# Monthly Archives: January 2015

## Dear Mr. Cuomo – A Fourth Grader Speaks Out

And it’s not as though passing these supertests indicates a great talent in the math that the common core espouses. Just jump through a different set of hoops.

This is going to be my third blog about a Student Hero. My newest hero is Josh – a fourth grader from NYC who has a legitimate beef with Governor Cuomo. Here is the very persuasive letter he wrote to the governor: He is my hero because he is bold and brave and not afraid to express himself in writing. He is also my hero because he wants his words shared with the world. Josh happens to be the son of a filmmaker. Josh expressed his desire to impact the world with his own words just as his father impacts the world with his movies. His dad agreed to post his letter on Facebook and included his own fatherly rant with the posting. In the rant, his father stated:

…for full disclosure josh will never take a standardized test not while im around to make sure he doesnt. but we…

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## The Distributive Law, again !

The formal statement of the distributive law should read as follows:

If a, b, c and d are numbers, or algebraic expressions (same thing really) and b = c + d then ab = ac + ad

It is a by-product of the law that it tells you how to expand an expression with a bracketed factor.

In any case, what’s the big deal ?

Filed under abstract, algebra, arithmetic, language in math, teaching

## The angle bisector theorem and a locus

The theorem states that if we bisect an angle of a triangle then the two parts of the side cut by the bisector are in the same ratio as the two other sides of the triangle. So I thought, what if we move the point, G in the gif below, so that GB always bisects the angle AGC, then where does it go ? Or, more mathematically, what is its locus. Geometry failed me at this point so I did a coordinate geometry version of the ratios and found that the locus was a circle ( a little surprised !). The algebra is fairly simple.

What was a real surprise was that if we take the origin of measurements to be the other point where the circle cross the line ABC, then point C is at the harmonic mean of points A and B. There is YOUR problem from this post.

Getting the construction to show all this was tricky. It is shown below.

I did the construction on my geometry program, Geostruct (see my Software page).

Line bb passes through A. Point D is on line bb. Point E is on this line and also on the line through C parallel to line cc joining B and D, So AD and DE are in the same ratio as AC and CB. When point D is moved the ratios are unchanged (feature of the program). So the point of interest is the intersection of a circle centre A, radius AD, and a circle centre C and radius DE (= FC)

Filed under algebra, geometry, teaching

## The mean ? Which mean ? With interesting ratios.

Playing around with the Harmonic Mean of two numbers I stumbled on an interesting ratio, and looked at the others as well.

Here are the definitions, for numbers a and b

If we use m for the mean, then

for the arithmetic mean we have the ratio (b-m)/(m-a) = 1

for the geometric mean we have b/m = m/a

for the harmonic mean we have (b-m)/(m-a) = b/a

and for the RMS mean we have (b2 – m2)/( m2 – a2) = 1

I am quite sure that there is a way of seeing these which ties them all together, perhaps Mr. Joseph Nebus can find it !

The harmonic mean can be used to explain the harmonic tuning of a keyboard instrument (as opposed to equal temper tuning). I am planning a post on this for later.

The formula I gave for the harmonic mean is not the usual one – use a bit of algebra ! – but it is easier to calculate with.

The RMS mean is used extensively in Statistics, Rigid Body Dynamics and Electrical Engineering. The well known 110 volts in your house electric system is the RMS mean of the alternating voltage actually supplied. The Standard Deviation is the RMS average of the distances of the data values from the arithmetic mean value.

A non formal view of these means (the first three) is that the arithmetic mean is about the positions of the two numbers, the geometric mean is about the absolute sizes of the numbers and the harmonic mean is about the relative sizes of the numbers.

if we take the zero, the two numbers, and the harmonic mean the four values have a cross ratio of -1 (see part 3 of the Christmas Tale)

Filed under abstract, engineering, statistics, teaching

## A “real life” geometry problem.

While designing a system for connecting “educational” cubes together I figured that the holes in the faces had to be positioned very carefully. To achieve what I wanted the holes had to be positioned with  length a equal to length b, and length c had to be twice the length a.

