SAT Quick Challenge  Week 3
Summer 2021
Saturday, July 24, 2021
Providence Baptist Church is concerned about you, your family, your friends, and the community. We are following the recommendations to limit the number of people in our inperson gatherings. For this reason, all SAT Preparation classes at the church are cancelled until further notice.
Although the current COVID19 pandemic abruptly ended the Spring 2020 SAT classes, technology offers an opportunity for students to refresh, retain, and/or acquire SAT knowledge and skills essential for answering various kinds of questions often found on the SAT. Therefore, the Providence Baptist Church SAT Preparation Program has provided this “SAT Quick Challenge” website. Each Saturday except those occurring during holiday weekends, this site will be updated with a set of activities that will help students acquire and maintain mastery of important skills that help lead to high SAT scores. Be sure to review the SAT Math Formulas and the SAT Math Operations each week. You can find them in the dropdowns at the bottom of this page.
The Providence Baptist Church Fellowship of Young Adults hosted the 2021 Virtual OfftoCollege Workshop on Saturday, July 31. Therefore, no new lessons were posted on July 31. The next set of lessons will be posted on Saturday, August 7.
Please stay healthy and safe. Remember the three Ws: Wear, Wait, Wash.
SAT Math  Questions from Week 3 (July 24, 2021)
Questions about the standard deviation are also asked on the SAT. The standard deviation is a measure of how closely clustered a data set is about the mean of the data set (how spread out the values are).
The standard deviation is low if most of the values are near the mean and close together (narrow spread).
The standard deviation is high if most of the values are spread out over the range of values (wide spread).
The SAT will not require you to calculate the standard deviation, but you need to understand the concept.
Consider the following example:
Arnold, Ronald, and David are the top three scorers on the Eastern High School basketball team. During the first five games of the season, they scored the following points:
Game 1  Game 2  Game 3  Game 4  Game 5  
Arnold  18  21  15  24  22 
Ronald  14  10  26  32  18 
David  20  20  20  20  20 
In each case the mean is 20:
Arnold: (18 + 21 + 15 + 24 + 22)/ 5 = 100/5 = 20
Ronald: (14 + 10 + 26 + 32 + 18)/5 = 100/5 = 20
David: (20+ 20 + 20 + 20 + 20)/5 = 100/5 = 20
Let’s compare Arnold and Ronald: The points scored by Arnold are close to the mean, while the points scored by Ronald are more spread out. The standard deviation of Arnold is lower than the standard deviation of Ronald. At the extreme, the points scored by David are all the same; there is no variation in the value of the points. Thus the standard deviation for David is zero.

Set A 37 42 58 19 66 97 22 Set B 48 63 55 59 42 51 65
 The standard deviation of Set A is lower than the standard deviation of Set B.
 The standard deviation of Set A is greater than the standard deviation of Set B.
 The standard deviation of Set A is equal to the standard deviation of Set B.
 The standard deviation of Set A and the standard deviation Set B cannot be compared.
 The tables below give the distribution of speeding tickets given by patrol officers in Greensboro and WinstonSalem for the 30 days in June.
Greensboro WinstonSalem Number of Speeding Tickets Given Frequency Number of Speeding Tickets Given Frequency 13 1 13 6 14 5 14 7 15 19 15 7 16 3 16 6 17 2 17 4
 The standard deviation of the number of speeding tickets given in Greensboro is smaller.
 The standard deviation of the number of speeding tickets given in WinstonSalem is smaller.
 The standard deviation of the number of speeding tickets given in Greensboro is the same as that in WinstonSalem.
 The standard deviation shows that the number of speeding tickets given in WinstonSalem is too large.
 13, 22, 29, 17, 9, 15, 27, 21
If the number 43 is added to the above set of numbers, how will the standard deviation of the set change?
 The standard deviation of the set will be higher.
 The standard deviation of the set will be lower.
 The standard deviation of the set will be unchanged.
 The standard deviation of the set cannot be compared.
 The weights, in pounds, for 45 players on the football team were reported, and the mean, median, range, and standard deviation were calculated for the data. The player for the lowest reported weight was found to actually weigh 15 pounds less than his reported weight. What value remains unchanged if the four values are reported using the corrected weight?
 Mean
 Median
 Range
 Standard deviation
SAT Math  Answers to Questions from Week 3 (July 24, 2021)

