I got confused after using the word “his” to explain the genitive case of the definite article (here).
It’s especially confusing because the possessive pronoun for 3rd person plural (ie “their”), is the same word (“τους”) as the accusative form of the definite article for male plural (here), but NOT the same as the genitive form of the definite article for male plural (which is “των”).
Anyway, just to note that the definite article and the possessive pronoun are often, but not always, the same.
I’m currently helping my 15-yr-old son revise for his maths GCSE, and one topic is “finding the nth term of a quadratic sequence”. I’m an ex high school maths teacher, but I had forgotten how to do this. I couldn’t find decent complex examples on either of my favourite GCSE maths revision sites (Maths Genie and BBC Bitesize), and when you’re doing the more complex examples, a step-by-step guide is really useful.
So I’m placing my notes here in case they’re any use to anyone else.
You’re aiming for a result of an2 + bn + c, but easier examples might have a solution of an2 + b, and even easier ones will just be an2.
Simplest Example (an2):
Find the nth term for the following quadratic sequence: 3, 12, 27, 48, …
First calculate the gaps between the numbers – these are 9, 15 and 21.
Then find the gaps between the gaps – these are 6 and 6. Like this:
Take that 6 and divide it by 2 (it’s easy to forget to divide by 2!), to get 3. This tells you that your final result will contain the term 3n2.
I’ve already told you that this is a simple example – we’ve reached our solution: 3n2. But you should always check your results:
n
1
2
3
4
n2
1
4
9
16
3n2
3
12
27
48
Yup, that’s our original sequence.
More Complex Example (an2 + b):
Find the nth term for the following quadratic sequence: 1, 10, 25, 46, …
First calculate the gaps between the numbers – these are 9, 15 and 21.
Then find the gaps between the gaps – these are 6 and 6. Like this:
Take that 6 and divide it by 2 (it’s easy to forget to divide by 2!), to get 3. This tells you that your final result will contain the term 3n2.
Create a grid, which starts with your original sequence. Below that, add whatever rows you need to help you calculate 3n2.
Now, subtract 3n2 from the original sequence. So in the below grid, we subtract the fourth row from the first row, and that gives us a new sequence, which we have placed in the fifth row:
start
1
10
25
46
n
1
2
3
4
n2
1
4
9
16
3n2
3
12
27
48
start minus 3n2
-2
-2
-2
-2
We now have a row of constant numbers. This tells us we can reach a solution. It tells us to add -2 to 3n2, and that will be our solution: 3n2 – 2.
We can easily check this by adding up the fourth and fifth rows, which gives us the first row (the original sequence).
Most Complex Example (an2 + bn + c):
Find the nth term for the following quadratic sequence: -8, 2, 16, 34, …
First calculate the gaps between the numbers – these are 10, 14 and 18.
Then find the gaps between the gaps – these are 4 and 4. Like this:
Take that 4 and divide it by 2 (it’s easy to forget to divide by 2!), to get 2. This tells you that your final result will contain the term 2n2.
Create a grid, which starts with your original sequence. Below that, add whatever rows you need to help you calculate 2n2.
Now, subtract 2n2 from the original sequence. So in the below grid, we subtract the fourth row from the first row, and that gives us a new sequence, which we have placed in the fifth row:
start
-8
2
16
34
n
1
2
3
4
n2
1
4
9
16
2n2
2
8
18
32
start minus 2n2
-10
-6
-2
2
We don’t have a row of constant numbers yet, so we need to keep working. We need to look at the gaps between the numbers in our new sequence (in the bottom row of the table):
Now we have found a constant difference. This tells us that there will be a 4n in our answer. Note that this is because we have found a linear sequence. Note also that in the case of a linear sequence, we do NOT divide the number by 2.
So now we add some more rows to our grid. First we calculate 4n, and then we calculate 2n2 + 4n. Finally we subtract (2n2 + 4n) from our original sequence (subtract the 7th row from the first row):
start
-8
2
16
34
n
1
2
3
4
n2
1
4
9
16
2n2
2
8
18
32
start minus 2n2
-10
-6
-2
2
4n
4
8
12
16
2n2 + 4n
6
16
30
48
start minus (2n2 + 4n)
-14
-14
-14
-14
We now have a row of constant numbers. This tells us we can reach a solution. It tells us to add -14 to 2n2 + 4n, and that will be our solution: 2n2 + 4n – 14.
We can easily check this by adding up the seventh and eighth rows, which gives us the first row (the original sequence).
