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Quadrilaterals

Geometry: Quadrilaterals 

Read the following SAT test question and then select the correct answer. 

Math questions should always be read carefully.  You will also avoid making errors by identifying the bottom line and assessing your options for solving the question.  Choose the most efficient method to attack the problem.  When you have finished, loop back to be sure that even if there were multiple steps, you reached the bottom line.

The length of a rectangle is increased by 20%, and the width of the rectangle is increased by 30%. By what percentage will the area of the rectangle be increased?

Bottom Line: % change = ?

Assess your Options:  You could work this problem without picking any numbers; however, picking easy numbers will allow you to think about the problem in a more concrete way and avoid errors.

Attack the Problem:  One of the easiest numbers to work with is one.  Think of your original rectangle as having a length of one and a width of one.  The formula for area of a rectangle is length times width.  If L × W = A, for your first rectangle you have 1 × 1 = 1.   The area of the original rectangle is one.

Then think about the changes that occur to that rectangle.  The length increases by 20%.  In order to find 20% of 1, all you have to do is move the decimal over twice to .2.  The new length is 1.2.  Use the same method to find the new width, and an increase of 30% becomes 1.3.  The area of the rectangle after the change is 1.2 × 1.3 = 1.56

The formula for percent change would require you to find the difference between these two areas and divide that by the original number.  You use the same formula whether you are looking for an increase or a decrease.  Notice that your original number is one, so dividing by one will not change your answer.  All you need to do is find the difference between the areas: 1.56 – 1 = .56.  What is .56 as a percent?  Your answer is 56%.

Loop Back:  You found the percent change, which was your bottom line.  Look down at your answer choices.

(A) 25%
(B) 36%
(C) 50%
(D) 56%
(E) 60%

The correct answer is (D).


On sat.collegeboard.org, 34% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Coordinate Geometry

Link of the Day

Today is Veterans Day, a day set aside to thank those who have risked their lives to serve our country.  As you gather historical, current, and literary examples for your SAT essay, consider including an example involving soldiers.  Think about the courage that it takes to be willing to serve in such a capacity, and the reasons behind the choice to enlist.  Take a look at this article and think about how our lives are different due to the sacrifices of many veterans.

11/11 Coordinate Geometry

Read the following SAT test question and then select the correct answer. 

Use the same process for every math problem.  Read the problem carefully.  Identify the bottom line.  Assess your options, then choose the most efficient method to attack the problem.  Once you have worked the problem, loop back to verify that you have solved for the bottom line.

 math image
In the figure above, which quadrants contain pairs (x, y) that satisfy the condition  x over y = 1?

Bottom line:  Where can you find an x and a y that work for this problem?

Assess your options:  This question concerns coordinate geometry, so you will have to use the facts that you know about graphing to answer the question.  You could pick specific points in each quadrant to see if they work, but simply knowing the properties of the graph should be enough to get you to the answer.

Attack the problem:  In order to divide a number by another number and get one, you need equal numbers.  To satisfy this condition, x and y must be equal to each other.  Ask yourself whether the x and y can be equal to each other in each quadrant.  In quadrant I, all the numbers are positive (+, +), so it is possible for x and y to equal each other and create a positive 1 after division.

Now think about the characteristics of quadrant II.  In quadrant II, all of the x values are negative and all of the y values are positive (-, +), so x and y cannot be equal.  When you divide a negative number by a positive number, you will get a negative number; there is no way to get a positive 1.  Quadrant II does not satisfy this condition.

In quadrant III x and y are both negative (-, -), so they could be equal.  If you divide a negative number by a negative number, the result will be positive.  You can get a positive number 1 in this quadrant.

In quadrant IV, the x values are positive while the y values are negative (+, -).  Once again x and y cannot be equal.  You cannot divide a positive by a negative and get a positive number, so quadrant IV does not satisfy this condition.

Loop Back:  You checked all four quadrants, so look down at your answer choices now.

(A) I only
(B) I and II only
(C) I and III only
(D) II and IV only
(E) I, II, III, and IV

The correct answer is (C).


On sat.collegeboard.org, 59% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Ratios

Link of the Day

The election results are in!  The amount of information after an election day can be overwhelming, but limit yourself to one story from the election, and you will have an excellent current event for your SAT essay.  Many people focus on the presidential election, but there are hundreds of other important issues that were brought before the nation.  One group of United States citizens who currently cannot vote for the president of the United States voted about the possibility of becoming the 51st state.  Read this article about what is happening in Puerto Rico, and think about how this prospective state differs from or is similar to other territories that have become states.  Think of the broad themes raised by this story that could relate this article to SAT essay questions.

11/8 Algebra:  Ratios

Read the following SAT test question and then select the correct answer.

