Mar 31 2010
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Mathematics > Standard Multiple Choice
Read the following SAT test question, then click on a button to select your answer.
My tutoring note: If you take out the figures, the question reads this way:
If the function is defined by , and if , what is the value of ?
Answer Choices (A) (B) (C) (D) (E) The answer choices are all one number. The numbers start with A.4 and go up in amount, and they are all whole numbers and positive. The answer will be whole and positive.
As usual, I do want you to go to the SAT site and to look at the problem, attempt to solve the problem, use your strategies, and look at the hint--even if you know how to do the problem.
Today's answer is explained on the SAT in a comprehensive way if you put the hint together with the final answer.
I want a test-taker (and student) to understand how to do this problem, and I want you to understand that if you can look at a problem you do not understand completely and reword it for yourself, you may be able to answer it correctly without immediately remembering how to do each traditional step.
The reason this is true more often on an SAT question than on some standardized tests is that the SAT questions are designed to test your aptitude.
Therefore, with the information given, you will always know more than you realize immediately. Part of your assistance in answering the question will be enclosed in the question and the format PLUS the fact that the calculations are not usually the point of the question, so they work out better than in 'normal' life.
Strategy One: Use your calculator if you do not do fast arithmetic and multiplication while figuring out how to do a problem. The numbers will often be the most manageable part of the problem when the question is about how to do a problem. For example, in this problem, the numbers are all low, whole, and positive.
The question here is:
If the function of f is defined by f(x)=2x+3, and if f(a)=11, what is the value of a?
Look at the explanation of how to do this to see the way to use the pattern to solve more complicated problems.
For this problem, I wrote it in the format to find the value: f(a)=2a+3 and merely plugged in the first answer, 4, because it was easy to do. f(a)=2(4)+3=11.
So I knew immediately that '4' is the answer. Also, all the other numbers are too big. The only way I would be able to do this in this way is if the answers are given, as they are on a multiple choice test.
If you study the answer on the SAT, you will learn how to answer even more mathematics questions that are written in this way.
If you study my strategy, you will learn one of the better mathematics strategies I learned from the Princeton Review. That is, when you are trying to visualize a mathematics solution, plug in some numbers that are easy for you to manage. In this case, one of the answers filled the bill for this strategy.
In some cases, the answers do NOT provide an easy way to 'plug in' numbers so you can see how the math problem is going.
Next time, I will show you how this strategy really helps in learning mathematics--as well as answering correctly on a standardized test.
When you 'teach' yourself, rather than just learn the steps, you reach a different area of your brain capacity. The results are an increased level of performance that reveals what you know to yourself as well as to others.
judiethcarol&rocketcatMarch2010
Thursday, April 1, 2010
Wednesday, March 31, 2010
WEDNESDAY MARCH 31 CONTINUED
My Personal : Hello? Strategy for Gifted, Analytical & Argumentative Thinkers
Read the directions for each section. Know them ahead of time. When you read them at the time of doing the questions, the actual reading of the directions will put your mind (your brain) into the correct sequencing strategy to do these questions if you have practiced them ahead of time.
This will become a time-saving and accuracy-producing strategy for answering the questions you know how to do correctly and for taking your best shot at the questions demanding more time from you.
The strategies relating to the questions and answers that you tend to do quickly, plus the questions and answers which may be examined quicker than others due to their brevity, are designed for you to practice to make better choices daily—using what you know to your advantage.
Directions:
Talk to your brain: Identify what this section is asking. Read the title.
Writing-Identifying sentence errors
(Note from me: The title lets you know what type of language arts function you are watching to find. In this case, you are detecting sentence errors.)
(Note from me: Remind yourself of the directions for this particular set of questions. You know it from practice. This is a reassurance to your brain and an adjustment of the mindset to the task.)
Directions
The following sentence contains either a single error or no error at all. If the sentence contains an error, select the one underlined part that must be changed to make the sentence correct. If the sentence contains no error, select choice E.