Set A 37 42 58 19 66 97 22 Set B 48 63 55 59 42 51 65
 The standard deviation of Set A is lower than the standard deviation of Set B.
 The standard deviation of Set A is greater than the standard deviation of Set B.
 The standard deviation of Set A is equal to the standard deviation of Set B.
 The standard deviation of Set A and the standard deviation Set B cannot be compared.
The numbers in Set A are spread out farther than the numbers in Set B. Thus Set A has a greater standard deviation than Set B. The answer is B.
 The tables below give the distribution of speeding tickets given by patrol officers in Greensboro and WinstonSalem for the 30 days in June.
Greensboro WinstonSalem Number of Speeding Tickets Given Frequency Number of Speeding Tickets Given Frequency 13 1 13 6 14 5 14 7 15 19 15 7 16 3 16 6 17 2 17 4
 The standard deviation of the number of speeding tickets given in Greensboro is smaller.
 The standard deviation of the number of speeding tickets given in WinstonSalem is smaller.
 The standard deviation of the number of speeding tickets given in Greensboro is the same as that in WinstonSalem.
 The standard deviation shows that the number of speeding tickets given in WinstonSalem is too large.
The number of tickets given out in Greensboro are much more tightly concentrated than those in WinstonSalem. Since most of the values in Greensboro are very closely clustered around 15 tickets, the number of tickets given in Greensboro has a lower standard deviation than that of WinstonSalem. The answer is A.
 13, 22, 29, 17, 9, 15, 27, 21
If the number 43 is added to the above set of numbers, how will the standard deviation of the set change?
 The standard deviation of the set will be higher.
 The standard deviation of the set will be lower.
 The standard deviation of the set will be unchanged.
 The standard deviation of the set cannot be compared.
Adding the number 43 to the set of numbers will increase the spread of the numbers. Thus the standard deviation will be higher. The answer is A.
 The weights, in pounds, for 45 players on the football team were reported, and the mean, median, range, and standard deviation were calculated for the data. The player for the lowest reported weight was found to actually weigh 15 pounds less than his reported weight. What value remains unchanged if the four values are reported using the corrected weight?
 Mean
 Median
 Range
 Standard deviation
The answer is B.
SAT Verbal  Questions from Week 3 (July 24, 2021)
SAT Quick Challenge R21C
The Semicolon
Using the Semicolon Correctly. You will recall that an independent clause consists of a subject and verb that work together to express a complete thought that stands alone as a complete sentence. SAT questions about the semicolon often focus on using that punctuation mark to create a new sentence by connecting two closely related independent clauses, as shown in sentences A, B, and C, which follow. (A) Elaine enjoys cooking. (B) She especially loves to create tasty desserts. (C) Elaine enjoys cooking; she especially loves to create tasty desserts. Since sentence B provides further insight into what was said in sentence A, the two clauses are closely related, and the semicolon in sentence C creates a new sentence by connecting A and B properly.
The Comma Splice Error. A common mistake tested on the SAT is the comma splice, an error created when a writer connects two independent clauses with just a comma. You may recall that a comma can help create a single sentence from two independent clauses if the comma comes just before an appropriate FANBOYS conjunction (for, and, nor, but, or, yet, and so). However, the comma cannot do the job without the FANBOYS conjunction. Note Sentences DH, which follow. (D) Elaine enjoys cooking. (E) She especially loves to create tasty desserts. (F) Elaine enjoys cooking, she especially loves to create tasty desserts. (G) Elaine enjoys cooking, and she especially loves to create tasty desserts. (H) Elaine enjoys cooking; she especially loves to create tasty desserts. Note the comma splice in sentence F and the corrections in sentences G and H.
List of Paired Items Connected with Commas. When the items listed in a sentence consist of paired words that must be connected to each other with a comma (such as a city and state), a semicolon instead of a comma must be placed between the listed pairs to make the writing easier to read, as in Sentence J. (J) During Lisa's vacation, she plans to visit Cape Town, South Africa; Atlanta, Georgia; Albany, New York; and Las Vegas, Nevada. As shown in Sentence K, which follows (and which is almost completely identical to sentence J), the semicolon just before the word "and" at the end of the listing in sentence J is optional. Therefore, Sentences J and K are both correct. (K) During Lisa's vacation, she plans to visit Cape Town, South Africa; Atlanta, Georgia; Albany, New York and Las Vegas, Nevada. Now, keeping in mind the information above, complete exercise R21C below.
QUICK CHALLENGE R21C: The Semicolon 

Directions. Replace the underlined words in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A  NO CHANGE. 