So what is length a, as a fraction of the side of the cube ?

There will be eight holes altogether, and all cubes are the same size.

Filed under education, geometry, teaching

## I didn’t know whether to laugh or to cry,

so I posted it without further comment (too many <expletive deleted>’s required).

Click the link for the full thing with theory, explanation and so on. The item is in .doc text format

Filed under arithmetic, education, teaching

## More bad language in math

Here is another horror which I found recently:

The distributive law of addition: a(b + c) = ab + ac (OK, it’s a definition)

The current school math explanation:
You take the a and distribute it to the b to get ab
and then you distribute the a to the c to get ac
and then you add them together to get ab + ac

I have come across this explanation in several places, and once again real damage is done to the language, and real confusion sown. “Distribute” means “to spread or share out” as in “The Arts Council distributed its funds unevenly, as usual. Opera got the lion’s share.” So it is NOT the a that is distributed. I tried to find a definition of the term in wordy form as it applies to algebra systems but failed. Heavy thinking produced the “answer”. What is being distributed is the second factor on the left.
Example:
Take 3 x 7. We know that the value of this is 21
Distribute, or spread out, the 7 as 2 + 5 . . . . . . . . the b + c
Then 3 x (2 + 5) has the value 21
But so does 3 x 2 + 3 x 5. To check, get out the blocks !
So 3 x (2 + 5) = 3 x 2 + 3 x 5 ……… The Law !

Regarding the “second” version of the distributive property, a(b – c) = ab – ac, this cannot just be stated, and you won’t find it in any abstract algebra texts. Since the students are looking at this before they have encountered the signed number system, a proof must not involve negative numbers, as a, b and c are all natural numbers. It can be done, and here it is:

set b – c equal to w (why not!)
then b = c + w
multiply both sides by a
ab = a(c + w)
expand the right hand side by the distributive law
ab = ac + aw
subtract ac from both sides
ab – ac = aw
replace w by b – c, and then
ab – ac = a(b – c)
done !

Filed under abstract, arithmetic, language in math, teaching

## Testing to destruction.

Testing to destruction.

“IIHS Hyundai Tucson crash test” by Brady Holt – Own work. Licensed under CC BY 3.0

A method used in manufacturing for product testing, where the product is
designed for a single action, and is used in practice as an insurance. The
best example is vehicle air bags. To see if they work a car is driven into
a wall, and the effect on the dummy people in the car is assessed.
Unfortunately some medical procedures can have a similar effect, of course
unintended. The classic case is amniocintesis, a procedure for assessing
the presence of Down’s Syndrome in the fetus. The reality was that the
probability of a fetus having the syndrome was way smaller than the
probability of the procedure itself causing a miscarriage. Initially, and
for quite a long time this was not realised. Eventually the test was only
offered to women who had a higher chance anyway of having a syndrome baby.
Could there be a connection between this stuff and the roll out of high
stakes testing in schools. Think about it.

Filed under education, statistics

## Out of the Mouths of Babes Comes Truth

So what happened to “education”?

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## The New GCSE #1 OCR Paper 4

Perhaps you all want to know what goes on in England. Here is a sample GCSE maths exam, taken at the end of year 10, or later, or still later. 3 exam papers in all. This is at the “higher”level and it shows what is expected by any student wanting to go on to higher education (University). Of course, more is required of those wanting to study maths, hard science or engineering. For the rest, maths stops here. These papers are taken by the upper 30% of students. There is a foundation level as well, in which the top grade is deemed OK for college.
Check this for the same stuff from another examining company:
http://solvemymaths.com/2015/01/16/the-new-gcse-3-aqa-paper3/

Interestingly, Pearson has got its sticky fingers into the UK game as well.