Don’t just read the question; read it carefully.  Make sure you know which labels apply to which numbers.  Identify the bottom line.  Assess your options for solving the problem so that you can choose the most efficient method to attack the problem.  Once you have solved the problem, loop back to make sure that you have solved for the bottom line.

In a class of 80 seniors, there are 3 boys for every 5 girls. In the junior class, there are 3 boys for every 2 girls. If the two classes combined have an equal number of boys and girls, how many students are in the junior class?

Bottom Line:  Number of juniors = ?

Assess your options:  You could work backwards by starting with the answer choices, but it might take you a long time to work through all of the possible answers.  Instead, start turning those ratios into actual numbers of students.

Attack the problem:  You know the most about the seniors, so start with them.  You are given a ratio of 3 boys to 5 girls, and you know that the total number of boys and girls must equal 80.  You know that 3 + 5 = 8, so all you have to do is multiply the 3 and the 5 each by 10 and you will have a total of 80 seniors.  There are 30 senior boys and 50 senior girls.



Now that you know the number of senior boys and senior girls, how does that help you find the number of juniors?  Remember that the two classes combined have an equal number of boys and girls.  That means that the senior boys plus the junior boys must be equal to the senior girls plus the junior girls. 



Plug in the numbers that you found for the senior boys and girls.  



What information do you know about the juniors?  You know that there are 3 boys for every 2 girls.  You do not know the total number of juniors, so use an x to represent this number.  What fraction of the total are the boys?  They are actually three fifths of the total number of juniors because you must add the boys and girls to find the total number of juniors.  That means that the girls are two fifths of the total number of juniors.  Plug this into your formula, remembering that anytime you have “of the total” that means that you must multiply by the unknown total. 



Now solve for x.  Rearrange the equation so that you have like terms on the same sides of the equation, and combine those like terms.  Start by subtracting the two fifths of x from each side.



Get those whole numbers together by subtracting 30 from each side.



To get rid of the fraction, you will need to multiply both sides by 5.  Your answer is x = 100.

Loop Back:  What does x represent?  It represents the total number of juniors, which matches your bottom line.  You are ready to look down at your answer choices.

(A) 72
(B) 80
(C) 84
(D) 100
(E) 120

The correct answer is (D).


On sat.collegeboard.org, 44% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Coordinate Geometry

Geometry:  Coordinate Geometry

Read the following SAT test question and then select the correct answer. 

Use the same method for all the math questions on the SAT.  First, read the question carefully to avoid making mistakes.  Identify the bottom line and assess your options for reaching it.  Next, choose an efficient method to attack the problem.  Finally, loop back to make sure that your answer addresses the bottom line.  Many problems have multiple steps.

If the graph of the function f in the xy-plane contains the points (0, -9), (1, -4), and (3, 0), which of the following CANNOT be true?

Bottom Line:  You are looking for something false.

Assess your Options:  You could try drawing an xy-plane and graphing the points to help you visualize the question, but your graph may be inaccurate without graph paper.  Instead, try to find the relationship between the three points.

Attack the problem:  To find the relationship between these points, you will need to find the slope of the line between each point.  The formula for slope is:  

 




Check the slope of the line between (0, -9) and (1, -4):


Then check the slope of the line between (1, -4) and (3, 0):


The function in this problem has a very steep slope between the first two points, but becomes less steep between the second two.  This is a “which of the following” question, so start with answer (E) as you work through your answer choices.

(A) The graph of f has a maximum value.
(B) y ≤ 0 for all points (x, y) on the graph of f.
(C) The graph of f is symmetric with respect to a line.
(D) The graph of f is a line.
(E) The graph of f is a parabola.

(E) The function could be a downward facing parabola if it continues to the right.  You are only given three points, but there could be many more points on this function. 

(D)  In geometry, a line is always straight, without any curves.  Notice that there are different slopes connecting the three points.  You cannot draw one straight line through all three of these points, so this choice cannot be true.

Loop Back:  Your goal was to find an answer choice that was false.  You did so, so you are finished!  If you have extra time, you can check the other answer choices and see that they are all possible, depending on how you draw the rest of the function.  (E), (C), (B), and (A) could all describe a downward facing parabola with the equation y = -(x – 3)².

The correct answer choice is (D).


On sat.collegeboard.org, 42% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Circle Graph

Read the following SAT test question and then select the correct answer.

When a math question involves a table or chart, read both the text of the question and the labels of the chart carefully.  Identify the bottom line, and assess your options for solving the problem.  Attack the problem to find the answer, and loop back to make sure that your answer addresses the bottom line.

graphic

In a survey, a group of students from Westville High School were asked about their favorite movie genre. Each student in the group selected exactly one movie genre, and the data collected are summarized in the circle graph above. If 40 more students chose Action than Fiction, how many students were surveyed in total?

Bottom Line:  Total students = ?

Assess your Options:  You could plug in answer choices for the total and then take percentages of those to find out which answer would produce a difference of 40 between Action and Fiction.  That would take a lot of steps!  Instead, start with what you know and use what you know to write an equation.

