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After page 3 formulas are given as geometric, quadratic and statistics formulas, sequence and series formulas, permutations and combinations formulas.
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Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a
resource for students and parents. Each nine weeksโ Standards of Learning (SOLs) have been identified and a
detailed explanation of the specific SOL is provided. Specific notes have also been included in this document
to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models
for solving various types of problems. A โ โ section has also been developed to provide students with the
opportunity to solve similar problems and check their answers. Supplemental online information can be
accessed by scanning QR codes throughout the document. These will take students to video tutorials and online
resources. In addition, a self-assessment is available at the end of the document to allow students to check their
readiness for the nine-weeks test.
The document is a compilation of information found in the Virginia Department of Education (VDOE)
Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE
information, Prentice Hall Textbook Series and resources have been used. Finally, information from various
websites is included. The websites are listed with the information as it appears in the document.
Supplemental online information can be accessed by scanning QR codes throughout the document. These will
take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the
document to allow students to check their readiness for the nine-weeks test.
The Algebra II Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number
of questions per reporting category, and the corresponding SOLs.
Sequences and Series
AII.5 The student will investigate and apply the properties of arithmetic and
geometric sequences and series to solve practical problems, including
writing the first n terms, determining the n
th
term, and evaluating
n
Arithmetic Sequence Geometric Sequence
A sequence where the difference between
consecutive terms is a constant.
(You add or subtract a constant value)
A sequence where the difference between
consecutive terms is a common ratio.
(You multiply or divide a constant value)
Examples: 3, 5, 7, 9, 11โฆ (constant is + 2 )
25, 20, 15, 10โฆ (constant is โ 5 )
Examples: 3, 6, 12, 24, 48โฆ (ratio is
2
1
6, 9, 13.5, 20.25, 30.375โฆ (ratio is
3
2
Formula
๐
๐
= ๐ +
( ๐ โ 1
) ๐
๐ is the starting value, ๐ is the common
difference and ๐ is the number of terms.
Formula
๐
๐
= ๐ โ ๐
๐โ 1
๐ is the starting value, ๐ is the common
ratio and ๐ is the number of terms.
Example 1: What is the 35
th
term of the arithmetic sequence that begins 7, 4โฆ
๐
๐
๐
Example 2: What is the 20
th
term of the geometric sequence that begins 1, 2, 4โฆ
๐
๐โ 1
๐
20 โ 1
๐
Example 3: What is the missing term in this geometric sequence 9 , โ , 1 โฆ
๐
๐โ 1
3 โ 1
2
The missing term is 9 โ
1
3
= 3
Substitute your values (๐ = 7 , ๐ = 35 , ๐ = โ 3 )
Simplify
Substitute your values (๐ = 1 , ๐ = 2 , ๐ = 20 )
Simplify
Substitute your values (๐
๐
= 1 , ๐ = 9 , ๐ = 3 )
Simplify
Solve for the common ratio, ๐.
8
A series is the sum of a geometric or arithmetic sequence.
Sum of a Finite
Arithmetic Series
Sum of a Finite
Geometric Series
Sum of an Infinite Geometric
Sequence
( Only applicable for |๐| < 1 )
๐
1
๐
Where ๐
1
is the first term, ๐
๐
is
the ๐
๐กโ
term, and n is the
number of terms.
๐
1
๐
Where ๐
1
is the first term, ๐ is
the common ratio, and n is the
number of terms.
๐
1
. Where ๐
1
is the first term, and ๐ is
the common ratio.
You may see series written in Summation Notation
You can write the series 7+9+11+ โฆ+89 as
42
๐ = 1
Example 4: Evaluate
Because the explicit formula is linear, this will be an arithmetic series. In order to evaluate an
arithmetic series we need to know the first and last term and number of terms.
๐
๐
2
1
๐
๐
๐
Sequences and Series
th
term of the geometric sequence that begins 2, 1, โฆ
๐
10
๐= 1
n = 1 is the lower limit
42 is the upper limit
2n +5 is the explicit formula
for each term in the series.
42
๐= 1
Substitute your values
(๐ = 42 , ๐
1
= 2 ( 1 ) + 5 = 7 , ๐
42
= 2 ( 42 ) + 5 = 89 )
Simplify
Scan this QR code to go to a
video tutorial on sequences
and series.
Example 2: In the finals of the diving meet referenced in Example 1, the top 3
finishers score points for their team. First place receives 10 points, 2
nd
place receives 8
points, and 3
rd
place receives 6 points. In how many ways can the 8 finalists finish in
the top 3?
Now, the order is important because 1
st
place gets more points than 2
nd
place. We will
use a permutation!
8 3
There are 336 possible ways that the top 8 divers can finish in the top 3.
Statistics
same questions, but have each student's questions appear in a different order. If
there are twenty-seven students in the class, what is the least number of questions
the quiz must contain?
ways can the coach choose the starters?
Standard Deviation
AII.11 The student will
a) identify and describe properties of a normal distribution;
b) interpret and compare z-scores for normally distributed data; and
c) apply properties of normal distributions to determine probabilities associated
with areas under the standard normal curve.
Standard Deviation and Variance
The standard deviation of a data set tells us how โspread outโ the data is, if the data is
very spread out, the standard deviation will be higher than if the data is all clumped
together. The variance is another measure of how spread out the data is.
Standard deviation is represented by ฯ (lowercase Greek letter sigma). The variance is
just the standard deviation squared, ฯยฒ.
There is a way to calculate these values in the graphing calculator.
Example 1: The height in inches of the Washington Wizards starting lineup is shown
below. Find the standard deviation and the variance of the data, round your
answer to the nearest hundredth.
Start by entering the data into L1 in your STAT menu.