1. Next summer, students from our College International Fellowship will have summer internships in New Delhi, India, London, England, and Paris, France.


2. Donnie Gray does not plan to get a COVID 19 shot, he is extremely afraid of needles.


3. We slept too late this morning; we need Mom to take us to school so we won't be late.

SAT Verbal  Answers to Questions from Week 3 (July 24, 2021)
 C
 C
 A
SAT Math  Questions from Week 2 (July 17, 2021)
It was noted in the last lesson that on the SAT the use of the word average usually refers to the mean and is indicated by “average (arithmetic mean).”
It was also noted that the key to solving any problem involving the average (mean) is to find the total of the items before you do anything else. There are two ways to find the total: (1) add the numbers and (2) multiply the average (mean) by the total number of items. The second method is frequently used on the SAT.
In some problems it will be necessary to calculate two or three totals. Some problems require maximizing or minimizing values. In order to minimize one value, you will need to maximize some other values. For example, if you have an upcoming test in a class and you are trying to determine the minimum score you can get on that test to give you a specific average for all of the tests in that class, you will need to assume that you will get the maximum score of 100 on the remaining tests.
Consider the following example:
An online store receives customer satisfaction ratings between 0 and 100, inclusive. In the first 10 ratings the store received, the average (arithmetic mean) of the ratings was 75. What is the least value the store can receive for the 11th rating and still be able to have an average of at least 85 for the first 20 ratings?
We begin by getting the total for the first 10 ratings. Total = 10 x 75 = 750.
Since we want an average of 85 for 20 ratings, we get the total for 20 ratings: Total = 20 x 85 = 1700.
We can now determine the total for the last 10 ratings: 1700  750 = 950.
Now if one rating, the 11^{th}, is as small as possible, the other 9 ratings must be as large as possible. The highest possible rating is 100. Thus the maximum total for the other 9 ratings = 9 x 100 = 900.
Thus the least possible rating for the 11^{th} rating = 950  900 = 50.
Keeping in mind the information above, answer the following questions.
 In a set of 15 integers, three of the integers are 12, 19, and 23. The mean of the 15 integers is 44. If 12, 19, and 23 are removed from the set, what is the mean of the remaining 12 numbers in the set?
 A new computer game receives critical reviews between 1 and 50, inclusive. In the first 6 ratings, the average (mean) of the ratings was 40. What is the least value the game can receive for the 8th rating and still be able to have an average of at least 42 for the first 10 ratings?
 Jackie took 6 tests in the fall semester of school. The mean score of the 6 tests was 84. If the mean score of the first four tests is 80, what is the mean score of the last 2 tests?
SAT Math  Answers to Questions from Week 2 (July 17, 2021)
 In a set of 15 integers, three of the integers are 12, 19, and 23. The mean of the 15 integers is 44. If 12, 19, and 23 are removed from the set, what is the mean of the remaining 12 numbers in the set?
We begin by getting the total for the 15 integers. Total = 15 x 44 = 660
We then subtract 12, 19, and 23 from 660 to get the new total: 660 – 12 – 19 – 23 = 606
This new total divided by 12 gives us the mean of the remaining 12 numbers:
606/12 = 50.5  A new computer game receives critical reviews between 1 and 50, inclusive. In the first 6 ratings, the average (mean) of the ratings was 40. What is the least value the game can receive for the 8th rating and still be able to have an average of at least 42 for the first 10 ratings?
We begin by getting the total for the first 10 ratings. Total = 6 x 40 = 240.
Since we want an average of 42 for 10 ratings, we get the total for 10 ratings:
Total = 10 x 42 = 420
We can now determine the total for the last 4 ratings: 420 – 240 = 180.
Now if one rating, the 8^{th}, is as small as possible, the other 3 ratings must be as large as possible. The highest possible rating is 50. Thus the maximum total for the other 3 ratings = 3 x 50 = 150.
Thus the least possible rating for the 8^{th} rating = 180  150 = 30.  Jackie took 6 tests in the fall semester of school. The mean score of the 6 tests was 84. If the mean score of the first four tests is 80, what is the mean score of the last 2 tests?
The total for the 6 tests: total = 6 x 84 = 504 The total for the first 4 tests: total = 4 x 80 = 320
Total for the last 2 tests = 504  320 = 184
Thus the mean of the last 2 tests = 184/2 = 92
SAT Verbal  Questions from Week 2 (July 17, 2021)
SAT Quick Challenge R21B
Routine Uses of the Comma, Part II
The Versatile Comma. As noted in previous lessons, the comma does different kinds of jobs. For instance, when two separate independent clauses (also sentences by definition) are combined to make a single sentence, a FANBOYS conjunction (for, and, nor, but, or, yet, and so) can be used to create that new sentence. Note Sentences A, B, and C, which follow. (A) We expected our cousins to arrive this morning. (B) They missed their flight and won't get here until tonight. (C) We expected our cousins to arrive this morning, but they missed their flight and won't get here until tonight. As Sentence C shows, a comma must be placed in front of "but," the FANBOYS conjunction that turns the two independent clauses into a single sentence. As you will note below, today's lesson identifies additional ways in which commas are used in writing.
Separating Items in a List. When a list of items is written as part of a sentence, a comma is often placed just before the word "and" to show that the last item is about to be named. However, that comma is optional. Note Sentences D and E, which follow. (D) People are advised to keep on hand masks, gloves, hand sanitizer, first aid kits, nonperishable foods, water, and other essential emergency supplies. (E) People are advised to keep on hand masks, gloves, hand sanitizer, first aid kits, nonperishable foods, water and other essential emergency supplies. Since the comma is optional, sentences D and E are both correct, so the SAT will not ask you to choose between the two; you will just need to remember that both are correct.
Introductory Words and Phrases. An introductory word/phrase comes at the beginning of a sentence and sets the tone for what will be said. As illustrated in Sentence F, which follows, a comma must be used after an introductory word/phrase (in fact, initially, nevertheless, however, etc.) in a sentence. (F) Initially, we were going to the beach, but we decided to cook out at home, instead.
The "Self" Words Exception. "Self" pronouns are used to emphasize the fact that a particular person or thing is being referred to. Every "object" pronoun (both direct and indirect objects) has a "self" counterpart, as in the following: me/myself, you/yourself, he/himself, she/herself, it/itself, etc. As illustrated in Sentence G, which follows, generally, a writer must not place a comma in front of or behind a self word. (G) Britney Spears herself will sing "America, the Beautiful" during the halftime show at the game on Friday. However, if a grammar rule requires a comma where a self word is used, you must follow that rule. (That is why commas follow the bold, italicized self words listed above in this paragraph.) Another example of the exception is the required comma for combining two independent clauses with a FANBOYS conjunction, as in Sentence H, which follows: (H) I will take Eric to school myself, but Mom will pick him up after school.
Keeping in mind the information above, complete QUICK CHALLENGE R21B below. Then use the dropdown in the next section to check your work.
QUICK CHALLENGE R21B: Routine Uses of the Comma, Part II 

