PSSA – Grade 11 - Math: PSSA – Grade 11 - Math Targeted Review of Major Concepts
The Pythagorean Theorem: 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
Sum of the Angles of a Polygon: 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
Famous Right Triangle Ratios: Famous Right Triangle Ratios Many right triangle problems will include references to these popular right triangle measures:
3 – 4 – 5
5 – 12 - 13
Positive vs. Negative Correlation: 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
Percentages…Percentages !: Percentages…Percentages ! 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 100 IS OF
Mean – Median - Mode: 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 !
y = mx + b: 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)
Exponential Functions: Exponential Functions y = ax
y = 2x y = 2(-x) The independent variable (x) is found in the exponent of the function Example from PSSA:
What does the graph of y = 2(.25x) look like? Growth Decay
Families of Functions: Families of Functions y = x
Linear function (Line)
y = x2
Quadratic Function (Parabola)
y = x3
Cubic Function
Systems of Equations Terminology: 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 + 4 y = 2x + 4
y = 2x - 3 y = 2x + 4
y = -3x – 3
Finding Maximum or Minimum Values: 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
Equations of Circles: Equations of Circles
(x – h)2 + (y – k)2 = r2
Center (h,k)
Radius r
Proportions: Proportions If 60 out of 370 people surveyed preferred Doritos over Tostitos, how many people out of 2400 would you expect to prefer Doritos?
60 370 2400 x = 370 x = 144,000
x = 389.19 = 389
Probability Calculations: 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?
9 17 P(G) =
Radians to Degrees…and Back !: Radians to Degrees…and Back ! To convert from radians to degrees: multiply by (180/π)
To convert from degrees to radians: multiply by (π/180)
The Guess & Check Strategy: The Guess & Check Strategy The main advantage to taking a multiple choice test is that the correct answer is right in front of you…you simply have to find it. Remember, sometimes the “most mathematical” way to get the answer may not be the easiest! Example:
What value of k makes the following true?
(53)(25) = 4(10k) 2
3
4
6 Plug your answer choices into the calculator until you find the one that works!!
Direct Variation: 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
Inverse Variation: 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. 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? (x)(y) = k
(2)(36) = 72
So, k = 72
(6)(y) = 72
Therefore, y = 12 workers
Sequences & Series: 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!
The Counting Principle: 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
The Counting Principle: 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 10 10 10 26 26 10*10*10*26*26 = 676,000
The Normal Curve: The Normal Curve An example of the normal curve as it relates to IQ scores An example of the standard normal curve with a mean of zero and a standard deviation of one
The Normal Curve: 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.
The Normal Curve: 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
Statistics - Continued: 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.
The Standard Deviation: 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
Finding the Vertex of a Parabola: Finding the Vertex of a Parabola Example:
What are the coordinates of the vertex of the parabola y = x2-8x+5?
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
Infinite Series: 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) 1 – (1/2) = (4/3)
Amplitude & Period / Trig: Amplitude & Period / Trig y = a sin(bx)
y = a cos(bx)
Amplitude = a
Period = (2π)/b
Example:
What is the amplitude of y = 8 sin(2x)
Solution: 8
Similar Triangles: Similar Triangles Corresponding sides of similar triangles are proportional ! Example: What is the measure of side BE?
Solution: 32 54 72 x = 72x = 1728
x = 24 By AAA, triangle ACD is similar to triangle ABE. Therefore, corresponding sides of the two will be proportional!
30 – 60 – 90 Triangle Ratios: 30 – 60 – 90 Triangle Ratios 30 – 60 – 90
1 - √3 – 2
x - x√3 – 2x
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 x√3 2x
Slide33: 45 – 45 – 90 Triangle Ratios
45 – 45 – 90
1 - 1 – √2
x - x – x√2
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. x x x√2
Linear Regression: 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
Linear Regression (TI83) - Process: Linear Regression (TI83) - Process Enter “x” values in L1 of calculator
Enter “y” values in L2 of calculator
QUIT to Home Screen
STAT – CALC – Opt #4 – L1, L2, Y1
“a” is the slope ; b the y-intercept
Equation has also been transferred to the Y= screen automatically
Linear Regression (TI83) - Process: 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= screen…graph and adjust window to view the intersection…use intersection command under 2nd – TRACE – Option #5
Linear Regression (TI82) - Process: 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
Linear Regression (TI82) - Process: 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 = screen…graph and adjust viewing window to see intersection…use intersection command under 2nd – TRACE – Option #5 – Then ENTER three times.
The Formula Sheet: 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 !
You Can Do It – Good Luck !!: You Can Do It – Good Luck !!