Transcript PSSA – Grade 11

PSSA – Grade 11 - Math
Targeted Review of Major Concepts
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
C
B
A
I f y ou are mis sing a leg of the t riangle, square t he ot her two sides , s ubtrac t, and take t he
s quare root.
I f y ou are mis sing t he hy potenus e of t he t riangle, square t he two legs, add, and t ak e the
s quare root.
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
Many right triangle
problems will include
references to these
popular right triangle
measures:
3–4–5
5 – 12 - 13
5
3
4
13
5
12
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 !
• 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
=
100
OF
Mean – Median - Mode
• Mean is the sum of
All three of these can be
the data divided by
interpreted as the average.
the total number in
Remember that the median is
the data set
unaffected by really large or
• Median is the middle
small data values. But, the
data point – average
mean can be drastically
the middle two if the
affected by such values.
set has an even
number
Your graphing calculator can
• Mode is the data
calculate the mean and the
point which occurs
median quite easily !
most
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
y = ax
The independent variable (x) is
found in the exponent of the
function
Growth
y = 2x
Example from PSSA:
What does the graph of y = 2(.25x)
look like?
y = 2(-x)
Decay
Families of Functions
y=x
Linear function (Line)
y = x2
Quadratic Function
(Parabola)
y = x3
Cubic Function
Systems of Equations Terminology
Inconsistent
(Parallel - same slope)
y = 2x + 4
y = 2x - 3
Consistent & Independent
(Intersect with one solution)
Consistent & Dependent
(Lines Coincide – Same Line)
y = 2x + 4
y = -3x – 3
y = 2x + 4
y = 2x + 4
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:
1)
2)
3)
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
Radius = 4 cm
(x – h)2 + (y – k)2 =
r2
Center (h,k)
Radius r
C: (-5, 3)
CD: (x + 5)2 + (y - 3)2 = 42
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
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
P(G) =
17
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 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:
a)
2
What value of k makes the following b) 3
true?
c) 4
3
5
k
(5 )(2 ) = 4(10 )
d) 6
Plug your answer
choices into the
calculator until
you find the one
that works!!
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
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
?
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
Arithmetic
Each term is increased by the same
value each time
(common difference)
Geometric
Each term is multiplied by the
same value (common ratio)
The graphing calculator can be a
useful resource on these as well!
an = a + (n-1)d
Sn = (n/2)[2a + (n-1)d)]
an = ar(n-1)
a - arn
Sn =
1-r
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
Example:
Digit
Digit
Digit
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:
10
10
10
26
26
Letter
Letter
10*10*10*26*26 = 676,000
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
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
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
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
• 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
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:
1)
Enter function on Y= screen
2)
Graph / Change window if
necessary
to view the parabola
3)
Use the “minimum” or
“maximum”
feature under 2nd - TRACE
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:
a
S=
1-r
a is the first term and r is the common
ratio, so:
(2/3)
1 – (1/2)
=
(4/3)
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
Corresponding sides of similar triangles are proportional !
C
40
54
B
x
32
D
E
BE is parallel to CD
A
Example:
Solution:
By AAA, triangle ACD
is similar to triangle
ABE. Therefore,
corresponding sides of
the two will be
proportional!
What is the measure of side BE?
32
72
=
x
72x = 1728
54
x = 24
30 – 60 – 90 Triangle Ratios
A
30 – 60 – 90
1 - √3 – 2
30°
x - x√3 – 2x
2x
x√3
90°
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.
60°
C
B
x
45 – 45 – 90 Triangle Ratios
A
45°
45 – 45 – 90
x√2
1 - 1 – √2
x
x - x – x√2
90°
45°
C
B
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.
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
1.
2.
3.
4.
5.
6.
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
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
1.
2.
3.
4.
5.
6.
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
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
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 !!