Directions. Replace the underlined words in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A  NO CHANGE. 

1. Mom said that you, yourself, must was the dishes after lunch today.


2. Little Andy wants to have cupcakes, pie ice cream: and candy for his birthday dinner.


3. The hotel overbooked its rooms; so our guests will be staying at our home during their visit.


4. Finally, Mom said, that we could go to the beach, but bad weather spoiled our plans.

SAT Verbal  Answers to Questions from Week 2 (July 17, 2021)
 D
 C
 C
 B
SAT Math  Questions from Week 1 (July 10, 2021)
There are three averages that are tested on the SAT: mean, median, and mode.
 Mean: the total of the items divided by the number of items
 Median: the number that is exactly in the middle of a group of numbers when the numbers are arranged from smallest to largest; the median is always the middle value in a data set
 Mode: the number that appears most often
Find the averages and range of these numbers: 6, 18, 12, 6, 8
 Mean = (6 + 18 + 12 + 6 + 8)/5 = 50/5 = 10
 Median; arrange the items in order: 6, 6, 8, 12, 18
 Median = 8
 Mode = 6
 Range = 18 – 6 = 12
 Arrange in order: 4, 7, 8, 15, 15, 20
 In this case, the median is midway between the two middle numbers:
Median = (8 + 15)/2 = 23/2 = 11.5
The key to solving any problem involving the average (mean) is to find the total of the items before you do anything else. There are two ways to find the total: (1) add the numbers and (2) multiply the average (mean) by the total number of items. The second method is frequently used on the SAT.
The average of 4 numbers is 5. If three of the four numbers are 3, 4, and 5, what is the fourth number? The first thing to do is to get the total. The total = 4 x 5 = 20.
Thus the sum of the four numbers must total 20.
3 + 4 + 5 = 12; to make the total = 20, the fourth number must be 8.
 1, 6, 4, 10, 16, 4, 10, 25, 4, 20
Calculate the mean, median, mode, and range for the above set of numbers. 
Weight (Pounds) Number of Dumbbells 5 6 10 10 20 4
David bought dumbbells in three different weights, in pounds. The table above shows the weight of the dumbbells, in pounds, and the number of dumbbells for each weight David bought. What is the mean weight of the dumbbells, in pounds? 11.67
 10.50
 8.50
 6.67

