# cset

Discussion in 'Multiple Subject Tests' started by 2stressed, Sep 16, 2005.

1. ### 2stressedRookie

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Sep 16, 2005

I feel very discouraged having taken this test 4 times. The Math is killing me. Someone school me because I need special ed. help....seriously!

3. ### TeacherGroupieModerator

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Sep 16, 2005

My sympathies, 2stressed. What have you done in the past to prepare?

4. ### MalcolmEnthusiast

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Sep 16, 2005

What kind of questions are you having trouble with?

5. ### TeacherGroupieModerator

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Sep 17, 2005

Are your issues in fractions? decimals? percents? working with variables? setting up equations? perimeter/circumference, areas, and volumes? probabilities? Pick one or more, or name something I haven't, and then we on this forum can brainstorm some Plausible But Not Actual Test Questions to work out the kinks with.

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Sep 17, 2005

How many lines of symmtery does a parallelogram have?
How many angles of rotation does a parallelogram have?
Define a polygon?
Is a square a rectangle?
Is rectangle a square?
What is the formula for getting the internal angles of an n-gon?

Just a few questions that one might ask a budding elementary teacher.

7. ### TeacherGroupieModerator

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Sep 18, 2005

Let's review some definitions, to begin with. A parallelogram is a quadrilateral (four-sided figure that can be drawn in a single plane) in which opposing pairs of sides are of equal length. In other words, if it were drawn on a map on which up is north, the north and south pairs of sides would be the same length, and the east and west sides would be the same length. What's more, its opposite angles are equal: on the map, the northeast and southwest angles are equal, and the northwest and southeast angles are equal.

Now what about symmetry? Reflectional symmetry is identity along an axis: that is, if you draw a line down the middle of a plane figure, that's an axis; if you flip the figure over along that line and it looks EXACTLY the same (all the lines and angles match up EXACTLY), it's symmetrical in that axis. A square has four axes of reflectional symmetry, two through the sides (north to south, east to west) and two through the corners (northwest to southeast, northeast to southwest). A rectangle that is not a square has two: north to south and east to west, but not through the corners. A circle has infinitely many axes of reflection. A human being has one, more or less, from head to feet.

So how many axes of symmetry does a parallelogram have? Make one, draw some axes on it, and try flipping it. You should find that the sides end up "leaning" in different directions than they did before you started flipping it: that is, a parallelogram has no axes of reflectional symmetry.

Rotational symmetry is symmetry when the figure is rotated in space: that is, if you draw an axis on the figure and then spin it around its center, at some point before the axis completes a 360-degree rotation, the figure will look identical. A circle has a 1-degree angle of rotation (or 360 axes of rotational symmetry): whether you spin it a little or a lot, it looks the same. A square has a 90-degree angle of rotation (or four axes of rotational symmetry): every time it turns a quarter circle, it looks exactly the same.

Try rotational symmetry with your parallelogram. You should find that you have to rotate it a full turn - 360 degrees - before it looks EXACTLY as it did when you started.

Try rotational symmetry with a rectangle.

As to the definition of a polygon:

(a) it is a plane figure (can be drawn on a piece of paper and we see the whole thing - no hidden side)

(b) it has three or more angles

(c) it has exactly as many sides as it has angles, and each side is a straight line (no curves nor waves allowed).

Last edited: Sep 18, 2005
8. ### TeacherGroupieModerator

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Sep 18, 2005

Is a square a rectangle? Is a rectangle a square?

First, let's define "rectangle" and "square". A rectangle has four sides and four angles; opposing pairs of sides are equal (north = south, east = west) and opposite angles are equal (northeast = southwest, northwest = southeast). So far this sounds exactly like a parallelogram. But a rectangle is called a RECTangle because each of its angles is a 90-degree angle, or RIGHT angle. So a rectangle is a special kind of parallelogram.

A square has four sides and four angles; opposite pairs of sides are equal and opposite angles are equal, as for the parallelogram; and its angles are right angles, as for the rectangle. But in a square, adjacent sides are equal too: that is, north = east = south = west. So a square is a special kind of rectangle.

