Category Archives: teaching

Political correctness in K-12 math

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:

Click to access 07carnegie.pdf

Link found on f(t)’s blog

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Filed under abstract, algebra, CCSS, Common Core, depth, education, K-12, math, standards, teaching, tests

MUSIC: tuning, scales, fractions, ratio, harmonics :MATH ! Part deux.

Recap from previous post. here is a readable table of the notes and relative frequencies for the scale of C:

beat harmonic scale
Notice that the whole tone intervals are not all the same size. There are two distinct sizes, with frequency ratios 9/8 and 10/9. Only the two semitone intervals EF and BC’ are the same. Check the others ! Use the slightly not obvious relationship between ratios  P/Q = (P/R)/(Q/R)

beat tone semitone

This was the situation faced by J. S. Bach when he wanted to write music in any key. He figured that if two intervals had the same relative size then the two notes from the first, played together, would have the same musical “feel” as the two notes from the second interval. He wanted the major chord Bflat-D-F to sound like the major chord C-E-G, and so on. Not only that, but the scale devised so far is lacking four of the five “black” notes, and hitherto these had been fudged in. If one pursues the harmonic approach one finds that for example F-sharp and G-flat are different. See later for details.

So he figured that as there were 12 semitones in the full octave the frequency of each note would have to be a fixed multiple of the previous note, so multiplying 1 by this fixed number 12 times must get you to 2. In other words the magic number is the twelfth root of 2. Its value is 1.059463094 from my calculator.

The table below shows the comparison between the new Equal Temper scale and the harmonic scale.

beat pitch comparison

In 1722 Bach composed twelve preludes and fugues for keyboard called Das Wohltemperierte Klavier

If the link doesn’t work here it is:

https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier

Part three will look at beat frequencies and reasons for preferring the harmonic system, though not for keyboard instruments.

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MUSIC: tuning, scales, fractions, ratio, harmonics :MATH !

beat spinet 1704A spinet, from1704

Nobody knows when or how humans began to make music, but as soon as they could make things that could be used to play “nice” sounds (call them “notes”) they wanted to make musical instments, objects that could produce a number of different notes. The ocarina was an early one of these, and a strange collection of notes it makes. Eventually it was found that some sequences of notes of increasing pitch sounded “better” than others, and the result was a “scale”. The one I am considering here is the well known eight note scale, which runs from a start note to a finish note one octave higher. Found on a piano as the “white” notes, they are C D E F G A B and C’. As usual, it’s one thing knowing what you want, it’s often quite another to find a way of achieving it. In this case it is “string to the rescue”.

A stretched string can be made to vibrate by plucking it. The result is a “nice” sound. This sound will have a “pitch”, which is musical jargon for the frequency of the vibrations, which is physics/engineering jargon for the rate at which the string repeats its vibrations. This rate is expressed in repeats or cycles per second (or other unit of time).
For the record the pitch of the middle C on a piano is 256 (movements of the string up and back down
per second).

Now it was observed millenia ago that if a finger is placed on a string halfway along and the string is then plucked the vibration rate or pitch is doubled, and the sound is described as one octave higher than from the unfingered string. This new note is also known as the first harmonic of the original. The second harmonic is what you get if the finger is placed one third of the way along the string, and its pitch is three times the original. The third harmonic, with pitch four times the original, two octaves up, comes from a quarter of the original string length, and so on …….
Taking the second harmonic, and halving the frequency, or pitch, we get a new note which is between the original and the first harmonic, and either of these played at the same time as the new note gives a “pleasant” sound.
Doing this for the the fourth harmonic (one fifth of the original string length) we get another new note and the original plus the two new notes together produce what is known as the “major chord”.
On the piano, with C the original note, the new notes described above are the G and the E.
Now we can do some math !
If the pitch of C is 256 then the (harmonic) pitch of G is 256 times 3/2. Why? Because the pitch of a harmonic from a plucked string is inversely proportional to the fraction of the string used to make the harmonic.
First harmonic – half the string – pitch 2 times 256 (one octave up)
Second harmonic – one third of the string – pitch 3 times 256 (to get the G)
Third harmonic – one quarter of the string – pitch 4 times 256 (two octaves up)
Fourth harmonic – one fifth of the string – pitch 5 times 256 (to get the E)
and continuing..
Fifth harmonic – one sixth of the string – pitch 6 times 256
Sixth harmonic – one seventh of the string – pitch 5 times 256
The fifth gives a note one octave up from the second, and the sixth gives a note which we call B-flat,
with pitch 256 times 7 divided by 4