Player Height Player Height Alice 77 Florence 73 Barbara 69 Geraldine 76 Carolyn 71 Helen 68 Denise 72 Ivey 70 Edith 67 Jane 74
The table above shows the heights of 10 players on the Greensboro High School women’s basketball team. If the coach takes Alice out of the game and substitutes Geraldine in her place, and makes no other substitutions, which of the following must be true? (In basketball, exactly five players from a team are allowed on the court at a time.)
 The median height of players on the court from Greensboro High School will not change.
 The median height of players on the court from Greensboro High School will increase.
 The median height of players on the court from Greensboro High School will decrease.
 A change in the median height of players on the court from Greensboro High School cannot be determined from the information given.
SAT Math  Answers to Questions from Week 1 (July 10, 2021)
 1, 6, 4, 10, 16, 4, 10, 25, 4, 20
Calculate the mean, median, mode, and range for the above set of numbers.
 Mean = (1+6+4+10+16+4+10+25+4+20)10 = 100/10 = 10
 Median: arrange the numbers in order from lowest to highest:
1, 4, 4, 4, 6, 10, 10, 16, 20, 25 The two middle numbers are 6 and 10.
Thus the median = (6 + 10)/2 = 16/2 = 8  Mode = 4 (the number that occurs most often)
 Range = highest  lowest = 25 – 1 = 24

Weight (Pounds) Number of Dumbbells 5 6 10 10 20 4
David bought dumbbells in three different weights, in pounds. The table above shows the weight of the dumbbells, in pounds, and the number of dumbbells for each weight David bought. What is the mean weight of the dumbbells, in pounds? 11.67
 10.50
 8.50
 6.67
Remember in an average problem, the first thing to do is to find the total.
 The total weight of 6 dumbbells that weigh 5 pounds each is 6 x 5 = 30.
 The total weight of 10 dumbbells that weigh 10 pounds each is 10 x 10 = 100.
 The total weight of 4 dumbbells that weigh 20 pounds each is 4 x 20 = 80.
 There are 20 dumbbells (6 + 10 + 4 = 20).
 The total weight of the 20 dumbbells = 30 + 100 + 80 = 210.
 Thus, the mean = 210/20 = 10.50 > B