In order to prepare for the speech he was given to all of the parents and teachers at the school, George practiced speaking in front of a group of his friends. No error.
My note: The underlined portions are labeled A, B,C,D, and E for you to choose the answer by choosing the letter.
To do this type of question, read until you see an error (was given). This is the correct answer. Mark it in your booklet.
Now, quickly, check the other underlined portions—sincerely looking for them to be in error because if you find a worse error, there is a possible way that ‘was given’ could be in a sentence like this. George could have been given a speech from someone else to prepare to present to others. This likelihood would involve too much explanation and would require another part of the sentence in need of correction.
ADDITIONALLY, this is the type of reasoning that causes gifted, analytical, and creative students to miss obvious correct answers on tests. Learn how to retain your gift for seeing the clues about incorrect answers that will always be on multiple choice tests while avoiding a tendency to choose an answer to ‘argue’ a reason to include.
I call this the ‘Hello?’ strategy, and I remind students I tutor one on one about it when I think the reminder itself will not arouse the tendency to bring out this trait.
My ‘Hello?’ strategy is this: If you must explain to someone who is looking at you with upraised eyebrows WHY you choose this answer, find the answer the test writer intends as the correct answer. You do not have it yet if you have to explain to surprised people.
This is different from a question and answer that makes you notice how you tend to a wrong answer at times (as when you are looking for the answer that is 20% more than 100 and 125 and 120 are possible answers). The difference is that you do know a way this answer can be true.
In the March 30 question, there is a way that the sentence can be written to say that George is practicing a speech that was given to him. That is not the correct answer for this question. This sentence is giving the information that George is practicing a speech that he will be giving in the future. Answer accordingly, and do not hesitate. Most importantly, if you are fatigued or stressed, your mind will wander to these other ‘arguments’ on questions. Do not allow this to happen.
Rest before the test.
My seasoned and reasoned advice to all students w ho are inclined to find answers you need to talk with the test writer to mark correctly is: Find the test writers’ answer and move on.
This is a good place to include the thought that writing test questions for any test is a difficult art. The answer is in front of you on a multiple choice test. The test is written so that you display the information that you can answer that question without having the answer in front of you—that the fact that it is there is for two reasons: accuracy in grading the test (by machine/computer) and standardization of the form of the answer considered acceptable.
Both of these reasons are excluded on the section of the math test in which you write in the answers. This is called the ‘grid-in’ section and is merely a section in which you are allowed to write down the answer rather than choosing it from a list.
Plan to answer every problem in the ‘grid-in’ section. No points are counted against you for an incorrect answer in this section. There is absolutely no risk and all gain—except for pacing. If the problem takes too long to do—still, try to guess, if possible. But keep going. Always be aware of the pacing of the test.judiethcarol&rocketcatMarch2010c
Read the directions for each section. Know them ahead of time. When you read them at the time of doing the questions, the actual reading of the directions will put your mind (your brain) into the correct sequencing strategy to do these questions if you have practiced them ahead of time.
This will become a time-saving and accuracy-producing strategy for answering the questions you know how to do correctly and for taking your best shot at the questions demanding more time from you.
The strategies relating to the questions and answers that you tend to do quickly, plus the questions and answers which may be examined quicker than others due to their brevity, are designed for you to practice to make better choices daily—using what you know to your advantage.
Directions:
Talk to your brain: Identify what this section is asking. Read the title.
Writing-Identifying sentence errors
(Note from me: The title lets you know what type of language arts function you are watching to find. In this case, you are detecting sentence errors.)
(Note from me: Remind yourself of the directions for this particular set of questions. You know it from practice. This is a reassurance to your brain and an adjustment of the mindset to the task.)
Directions
The following sentence contains either a single error or no error at all. If the sentence contains an error, select the one underlined part that must be changed to make the sentence correct. If the sentence contains no error, select choice E.