The result of all this is that we now have five notes in the scale:
C 256 … E 320 … G 384 … B-flat 448 … C’ 512
but it is easier to see what is going on if we just look at these as proportions of the “home” note C
Then we get
C 1 … E 5/4 … G 3/2 … B-flat 7/4 … C’ 2
or even better
C 4/4 … E 5/4 … G 6/4 … B-flat 7/4 … C’ 8/4

There do seem to be some holes in this, when comparing with the piano.
Where are D, F, A and B ?
We could go up in eighths, which does produce good values for D and B:
C 8/8 … D 9/8 … E 10/8 … G 12/8 … B-flat 14/8 … B 15/8 … C’ 8/4
but F at 11/8 and A at 13/8 don’t look good.
So we look at G, the first of the “new” notes, and think of it as a new home note. Then we can figure out
the B and the D in relation to the G.

The three notes C, E and G form a major triad (the chord of C major), and have the ratios 1, 5/4, 3/2
So to get the ratios of B and D to G just divide all three by the G to C ratio:
(3/2)/(3/2), (15/8)/(3/2) and (9/4)/(3/2) (using 2 times 9/8 as the ratio for D ‘)
and get 1, 5/4 and 3/2, which shows that G B D’ forms a major triad, just like C E G

We still have a problem with F and A, so let us try to set up F A C’ as a major triad by using some simple fraction manipulation.
F to C’ is to be the same ratio interval as C to G, so using the letter F to stand for its ratio to C we want
C’/F = G/C = 3/2
but C’ stands for the ratio of C’ to C, which is 2
So 2/F = 3/2, and solving we get F = 4/3

You can now do the calculations for A, which ends up as A = 5/3, so our full scale, including B-flat, is
C 1 … D 9/8 … E 5/4 … F 4/3 … G 3/2 … A 5/3 … B-flat 7/4 … B 15/8 … C’ 2

Observe that these frequency ratios determine the notes for the scales of C and F, and the scale of G without the F-sharp.
More ratios of interest can be found, as we can see that the intervals CD, DE, FG, GA, AB are whole tones, and EF and BC’ are half or semitones.

Up to the time when Johann Sebastian Bach intruduced the equal tempered scale together with a set of pieces written in each of the possible keys the methods of tuning keyboard instruments (harpsichord, clavichord, spinet) were based on some variant of the harmonic approach.

Here is a link for some heavier stuff:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Next post: The equal tempered scale and some more math.

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Gross misuse of + and – and x and the one that’s not on my keyboard

Arithmetic is the art of processing numbers.
We have ADD, SUBTRACT, MULTPLY and DIVIDE
In ordinary language these words are verbs which have a direct object and an indirect object.

“Add the OIL to the EGG YOLKS one drop at a time”.
“To find the net return subtract the COSTS from the GROSS INCOME”.

In math things have got confused.
We can say “add 3 to 4″or we can say “add 3 and 4”.
We can say “multiply 3 by 4” or we can say “multiply 3 and 4”.
At least we don’t have that choice with subtract or divide.

The direct + indirect form actually means something with the words used,
but when I see “add 3 and 4” my little brain says “add to what?”.

There are perfectly good ways of saying “add, or multiply, 3 and 4” which do not force meanings and usages onto words that never asked for them.
“Find the sum of 3 and 4” and “Find the product of 3 and 4” are using the correct mathematical words, which have moved on from “add” and “multiply”, and incorporate the two commutative laws.