Player Height Player Height Alice 77 Florence 73 Barbara 69 Geraldine 76 Carolyn 71 Helen 68 Denise 72 Ivey 70 Edith 67 Jane 74
The table above shows the heights of 10 players on the Greensboro High School women’s basketball team. If the coach takes Alice out of the game and substitutes Geraldine in her place, and makes no other substitutions, which of the following must be true? (In basketball, exactly five players from a team are allowed on the court at a time.)
 The median height of players on the court from Greensboro High School will not change.
 The median height of players on the court from Greensboro High School will increase.
 The median height of players on the court from Greensboro High School will decrease.
 A change in the median height of players on the court from Greensboro High School cannot be determined from the information given.
 This problem does not require any calculation or rearranging of data; it requires only an understanding of the concept of the median.
 If Alice is playing and Geraldine is not, it does not matter who the other four players on the team are playing with her; Alice is the tallest player.
 If Alice is taken out of the game and is replaced by Geraldine, then Geraldine is the tallest player on the court. All that has happened is that the tallest player on the court from Greensboro High School has changed.
 Since the median is the middle value in a data set, changing only the greatest value has no impact on the median. Thus the answer is A.
SAT Verbal  Questions from Week 1 (July 10, 2021)
SAT Quick Challenge R21A
Routine Uses of the Comma, Part I
Punctuating Nonessential Information (NESI). A nonessential word or word group provides extra information (NOT the main idea) about another word or word group in a sentence  generally the word that the nesi follows. If the nesi is removed, we will still have the main idea of the sentence, and the sentence will still make sense. Therefore, the nesi should be enclosed in commas. However, if the word or word group is needed to express the main idea clearly, do not use commas.
When deciding whether to place commas around a word group, and where to place them if they are needed, draw a line under the entire possible nesi. The underlining can help you remember (1) to omit the nesi when you are reading to see if the sentence makes sense without it and (2) to read the entire nesi as you decide whether or not to use commas when an SAT question underlines only a small part of a long nesi.
A relative pronoun (such as who, which, or that) is sometimes used to introduce a relative clause that modifies the word the clause follows. If the clause provides nonessential information (nesi) that does not help the reader understand clearly the main idea the writer is trying to state, use commas to separate the nesi from the rest of the sentence. If the clause is needed to help convey the main idea, do not separate it.
Comparing NESI Punctuation. The three sentences which follow show how relative pronouns are sometimes used to introduce nesi. (A) Nan Epps, who just graduated from Northwest High School, has accepted a scholarship to Salem College. (B) Nan Epps has accepted a scholarship to Salem College. (C) The student who has accepted a scholarship to Salem College is Nan Epps.
Commas separate the underlined relative clause in sentence A from the main idea because the reader does not need the information in the clause in order to understand the main idea. Sentence B simply states its main idea; it has no relative clause. In sentence C, commas do not separate the underlined relative clause from the rest of the sentence because the reader needs that clause in order to understand who Nan Epps is.
The Interrupter. An interrupter comes within a sentence and creates emphasis or shows emotion by temporarily breaking the flow of thought in that sentence. Expressions routinely used as introductory words/phrases (in fact, however, initially, etc.) are also used as interrupters. Other common interrupters include expressions such as as you know and for example. Even a person's name can be used as an interrupter.
You can use two commas, two dashes, or two parentheses to separate an interrupter from the rest of the sentence, as in sentences C and D, which follow. (C) What, Kenny, did you think would happen after you broke that window? (D) The heroic teenagers  as we had expected  received awards for their bravery.
Keeping in mind the information above, complete QUICK CHALLENGE R21A below. Then use the dropdown in the next section to check your work.
QUICK CHALLENGE R21A: Routine Comma Uses, Part I 

Directions. Replace the underlined words in each question below with the answer choice that corrects the error in the sentence. If there is no error, select choice A  NO CHANGE. 

