This is definitely worth a read.
Here is a quote:
“High schools focus on elementary applications of advanced mathematics whereas most people really make more use of sophisticated applications of elementary mathematics. This accounts for much of the disconnect noted above, as well as the common complaint from employers that graduates don’t know any math. Many who master high school mathematics cannot think clearly about percentages or ratios.”
And here is the link:
Link found on f(t)’s blog
Filed under abstract, algebra, CCSS, Common Core, depth, education, K-12, math, standards, teaching, tests
To show that the angle bisector of an angle in a triangle splits the opposite side in the ratio of the two adjacent sides.
My first proof used angles in the same segment. See
Several tries later (today) and i came up with this annoyingly simple proof:
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.
There is a website with 100 proofs of the famous theorem of Pythagoras, but when I trawled the net looking for a proof of the converse, they all assume the basic theorem.
Here’s how to do it from scratch, which is considerably more satisfying, and also a simple application of similar triangles and basic algebra:
They tell you that conic sections are exactly what the words describe : Slice a double cone, the edge of the slice is a conic section, parabola, hyperbola, ellipse.
Then they tell you that y = x^2 is a parabola, or that all second degree equations in x and y are conic sections, or worst of all, they come up with the focus/directrix definition.
NOBODY shows you how to get the equation from the sliced cone !!!!!!!!!!!!
Well, here goes – (the math is after the picture, and it is so simple)
Here are some gems from the CCSS math document.
CCSS grade 6
Apply and extend previous understandings of multiplication and
division to divide fractions by fractions.
How many 3/4-cup servings are in 2/3 of a cup of yogurt?
Well, I would say “None”
CCSS grade 7
Analyze proportional relationships and use them to solve real-world
and mathematical problems.
1. Compute unit rates associated with ratios of fractions, including ratios
of lengths, areas and other quantities measured in like or different
For example, if a person walks 1/2 mile in each 1/4 hour, compute
the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
And here we are asked to crack a nut with a sledgehammer.
Common sense to the rescue!
Half a mile in a quarter of an hour is one mile in half an hour and so 2 miles in one hour, or 2 miles per hour.
is a complex fraction …. really? Only if fractions are not numbers!!!!!!!
More on fractions in a later post…………………………………….