If we were to view operations with numbers as actions, so that an operation such as “add” has a number attached to it, eg “add 7”, then meaningful arithmetical statements can be made, like

“start with 3 and then add 5 and then add 8 and then subtract 4 and then add 1”

which with the introduction of the symbols “+” and “-“, used as in the statement above allows the symbolic expression 3+5+8-4+1 to have a completely unambiguous meaning. It uses the “evaluate from left to right” convention of algebra, and does not rely on any notion of “binary operation” or “properties of operations”.

If we want to view “+” as a binary operation, with two inputs then, yes, we can ascribe meaning to “3+4”, but not in horrors such as the following (found in the CCSSM document):

To add 2 + 6 + 4, the second two numbers can be added to make a ten,
so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

If + is a binary operation, which are the two inputs for the first occurrence of + and which are the inputs for the second occurrence of + ?
The combination of symbols 2 + 6 + 4 has NO MEANING in the world of binary operations.

See A. N. Whitehead in “Introduction to Mathematics” 1911.
here are the relevant pages:
whitehead numbers 1
whitehead numbers 2a
whitehead numbers 2b
whitehead numbers 3a
whitehead numbers 3b

And here are two more delights from the CCSSM document
subtract 10 – 8
add 3/10 + 4/100 = 34/100

In addition I would happily replace the term “algebraic thinking” in grades 1-5 by”muddled thinking”.

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Filed under arithmetic, big brother, Common Core, language in math, operations, subtraction, teaching

Vertex of a parabola – language in math again

Here are some definitions of the vertex of a parabola.

One is complete garbage, one is correct  though put rather chattily.

The rest are not definitions, though very popular (this is just a selection). But they are true statements

Mathwarehouse: The vertex of a parabola is the highest or lowest point, also known as the maximum or minimum of a
parabola.
Mathopenref: A parabola is the shape defined by a quadratic equation. The vertex is the peak in the curve as shown on
the right. The peak will be pointing either downwards or upwards depending on the sign of the x2 term.
Virtualnerd: Each quadratic equation has either a maximum or minimum, but did you that this point has a special name?
In a quadratic equation, this point is called the vertex!
Mathwords: Vertex of a Parabola: The point at which a parabola makes its sharpest turn.
Purplemath: The point on this axis which is exactly midway between the focus and the directrix is the “vertex”; the vertex is the point where the parabola changes direction.
Wikibooks: One important point on the parabola itself is called the vertex, which is the point which has the smallest distance between both the focus and the directrix. Parabolas are symmetric, and their lines of symmetry pass through the vertex.
Hotmath: The vertex of a parabola is the point where the parabola crosses its axis of symmetry

Scoring is 10 points for finding the garbage definition and 5 points for the correctish definition !!!! Go for it!

When I studied parabolas, back in 1958 or so (!) the parabola had an apex. So I checked the meaning of vertex, and found that the word was frequently misused.

Here is a good account: https://en.wikipedia.org/wiki/Vertex_(curve)

Basically a vertex of a curve is a point where the curvature is a maximum or a minimum (in non math terms, most or least curved).

Here are two fourth degree polynomials, one has three vertices and the other has five. The maximum curvature points are indicated. The minimum curvature points are at the origin for the first curve, and at the points of inflexion for the second curve (curvature = zero)

01Gquartic201Gquartic0

A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis.

For a circle, which has constant curvature, every point is a vertex.

The center of curvature at a (nice) point on a curve is the center of the closest matching circle at that point. This circle will usually lie “outside” the curve on one side of the point, and “inside” the curve on the other side. Look carefully at the picture. It is called the osculating or kissing circle (from the Latin).

The center of curvature can be estimated by taking two point close to the point of interest, finding the tangents at these points, and then the lines at right angles to them and through the points. the center of curvature is roughly at the point of intersection of these two lines

01center of curvature

The diagram below shows this estimate, for the blue parabola, at the vertex.

02center of curvature

Finally (this has gone on further than expected!) I found this delightful gif.

01Lissajous-Curve+OsculatingCircle+3vectors_animated

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Filed under conic sections, conics, construction, geometry, language in math, teaching

Mathematical discrimination in the brave new world of Common Core

Merriam Webster
discriminating
adjective dis·crim·i·nat·ing

: liking only things that are of good quality

: able to recognize the difference between things that are of good quality and those that are not

Full Definition of DISCRIMINATING

1: making a distinction : distinguishing
2: marked by discrimination:
a : discerning, judicious
b : discriminatory

Take your pick !

