PSSA

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PSSA Grade 11 - Math

• Targeted Review of Major Concepts

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The Pythagorean Theorem

• This theorem applies to all right triangles and
  can be used to find the missing measure of a side
  of the right triangle. Remember, that c is
  always the side opposite the right angle.
• C2 A2 B2

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Sum of the Angles of a Polygon

• Triangle 180
• Quadrilateral 360
• Any other polygon
• 180 (n 2)
• Where n is the number of sides of the polygon

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Famous Right Triangle Ratios

• Many right triangle problems will include
  references to these popular right triangle
  measures
• 3 4 5
• 5 12 - 13

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Positive vs. Negative Correlation

• A positive correlation in a set of data points is
  indicated by a positively sloping (up left to
  right) line
• A negative correlation is indicated by a
  negatively sloping (down left to right) line
• Some data display no correlation

Positive
Negative
None
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PercentagesPercentages !

• A percentage indicates a part of the whole
• Percentages can be expressed as fractions and
  decimals as well
• 75 .75 ¾
• 25 .25 ¼
• 10 .10 1/10

REMEMBER
P
IS
OF
100
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Mean Median - Mode

• Mean is the sum of the data divided by the total
  number in the data set
• Median is the middle data point average the
  middle two if the set has an even number
• Mode is the data point which occurs most

All three of these can be interpreted as the
average. Remember that the median is unaffected
by really large or small data values. But, the
mean can be drastically affected by such values.
Your graphing calculator can calculate the mean
and the median quite easily !
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y mx b

• Slope-intercept form of a linear function
• m is the slope
• b is the y-intercept
• ax by c can easily be converted to this form
• Know the positive and negative y-axis
• Know slope relationships

Y 2x 1
Y -3x - 4
2x 3y 19 3y -2x 19 y (-2/3)x (19/3)
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Exponential Functions

• y ax

Growth
The independent variable (x) is found in the
exponent of the function
y 2x
Example from PSSA What does the graph of y
2(.25x) look like?
y 2(-x)
Decay
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Families of Functions

• y x
• Linear function (Line)
• y x2
• Quadratic Function (Parabola)
• y x3
• Cubic Function

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Systems of Equations Terminology

• Inconsistent
• (Parallel - same slope)
• Consistent Independent
• (Intersect with one solution)
• Consistent Dependent
• (Lines Coincide Same Line)

y 2x 4 y 2x - 3
y 2x 4 y -3x 3
y 2x 4 y 2x 4
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Finding Maximum or Minimum Values

• Example
• Find the maximum height of a projectile whose
  height at any time, t, is given by h(t) 160
  480t 16t2
• Strategy
• Enter the function of the Y screen of calculator
• Graph and adjust window to view the function
• Use 2nd TRACE option 3 or 4 to find desired
  value

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Equations of Circles

• (x h)2 (y k)2 r2
• Center (h,k)
• Radius r

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Proportions

• If 60 out of 370 people surveyed preferred
  Doritos over Tostitos, how many people out of
  2400 would you expect to prefer Doritos?

60
x

370
2400
370 x 144,000 x 389.19 389
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Probability Calculations

• The probability of an event happening is the
  number of successes over the total number of
  possible outcomes.

Example A box contains 8 red and 10 green
marbles. A green marble is drawn out of the box
and set aside. What is the probability that the
next marble drawn out is a green marble?
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P(G)
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Radians to Degreesand Back !

• To convert from radians to degrees multiply by
  (180/p)
• To convert from degrees to radians multiply by
  (p/180)

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The Guess Check Strategy
The main advantage to taking a multiple choice
test is that the correct answer is right in front
of youyou simply have to find it. Remember,
sometimes the most mathematical way to get the
answer may not be the easiest!
Plug your answer choices into the calculator
until you find the one that works!!
Example What value of k makes the following
true? (53)(25) 4(10k)

1. 2
2. 3
3. 4
4. 6

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Direct Variation

• When y varies directly as x, this means that y
  always equals the same number multiplied by x,
  that is y kx, where k is the constant of
  variation.
• k can always be found by taking (y/x) !

Example When traveling at 50 mph, the number of
miles traveled varies directly with the time
driven. Find the miles traveled in 4.5 hours.
y 50x y 50(4.5) y 225 miles
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Inverse Variation
When y varies inversely as x, this means that the
x multiplied by the y will always equal the same
number, that is xy k, where k is the constant
of variation.
Workers 2 4 5 6
Hours 36 18 14.4 ?
(x)(y) k (2)(36) 72 So, k 72 (6)(y)
72 Therefore, y 12 workers
Example The number of hours it takes to paint a
room varies inversely as the number of workers
according to the chart above. How long would it
take 6 workers to complete the room?
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Sequences Series

• Arithmetic
• Each term is increased by the same value each
  time
• (common difference)
• Geometric
• Each term is multiplied by the same value (common
  ratio)

an a (n-1)d Sn (n/2)2a (n-1)d)
an ar(n-1) Sn
a - arn
1 - r
The graphing calculator can be a useful resource
on these as well!
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The Counting Principle

• How many different sandwiches can be made using
  exactly one cheese, one meat, and one bread if
  there are 6 cheeses, 3 meats, and 4 breads
  available?
• (6)(3)(4) 72 sandwiches

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The Counting Principle

• Example
• Every digit (0-9) or letter of the alphabet can
  be used to create the above license plate. How
  many different plates can be produced in each
  state?
• Solution