But a rectangle is not a square, or at least not necessarily, because ITS adjacent sides (north to east, for example) aren't necessarily equal. So a rectangle is not a square.

9. ### TeacherGroupieModerator

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Sep 18, 2005

As to the sum of the interior angles of a polygon, take a look at this:

http://www.coolmath4kids.com/interior.html

I just found this by following a tip I owe to a math teacher on one of the other forums: I launched Google and typed in "interior angles polygon applet". This brought up a whole raft of Java applets and other material illustrating properties of plane figures and angles of all descriptions.

Whether this would also work for science, I don't know, but it probably wouldn't hurt to try.

10. ### Calif StudentRookie

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Sep 18, 2005

The measure of "an" interior angle of a "regular" n-gon is 180(n-2)/n.

The "sum" of the interior angles of a polygon, whether it is a "regular" ( meaning it's equilateral) n-gon or not (meaning all sides could be different), is 180(n-2).

11. ### TeacherGroupieModerator

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Sep 18, 2005

Another version of the formula for one interior angle of a regular n-gon (or, for that matter, the average interior angle of a non-regular n-gon) is 360/n. In case you wonder, n stands for the number of sides: a standard red STOP sign is an 8-gon.

12. ### innovationguyCohort

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Sep 18, 2005

Um, this almost seems to be an intrusion, straying into another's terrain - I normally 'concern myself' with Single Subject Math issues, whatever that means! - but I couldn't restrain myself in this instance! (764 generations of the family have been comporting themselves with rank officiousness, without shame nor embarrassment, and it's all about something falling not far from the tree, you know!...)

Be that as it may, a small clarification is in order: the above ought to read

"the formula for ONE (or EACH) exterior angle of a regular n-gon (or, for that matter, the average exterior angle of a non-regular n-gon) is 360/n".

As the other chap says:
The measure of ONE (or EACH) interior angle of a regular n-sided (3, 4, 5, ...) polygon is 180(n-2)/n.

For instance, for an equilateral triangle:

EACH interior angle = (3-2)*180 / 3 = 60
=> EACH exterior angle = 360/3 = 120

Likewise, for a regular pentagon (like the building that houses the U.S. Dept. of Defense...):

EACH interior angle = (5-2)*180 / 5 = 108
=> EACH exterior angle = 360/5 = 72.

Succinctly, then:

* n-gon = n-sided polygon ~ n-sided figure on a surface

* Regular polygon ~ all sides congruent, all interior angles angles congruent, all exterior angles congruent (one follows from the other for a 'convex' polygon, actually: one that 'folds in' throughout...)

* Exterior Angle = 180 degrees - Interior Angle

* For ANY 'convex' polygon,

Sum of Interior Angles = (n-2)*180
Sum of Exterior Angles = 360

Consequently,

EACH interior angle of a REGULAR polygon = (n-2)*180/n [Basically, EACH angle = Sum/n...and that makes sense!]

EACH exterior angle of a REGULAR polygon = 360/n [Again, basically, EACH angle = Sum/n...and that makes sense!]

From above, if just one, say the interior OR exterior angle, is known, the other can be derived by subtracted the first from 180...they form a 'linear pair' or a straight line, you see!

Here's how the (n-2) quantity arrives:

1. Take ANY n-sided polygon (say, n = 8 => an 8-sided polygon: octogon).

2. Select ANY vertex (corner) and you can draw (n-3 = 5) DIAGONALS from that vertex to the others, thereby forming (n-2) = 6 [ah, that's where the (n-2) came from!!] - TRIANGLES.

Since the Triangle Sum = 180, the Sum of the Angles of (n-2) Triangles = (n-2)*180...

I personally remember the formula NOT as (n-2)*180, but as (n-2) TRIANGLES, since the visual is branded in my head in segments of fire!

[teachergroupie's excellent link offers a display of this 'concept'...for a pentagon and hexagon!]

Jay.
innovationguy@yahoo.com

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Sep 18, 2005

Wow what a response..thanks.

A great job guys!!

14. ### 2stressedRookie

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Sep 18, 2005

I've been trying to rememer formulas and just going throught the practice tests. Geometry and essay problems is giving me problems.

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