In the UK we used to call it prejudice, as in “racial prejudice”.

Today the perfectly good word “discrimination” has been hijacked and the old meaning has all but disappeared.

Now you ask “What’s this got to do with math?”.

We have to look at quadratic equations for an answer.

The standard quadratic equation is ax2 + bx + c = 0
It may or may not have real roots
and the sign of D = b2 – 4ac resolves this uncertainty.

D < 0 … no real roots, D = 0 … equal real roots D > 0 … different real roots

What is this D ? It is the “discriminant” of the equation.

It is used to discriminate between equations with real roots and equations without real roots.

So WHY isn’t it in the Common Core math standards. It’s about as standard a thing as is possible?

Clearly the CCSSM authors are guilty of serious discrimination in discriminating against the word “discriminant”, in order not to be accused of discrimination language. Neither “discrimination” nor “discriminant” appear in the CCSSM doc, and the result is that the poor kids ( in the little darlings sense) have to complete the square from scratch every time they have a quadratic equation ( instead of once only in order to get the discriminant formula).

So sad !

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Real problems with conic sections (ellipse, parabola) part two

So suppose we have a parabolic curve and we want to find out stuff about it.

Its equation … Oh, we have no axes.

Its focus … That would be nice, but it is a bit out of reach.

Its axis, in fact its axis of symmetry … Fold it in half? But how?

Try the method of part one, with the ellipse. (previous post)

parabola find the focus1

This looks promising. I even get another axis, for my coordinate system, if I really want the equation.

Now, analysis of the standard equation for a parabola (see later) says that a line at 45 deg to the axis, as shown, cuts the parabola at a point four focal lengths from the axis. In the picture, marked on the “vertical”axis, this is the length DH

parabola find the focus2

So I need a point one quarter of the way from D to H. Easy !

parabola find the focus3

and then the circle center D, with radius DH/4 cuts the axis of the parabola at the focal point (the focus).

Even better, we get the directrix as well …

parabola find the focus4

and now for the mathy bit (well, you do the algebra, I did the picture)

parabola find the focus math

Yes, I know that this one points up and the previous one pointed to the right !

All diagrams were created with my geometrical construction program, GEOSTRUCT

You will find it here:

www.mathcomesalive.com/geostruct/geostructforbrowser1.html

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Real problems with conic sections (ellipse, parabola)

So there is an oval hole in a metal casting. It’s supposed to be an elliptical hole. Is it ????? How can we find out ?????
A good start would be to find the line which would be the major axis if it was elliptical. This turns out to be an engineering problem, not a mathematical one (I cannot see a way!). If the oval curve has an axis of symmetry then the method below will find it:

Firstly, get a computer picture of the oval.
Take two circles, of different radii, and push them along until each one touches the oval in two places.

ellipse12

The line joining the two centers will be the axis of symmetry if there is one (this can be shown mathematically).

ellipse34

ellipse5

The equation of an ellipse uses the lengths of the major and minor axes. Do it !

The closeness to elliptic can be assessed in various ways. Think of one.

next…..finding the focus of a parabolic shape

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GEOSTRUCT, a program for investigative geometry

I have been developing this computer software / program / application for some years now, and it is now accessible as a web page, to run in your browser.

It provides basic geometric construction facilities, with lines, points and circles, from which endless possibilities follow.

Just try it out, it’s free.

Click on this or copy and paste for later : www.mathcomesalive.com/geostruct/geostructforbrowser1.html

.Here are some of the basic features, and examples of more advanced constructions, almost all based on straightedge and compass, from “make line pass through a point” to “intersection of two circles”, and dynamic constructions with rolling and rotating circles.

help pic 1
Two lines, with points placed on them
help pic 3
Three random lines with two points of intersection generated
help pic 6
Five free points, three generated circles and a center point
help pic 7
Three free points, connected as point pairs, medians generated
help pic 5
Two free circles and three free points, point pairs and centers generated
gif line and circle
GIF showing points of intersection of a line with a circle
hypocycloid locus
Construction for locus of hypocycloid
circle in a segment
gif002
GIF showing a dilation (stretch) in the horizontal direction
gif piston cylinder
Piston and flywheel
gif touching2circles
Construction for circle touching two circles
gif parabola
Construction for the locus of a parabola, focus-directrix definition.