Attack the Problem:  You know that 40 more students chose Action than Fiction.  That means that Action – Fiction = 40.  You also know the percentages for both Action and Fiction.  Plug in the percentage from the chart and you will see that 30% (Action) – 14% (Fiction) = 16%.  Now you need to combine the two things that you know, percents and actual numbers, into a single equation.  You can write percents as decimals by moving the decimal two times to the left.  What are these percents of?  The uknown total number of students.  For any unknown number, you can plug in the variable x.  Now you have the equation .16x = 40.  Solve for x by dividing each side by .16 and you will get the answer 250.

Loop Back:  What did x represent?  The total number of students.  That matches your bottom line, so you are ready to look down at your answer choices.

(A) 100
(B) 150
(C) 200
(D) 250
(E) 300

The correct answer is (D).


On sat.collegeboard.org, 50% of the responses were correct.

For more help with SAT writing, visit www.myknowsys.com!

Algebra

Link of the Day

Many of you have been following the news about Hurricane Sandy, and our thoughts and prayers are with those affected by the storm.  For others, it may be easy to hear things like “school is out” and “the power is out” and wish you were in the same situation.  You may think of enjoyable times spent wrapped in blankets and telling stories as a candle or flashlight flickers.  Think for a moment about how important power is for a hospital.  Here is an article about how one hospital responded to the storm.  Think about broad themes such as courage, the fight for life, and the response to danger as you read about this current event.

10/30 Algebra

Read the following SAT test question and then select the correct answer. 

If you approach all math problems the same way, you are less likely to make a careless mistake.  Start by reading the problem carefully and identifying the bottom line.  Assess your options for solving the problem so that you do not do more work than you need to.  Then attack the problem and solve it.  Loop back after you have finished to make sure that you found the bottom line.

If  y = ((x +1) times (x minus 2)) over ((x + 3) times (x minus 4)), for which of the following values of x is y NOT defined?

Bottom line:  Although the problem includes an equation for y, you need an x value.  Your answer will be an x value that does something specific to this equation to produce a y that is not defined.  So make a note: x = ?

Assess your options:  You could work backwards and plug in answer choices to find a value that produces a y that is not defined.  This might require you to work the problem numerous times.  Instead, think about your knowledge of number properties.

Attack the problem: Any time a number is divided by zero, it is not defined.  If y is not defined, then it must be equal to something over zero.   Take the bottom part of your fraction and set it equal to zero: (x + 3)(x – 4) = 0.  For which values is this true?  Well, anytime zero is multiplied by a number, the answer is zero.  If either of these binomials is equal to zero, then there will be a zero on the bottom.  So set each binomial equal to zero: x + 3 = 0 and x – 4 = 0.  When you solve both of these equations, you will get two answers: x = -3 and 4.  Both answers will create a zero in the bottom of your fraction.

Loop back:  You found two answers for x that will create a zero on the bottom of your fraction.  Look down at your answer choices to see which one is present.

(A) -4
(B) -3
(C) -1
(D) 2
(E) 3

The correct answer is (B).


On sat.collegeboard.org, 55% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Algebra Equations

Read the following SAT test question and then select the correct answer. 

Use the same method for all SAT math questions.  Read the question carefully, identify the bottom line, and assess your options for solving the problem.  Choose the most efficient way to solve the problem, and attack it!  Finally, loop back to make sure that you solved for the bottom line.


If s = 1 + (1 over 2) + (1 over 4) + (1 over 8) + (1 over 16) + (1 over 32) and t =1 + (1 over 2) times s, then t exceeds s by

Bottom line:  You need to know how much t exceeds s.  So the question you are asked is really, “How much bigger is t than s?”  Your bottom line is t – s.

Assess your options:  Normally you would simplify s before plugging the s value into the equation for t, then find the difference between t and s.  However, doing this will give you some ugly fractions and take a lot of time.  Instead, try starting with your bottom line and plugging in everything that you know.

Attack the problem:  Start with the t – s and plug in the equations for both of these:



Then plug in s one more time:



This looks terribly ugly, but keep calm and use the order of operations.  You always multiply or divide before you add or subtract, so your first job is to distribute the half within the brackets by multiplying it by every number within the first set of parentheses:



Look at the new equation that you have.  How many things cancel out when you start subtracting the second group of numbers from the first?  Almost everything!  You are left with a single fraction:



Loop Back:  You solved for t - s, your bottom line, so you are ready to look at the answer choices.


(A)1 over 4
(B)1 over 8
(C)1 over 16
(D)1 over 32
(E)1 over 64

The correct answer is (E).


On sat.collegeboard.org, 41% of the responses were correct.

For more help with math questions, visit www.myknowsys.com!

Number Line

Read the following SAT test question and then select the correct answer. 

