March 20, 2012 Aim: How Do We Find The Area Of A Circle?

Circle Area Conjecture:

-The area of a circle is given by the formula__,where "A" is the area and "r" is the radius of the circle.

  

Examples:

Example 1:	The radius of a circle is 3 inches. What is the area?
Solution:
	= 3.14 · (3 in) · (3 in)
	= 3.14 · (9 in2)
	= 28.26 in2

http://www.mathgoodies.com/lessons/vol2/circle_area_part3.html 

Now You Try It:

1.The radius of a circle is 9 centimeters. What is the area?

ANSWER BOX:   = cm²  

RESULTS BOX: 

2.The diameter of a circle is 12 inches. What is the area?

ANSWER BOX:   = in²  

RESULTS BOX: 

http://www.mathgoodies.com/lessons/vol2/circle_area_part3.html  

March 19 , 2012 Aim: How do we find the area of regular polygons?

Area of a Pentagon: 5(1as)
                                  2
Apothem: the variable "a" appears in the formula for regular polygons. The apothem is the perpendicular segment from the center to the side of the polygon.

Regular Polygon Area Conjecture
A=area    a=apothem  s=length of each side
n= number of sides  P=perimeter

Examples:



http://www.cliffsnotes.com/study_guide/Regular-Polygons.topicArticleId-18851,articleId-18805.html

Now You Try It:

1. pentagon
apothem = 7.3
side = 10.6

2.octagon
apothem = 14.1
side = 11.7

http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/6-Area%20of%20Regular%20Polygons.pdf

March 15,2012, Aim: How do we find the area of Parallelograms, Kites and Trapezoids?

Areas:
-Parallelogram: Base x Height
-Kite: D1 x D2
                2
-Trapezoid: 1/2(B1 + B2)Height


Examples:

Example 1:	Find the area of a parallelogram with a base of 12 centimeters and a height of 5 centimeters.
Solution:
	= (12 cm) · (5 cm)
	= 60 cm2

                                          http://www.mathgoodies.com/lessons/vol1/area_parallelogram.html

Example 2

Find the area of this kite shape.

This time the two diagonal lengths of the kite are 9mm and 7mm.

First of all, multiply these two side length together.

9 × 7 = 63

Now all you need to do next is halve this number.

63 ÷ 2 = 31.5 mm²

So the area of the kite is 31.5 mm².

http://www.bukisa.com/articles/330258_calculating-the-area-of-a-kite-shapeeasy-guide-with-diagrams 

Example 3:	Find the area of a trapezoid with bases of 9 centimeters and 7 centimeters, and a height of 3 centimeters.
Solution:
	=  · (9 cm + 7 cm) · 3 cm
	=  · (16 cm) · (3 cm)
	=   · 48 cm2
	= 24 cm2 http://www.mathgoodies.com/lessons/vol1/area_trapezoid.html

Now you try it:

1.Find the area of a parallelogram with a base of 8 feet and a height of 3 feet. 

ANSWER BOX:   =  ft²  

RESULTS BOX: 

http://www.mathgoodies.com/lessons/vol1/area_parallelogram.html 

2.Find the area of the following kite.

http://www.mathsteacher.com.au/year8/ch12_area/06_kite/kite.htm


3.Find the area of a trapezoid with bases of 17 centimeters and 19 centimeters, and a height of 2 centimeters. 

ANSWER BOX:   =  cm²  

RESULTS BOX: 

http://www.mathgoodies.com/lessons/vol1/area_trapezoid.html 

March 12,2012, Aim: How do we calculate the area of Rectangles and Triangles?

Areas:
-Rectangle:  Base x Height or Length x Width
-Triangle: Base x Height
                        2
Examples:
1) Consider a rectangle of length 5 cm and width 3 cm.

Using the method of counting squares, we find that the area of the rectangle is 15 cm².
Clearly, the rectangle contains 3 rows of 5 squares.  Therefore:

                                        http://www.mathsteacher.com.au/year7/ch13_area/03_rect/rect.htm

2)

	Area = Length x Width Area = 3 x 2 = 6 square units
	Area = Length x Width Area = 8 x 6 = 48 square units

                                              http://www.helpingwithmath.com/by_subject/geometry/geo_area.htm

Now you try it:

1.

