Tag Archives: rigid

Quadrilaterals – a Christmas journey – part 1

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
rhombus”.

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:

quadrilaterals 1 pic

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:
arect2
It consists of four line segments, AB, BD, DC and CA

Let us see what the full extended lines look like:
arect3
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.

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Congruence, proof, and rigid motions: The Common Core says WHAT, not HOW

With all the stuff in the high school geometry about proving congruence by rigid motions we get this sample geometry question from PARCC
PARCC geom test proofdef
(get the rest from numberwarrior here)

Numberwarrior’s concerns are about the language and the formal properties of congruence and I agree with him on this.
My concerns are about the stated claims of the CCSS to specify the “What do the need to know/understand/be able to do”, and the PARCC test which says “This is HOW you do a proof”.
In this particular example there are other ways of proving the assertion, not least those using the definition of congruence by rigid motions.

Let us do it this way:
1: vertical angles are equal, as there is a rotation of line AD to GC through the angle CBD, and then AD is on top of GC, so angle ABD ABF is also the angle of rotation, and is therefore congruent to angle CBD
2:There is a translation of line HE to line AD, as they are parallel. So the translation of H to H’ puts H’ on the line AD, and so angle H’BF is congruent to angle ABF.
3: But angles are preserved by rigid motions, so angle H’BF is congruent to HFG, and therefore angle ABF and HFG are congruent.

So, if I chose to teach about proof using this approach (my “HOW”) the students won’t even understand the question. Test items MUST be “Method Free”.

Also, the so called Reflexive, Symmetric and Transitive properties of congruence are no different from a=a, if a=b then b=a, and if a=b and b=c then a=c for numbers, and in both situations these are so STUNNINGLY obvious that it is cluttering up the minds of the learners to burden them with this sort of stuff. It is clear to me that this is a contribution to the CCSS from the sole pure mathematician on the committee.

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Rigid Transformations – Coordinate axes

A simple diagram with original axes in blue.
The coordinates of point E are (1,1)
A translation defined by x -> x + 2, y -> y + 1 moves point E to point D, with coordinates (3,2)
translation v moving axes

If the x axis is moved 2 steps left and the y axis is moved one step down then the coordinates of the original point E in the moved axes are (3,2)

This will be the case for any original point – the coordinates of each one of them will be the same as the coordinates of their new positions under the translation (in the original coordinate system).

This can be seen to be true for the other rigid motions, for example
rotation about the origin through an angle theta is equivalent to a rotation about the origin of the axes through an angle minus theta. So there is a one to one correspondence between rigid motions and change of axes (scales preserved).

On a lighter note, it does seem easier to rotate a pair of axes than rotating the whole plane ! ! !

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Common sense versus logic and math: Congruence again

I thought I would write a computer routine to check if two figures were congruent by the CCSS definition (rigid motions). One day I will post it.

The most important thing was to be specific as to what is a geometrical figure. You can read the CCSS document from front to back, back to front, upside down and more, but NO DEFINITION of a geometrical figure. For the computer program I decided that a geometrical figure was simply a set of points. My diagram may show some of them joined, but any two points describe a line segment (or a line). So a line segment “exists” for any pair of points.

The question is “Are the two figures shown below congruent or not?

congruent or not

I rest my case…..

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Congruence. Transformations. Definitions. Unnecessary.

Once again I take the Common Core to task.

This time their obsession with transformations.

There is nothing wrong with learning about transformations but it is silly to attempt a definition of congruence in terms of transformations, as we will see. Rigid transformations of the plane (translations and rotations) preserve lengths and angles, and so a rigid transformation of a figure gives a new figure which is “the same”. Also, and ignored by CCSS is the vital fact that such a transformation can take an image to ANY position in the plane, where position is identified as “point + direction”.

So this is what is written:

High school geometry: Congruence
Preamble
The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here
assumed to preserve distance and angles (and therefore shapes generally).
In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. (me – This last is garbage – see at end)

So here goes, an attempt to see if line segment FQ is congruent to BA

congruent line segs rigid transf

We slide BA in the direction B to F until B coincides with F – this is our translation.  A is now at the point K

We then rotate BA around point F until it is on top of  FQ, it is now in position FL. This is all possible as the position of the (simple) figure FQ must be known.

Have we carried BA onto FQ ? Not necessarily. It depends on where point Q is. The only way that L and Q can be the same point is a) if we know the length of segment FQ and b) this length is equal to the length of FL, which is the length of BA.

Therefore the only way that BA can be moved and placed on FQ is if they are the same length.

What have we done?  We have shown that if BA and FQ (or any two line segments) have the same length then there is a rigid transformation carrying one to the other. Hence by the definition of “congruent” BA and FQ are congruent.

In simple words, If BA and FQ have the same length then they are congruent, so simplicity (a highly desirable aim in math) is better served by the simpler definition:

TWO LINE SEGMENTS ARE DEFINED TO BE CONGRUENT IF THEY HAVE THE SAME LENGTH (AND not OTHERWISE)

The dragging in of transformations really muddies the water.
Here is part of the High School geometry detail

Understand congruence in terms of rigid motions
6. …..; given two figures, use the definition of congruence in terms of rigid motions
to decide if they are congruent.

and here is some dictionary stuff on superposition

http://www.thefreedictionary.com/superposition
 2.  The principle by which the description of the state of a physical system can be broken down into descriptions that are themselves possible states of the system. For example, harmonic motion, as of a violin string, can be analyzed as the sum of harmonic frequencies or harmonics, each of which is itself a kind of harmonic motion; harmonic motion is therefore a superposition of individual harmonics.
3. superposition – (geometry) the placement of one object ideally in the position of another one in order to show that the two coincide
locating, positioning, emplacement, location, placement, position – the act of putting something in a certain place
 4. superposition – the placement of one thing on top of another
locating, positioning, emplacement, location, placement, position – the act of putting something in a certain place

There is no Principle of Superposition in geometry!

 

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