Use the Knowsys Method for every question on the SAT.  The math method is always the same.  Read the question carefully, identify the bottom line, assess your options for solving the problem, select the most efficient method for solving the problem, and attack the problem.  Once you have finished your work, loop back to make sure that you solved for the bottom line.  Many problems have multiple steps, so you want to be sure that you are answering the question that was asked!

math image

Which of the following statements must be true of the lengths of the segments on line m above?

I.  AB + CD = AD
II.  AB + BC = AD – CD
III.  AC – AB = AD – CD

Bottom Line:  You must find out which of the statements above must be true.  You will need to mark each one true or false in order to find the correct answer.

Assess your Options.  You could plug in numbers for these spaces, but you are given no numerical information about the line.  You run the risk of finding answers that can be true rather than answers that must be true if you use this method.  Instead, keep the information abstract and use the line to evaluate the equations that you are given in order to see which ones work.

Attack the Problem:

I.  You are given AB + CD = AD.  You can tell from the line what AD must be equal to if you add up all the parts within AD.  From the line you can see that AD = AB + BC + CD.  Now plug that information into the given equation for AD.  Your new equation is AB + CD = AB + BC + CD.  One side has no BC while the other side has a BC.  You know that BC cannot be equal to zero because it is allotted a certain measure of space on the line.  This equation is not true.

II.  You are given AB + BC = AD – CD.  Look up at the line and see which parts of the line are still included if you take AD and subtract CD.  You are left with AB and BC.  Your new equation is AB + BC = AB + BC.  If you cannot get that information from looking at the line, think about this problem a little differently.  Substitute what you know about AD into the problem.  AD = AB + BC + CD.  Plug that in and your given equation is AB + BC = AB + BC + CD – CD.  The CD will cancel when you subtract it from itself, and you are left with AB + BC = AB + BC.  Will that always be true?  Yes, this equation is true.

III.  You are given AC – AB = AD – CD.  Look up at the line.  If you consider all of AC and then subtract out AB, what are you left with?  BC.  If you consider all of AD and then take out CD, what remains?  AB + BC.  Will BC = AB + BC?  Again, a space on a line cannot be equal to zero, so this equation is false. 

Loop Back:  You broke down each of these equations and determined whether they were true or false, so look down at your answer choices.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

 The correct answer is (B).


On sat.collegeboard.org, 55% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Algebra Equations: Translate

Link of the Day

A recent SAT Question of the Day involved the architect Frank Lloyd Wright.  On this date in history, one of Wright’s most famous buildings opened to the public.  The Guggenheim Museum opened its doors in 1959 and is now one of the wealthiest museums devoted to Modern art in the world.  If you are at all interested in art or architecture, consider using the Guggenheim opening as one of your excellent historical examples for the SAT essay.  You can find more information about the museum opening here and see photos from the event here.

10/21 Algebra

Read the following SAT test question and then select the correct answer. 

Use the same method for every math question on the SAT.  Read the question carefully, identify the bottom line, and assess your options for solving the problem.  Once you have identified an efficient method to solve the problem, attack it!  Before you choose an answer, loop back to verify that the answer addresses the bottom line.

If x + 2x is 5 more than y + 2y, then x – y =  

Bottom Line: x – y = ?

Assess your Options:  It would not be easy to work backwards and plug in answer choices for this problem.  Instead, translate the written words into a mathematical equation and solve for the bottom line.

Attack the problem:  Identify the terms in the original sentence that easily translate from English into math terms.  The word “is” translates to “equals,” and you know that if you need “more than” the original, you will be adding the specific number.  You can now write the information that you are given as a single equation: x + 2x = y + 2y + 5.  You can simplify this equation by combining like terms: 3x = 3y + 5.  With many other problems, you would want to solve the problem by isolating a variable.  However, you are only working with one equation, so you will need to solve for the bottom line and not a single variable.  Notice that your bottom line includes a positive x and a negative y on the same side of the equation.  You can rearrange your equation to create this: 3x – 3y = 5.  Now notice that the x and the y both have a three in front of the variable.  Factor out that three so that you have 3(x  y) = 5.  If you divide both sides by 3, you will find that x – y 5 over 3.

Loop back You solved for your bottom line, so you are ready to look at the answer choices.

(A) -5
(B)negative (5 over 3)
(C)3 over 5
(D)5 over 3
(E) 5

The correct answer is (D).


On sat.collegeboard.org, 44% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Functions

Link of the Day

Current events that make excellent examples for SAT essays often hold two ideas in tension.  Read this article about a Turkish pianist who is being prosecuted for irreverent tweets.  Think about the conflicting themes in this article.  One side argues for respect while the other side argues for freedom of expression.  These are broad themes that apply to people around the world, so they are likely to show up on the SAT.  Two previous SAT essay prompts are "Should heroes be defined as those who say what they think when we ourselves lack the courage to say it?" and "Is it important to question the ideas and decisions of people in positions of authority?"  Think also about the modern use of technology.  Which facts from this article would you need to memorize if you wanted to relate this current event to an SAT essay prompt?

