the 'integer part of x'

Wikipedia gets round this by saying For nonnegative x, a more traditional name for floor(x) is the integral part or integral value of x. MathWorld wants to abandon [x] (due to its relationship with Mathematica), and PlanetMath gets it all wrong, mistaking 'integral part' for 'integer part'.

beans said...

Yes, I recall [x] being used by my supervisor (in the first semester) in the traditional way,and I did get confused with the negative number issue,(stil am as a matter of fact!). I normally take [x] to mean the integer part of x you see.

I think I'll need to read the Planet Math description again in the morning (the only thing that made sense was a lecturers name from The Uni. of Manc.!) Hmmm, I think we should specify what we want [x] to be, since there seems to be a lot of different things being said.

gah- sorry I'm slightly confused in my head now. (I'd been doing things a weird way before and had taken floor and [...] to mean integer part!) That served me well but you're right in saying negative numbers 'spoil' it.

BTW, thanks to your list I've got that book. :D I best leave having a look at it for the morning though! (Just realised it's been a while since I've added a book to my collection).

Joshua Issac said...

Thanks. MathsNet.net gave me an incorrect algorithm (for Zeller's).

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