“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.”
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.
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 units. 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.
1/2 —- 1/4 is a complex fraction …. really? Only if fractions are not numbers!!!!!!!