It all started with an aside in a blog post in which the author said how
some students have a real problem with statements such as “A square is a
From early years kids naturally like exclusive definitions, and have to be weaned off this. This would be easier if we were more careful with the word “is”. Even to me the statement ” a square is a rhombus” sounds weird, if not actually wrong. It would be better to be less brutal, and say “a square is also a rhombus” (and all the other such statements).
Even better, and quite mathematical, is the phrasing “a square is a special case of a rhombus”, as the idea of special cases is very important, and usually overlooked.
It is odd that the classification of triangles is done entirely with adjectives and the difficulty is thus avoided (but see later).
After fighting with a Venn Diagram I did a tree diagram to show the relationships:
I then got thinking about the words “triangle”, “quadrilateral”, “pentagon” etcetera.
“triangle” : three angles
“quadrilateral” : four sides
“pentagon” and the rest : … angles
The odd one out is the quadrilateral.
Take a look:
It consists of four line segments, AB, BD, DC and CA
Let us see what the full extended lines look like:
Let ab be the name for the full line through A and B
Likewise ac, bd and dc
Then we can see that the quadrilateral is determined by the points of intersection of the two pairs of lines ab,cd and ac,bd.
ab and cd meet at point E; ac and bd meet at point F
But if we consider the four lines then there are three ways of pairing them up. The two others are ab,ac with bd,cd and ab,bd with ac,cd.
This gives us two more quadrilaterals, and they all have the property that each side falls on one only of the four lines.
The three quadrilaterals are ABCD, FCEB and FDEA
ABCD is convex, FCEB is twisted and FDEA is not convex (concave at A)
Not only that, but also the first two are fitted together to give the third one.
This arrangement is called the “complete quadrilateral”, and has four lines and six points.
More next time.
a, the name for area
b, a letter much less used
c, a constant now called b
d, in calculus abused
e, for exponential growth
f and g, they’re functions both
and so we get to h
– – – – – – – – – – – – – –
h, the one that tends to 0
but it never quite gets there
i and j, are indices
k, with h a jolly pair
then a lot we will skip past
x and y are here at last
and we’re learning math so fast.
and in case you don’t get the tune, here it is backwards
em, yar, hod
Merry Christmas to all my followers
Reblog if you do.
(Note: I typed this in the old editor, too.)
Let’s talk about your new stats screen for a bit. I put up a one-sentence post a few hours ago to confirm that other people feel the same way I do, and it’s amassed eighteen comments and twenty likes in that time, so I’m pretty sure I’m not on my own here. I’ve been actively blogging on your site for about a year and a half, although I’ve had the account for several years longer than that, and I spend a lot of time obsessing about my stats. An unhealthy amount of time, in fact.
You recently changed your stats page, and by a number of indications you seem to be interested in user feedback on it. However, using your feedback form really didn’t give me a chance to explain what I actually dislike about it. It could be…
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A drawing is worth a thousand symbols !
And do get as far as this line and what follows:
“I find mixed success when inflicting drawings like these on my students.”
Math with Bad Drawings
Or, Seeing Arrays (Less Cinematic Than Seeing Dead People, But More Useful)
This year, I’m teaching younger students than I’ve ever taught before. These guys are 11 and 12. They’re newer than iPods. They watched YouTube before they learned to read.
And so, instead of derivatives and arctangents, I find myself pondering more elemental ideas. Stuff I haven’t thought about in ages. Decimals. Perimeters. Rounding.
And most of all: Multiplication.
It’s dawning on me what a rich, complex idea multiplication is. It’s basic, but it isn’t easy. So many of the troubles that rattle and unsettle older students (factorization, square roots, compound fractions, etc.) can be traced back to a shaky foundation in this humble operation.
What’s so subtle about multiplication? Well, rather than just tell you, I’ll try to show you, by using a simple visualization of what it means to multiply.
Multiplication is making an array.
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