Digit
Digit
Digit
Letter
Letter
1010102626 676,000

10
10
10
26
26
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The Normal Curve
An example of the standard normal curve with a
mean of zero and a standard deviation of one
An example of the normal curve as it relates to
IQ scores
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The Normal Curve
Characteristics of the Normal Curve Some of the
important characteristics of the normal curve
are The normal curve is a symmetrical
distribution of scores with an equal number of
scores above and below the midpoint of the
abscissa (horizontal axis of the curve). Since
the distribution of scores is symmetrical the
mean, median, and mode are all at the same point
on the abscissa. In other words, the mean the
median the mode. If we divide the distribution
up into standard deviation units, a known
proportion of scores lies within each portion of
the curve.
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The Normal Curve

• MEAN MEDIAN MODE
• On the Normal Curve !
• Example
• A random sample of 10,000 people was taken to
  determine the number of hours of TV watched per
  week. The results of the survey showed a normal
  distribution with a mean of 4.5 hours and a
  standard deviation of .5 hours. What is the
  median number of hours of TV watched!
• Solution
• This is a no-brainer if you realize that the
  mean, median, and mode all equal the same number
  in the normal distribution!
• Answer 4.5 hours

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Statistics - Continued

• Example
• Mrs. Jackson decided to add 5 points to each of
  the scores on her period 5 AMC test. She had
  already calculated the mean, median, mode, and
  range of the original scores. Which of the
  following would not be changed by the addition of
  the 5 points?
• Solution
• The mean, median, and mode would all change.
  But, the range would not. For instance, if the
  low score was 80 and the high 90 prior to the
  change, the range would be 10. But, after the
  addition of 5 points to every grade, the low
  would now be 85 and the high 95, resulting in a
  range of 10! The range would remain unchanged in
  this case.

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The Standard Deviation

• A measure of the spread of the data
• 68.3 of the data lies within one standard
  deviation of the mean on the normal curve
• Example
• The lifetime of a wheel bearing produced by a
  certain company is normally distributed. The
  mean lifetime is 200,000 miles and the standard
  deviation is 10,000 miles. How many bearings in
  a 3000 lot sample will be within one standard
  deviation of the mean?
• Solution
• .683(3000) 2049

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Finding the Vertex of a Parabola

• Example
• What are the coordinates of the vertex of the
  parabola y x2-8x5?
• Solution The fastest way to find this is on the
  graphing calculator
• Enter function on Y screen
• Graph / Change window if necessary
• to view the parabola
• Use the minimum or maximum
• feature under 2nd - TRACE

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Infinite Series

• Example
• What is the sum of the following series?
• (2/3) (1/3) (1/6) (1/12)
• Solution
• This is an example of an infinite geometric
  series with a common ratio of (1/2). According
  to the PSSA formula sheet, the formula for this
  sum is
• S

a
1 - r
a is the first term and r is the common ratio, so
(2/3)
(4/3)

1 (1/2)
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Amplitude Period / Trig

• y a sin(bx)
• y a cos(bx)
• Amplitude a
• Period (2p)/b

Example What is the amplitude of y 8
sin(2x) Solution 8
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Similar Triangles

• Corresponding sides of similar triangles are
  proportional !

By AAA, triangle ACD is similar to triangle ABE.
Therefore, corresponding sides of the two will be
proportional!
Example What is the measure of side
BE? Solution
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x
72x 1728 x 24

54
72
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30 60 90 Triangle Ratios
30 60 90 1 - v3 2 x - xv3 2x
2x
xv3
If you know the measure of any one side of a
30-60-90 triangle, you can use these ratios to
find the other two.
x
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45 45 90 Triangle Ratios
45 45 90 1 - 1 v2 x - x xv2
xv2
x
x
If you know the measure of any one side of a
45-45-90 triangle, you can use these ratios to
find the other two.
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Linear Regression

• The process of fitting a linear function,
• y mx b
• to a particular data set
• This process is most efficiently and effectively
  carried out on a graphing calculator

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Linear Regression (TI83) - Process

1. Enter x values in L1 of calculator
2. Enter y values in L2 of calculator
3. QUIT to Home Screen
4. STAT CALC Opt 4 L1, L2, Y1
5. a is the slope b the y-intercept
6. Equation has also been transferred to the Y
   screen automatically

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Linear Regression (TI83) - Process

• To evaluate your regression model at a specific
  x-value
• From the home screen, enter Y1(x-value) on the
  home screen and ENTER
• To evaluate your regression model at a specific
  y-value
• Enter the y-value for Y2 on the Y screengraph
  and adjust window to view the intersectionuse
  intersection command under 2nd TRACE Option
  5

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Linear Regression (TI82) - Process

• Enter x values into L1
• Enter y values into L2
• QUIT to home screen
• STAT CALC Option 5 L1,L2 ENTER
• a is the slope b the y- intercept
• Y - VARS - 5 EQ - 7 will cut and paste the
  equation to the y screen

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Linear Regression (TI82) - Process

• To evaluate your regression model at a specific
  x-value
• 2nd VARS FUNCTION Option 1 Then put
  x-value in parentheses
• To evaluate your regression model at a specific
  y-value
• Enter the y-value for Y2 on the Y screengraph
  and adjust viewing window to see intersectionuse
  intersection command under 2nd TRACE Option
  5 Then ENTER three times.

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The Formula Sheet

• Please be aware that you are permitted the use of
  the formula sheet provided it is very important
  that you familiarize yourself with this formula
  sheet ahead of time!
• If you do not currently have a formula sheet,
  please ask your math teacher for one !

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You Can Do It Good Luck !!