10/18 Functions

Read the following SAT test question and select the correct answer.

For every SAT math problem, read the problem carefully so that you know exactly what information you are given.  Then identify the bottom line, the information that you must find.  Assess your options for solving the problem, and choose the most efficient method to get to the answer.  Attack the problem to find the answer, and loop back to your bottom line to make sure that your answer matches what you were supposed to find.

A manager estimates that if the company charges p dollars for their new product, where 0 ≤ p ≤ 200, then the revenue from the product will be r(p) = 2,000p – 10p² dollars each week. According to this model, for which of the following values of p would the company’s weekly revenue for the product be the greatest?

Bottom Line:  Which of the following values of p will result in the greatest revenue?

Assess your options:  You could work backwards by plugging in all of the answer choices to r(p) = 2,000p – 10p², but that will take time.  Instead, use what you know about functions to determine the answer.

Attack the problem:  You know what the graph of x² looks like: a parabola that makes a “u.”  What happens to that graph when it is -x²?  That “u” turns upside-down and the parabola looks like a hill.  That is what you have for your function r(p) = 2,000p – 10p².  Now simplify your function by pulling out the numbers and variables that your two terms have in common so that r(p) = 10p(200 – p).  If you set each part of this equation equal to zero, you will find where the parabola crosses the x-axis.  If 10p = 0 and 200 – p = 0, then p = 0 and 200.  The parabola crosses the x-axis at 0 and 200.  That makes sense because you were told in the problem that 0 ≤ p ≤ 200.  Think about the characteristics of parabolas once more.  All parabolas are symmetrical.  Where will your greatest value for the revenue be?  It will be at the top of that “hill” exactly between 0 and 200.  What is the midpoint between 0 and 200?  100.

Loop back: Your bottom line was the value of p that would have the greatest revenue.  Although your function used r(p) rather than f(x),  that p value had to be on the x-axis.  You solved for the bottom line, so you are ready to look down at the answer choices. 

(A)  10
(B)  20
(C)  50
(D)  100
(E)  200

The correct answer is (D).


On sat.collegeboard.org, 37% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Algebra Equations: Substitution

Read the following SAT test question and then select the correct answer. 

For every SAT math problem, read the problem carefully so that you know exactly what information you are given.  Then identify the bottom line, the information that you must find.  Assess your options for solving the problem, and choose the most efficient method to get to the answer.  Attack the problem to find the answer, and loop back to your bottom line to make sure that your answer matches what you were supposed to find.

If   and x = 12, then x – y =

Bottom Linex – y = ?

Assess your options:  You could try to work backwards from the answer choices, but that would require you to know the values for and y.  Instead, use arithmetic to solve for and y, then plug those numbers into the last equation to find the bottom line.

Attack the problem: You are given the fact that x = 12.  You can create another equation to solve for x: if you multiply both sides of the first equation by y, you have = 3y. Now plug in your x value and you will have 12 = 3y. When you divide both sides by 3, you will find that 4 = y.  Return to the last equation and plug in your values for x and yx – y becomes 12 – 4 = 8.

Loop back: The last equation was your bottom line, so you are ready to look at the answer choices.

(A)  3

(B)  5

(C)  6

(D)  8

(E)  9

The correct answer is (D).

On sat.collegeboard.org, 78% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Fractions

Arithmetic: Fractions

Read the following SAT test question and then select the correct answer. 

Read each question carefully.  Then identify the bottom line and assess your options for finding it.  Choose the most efficient method to attack the problem.  Before selecting your answer, loop back to make sure that you solved for the bottom line.

If  then N = 

Bottom Line:  N = ?

Assess your options: Normally, you would begin working this problem by multiplying the two fractions on the right and then multiplying them by the reciprocal of the fraction on the left in order to find N.  Before you jump into the problem, think about the properties of multiplication and you will see that there is a much faster way to solve the problem.

Attack the problem:  The commutative property of multiplication tells you that order is not important when you are multiplying; 3 × 5 = 5 × 3.  If you rearrange your equation, you will see that  .   When you see the same thing, such as a fraction with 5 over 14, on both sides of the equation, you know that you can ignore that information.  No matter what number or variable you have, if it is the same on both sides of the equation, you will eliminate one side when you eliminate the other.  You can check this fact by multiplying both sides by the reciprocal of .  If you multiply both sides by , the  will cancel on each side and you are left with N = .


Loop back:  You solved for your bottom line, N, so you should look down at your answer choices.

(A) 
(B) 

(C) 5

(D) 7`

(E) 14

The correct answer is (A).


On sat.collegeboard.org, 82% of the responses were correct.