In order to prepare for the speech he was given to all of the parents and teachers at the school, George practiced speaking in front of a group of his friends. No error.
My note: The underlined portions are labeled A, B,C,D, and E for you to choose the answer by choosing the letter.
To do this type of question, read until you see an error (was given). This is the correct answer. Mark it in your booklet.
Now, quickly, check the other underlined portions—sincerely looking for them to be in error because if you find a worse error, there is a possible way that ‘was given’ could be in a sentence like this. George could have been given a speech from someone else to prepare to present to others. This likelihood would involve too much explanation and would require another part of the sentence in need of correction.
ADDITIONALLY, this is the type of reasoning that causes gifted, analytical, and creative students to miss obvious correct answers on tests. Learn how to retain your gift for seeing the clues about incorrect answers that will always be on multiple choice tests while avoiding a tendency to choose an answer to ‘argue’ a reason to include.
I call this the ‘Hello?’ strategy, and I remind students I tutor one on one about it when I think the reminder itself will not arouse the tendency to bring out this trait.
My ‘Hello?’ strategy is this: If you must explain to someone who is looking at you with upraised eyebrows WHY you choose this answer, find the answer the test writer intends as the correct answer. You do not have it yet if you have to explain to surprised people.
This is different from a question and answer that makes you notice how you tend to a wrong answer at times (as when you are looking for the answer that is 20% more than 100 and 125 and 120 are possible answers). The difference is that you do know a way this answer can be true.
In the March 30 question, there is a way that the sentence can be written to say that George is practicing a speech that was given to him. That is not the correct answer for this question. This sentence is giving the information that George is practicing a speech that he will be giving in the future. Answer accordingly, and do not hesitate. Most importantly, if you are fatigued or stressed, your mind will wander to these other ‘arguments’ on questions. Do not allow this to happen.
Rest before the test.
My seasoned and reasoned advice to all students w ho are inclined to find answers you need to talk with the test writer to mark correctly is: Find the test writers’ answer and move on.
This is a good place to include the thought that writing test questions for any test is a difficult art. The answer is in front of you on a multiple choice test. The test is written so that you display the information that you can answer that question without having the answer in front of you—that the fact that it is there is for two reasons: accuracy in grading the test (by machine/computer) and standardization of the form of the answer considered acceptable.
Both of these reasons are excluded on the section of the math test in which you write in the answers. This is called the ‘grid-in’ section and is merely a section in which you are allowed to write down the answer rather than choosing it from a list.
Plan to answer every problem in the ‘grid-in’ section. No points are counted against you for an incorrect answer in this section. There is absolutely no risk and all gain—except for pacing. If the problem takes too long to do—still, try to guess, if possible. But keep going. Always be aware of the pacing of the test.judiethcarol&rocketcatMarch2010c
My Hello? Strategy for Gifted, Analytical Thinkers
Today's question and answer is my 'Racecar' style, yet it offers an opportunity for me to mention another mathematical strategy, as well. This is the strategy I call the 'Hello?' strategy for those of us who analyze, think again, and create new possibilities. On a standardized test, to be successful, use that creative ability to think as though you are the test writer. The results of your choices will not always be the same, but you will know which answer is correct.
Today's language arts, sentence correction, question and possible answers is a classic example.
These are the easiest questions for me. My strategy includes eliminating all the short answers.after choosing the correct one. I also must NOT be so 'creative' and 'analytical' that I see a way for the obvious error to be correct, as well. See this blog entry continued today.judiethcarol&rocketcatmarch2010
Today's language arts, sentence correction, question and possible answers is a classic example.
These are the easiest questions for me. My strategy includes eliminating all the short answers.after choosing the correct one. I also must NOT be so 'creative' and 'analytical' that I see a way for the obvious error to be correct, as well. See this blog entry continued today.judiethcarol&rocketcatmarch2010
MATH & REASONING & FAMILY & THINKING
I will give you strategies about mathematics to improve your confidence about all things mathematical—plus all things logical. This pretty much covers everything. That is the nature of mathematics. The structure is ancient, and the scope is comprehensive.