More worked complex examples
Note that in this next one there is a NEGATIVE difference between the terms of the sequence on row 5. This one can easily catch you out. Rather than thinking of the difference between the numbers, it helps to ask yourself, “how do I get from each term to the next one?” The answer in this case is, “subtract one”. This one can also look a little tricky because it contains fractional numbers, but you just follow the same rules as before:
Telling the Difference Between a Linear Sequence (an + b) and a Quadratic Sequence (an2 + bn + c).
When we calculate gaps between the numbers in the sequence, if the first level of gaps is constant, this means it is a linear sequence:
If the second layer of gaps is constant, it is a quadratic sequence:
This is one in a series of cheatsheets. Full list here.
PRONOUNS
If “they” refers to a group all males or male and female or its gender composition is unknown, αυτοί is used.
Εγώ
I
Εσύ
you (singular)
Εσείς
you (plural)
Εμείς
we
Αυτός
he
αυτή
she
αυτό
it
Αυτοί
they (male)
αυτές
they (female)
αυτά
they (neuter)
TO BE
Important note: the pronoun (Εγώ, εσύ) …is not always needed.
Εγώ είμαι
I am
Εσύ είσαι
you (singular) are
Εσείς είσαστε
you (plural) are (or είστε)
Εμείς είμαστε
we are
αυτή είναι
she is (or he, or it)
αυτές είναι
they (female) are
TO HAVE
Singular
Plural
First Person
I have – έχω (“echo”)
we have – έχουμε, έχομε
Second Person
you have – έχεις
you have – έχετε
Third Person
she has – έχει
they (f) have – έχουν, έχουνε
First Conjugation Verbs
Many Greek verbs fall into this same pattern for changing their endings (or conjugating.)
We call this group of verbs the first conjugation verbs.
Here are a few more of them, given, as always, in the first person form:
I see
βλέπω
I buy
αγοράζω
I drink
πίνω
I know
ξέρω
I take
παίρνω
I give
δίνω
I eat
τρώω
POSSESSIVE PRONOUNS:
Person
Pronoun (own one thing)
Pronoun (own many things)
1st person singular
(Δικός/Δική/Δικό) μου
(Δικοί/Δικές/Δικά) μου
2nd person singular
(Δικός/Δική/Δικό) σου
(Δικοί/Δικές/Δικά) σου
3rd person singular (masculine)
(Δικός/Δική/Δικό) του
(Δικοί/Δικές/Δικά) του
3rd person singular (feminine)
(Δικός/Δική/Δικό) της
(Δικοί/Δικές/Δικά) της
3rd person singular (neuter)
(Δικός/Δική/Δικό) του
(Δικοί/Δικές/Δικά) του
1st person plural
(Δικός/Δική/Δικό) μας
(Δικοί/Δικές/Δικά) μας
2nd person plural
(Δικός/Δική/Δικό) σας
(Δικοί/Δικές/Δικά) σας
3rd person plural
(Δικός/Δική/Δικό) τους
(Δικοί/Δικές/Δικά) τους
(masc/fem/neuter)
(masc/fem/neuter)
EXAMPLES:
Ο άντρας μου=My husband
Ο δικός μου άντρας= My own husband (emphatic).
WHAT’S THE DIFFERENCE BETWEEN Δικός, δική, δικό?
Δικός is used if the owned object is of masculine gender: Ο άντρας είναι δικός μου=The man is mine.
Δικός becomes δικοί when the owned object of masculine gender is in plural.
So, οι άντρες είναι δικοί μου=the men are mine.
Δική is used if the owned object is of feminine gender: Η γυναίκα είναι δική μου=The woman is mine.
Δική becomes δικές when the owned object of feminine gender is in plural.
So, οι γυναίκες είναι δικές μου=the women are mine.
Δικό is used if the owned object is of neuter gender: Το παιδί είναι δικό μου=The kid is mine.
Δικό becomes δικά when the owned object of neuter gender is in plural.
So, τα παιδιά είναι δικά μου=the children are mine.
THE DOUBLE ACCENT RULE
When μου,σου,του,της,μας,σας,τους comes after a word that is accented on the antepenult (second syllable from the end e.g. αυτοκίνητο), then it is accented also on the last syllable.
Example:
το αυτοκίνητό μου=my car
το ραδιόφωνό της= her radio
η τσάντα του=his bag (no double accent here because the word τσάντα is not accented on the antepenult!)