For more help with math, visit 
www.myknowsys.com!


Algebra

Link of the Day

One of the previous essay questions asked on the SAT reads, “Is there always another explanation or another point of view?”  Rather than talking loosely about your own opinion, use a specific example from current events to back up your opinion.  For example, read this article about an incident that occurred in a London art gallery.  This current event could serve as an excellent example for other broad SAT questions as well.  Can you find themes in this article that could relate it to other questions about feelings, originality, creativity, change, authority, respect, and courage?  Which facts would you need to memorize to be able to write knowledgeably about this current event?

10/9 Algebra

Read the following SAT test question and then select the correct answer. 

Read each question carefully.  Then identify the bottom line and assess your options for finding it.  Choose the most efficient method to attack the problem.  Before selecting your answer, loop back to make sure that you solved for the bottom line; you do not want to give the answer for x when your problem asks for something else entirely!

If x + y = 3 and x – y = 5, then x² - y² =

Bottom Linex² - y² = ?

Assess your options: For this problem, noting your bottom line can save you a lot of time!  Most people are going to solve for x and y, but your bottom line is really x² - y².  You do not need to know what x and y equal to solve this problem.  If you solve for your bottom line rather than solving for x and y, you can save time on this problem and have more time for other problems during your SAT test.

Attack the problemx² - y² is one of three key quadratic equations that you should have memorized.  It shows up on your Knowsys Formula chart as a² - b² = (a + b)(a – b).  If you have this memorized, you know that x² - y² = (x + y)(x – y).  You have already been given the values for both (x + y) and (x – y) in the question!  All you have to do is plug them in.  Then you have x² - y² = (x + y)(x – y) = (3)(5).  Your answer is 15. 

Loop back:  This is the process that you just used:  x² - y² = (x + y)(x – y) = (3)(5) = 15.  You solved for the bottom line, so you are ready to look down at your answer choices.

(A) 4
(B) 8
(C) 15
(D) 16
(E) 64

The correct answer is (C).


On 
sat.collegeboard.org, 62% of the responses were correct.

For more help with math, visit 
www.myknowsys.com!

Number Line

Read the following SAT test question and then select the correct answer. 


For all math problems, read the problem carefully.  Identify your bottom line, and assess your options for reaching that bottom line.  Select the most efficient method to work the problem, and attack the problem.  Your last step is to loop back and make sure that the answer addresses the bottom line.

math image
On the line above, if AB < BC < CD < DE, which of the following must be true?

(A) AC < CD
(B) AC < CE
(C) AD < CE
(D) AD < DE
(E) BD < DE

You must decide which of the answer choices is true.  This is a “which of the following” question, and it would be really difficult to predict an answer choice, so start with answer choice (E).

(E) BD < DE

Normally you cannot depend on a drawing that has the words "Note: Figure not drawn to scale" underneath it, but this particular line follows the rule that each line segment is longer than the last.  Once you have ascertained that the image matches the information that you have been given, you can use the image to draw conclusions.  You can look at the line provided and see that this does not have to be true:  BD actually looks longer than DE.  You can also think of BD as BC + CD.  You don’t have any information to compare BC + CD and find out whether it is less than DE.  One way of proving this is to imagine that AB is 1, BC is 2, CD is 3 and DE is 4.  That fits AB < BC < CD < DE.  However, 2 + 3 is not less than 4.  Eliminate (E).

(D) AD < DE

Look back at the line.  You can clearly see that AD is longer than DE.  Eliminate (D).

(C) AD < CE

This one is not obvious from a glance at the line.  Think of AD as AB + BC + CD and CE as CD + DE.  Both equations share CD, but do we know that AB + BC is smaller than DE?  No.  Think about what would happen if AB had a high value.  Suppose that AB = 10, BC = 11, CD = 12 and DE = 13.  In that case AD would be 33 while CE would only be 25, and this answer choice would be false.  Eliminate (C).

(B) AC < CE

Think of AC as AB + BC and CE as CD + DE.  Compare the two.  You know that AB must be smaller than CE and BC must be smaller than DE.  What you really have is: small number + small number < big number + big number.  Is that always true?  Yes.  You do not have to check the last answer choice.

The correct answer is (B).


On sat.collegeboard.org, 70% of the responses were correct.

For more help with math, visit 
www.myknowsys.com!

Roots and Radicals

Read the following SAT test question and then select the correct answer.

Read the question carefully to insure that you understand all of the information that you are given.  Then make a note of the bottom line so that you will be sure to solve for the correct information.  Assess your options for solving the problem and choose the most efficient method.  Attack the problem, solve it, and loop back to verify that your answer matches the bottom line.

If  , what is the value of ?

Bottom Line: You must find what  is equal to.

Assess your Options:  You could solve for x.  However, that is actually the long way to do this problem.  You are taking a timed test, so the long way is the wrong way!  Your bottom line is , so you can use what you know about radicals to solve the problem without solving for x.