The tips, instructions, techniques, and patterns I teach are what I use to do well on mathematics tests—despite my stronger background in language arts. As I mention in many of these columns, most of the strategies I use in language arts sections of SAT and other tests relate to checking the answers by eliminating all the other answers. There are tests related to aptitude and math questions that have nothing to do with eliminating the incorrect answers.
First, a review of the OTHER type of strategy—the one involving connotations and emotional intelligence, as well as logic:
I call the questions and answers related to the language arts sections on the SAT ‘racecar’ questions and answers for three reasons:
1) The procedure for answering, including reading the passage, is usually fast and streamlined.
2) The choices are brief—one word, a pair of words, or the letter of the answer.
3) The word ‘racecar’ is a palindrome to remind me that my scores will be higher on the language arts section if I use the fact that these are easier for me to pace to read them ‘frontwards and backwards’—like a palindrome.
Second, a review of the general method to manage mathematics and quantitative questions (such as the ones on the Graduate Records Examination):
Reword the question in your mind while jotting down an illustration for yourself on your ‘scratching’ area. Now, attempt to find the correct answer and to eliminate two wrong answers.
Eliminate two wrong answers. I advise NOT eliminating every answer you believe to be incorrect. That is too time consuming. Instead, the elimination strategy relates to knocking out the obviously incorrect answers –especially when you are somewhat lost in a more difficult question.
Because most of the elimination of ‘wrong’ answers as a strategy has to do with moving on and the time pressure of the test, I am not going to begin this section about mathematics with all of the ‘elimination’ tips especially for math. To be sure to get the answers you know correctly chosen and marked, let’s start with some predictable aspects of this section.
As you plow through the minutes of this section, you will encounter terms for numbers. Keep looking at some of those terms.
Integers is the term you will see again and again: Integers include zero, whole numbers, negative and positive numbers…--Keep noticing more things about integers and why this term is used instead of one of the other definitive ‘sets of numbers’ such as:
Natural or counting numbers {1,2,3,4,5,…11,12,…}
Whole numbers {0,1,2,3…10,11,12,13…}
This would be a good time to go to a text and to find examples and definitions to make cards, PowerPoint presentations, and games from the definitions, spellings and examples for :
Integers
Natural or counting numbers
Whole numbers
Rational numbers
Irrational numbers
Real numbers
Imaginary numbers
Complex numbers
Today, the only other overview of EVERYTHING that I am going to mention is the Order of Operations. There will be questions on the math test that are only about whether you know which operation to do first. Otherwise, the computation will be simple for those questions. The answer will be correct ONLY if you do the operations in order.
From left to right, you look to see if there are
operations to do within parentheses,
then you handle the exponents, if any
Multiply
Divide
Add
Subtract
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
PEMDAS—Please Excuse My Dear Aunt Sally is the mnemonic often used. Put this on the same page with the formula information on your test booklet.
When I begin a mathematics test, I look for the page with the formulae. And I write this mnemonic for my quick review—even for a short problem like : 5+2x4= Is the answer 13 or 28?
(The answer is 13 because you multiply before you add.)
Commutative and associative properties are rules reminding you to realize that order of operations is paramount; HOWEVER, numbers may be added in any sequence; and they may be multiplied in either order to get the same result.
(See more about this later.)
Today’s Summary: Mathematics strategy is different from other strategies.
Keep practicing exercises with the terms. One term can change in meaning across the subject areas and even within an area.
Practice doing the operations (multiplication, division, addition, and subtraction) in the proper sequence. Before anything, do the operations within parentheses, then do the multiplication of the exponents—then: multiplication, division, addition, subtraction.
The mnemonic is PEMDAS: Please Excuse My Dear Aunt Sally.
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
(Come back to do more mathematics. We will return to the ‘racecar’ questions and answers, too.
