Module 4

 

Fractal:  an object that contains smaller and smaller copies of itself

Self-Similar:  fractals that are made up of pieces that are similar to the whole fractal

Term: The numbers or objects in a sequence

Congruent:  figures are congruent if they have the same shape and the same size.  If the sides of two triangles are the same length they are congruent.  A Sierpinski triangle has many congruent triangles.

(scan triangles on p. 247)  they symbol between them means congruent too)

Triangle Inequality:  The sum of the lengths of two sides of a triangle are greater than the length of the third side. 

Sequence:  an ordered list of numbers or objects.

Midpoint of a Segment:  divides a segment into two congruent segments.  Midpoints are found by constructing a perpendicular bisector.

 

 

Term # (n)

1

2

3

4

5

20

 

Term    (t)

2

4

6

8

10

40

 

 

 

Second table just adds #20  substitute 20 into the equation and you get 40 for your answer.

Rule t=2n

 

 

 

More Examples

 

Term # (n)

1

2

3

4

5

20

 

Term    (t)

4

9

14

19

24

 

 

 

 

(t) = n x 5 – 1  So for 20 the answer is t = 20 x 5 – 1 or 99

 

 

 

Term # (n)

1

2

3

4

5

20

 

Term    (t)

2

8

14

 

 

 

 

 

 

The terms are increasing by 6.  Under 4 you would have 20 and 5 you would have 26. 

(t) = n * 6 – 4.  In the 20th place t = 20 * 6 – 4.  The answer is 116.

 

 

Term # (n)

1

2

3

4

5

20

 

Term    (t)

9

21

33

45

57

 

 

 

 

These terms are increasing by 12.

Rule is t = 12n-3  So for 20 it is t = 12 * 20 –3 = 237

 

Another way to solve term numbers and sequences.  Use one of these two formulas. 

If the terms are increasing arithmetically (same number is being added or subtracted to get the next number in the sequence) use this formula. 

a = 1st term

d = the common difference

n = the nth term

 

a + (n-1) * d

 

 

Term # (n)

1

2

3

4

5

20

 

Term    (t)

5

10

15

20

25

 

 

 

 

5 + (20 –1) *5  is 5 + 19 * 5 = 100

 

Rule is t = 5n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

If the terms are increasing geometrically (same number is being multiplied to get the next number in the sequence) use this formula.

 

a = 1st term

r = common ratio

n = nth term

 

a(rn-1)

 

 

Term # (n)

1

2

3

4

5

20

 

Term    (t)

125

25

5

1

1/5

6.55 x10-12

 

 

 

The terms are 1/5 of the previous term. 

 

125 x (1/520-1)

 

Fibonacci Sequence:  Each term is the sum of two previous terms. 
Example:  1, 2, 3, 5, 8, 13, 21, 34, 55

 

Rotational Symmetry:  An object has rotational symmetry if you rotate it less than 360 degrees around a center point and it fits exactly on itself.

 

 

 


The cross has rotational symmetry at 90, 180, 270, and 360 degrees.

The triangle has rotational symmetry at 120, 240, and 360 degrees

 

In problems if they give you 3 points, always add 360 as the fourth point.

Minimal Rotational Symmetry:  smallest number of degrees a figure can be rotated and fit exactly on itself

 

Line Symmetry:  a line that divides a figure into two parts that are exact copies of each other. 

Rational Number:  rational number can be written as a/b where a and b are integers and b is not zero.  3/10 is a rational number 2 and 1/3 is a rational number.  3.8 is a rational number.  Rational numbers can be written as a fraction.

 

Irrational Number:  Irrational numbers cannot be written as the quotient of two integers.  Pi and the square root of 10 are irrational numbers.  The square roots of integers that are perfect squares are not irrational numbers.  Example square root of 25 is not an irrational number because it is a perfect square. 

 

Terminating Decimal:  A decimal that contains a limited number of digits.  Example 0.26 or 1.375

 

Repeating Decimal:  a decimal that contains a number or group of numbers that repeats forever. 

 

 

Changing a Repeating Decimal into a Fraction

 

Step 1:  Take the Decimal and make an equation

                              __                           __

Example:  0.36      becomes x = 0.36

 

_                             _

Example:  0.5      becomes x = 0.5

 

Step 2:  Multiply by the place value of the last digit shown in the decimal.  (If it is in the tenths place multiply by 10, if it is in the hundredths place multiply by 100, if it is in the thousandths place multiply by 1,000 and so on.

                        __                                __

Example: x = 0.36  becomes 100x  = 36.36 because both sides multiplied by 100

                        _                               _

Example: x = 0.5  becomes 10x  =  5.5 because both sides multiplied by 10

 

 

 

 

 

 

 

 

Step 3 Subtract the original equation from the step 2 equation

                   __                                        _

100x  =  36.36                          10x  =  5.5

_    x  =      .36                         -   x  =    .5

 

  99x  =  36                                 9x  =  5

 

     x  =  36/99 or 4/11                   x  = 5/9

 

 

Using reciprocals in equations with fractions.

 

These problems are solved by getting the variable by itself.

 

Step 1 if the equation also has subtraction switch it to an addition problem.  Then use the inverse operation to complete the first step of isolating the variable.

