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STANDARDS OF LEARNING

CONTENT REVIEW NOTES

ALGEBRA II

th

Nine Weeks, 2018 - 2019

OVERVIEW

Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a

resource for students and parents. Each nine weeks’ 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 “ ” 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

summation formulas. Notation will include  and a

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, 11… (constant is + 2 )

25, 20, 15, 10… (constant is − 5 )

Examples: 3, 6, 12, 24, 48… (ratio is

2

1

6, 9, 13.5, 20.25, 30.375… (ratio is

3

2

Formula

𝑎

𝑛

= 𝑎 +

( 𝑛 − 1

) 𝑑

𝑎 is the starting value, 𝑑 is the common

difference and 𝑛 is the number of terms.

Formula

𝑎

𝑛

= 𝑎 ∙ 𝑟

𝑛− 1

𝑎 is the starting value, 𝑟 is the common

ratio and 𝑛 is the number of terms.

Example 1: What is the 35

th

term of the arithmetic sequence that begins 7, 4…

𝑛

𝑛

𝑛

Example 2: What is the 20

th

term of the geometric sequence that begins 1, 2, 4…

𝑛

𝑛− 1

𝑛

20 − 1

𝑛

Example 3: What is the missing term in this geometric sequence 9 , ⎕ , 1 …

𝑛

𝑛− 1

3 − 1

2

The missing term is 9 ∙

1

3

= 3

Substitute your values (𝑎 = 7 , 𝑛 = 35 , 𝑑 = − 3 )

Simplify

Substitute your values (𝑎 = 1 , 𝑟 = 2 , 𝑛 = 20 )

Simplify

Substitute your values (𝑎

𝑛

= 1 , 𝑎 = 9 , 𝑛 = 3 )

Simplify

Solve for the common ratio, 𝑟.

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 |𝒓| < 1 )

𝑛

1

𝑛

Where 𝑎

1

is the first term, 𝑎

𝑛

is

the 𝑛

𝑡ℎ

term, and n is the

number of terms.

𝑛

1

𝑛

Where 𝑎

1

is the first term, 𝑟 is

the common ratio, and n is the

number of terms.

𝑛

1

. Where 𝑎

1

is the first term, and 𝑟 is

the common ratio.

You may see series written in Summation Notation

You can write the series 7+9+11+ …+89 as

42

𝑛 = 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.

𝑛

𝑛

2

1

𝑛

𝑛

𝑛

Sequences and Series

1. What is the 9

th

term of the geometric sequence that begins 2, 1, …

2. What is the missing term in this arithmetic sequence 12 , ⎕ , 25 …
3. Write the series in summation notation 120 + 115 + 110 + 105 + 100 + 95
4. Find the sum of the geometric series

𝑛

10

𝑛= 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

𝑛= 1

Substitute your values

(𝑛 = 42 , 𝑎

1

= 2 ( 1 ) + 5 = 7 , 𝑎

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

1. A teacher is making a multiple choice quiz. She wants to give each student the

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?

2. A coach must choose five starters from a team of 12 players. How many different

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 “spread out” 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 σ (lowercase Greek letter sigma). The variance is

just the standard deviation squared, σ².

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 σ, therefore our standard

deviation is 2.32.The variance is just the standard deviation squared = (2.32)² = 5.

Example 2: Using the data from Example 1 , how many of the starting lineups’ 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 – 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 class’s 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: Jessie’s teacher wouldn’t tell her the actual score that she received; only

that she had a z-score of 1.26. Determine Jessie’s 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.

𝑧 − 𝑠𝑐𝑜𝑟𝑒 =

𝐷𝑎𝑡𝑎 𝑃𝑜𝑖𝑛𝑡 − 𝑀𝐸𝐴𝑁

𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

𝑧 − 𝑠𝑐𝑜𝑟𝑒 =

𝐷𝑎𝑡𝑎 𝑃𝑜𝑖𝑛𝑡 − 𝑀𝐸𝐴𝑁

𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

=

78 − 83. 47

9. 16

= − 0. 597

This shows that a score of 78 is a little more than ½ standard deviation below the mean.

𝑧 − 𝑠𝑐𝑜𝑟𝑒 =

𝐷𝑎𝑡𝑎 𝑃𝑜𝑖𝑛𝑡 − 𝑀𝐸𝐴𝑁

𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

1. 26 =

𝑋 − 83. 47

9. 16

∙ 9. 16 ∙ 9. 16

11. 542 = 𝑋 − 83. 47

• 83. 47 + 83. 47

95. 01 = 𝑋

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 isn’t 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:

 68% of the data will fall within 1 standard deviation of the mean

 95% of the data will fall within 2 standard deviations of the mean

 99.7% of the data will fall within 3 standard deviations of the mean

𝑧 − 𝑠𝑐𝑜𝑟𝑒 =

𝐷𝑎𝑡𝑎 𝑃𝑜𝑖𝑛𝑡 − 𝑀𝐸𝐴𝑁

𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

1. 26 =

𝑋 − 83. 47

3. 60

∙ 3. 60 ∙ 3. 60

4. 536 = 𝑋 − 83. 47

• 83. 47 + 83. 47

88. 01 = 𝑋

Jessie’s 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 ∙ 500 = 113. 3 ≈ 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

“greater than” 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.

5. What is the z-score for 99 inches?
6. What is the z-score for 107 inches?
7. The tallest woman ever confirmed would have had a z-score of - 1.13. How tall was

she?

8. The means and standard deviations for two schools’ SAT scores is shown below.

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

9. The height of the men in the United States is normally distributed as shown in the

graph. The mean is 69.25” with a standard deviation of 2.5”. What percent of the

heights are between 66.75” and 74.25”?

10. The length of time that people can hold their breath under water is normally

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.)