Judiethcarol&Rocketcat march 2010
The tips, instructions, techniques, and patterns I teach are what I use to do well on mathematics tests—despite my stronger background in language arts. As I mention in many of these columns, most of the strategies I use in language arts sections of SAT and other tests relate to checking the answers by eliminating all the other answers. There are tests related to aptitude and math questions that have nothing to do with eliminating the incorrect answers.
First, a review of the OTHER type of strategy—the one involving connotations and emotional intelligence, as well as logic:
I call the questions and answers related to the language arts sections on the SAT ‘racecar’ questions and answers for three reasons:
1) The procedure for answering, including reading the passage, is usually fast and streamlined.
2) The choices are brief—one word, a pair of words, or the letter of the answer.
3) The word ‘racecar’ is a palindrome to remind me that my scores will be higher on the language arts section if I use the fact that these are easier for me to pace to read them ‘frontwards and backwards’—like a palindrome.
Second, a review of the general method to manage mathematics and quantitative questions (such as the ones on the Graduate Records Examination):
Reword the question in your mind while jotting down an illustration for yourself on your ‘scratching’ area. Now, attempt to find the correct answer and to eliminate two wrong answers.
Eliminate two wrong answers. I advise NOT eliminating every answer you believe to be incorrect. That is too time consuming. Instead, the elimination strategy relates to knocking out the obviously incorrect answers –especially when you are somewhat lost in a more difficult question.
Because most of the elimination of ‘wrong’ answers as a strategy has to do with moving on and the time pressure of the test, I am not going to begin this section about mathematics with all of the ‘elimination’ tips especially for math. To be sure to get the answers you know correctly chosen and marked, let’s start with some predictable aspects of this section.
As you plow through the minutes of this section, you will encounter terms for numbers. Keep looking at some of those terms.
Integers is the term you will see again and again: Integers include zero, whole numbers, negative and positive numbers…--Keep noticing more things about integers and why this term is used instead of one of the other definitive ‘sets of numbers’ such as:
Natural or counting numbers {1,2,3,4,5,…11,12,…}
Whole numbers {0,1,2,3…10,11,12,13…}
This would be a good time to go to a text and to find examples and definitions to make cards, PowerPoint presentations, and games from the definitions, spellings and examples for :
Integers
Natural or counting numbers
Whole numbers
Rational numbers
Irrational numbers
Real numbers
Imaginary numbers
Complex numbers
Today, the only other overview of EVERYTHING that I am going to mention is the Order of Operations. There will be questions on the math test that are only about whether you know which operation to do first. Otherwise, the computation will be simple for those questions. The answer will be correct ONLY if you do the operations in order.
From left to right, you look to see if there are
operations to do within parentheses,
then you handle the exponents, if any
Multiply
Divide
Add
Subtract
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
PEMDAS—Please Excuse My Dear Aunt Sally is the mnemonic often used. Put this on the same page with the formula information on your test booklet.
When I begin a mathematics test, I look for the page with the formulae. And I write this mnemonic for my quick review—even for a short problem like : 5+2x4= Is the answer 13 or 28?
(The answer is 13 because you multiply before you add.)
Commutative and associative properties are rules reminding you to realize that order of operations is paramount; HOWEVER, numbers may be added in any sequence; and they may be multiplied in either order to get the same result.
(See more about this later.)
Today’s Summary: Mathematics strategy is different from other strategies.
Keep practicing exercises with the terms. One term can change in meaning across the subject areas and even within an area.
Practice doing the operations (multiplication, division, addition, and subtraction) in the proper sequence. Before anything, do the operations within parentheses, then do the multiplication of the exponents—then: multiplication, division, addition, subtraction.
The mnemonic is PEMDAS: Please Excuse My Dear Aunt Sally.
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
(Come back to do more mathematics. We will return to the ‘racecar’ questions and answers, too.
Judiethcarol&Rocketcat march 2010
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