 

Step 2 Use the reciprocal to isolate the variable and solve for x.

 

Step 3 Put your answer back into the original equation to make sure your answer is correct. 

 

Example

 

3/5n – 7 = 56   becomes  3/5n + (-7)  =  56

 

3/5n  +  (-7)  =  56

         +    7    = + 7

 

3/5n  =  63  Then multiply by the reciprocal to get the variable alone and solve the equation.

 

5/3 (x) 3/5n  =  63 (x) 5/3

 

n  =  105

 

 

 

 

 

Quadrilaterals:  (Quad means 4) a polygon with 4 sides.

Classified by:

1.  angle measures

2.  parallel sides

3.  side lengths

 

 

scan the figure from p. 275

 

parallelogram:  a quadrilateral that has two pairs of parallel sides (crooked rectangle)

2 acute and 2 obtuse angles

opposites sides congruent or 2 sets of 2 congruent sides

opposite sides parallel or 2 sets of parallel sides

 

rectangle:  a quadrilateral that has 4 right angles

four 90 degree angles

opposite sides congruent or 2 sets of 2 congruent sides

opposite sides parallel or 2 sets of parallel sides

 

rhombus:  a quadrilateral that has four sides of equal length (crooked square)

2 acute, 2 obtuse angles  (obtuse – the angle measures between 90 and 180)

                                      (acute – angle measures greater than 0 but less than 90)

4 congruent sides

opposite sides parallel 2 sets of parallel sides

 

trapezoid:  a quadrilateral that has exactly one pair of parallel sides

angle measures (vary)

2 sides are congruent

has exactly one pair of parallel sides

 

square:  a quadrilateral that has four right angles and four sides of equal length

four 90 degree angles

four congruent sides (all the same length)

opposite sides parallel 2 sets of parallel sides

Convex: a polygon is convex when all of its diagonals lie in the interior of a polygon

 

Concave:  a polygon that is not convex

 

Regular Polygon:  A polygon with all sides of equal length and all angles of equal measure

 




Interior Angles of Polygons
 
The sum of the measures of the interior angles depends on the number of sides

 

A circle has 3600

 

A triangle has 1800

 

A quadrilateral has 3600

 

 

Text Box: S = 1800 (n –2)S = Sum of the measure of the interior angles  N = number of sides.




Example:  stop sign has 8 sides Thus S =  1800 (8 –2)

                                                          S =  1800  * 6

                                                          S =  10800

Solving Equations with Fractions

 

1.  Change subtraction problems to addition

2.  Add the opposite to both sides

3.  Change mixed numbers to improper fractions

4.  Multiply by the reciprocal

5.  Cross reduce

6.  Solve for the variable

 

-3y – 6 =  1

 4              2

 

-3y + (- 6) =  1

 4                   2

       +    6   = +6

 

-3y  =  6 1

 4            2

 

4  *  -3y  =  6    1   *  4

3        4             2       3

 

y =  13   *  4                   y = 52                             y = 8  2

        2         3                         6                              3

 

 

 



Pythagorean Theorem Formula

Formula a2 + b2 = c2
 
 

 

 


a right triangle has a 90 degree angle.  A right triangle has two legs and a hypotenuse.  The hypotenuse is the side opposite of the 90 degree angle, and it is always the longest side of the triangle.

 

a2 + b2 = c2
 
          right triangle


a = 3

b = 4

c = 5                     32 + 42 = 52            9 + 16 = 25

 

                   acute triangle – a triangle that has three acute angles (angles between 00 and 900

a2 + b2 > c2
 



a = 6

b = 7

c = 8                     62 + 72 > 82            36 + 49 > 64       85 > 64

 

obtuse triangle    (an triangle that has one obtuse angle – an angle between 900 and 1800 )

                             a2 + b2 < c2

 

a = 2

b = 4

c = 5           22 + 42 < 52                        4 + 16 < 25         20  < 25

 

 

 

 

 

Finding the length on an unknown side in a right triangle

 

a2 + b2 = c2

 

x2 + 72 = 122                     x2 + 49 = 144      x2 = 95       x = square root of 95

 

x = 9.75

 
 
 


Steps

C     square the lengths of the sides

C     get the variable alone

C     find the square root of the variable

 

 

 

 

 
 
 


Formulas for Area

Area:  L x W  Length x width

 

Area of a triangle:  1/2b(x)h

 

Area of a circle pi x the radius squared

 

 

 

 

 

Geometric Probability


 

 

 

 

 

 

 

 

 

 

 

Module 5

 

Surface area (of a cylinder) 2Pr2  +  2Prh

 

Volume of cylinder: v = Pr2h

 

Comparing Surface Area to volume:  For a container like a can, volume is the measure of the container’s efficiency.

 

Efficiency =        SA

                               V

 

 

 

The radius of this cylinder is 1 foot, and its height is 3 feet.  Use the formula for surface area:

 

2P12  +  2P(1)(3)  = 25.12 ft2

 
 
 

 

 

 

 

 

 

 

 

 

 


Area is always expressed in square feet

Volume of cylinder: v = P12(3) = 9.42 ft3

This is the volume of the above cylinder