Attack the Problem:  Look first at .  This is actually the same as  times 
.  Finding the square root of 4 is easy: .  Now you have .  Look back at the problem.  You are already given the square root of x!.  What is 2 times 16?  The answer is 32.  This is the process that you just followed:

Loop Back: Did you find the bottom line? Yes; .  Look down at your answer choices.

(A) 16
(B) 32
(C) 64
(D) 128
(E) 256

The correct answer is (B).


On sat.collegeboard.org, 36% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Exponents

Link of the Day

On September 30, 1868, Louisa May Alcott published the first volume of Little Women.  The story was extremely successful and has been beloved by readers ever since.  Louisa May Alcott would make an excellent historical figure to use as an example on your SAT writing section.  You can read more about her life here, and if you have read Little Women, remember that it could make an excellent literary example too!

9/30 Exponents

Read the following SAT test question and then select the correct answer. 

Math questions need to be read just as carefully as reading questions. Avoid incomplete answers by making a note of the bottom line. Are you solving for x, or do you need the answer to 2x + 3? Assess your options for solving the problem, choose the most efficient method, and attack the problem! Once you have the answer, loop back to verify that it addresses the bottom line.

If , which of the following expresses a in terms of b?

Bottom line: This question asks you to solve for the variable a.

Assess your options: You could try to plug in the answer choices for a and choose a number for b to try to find the answer. However, that method requires you to work several problems and includes multiple steps. Instead, use algebra to rearrange the equation.

Attack the problem: You see two numbers with exponents. When two bases are the same, then the exponents can be set equal to each other. Your two bases are 2 and 4. How can you make both bases the same? Use the fact that 2² = 4 by plugging that into your equation.





Now you can ignore the bases and set the exponents equal to each other. You now have:

a = 2b

Loop back: The question asked you to find a in terms of b, and that is just what you did. Look down at your answer choices


(A)

(B) b

(C) 2b

(D) 4b
(E)


The correct answer is (C).


On sat.collegeboard.org, 54% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Graphs

Link of the Day

Your SAT essay should include a current event as one of your excellent examples to show that you are thinking critically about reality as you respond to the prompt. Sometimes "an event" can simply be the way that you use technology. Today Google is celebrating its 14th birthday. Does that seem bizarre to you? Read this article and think about how you could relate the current uses of Google to SAT writing questions. Previous questions have concerned the themes of change (whether it is for the better), success, mottos and motivation, discovery, questioning ideas, the overabundance of knowledge, planning, and creativity. If you select Google as one of the current events you want to write about, be sure to copy a few facts from this article and review them before you take the test.

9/27 Graphs

Read the following SAT test question and then select the correct answer.

When a question includes a graph, it is especially important to read both the text under the graph and the labels on the graph. Identify the bottom line and assess your options for reaching it.  Ask yourself, "What could I do?" and then "What should I do?"  Once you have selected an efficient method to solve the problem, attack the problem!  Loop back to make sure that your answer addresses the bottom line.

graphic

The histogram above shows the distribution of 31 black cherry trees, by height. For example, the leftmost bar represents the black cherry trees that are at least 60 feet, but not more than 65 feet, in height. Based on the histogram, which of the following can be the average (arithmetic mean) height of the 31 black cherry trees?


Your bottom line is the average height of 31 trees, not the exact average, but what it could be.  This histogram does not tell you the exact height of any of the trees, so how can you find their average heights?  Look at that first  bar.  There are three trees that must be between 60 and 65 feet in height.  If you assume that all of those trees are as short as possible (60 feet), you will find the lowest value that their  average could possibly be.  Find the lowest height that all of the trees could possibly be and then average those heights together.




The lowest possible average for the heights of these trees is 72.74.  Any answer lower than this will be wrong.  Now, you could go back into your equation and plug in the highest possible value for each tree and average them again, but that will take a lot of time to retype into your calculator.  Instead, you should think logically about the height of the trees.  If you use the highest height that any tree can be, you are adding 5 to every single tree on the chart.  That means that your final average will be 5 feet higher than your current average.




You now have the highest and lowest possible averages of the heights of the trees.  Since your bottom line asks which of the following answers could be the average, you must eliminate any answers that are not between 72.74 and 77.74.


(A) 70 feet

(B) 72 feet
(C) 74 feet
(D) 78 feet
(E) 80 feet

The correct answer is (C).



On sat.collegeboard.org, 49% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Number Properties

Arithmetic: Number Properties

Read the following SAT test question and then select the correct answer.

Every time you work a math problem, read the problem carefully.  Identify the bottom line and think about the most efficient method to solve for the bottom line.  Choose a method to solve the problem and attack the problem without hesitation.  When you think you have the answer, loop back to make sure that the answer addresses the bottom line because questions often require multiple steps to get to the answer.