 

 

 

 

 

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Duality, fundamental and profound, but here’s a starter for you.

Duality, how things are connected in unexpected ways. The simplest case is that of the five regular Platonic solids, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. They all look rather different, BUT…..

take any one of them and find the mid point of each of the faces, join these points up, and you get one of the five regular Platonic solids. Do it to this new one and you get back to the original one. Calling the operation “Doit” we get

tetrahedron –Doit–> tetrahedron –Doit–> tetrahedron
cube –Doit–> octahedron –Doit–> cube
dodecahedron –Doit–> icosahedron –Doit–> dodecahedron

The sizes may change, but we are only interested in the shapes.

This is called a Duality relationship, in which the tetrahedron is the dual of itself, the cube and octahedron are duals of each other, and the dodecahedron and icosahedron are also duals of each other.

Now we will look at lines and points in the x-y plane.

3x – 2y = 4 and y = (3/2)x + 2 and 3x – 2y – 4 = 0 are different ways of describing the same line, but there are many more. We can multiply every coefficient, including the constant, by any number not 0 and the result describes the same line, for example 6x – 4y = 8, or 0.75x – 0.5y = 1, or -0.75x + 0.5y + 1 = 0

This means that a line can be described entirely by two numbers, the x and the y coefficients found when the line equation is written in the last of the forms given above. Generally this is ax + by + 1 = 0

Now any point in the plane needs two numbers to specify it, the x and the y coordinates, for example (2,3)

So if a line needs two numbers and a point needs two numbers then given two numbers p and q I can choose to use then to describe a point or a line. So the numbers p and q can be the point (p,q) or the line px + qy + 1 = 0

The word “dual” is used in this situation. The point (p,q) is the dual of the line px + qy + 1 = 0, and vice versa.

dual of a rotating line cleaned up1

The line joining the points C and D is dual to the point K, in red.  The line equation is 2x + y = 3, and we rewrite it in the “standard” form as  -0.67x – 0.33y +1 = 0  so we get  (-0.67, -0.33) for the coordinates of the dual point K.

A quick calculation (using the well known formula) shows that the distance of the line from the origin multiplied by the distance of the point from the origin is a constant (in this case 1).

The second picture shows the construction of the dual point.

dual of a rotating line construction1

What happens as we move the line about ? Parallel to itself, the dual point moves out and in.

More interesting is what happens when we rotate the line around a fixed point on the line:

gif duality rotating line

The line passes through the fixed point C.  The dual point traces out a straight line, shown in green.

This can be interpreted as “A point can be seen as a set of concurrent lines”, just as a line can be seen as a set of collinear points (we have fewer problems with the latter).

It gets more interesting when we consider a curve. There are two ways of looking at a curve, one as a (fairly nicely) organized set of points ( a locus), and the other as a set of (fairly nicely) arranged lines (an envelope).

A circle is a set of points equidistant from a central point, but it is also the envelope of a set of lines equidistant from a central point (the tangent lines).

So what happens when we look for the dual of a circle? We can either find the line dual to each point on the circle, or find the point dual to each tangent line to the circle. Here’s both:

dual of a circle4

In this case the circle being dualled is the one with center C, and the result is a hyperbola, shown in green.  The result can be deduced analytically, but it is a pain to do so.

dual of a circle3

The hyperbola again.  It doesn’t look quite perfect, probably due to rounding errors.

The question remains – If I do the dualling operation on the hyperbola, will I get back to the circle ?

Also, why a hyperbola and not an ellipse ? Looking at what is going on suggests that if the circle to be dualled has the origin inside then we will get an ellipse. This argument can be made more believable with a little care !

If you get this far and want more, try this very heavy article:

http://en.wikipedia.org/wiki/Duality_(mathematics)

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