I’ve been using DuoLingo, which is great in some ways, but utterly bewildering in others. The app asks you to remember random sentences and words with no apparent attempt to explain any basic grammatical rules or word endings. In fact that info is available on the DuoLingo website, but is still a bit haphazard even there, so I’ve created some cheat sheets with some useful basic rules:
Up until last week, my ten-year-old couldn’t quite tie a shoelace. These days you can easily get away with it, because apart from walking boots, most kids’ shoes don’t have laces. But his new school shoes have laces. So it’s time for us to get this sorted.
I enjoy teaching, but there is this thing called “Expert-Induced Amnesia“. It’s about that skill you’ve had for so long, you don’t know how you do it. When you try to teach it to other people, you struggle. As a parent, I’ve been starkly reminded of this in two examples: One is riding a bike, and the other is tying your shoelaces.
Teaching a child to ride a bike is extra hard, because you know at some point you’ll have to let go and hope for the best. And have the band-aid ready. But the shoelaces thing… maybe it’s just me, but wow, I found it hard to explain what I do and how I do it. My fingers just know. I don’t think about it. It’s muscle memory. And as soon as I try to slow it down and explain it, I can’t even remember what to do. I can only do it quickly, in a blur.
Also, learners need to practice repeatedly to get it right. Explaining this repeatedly can get wearing, particularly as most of the teaching opportunities come when you’re in a hurry to get out the door, and really you just want the shoes on the feet, with minimal fuss.
One of the hardest parts of teaching is patience. Resisting the urge to do it for them. “Oh, I’ll show you,” you say. You think you’re being helpful, but really you’re just being impatient. They need to do it for themselves.
That was a ridiculously long preamble.
The original version of this post was based on a series of diagrams I drew for my sons, to teach them how to tie their shoelaces the way I learnt. But then people showed me two videos which claim to teach revolutionary new shoelace-tying methods which are super-easy to learn.
A video my son and I failed to follow
The second video was supposedly even better than the first, and therefore my 10-yr-old son and I started there:
We played this video, rewound the crucial bit, played it again, rewound it…
Half an hour later, we were none the wiser. The crucial instruction, “pinch both of them and pull” is accompanied by a blur of activity, and no matter how much we watched it, we couldn’t make head or tail of it. My main problem was that if both hands were holding the bits of lace we were directed to hold, then the fingers got in each other’s way and it wasn’t possibly for both of them to pull. Even if we briefly let go with one hand, we got nowhere. It was frustrating as hell but in the end we gave up and had a look at this video instead:
A video my son and I got our heads around in the end
This one had a similar sticking point – “Feed them through each other’s loops”. Again, we had to rewind a few times and replay. Neither of us could work out which bit went through which bit. One problem is that both of these videos are shot from the point of view of a spectator, NOT from the view of the person tying the knot.
In the end I worked out what was going on and added this one extra instruction for myself and my son, and finally we both got it: After you have created the two loops, when you get to the “pull the loops through each other” part, you will have two loops facing you. At this point, one pair of fingers should grab the front of the other hand’s loop. The other hand should grab the back part of the opposite loop. Then each hand pulls the part it holds, through its own loop.
Oh, and that first video is describing the same technique. If you imagine yourself grabbing the front and the back of each loop and pulling it through, and if you turn the “zigzag” into two loops, you can make it work. That’s the only way I managed to make sense of it, but maybe my brain is just broken.
Some really good diagrams
This set of diagrams here is really good, and shows the knot from the knot-tyer’s perspective: https://www.fieggen.com/shoelace/ianknot.htm
My original diagrams
The original version of this post was based on the diagrams I drew below, and I have to confess I’m rather proud of my lovely diagrams. So I’m going to leave the rest of this post in place, despite the fact that I now think the Ian Knot (above) is a better technique. Oh well.
I forced myself to sit down and work it out, step by step. Then I drew diagrams and stuck them to a piece of card with laces attached. Then I gave it to my eldest son and left him to practice on his own. And now it’s my youngest son’s turn, so I dug the card out again (which is why it looks all tatty and old).
And here it is. Apologies for the tattiness. But just in case you’re teaching somebody to tie shoelaces, or learning to do it yourself… here are some diagrams that just might help.
(Also, apologies if you thought this was going to culminate in some fantastic metaphor, where the laces represent the meaning of life, the universe and everything. It really is a post about shoelaces.)
This was originally the submission for a talk which I’ve now delivered in a few places. There’s a recording of the talk here.
“I know nothing.”
“I know less than nothing.”
“I am an impostor.”
“The more I learn, the more I realise how little I know.”
These are sentences that will be familiar to the vast majority of IT professionals. But how about these?
“They know nothing about X.”
“They have no relevant experience.”
“Wow, I just discovered my colleague doesn’t understand Y. I’m shocked.”