1. Our Business Honor Society speaker is a lady who, we were told  ran a business when she was 11 years old.


2. A musical instrument, that, has been mishandled frequently, may never work properly again.


3. Crystal Beach, which attracts huge crowds to its annual Summer Fest events have great restaurants.

SAT Verbal  Answers to Questions from Week 1 (July 10, 2021)
 D
 D
 C
SAT Math Formulas
 At the beginning of each math section these formulas are given in the test booklet. If you haven’t memorized them, you should be familiar with what they mean.
 the length of the hypotenuse = twice the length of the side opposite the 30° angle.
 the length of the side opposite the 60° angle = the length of the side opposite the 30° angle times √3
 the length of the side opposite the 30° angle = ½ the length of the hypotenuse
 The two legs are equal
 the length of the hypotenuse = the length of the either leg times √2
 Area of a circle: A = π r^{2}
 Circumference of a circle: c = 2 π r
 Area of a rectangle: A = lw
 Area of a triangle: A = ½ bh
 Pythagorean theorem: c^{2} = a^{2} + b^{2}
 30° – 60° right triangle:
 45° – 45° right triangle:
 The volume of a rectangular solid: V = lwh
 The volume of a cylinder: V = π r^{2}h
 The volume of a sphere: V = (4/3) π r^{3}
 The volume of a cone: V = (1/3) π r^{2}h
 The volume of a pyramid: V = (1/3)lwh
 The number of degrees in a circle = 360
 The number of degrees in a triangle = 180
 The number of radians in a circle = 2π
You are given these 12 formulas and three geometry laws on the test itself. It can be helpful and save you time and effort to memorize the given formulas, but it is ultimately unnecessary, as they are given on every SAT math section.  The following formulas are not printed on the test booklet; you will have to memorize them.
 Slope = rise (vertical change)/run (horizontal change)
 Given two points on a line, (x_{1}, y_{1}) and (x_{2}, y_{2}), the slope = (y_{2} – y_{1})/(x_{2} – x_{1}).
 If the equation of the line is in the slope/intercept form, y = mx + b, the slope = m
 If the equation of the line is in standard form, Ax + By = C, the slope = A/B
 Total = sum of the items
 Total = the average times the number of items. This method is usually required on SAT problems.
 sine of an angle = side opposite the angle over the hypotenuse (SOH)
 cosine of an angle = side adjacent to the angle over the hypotenuse (CAH)
 tangent of an angle = side opposite the angle over the side adjacent to the angle (TOA)
 i = √1
 i^{2 }= 1
 i^{3 }= i
 i^{4 }= i
 i^{5 }= i
 i^{6 }= i
 i^{7 }= i
 i^{8 }= i
etc.
 The volume of a square: A = s^{2}
 The perimeter of figure = the sum of all of the sides
 Area of a parallelogram: A = lw
 Area of a trapezoid: A = ½ h(b_{1} + b_{2})
 Given a radius and a degree measure of an arc from the center of a circle, find the area of the sector that is defined by the angle and the arc:
Area of a sector of a circle: A = (t/360) π r^{2} when t = the number of degrees in the central angle  Given a radius and a degree measure of an arc from the center, find the length of the arc:
Length of an arc: L = (t/360) (2 π r) when t = the number of degrees in the central angle  When the angles of triangle A are equal to the angles of triangle B, the sides of triangle A are proportional to the sides of triangle B.
 x^{2} – y^{2} = (x + y)(x – y)
 (x + y)^{2} = x^{2 }+ 2xy + y^{2}
 (x  y)^{2} = x^{2 } 2xy + y^{2}
 A function in the form of f(x) = 3x + 12 is the same as y = 3x + 12.
 The equation of the line in the slope/intercept form: y = mx + b, where the slope = m, and the yintercept = b.
 The equation of the line in standard form: Ax + By = C, where the slope = A/B and the
yintercept = C/B  Slope – four ways to determine the slope:
 The standard form of a parabola equation: y = ax^{2} + bx + c
 Vertex form of the parabola equation: y = a(x – h)^{2} + k, where the vertex is the point (h,k).
 Equation of a circle? (x – h)^{2} + (y – k)^{2} = r^{2} where the center of the circle is the point (h,k)
and the radius of the circle is r.  The quadratic formula:
For ax^{2} + bx + c = 0, the value of x is given by:
 The key to solving average problems is to find the total of the items before doing anything else. There are two ways to find the total:
 Average speed = total distance / total time; Distance = (speed) x (time)
 SOHCAHTOA (applies to a right triangle)
 180 degrees = π radians
 Imaginary Numbers
 A present amount P increases at an annual rate r for t years. The future amount F in t years is:
F = P(1 + r) ^{t }  A present amount P decreases at an annual rate r for t years. The future amount F in t years is:
F = P(1  r) ^{t }  Item sold at discount: discount amount = original price x discount percent
 Item sold at discount: reduced price = original price x (1discount percent)
 Given two points, A(x_{1},y_{1}) , B(x_{2},y_{2}), find the midpoint of the line that connects them:
Midpoint = the average of the x coordinates and the y coordinates: (x_{1} + x_{2}) / 2 , (y_{1} + y_{2}) / 2  Given two points, A(x_{1},y_{1}) , B(x_{2},y_{2}), find the distance between them:
Distance= √[ (x_{2}  x_{1})^{2} + (y_{2}  y_{1})^{2 }]  Probability of x = (number of outcomes that are x) / (total number of possible outcomes)
SAT Math Operations
 Substitute values for a variable and simplify.