We want to use the standard deviation that is represented by ฯ, therefore our standard
deviation is 2.32.The variance is just the standard deviation squared = (2.32)ยฒ = 5.
Example 2: Using the data from Example 1 , how many of the starting lineupsโ heights
are within one standard deviation of the mean?
The heights were 75, 80, 76, 79, 81
This question is referring to players who are both one standard deviation above the
mean and one standard deviation below the mean.
The mean was 78.2 inches, and the standard deviation was 2.32 inches.
78.2 + 2.32 = 80.52 inches 78.2 โ 2.32 = 75.88 inches
Then go to STAT, scroll over to CALC, and
select 1: 1-Var Stats
When you press ENTER twice, your
calculator will display the single
variable statistics.
MEAN
SUM of the DATA
SUM Squared
Sample Standard Deviation
Population Standard Deviation
Sample Size
KEY
Standard Deviation = 2.32 inches
Variance = 5.38 inches
13
The formula for calculating a z-score is on the Algebra II formula sheet.
Example 4: A classโs history midterm grades are shown below. What is the z-score for
a score of 78?
Grades: 81, 62, 90, 77, 82, 86, 98, 100, 90, 75, 83, 88, 79, 76, 85
First calculate the 1-Var Stats in your calculator.
We need the Mean and Standard Deviation.
We want to know the z-score for a 78.
Example 5: Jessieโs teacher wouldnโt tell her the actual score that she received; only
that she had a z-score of 1.26. Determine Jessieโs score.
Now we are given the z-score and asked to find the data point. Plug everything that
you know into the formula, and then solve the equation for the missing piece.
๐ง โ ๐ ๐๐๐๐ =
๐ท๐๐ก๐ ๐๐๐๐๐ก โ ๐๐ธ๐ด๐
๐๐ก๐๐๐๐๐๐ ๐ท๐๐ฃ๐๐๐ก๐๐๐
๐ง โ ๐ ๐๐๐๐ =
๐ท๐๐ก๐ ๐๐๐๐๐ก โ ๐๐ธ๐ด๐
๐๐ก๐๐๐๐๐๐ ๐ท๐๐ฃ๐๐๐ก๐๐๐
=
78 โ 83. 47
= โ 0. 597
This shows that a score of 78 is a little more than ยฝ standard deviation below the mean.
๐ง โ ๐ ๐๐๐๐ =
๐ท๐๐ก๐ ๐๐๐๐๐ก โ ๐๐ธ๐ด๐
๐๐ก๐๐๐๐๐๐ ๐ท๐๐ฃ๐๐๐ก๐๐๐
๐ โ 83. 47
โ 9. 16 โ 9. 16
We know the
z-score, mean and
standard deviation.
Now we just need to
solve for X!
Scan this QR code to
go to a video tutorial on
z-scores.
Example 6: Another class took the exam and had the same class average but a
standard deviation of 3.60. If Jessie had been in this class (and still had a z-score of
1.26), would her score be lower or higher? Explain.
We could calculate the answer to this problem, but this isnโt necessary. The class
average is exactly the same, but the standard deviation is lower. Jessie still scored
1.26 standard deviations above the mean, but now those standard deviations are
smaller, therefore her grade will be lower.
Normal Curve
In a normal distribution:
๏ท 68% of the data will fall within 1 standard deviation of the mean
๏ท 95% of the data will fall within 2 standard deviations of the mean
๏ท 99.7% of the data will fall within 3 standard deviations of the mean
๐ง โ ๐ ๐๐๐๐ =
๐ท๐๐ก๐ ๐๐๐๐๐ก โ ๐๐ธ๐ด๐
๐๐ก๐๐๐๐๐๐ ๐ท๐๐ฃ๐๐๐ก๐๐๐
๐ โ 83. 47
โ 3. 60 โ 3. 60
Jessieโs score was an 88%
We know the
z-score, mean and
standard deviation.
Now we just need to
solve for X!
Example 1:
Use the z-table to determine the probability that a data value will fall below a data
value associated with a z-score of 0.53.
Example 2:
The length of the life of an instrument is 12 years with a standard deviation of 4 years.
Out of 500 instruments, how many can be expected to last 15 or more years?
First, the z-score of a length of 15 must be found.
Next, use the z-table above to find the probability value of a z-score of 0.75. You
should find 0.7734 (or 77.34%).
Remember, subtract this value from 1, because we are looking for the area under
the curve to the right since we were asked about 15 or more years!
Multiply 0. 2266 by 500. 0. 2266 โ 500 = 113. 3 โ 113
113 instruments represents 22.66% of 500 instruments.
Look at what the
value represents.
Remember, that this
value represents
area to the left of the
curve! If you were
asked to find
probability of values
โgreater thanโ the
value associated
with 0.53, you would
have to subtract the
value from 1.
1 - .7019 = 0.
Therefore, the probability or area
under the curve is 0.7019 ( or 70.19%).
Standard Deviation
The heights of the tallest 7 men ever recorded are shown below (in inches). Use these
to answer the questions.
she?
The z-score for the 95
th
percentile is 1.598. By how many points do the 95
th
percentile scores differ for each school? (Round to the nearest whole number.)
School A: Mean = 1520, Standard Deviation = 110
School B: Mean = 1490, Standard Deviation = 155
graph. The mean is 69.25โ with a standard deviation of 2.5โ. What percent of the
heights are between 66.75โ and 74.25โ?
distributed with a mean of 32 seconds and a standard deviation of 12 seconds.
Out of 750 people, about how many people would be expected to hold their breath
for 42 seconds or longer? For 35 seconds or less? (Hint: Use the part of the z-table
shown on page 15.)