“Can you believe, I just interviewed this dev, and they didn’t even know what a Z was?”
Over my 18-year software career, those last two have been said to me countless times. They are said derisively, scornfully, impatiently. And every time those words are said, we lose both existing and potential members of our profession. We lose them because they feel stupid; because they believe they can’t keep up; because they have given up on ever really knowing what they’re doing; because they’re terrified that people are saying the same things about them.
We work in an industry where knowledge is highly valued, and where every time we look for a new job we have to prove how much we know. We find ways of posturing to one another, of proving how well-informed we are. Sometimes we join in when others’ knowledge is criticised – relieved that we are not the target ourselves.
And yet, we know that everybody has gaps. There are a million different paths through software development, touching a million different combinations of technologies and skills. On a day-to-day level we have to specialise on one task at a time. The skills we don’t need right now are necessarily forgotten, or delegated to someone else. And that’s fine.
If somebody already feels like they don’t “fit in”, then this kind of pressure and insecurity can be the final shove that persuades them to leave the profession or not try and join in the first place. Women and non-binary people, people of colour, older people, LGBT people and many other under-represented groups are strongly impacted by intellectual elitism. But of course, ALL software professionals are impacted.
Let’s stop putting pressure on individuals to know everything, and focus instead on how teams can work together to build and provide the unique combination of skills required to deliver their current project – in the certain knowledge that whatever that combination was today, it will be different tomorrow.
Instead of knowledge, let’s focus on aptitude. Instead of judging people about what they don’t know, let’s help them to feel excited about all the new things they’re going to discover.
Instead of saying “For God’s sake, do you really not know about X?” let’s say “Fantastic, you don’t know about X! Lucky you. That means you get to learn it. What can I do to help?”
(I’m hoping to do talks on this topic at events in 2018 – let me know if you have an event you’d like me to talk at).
I used to be a high school maths teacher. It didn’t go well. I miss the energy and creativity of the young people I worked with, but apart from that I have no regrets about leaving that profession behind me.
A friend of mine has just completed his first term of teacher training, and it’s HARD. You would think that now the Christmas holiday period has started, he’d be able to have a rest. Nuh-uh. Too much work to do.
Coincidentally I’ve also been facing a giant to-do list this week, and with the festive season fast approaching I’ve struggled to stay focused. But I have various coping mechanisms, which I just shared with my teaching friend and I thought were worth preserving for my own benefit too. And if they can help anyone else (particularly teachers), so much the better:
“What you have to remember is that you’ve been working flat out for months without even the option of slowing down. Whether you like it or not, your mind and body need a rest and what’s happening right now is that your subconscious is forcing that rest on you whether you like it or not. Sadly your conscious mind knows that there isn’t really time for that rest, and that’s preventing you from enjoying it – which is a shame.
So…
1) Accept right now that the workload is impossible. Therefore it’s either not all going to get done, or some of it will have to be done to a poor standard. There’s no point lying to yourself about that. You can’t bend time. You and thousands of other teachers are in the same position and you just have to accept it.
2) Given that you’re not going to get all of it done anyway, just pick TWO OR THREE SMALL THINGS and do those. Be unambitious. Don’t tell yourself “today I’m going to do the top 50 things on my to-do list,” because that’s a lie. You’re not going to achieve that, no matter how much you hope. And you already feel shit about that, and the whole thing is so depressing that it’s preventing you from doing anything.
Instead, aim low. Tell yourself you’re going to do a small number of things. Then do them, and feel good about having achieved your goals for the day. If that glow of success gives you the energy to do a little more, fantastic. But don’t plan for that, otherwise you’re back to square one.
Baby steps. You need to get back into a rhythm. If somebody gives you a mountain to climb when you’ve just eaten ten mince pies, you’re going to tell them to sod off. But a flight of stairs? Yeah. That’s a possibility.”
POSTSCRIPT:
I just did something which I realised I often do, and may also be good advice:
I scrolled down my to-do list until I found something I could easily do. The thing in question is pretty pointless and not at all urgent. It can easily wait til after Xmas and the world wouldn’t end if I never got round to it at all. But it’s a gateway drug. It’s not only easy, it’s vaguely enjoyable. And I know that it represents an “in”, ie once I’ve done that, I’ll be in the swing of Getting Things Done and I’ll find it easier to tackle a more worthwhile item from the list.
I know from experience that if I don’t do this, I’ll just arse around on the internet and do nothing from my to-do list. So it’s better to do something slightly pointless, but knowing I’ll find it much easier to follow it with something pointful… than to do nothing at all.