 Add fractions with different denominators, where the denominators are numbers.
 Add fractions with different denominators, where the denominators are variables.
 Know how to simplify complex fractions.
 In a fraction, the denominator cannot equal zero. If an equation is solved and the value of the variable makes the denominator = zero, then that value cannot be a solution to the problem.
 When picking numbers, consider positive numbers, negative numbers, zero, decimals, and extreme numbers.
 Understand the definitions of the terms digit, integer, number, prime number, factor, multiple, divisible, reciprocal of a number, absolute value of a number.
 Know the absolute value sign.
 Know the common fractiondecimalpercent equivalents.
 Know how to change a fraction to a decimal or to a percent.
 Know how to change a decimal to a percent.
 Know how to change a percent to a decimal.
 Understand the factorial concept.
 Know how to compute permutations and combinations: n items taken x at a time.
 Know when to use Venn diagrams.
 When angles are formed when a line crosses parallel lines, several equal angles are created.
 When a diagram is given in a geometry problem, consider adding one or more lines to create another figure.
 Geometric figures are not necessarily drawn to scale; lines that look equal may not be equal; angles that look equal may not be equal.
 In a triangle, the length of sides opposite equal angles are equal.
 In a triangle, the length of a side opposite a larger angle is greater than the side opposite a smaller angle.
 Know the third side rule for triangles: the length of any one side of a triangle must be less than the sum of the other two sides, and greater than the difference between the other two sides.
 Two triangles are congruent if the sides of one triangle are equal to the corresponding sides of the other triangle and the angles of one triangle are equal to the corresponding angles of the other triangle.
 Two triangles are similar if the angles of one triangle are equal to the corresponding angles of the other triangle and the sides of one triangle are not equal to the corresponding sides of the other triangle.
 If two triangles are similar, their corresponding sides are proportional.
 The measure of an angle inscribed in a circle is half the measure of the central angle that intercepts the same arc.
 The length of an arc is a fraction of the circumference of a circle.
 A line tangent to a circle produces a right angle at the point of tangency between the line and another line that connects the point of tangency to the center of the circle.
 Know the exponent rules.
 Know how to express a number with alternative bases using appropriate exponents; the most common problems involve changing a number to a base of 2 or a base of 3.
 Know how to determine the three averages: mean, median, and mode.
 Know how the normal curve, mean, and standard deviation interact.
 Read ratio problems carefully,
 A ratio can express a part to part relationship.
For example, a ratio of 1 to 2 = 1:2 = ½.  A ratio can express a part to whole relationship.
For example, a ratio of 1 to 2 has two parts and a whole (1 + 2 = 3). One part is ⅓, the other part is ⅔.  Solve linear equations when the answer is a number (one equation and one unknown.)
 Solve linear equations when one variable is in terms of other variables (one equation with all variables.)
 Solve simultaneous equations (two equations in 2 unknowns.)
 Solve quadratic equations by factoring, by using the quadratic equation, and by completing the square.
 Find the radius of a circle from the formula of a circle.
 Know how to write the formula of a circle in standard form.
 Factor an expression.
 Type 1: 3xy + 7x = x(3y + 7)
 Type 2: 2x^{2} + 13x + 15 = (2x + 3)(x + 5)
 Type 3: 2^{13} – 2^{11} = 2^{11}(2^{2} – 1) = 2^{11}(41) = 2^{11}(3)
 Solve inequalities.
 Find the price of an item after a sales tax is added.
 Find the price of an item after a percent increase.
 Find the price of an item after a percent decrease.
 Find the percent of a number.
 When one number is greater than another, find the percent greater.
 When an amount changes, find the percent change.
 Know the three averages: mean, median, and mode.
 In an average problem, the first thing to do is to find the total; there are two ways to find the total.
 Find the average of a set of numbers.
 Find the missing number in a set of numbers when the mean is known.
 From the equation of a line, determine the y intercept, x intercept, and slope.
 Understand positive slopes, negative slopes, and slope = zero.
 Equation of a parabola.
 Coordinate geometry: locate points in the xy plane.
 Know the I, II, III, and IV quadrants.
 Evaluate information in a chart.
 Word problems: write down each detail; proceed in a step by step fashion.
 Trigonometry: find the sine of an angle; find the cosine of an angle; find the tangent of an angle (remember SOHCAHTOA.)
The sine of an angle = the side opposite the angle divided by the hypotenuse (SOH)
The cosine of an angle = the side adjacent to the angle divided by the hypotenuse (CAH)
The tangent of an angle = the side opposite the angle divided by the side adjacent to the angle (TOA)  In right triangle ABC, if angle B is the right angle, then the sine of angle A = the cosine of angle C.
February 2021 Lessons
March 2021 Lessons
April 2021 Lessons
May 2021 Lessons