When the positive integer n is divided by 5, the remainder is 0. What is the remainder when 3n is divided by 5?

Make a note that your bottom line is the remainder of 3n.  Then think carefully about the first portion of information that you are given.  Some number, n, is divided by 5 and there is no remainder.  That means that n must be a multiple of 5.  If you do not immediately see this, think about a concrete number that will not result in a remainder:

 or    or  
All of these values for n result in a whole number with no remainder.

If n is a multiple of 5, what will 3n be?  It will still be a multiple of 5!  It will still result in a remainder of 0.  If you cannot see this, look back at the examples above using 5 and 15.  If 5 is your n value, 3 times 5 is 15, and when you divide 15 by 5 the answer is 3 with a remainder of 0.  Now that you have found your bottom line, look down at your answer choices.  

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

The correct answer is (A).


On sat.collegeboard.org, 70% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Parabolas

Read the following SAT test question and then select the correct answer. 

Don’t let this question intimidate you just because it has a parabola.  Use the same method that you would use with any other math problem.  Read the question carefully, identify the bottom line, and choose an efficient method to solve the problem.  Then attack the problem and loop back to make sure that you solved for the bottom line.

math image

The quadratic function f is graphed in the xy-plane above. If f(x) ≤ u for all values of x, which of the following could be the coordinates of point P?

Your bottom line is which values could be the coordinates of point P, so make a note of the bottom line on your paper, and start with what you know about this point.  You are told that f(x) ≤ u for all the values of x.  That is your y value, so that is just letting you know that nothing can be higher than u, which is on point P.  If you are looking for the highest point on a downward opening parabola, what are you actually looking for?  The vertex!


Think about it this way: as the parabola extends outward from the vertex, both sides stay an equal distance from the vertex. You have just examined the information given about the y-axis, so turn your attention to the x-axis.  You are given two x values that are of equal height on your parabola, so the x value of the vertex, P, must be exactly between them.  Your highest value is 4, so you might be tempted to halve 4 and get 2.  Just be sure to remember that the first point is not at zero, but at 1.  That means that your parabola has been shifted 1 unit to the right.  To find the midpoint, use the midpoint formula, which is simply an average of the two numbers that you have. 

 

You now have the x value of 2.5. You are not given any additional information about the limits of the y-axis, so loop back to the bottom line.  The question is not actually asking you to find both x and y coordinates.  Remember that your bottom line is what “could” be the coordinates, so this is probably enough information to find the correct answer.  Look down at your answer choices now.

(A) (2, 3.5)
(B) (2.25, 3.25)
(C) (2.5, 3)
(D) (2.75, 4)
(E) (3, 2.5)

All you know about the y value is that it must be greater than 0, so all of the y values will work, but only one of the answers has the x value of 2.5. 

The correct answer is (C).


On sat.collegeboard.org, 53% of the responses were correct.

For more help with math, visit www.myknowsys.com!

Circles

Link of the Day

Would you like a break from school right around now?  Students in Chicago are getting just that.  They have had over a week without school as teachers strike over conflicts with the mayor concerning the best methods to improve underperforming schools.  Take a look at this current event and think about how you could use details from this story in an excellent example for your essay that shows SAT graders that you are well informed.  Make a list of all of the broad themes that this story illustrates, such as change, education, other points of view, adversity, success (Can it be disastrous?), how to question those in authority, and many many more. 

9/18 Circles

Read the following SAT test question and then select the correct answer. 

 math image

The circle above has center P. Given segments of the following lengths, which is the length of the longest one that can be placed entirely inside this circle?

Your bottom line is the longest length that can be placed inside a circle.  Before you look at any of the numbers in this particular problem, think about circles in general.  You know that a line segment, within the circle, with both end points on the circle is called a chord.  The longest chord will always be the diameter of the circle.

You have enough information to find the diameter of this circle.  The line labeled 4 is actually the radius of the circle.  A radius is half of the diameter, so multiply the radius by 2 in order to find the diameter: 2 × 4 = 8. 

This is the part of the problem where you loop back and see whether your answer matches the bottom line.  You found the diameter of the circle, but that is not actually the bottom line.  A diameter is a chord that touches the circle’s edges.  You were asked to find the longest length that can be placed entirely in the circle.  In order for the length you found to not touch the circle, it must be slightly smaller than 8, perhaps 7.9999999.  Now look down at your answer choices.

(A) 6.99
(B) 7.00
(C) 7.99
(D) 8.10
(E) 14.00

The correct answer is (C).  
Notice that you did not need to use all of the information that you were given in this problem.  Always take the time to read the question carefully so that you will not be confused or distracted by extra information, such as the chord labeled 7 in this problem.


On sat.collegeboard.org, 62% of the responses were correct.

For more help with writing, visit www.myknowsys.com!