 

Find the area.

Choose:

 32.5 sq. units
 400 sq. units 500 sq. units 600 sq. units 

                         Explanation                         Area = (1/2) base times height                         The base is 40.                         The height is 25.                       

                                        http://www.regentsprep.org/Regents/math/ALGEBRA/AS1/PracArea.htm

2.

(a)	Area =  cm²

                                           http://www.cimt.plymouth.ac.uk/projects/mepres/book7/bk7i9/bk7_9i3.htm

March 7, 2012 , Aim: How do we find Compound Loci?

Compound Loci:
- Involves two or more, locus conditions occurring at the same time.
-The different conditions in a compound locus problem are separated by the word "AND" or the words "AND ALSO"


Examples:

Consider:  A treasure is buried in your backyard.  The picture below shows your backyard which contains a stump, a teepee, and a tree.  The teepee is 8 feet from the stump and 18 feet from the tree.  The treasure is equidistant from the teepee and the tree AND ALSO 6 feet from the stump.  Locate all possible points of the buried treasure.

Description:

The red line represents the locus which is equidistant from the teepee and the tree (the perpendicular bisector of the segment).  The blue circle represents the locus which is 6 feet from the stump.  These two loci intersect in two locations.  The treasure could be buried at either "X" location.



http://www.regentsprep.org/Regents/math/geometry/GL3/LocusCom.htm

Questions:

1.

What is the number of points in a plane two units from a given line and three units from a given point on the line? Explanation

Choose:  1  2   3 4

2.

Parallel lines r and s are 8 meters apart, and A is a point on line s.  How many points are equidistant from r and s and also 4 meters from A?  Answer

Choose:  0  1 2 3

http://www.regentsprep.org/Regents/math/geometry/GL3/PracLocC.htm

March 5,2012 , Aim: How do we find the locus of points?

Locus:
-The set of all points that satisfy a given condition
-A locus is a path of points that satisfy a certain condition

Types of Locus:
1.Two fixed Points
2.One line
3.Two parallel lines
4.Two intersecting lines
5.locus of points from a fixed distant (a Point)

Examples:

Example: A Circle is "the locus of points on a plane that are a certain distance from a central point".

As shown below, just a few points start to look like a circle, but if you collect ALL the points, you will actually have a circle.







http://www.mathsisfun.com/definitions/locus.html





Questions:
1.

Two buoys are located at coordinates (-10,-3) and (-2,-3). A scuba diver swims so that he is always equidistant from both buoys. What equation represents the line along which the diver swims?      [1]  y = -8              [2]  y = -6                          [3]  x = -6              [4]  x = -8

2.

Jason jogs on a path equidistant from the parallel sides of two buildings. His jog path is represented by y = 4 and the side of building 1 is represented by y = 1. Which equation represents the side of building 2?       [1] y = 5       [2] y = 7       [3] y = 8       [4] x = 3

http://www.regentsprep.org/Regents/math/geometry/MultipleChoiceReviewG/Locus.html

                                                             Conditonals & Converse

Today is Friday.(hypothesis)
Tomorrow is Saturday.(conclusion)
-Conditonal: If today is Friday, then tomorrow is Saturday.
-Converse: If tomorrow is Saturday, then today is Friday.
*when the conditional AND the converse are both true, it is called a BICONDITIONAL


-From my notes

                                                            What is an Inverse?

Inverse:
-formed by negating the hypothesis and conclusion


Examples:
-If the cat will ,then the dog will chase the cat
-Inverse: If the cat will not run, then the dog will not chase the cat


http://answers.yahoo.com/question/index?qid=20090503105502AAKR71n

                                           What is a Mathematical Statement?

Mathematical Statement:

• A statement that can be judged to be true or false

Type of Statements:

• Negations=Always have the opposite truth value
• Conditional=there are four types of conditionals

           -conditional
           -converse
           -inverse
           -contrapositive

Examples:


Conditional:"If I found her address, I would send her an invitation".
Converse:"If I send her an invitation, then i found her address".
Inverse: "If its not her address, then I will not send her an invitation".
Contrapositive:"If I don't send her an invitation,then its not her address".


http://www.ego4u.com/en/cram-up/grammar/conditional-sentences