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SAT Math: What to Expect

By posted February 7, 2012

Get to know the structure of SAT Math before test day so there are no surprises!  It’s a great idea to know the directions ahead of time so you can get started answering questions right away on the real thing.

Timing: SAT Math is comprised of two 25 minute sections with approximately 20 questions and one 20 minute section with 16 questions.  Some questions will be strictly computational and take only seconds to complete; others could take up to a few minutes.  If you are totally baffled by a question or have been working on it for too long, move on to an easier question and come back to it at the end if you have time.  Each question is worth the same amount, so don’t waste your time on one question you might not even answer correctly.

Try this SAT math practice question and test your quantitative skills!

Format: There are two question type formats, multiple choice and student-produced response.  For multiple choice you will select one out of five answers.  For student-produced response, you will not be given any choices and will have to write your answer in a grid on the answer sheet, as well as fill in the corresponding bubbles.

Content Overview: The math section covers arithmetic, algebra, geometry, statistics, and probability.  So don’t sweat it if you never made it to calculus!  You will be given some basic formulas to refer to, such as the area of a triangle and the volume of a cylinder.  Refer to the collegeboard.com website to see exactly which formulas appear on the test so you don’t have to worry about memorizing them.

SAT Math has the least variety of question formats for you to learn, so you can focus on brushing up on math concepts, practicing solving problems, and learning relevant strategies.

Find out how you can get customized tutoring on SAT to hone your skills with a Grockit tutor.

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The SAT Essay: What to Expect

By posted February 1, 2012

The thought of writing an essay before delving into three hours of multiple-choice testing might send shudders down your spine, but the more you know about the SAT essay, the less daunting it will seem.  Read on to learn all the basics of the SAT essay.

Timing: The essay portion of the SAT comes first on the test.  You will have 25 minutes to read the prompt, decide on your viewpoint, brainstorm, outline, write, and proofread your essay.  Whew, that’s a lot in such a short time!  Obviously steps such as outlining are going to be very condensed versions of what you would do with a take-home essay for school, and some steps you may have to skip altogether.  Definitely write timed practiced essays at home before the big day so you’re prepared for what a time crunch it can be.

Format: You will be given a short paragraph relating to the prompt, usually a quotation from a historical figure, literature, etc.  Don’t ignore this information!  It can give you valuable ideas for your essay.  This will be followed by the prompt itself, which will ask you to formulate a point of view on an issue and support that viewpoint with examples and analysis.  You will be writing your essay on the lined pages provided.

Content Overview: Read through old SAT essay prompts to get an idea of the type of topics the test makers typically use.  You can find the most recent ones at collegeboard.com.  You will find a common thread through the prompts of “life’s big questions,” covering everything from ambition to honesty.  Every prompt will tell you to use examples from “your reading, studies, experience, or observations.”  Go into the essay armed with several examples from these areas that you feel comfortable writing about to support a thesis.

Now that you know the basics of the SAT essay, start writing!  Find out how Grockit’s expert tutors can help you to critique your practice essays so that you can learn from your mistakes.

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Fractions & Rational Numbers on the SAT

By posted October 17, 2011

Fractions are one of the important building blocks of SAT Math. All fractions are rational numbers. A rational number is a number that can be expressed as a ratio of two integers. Therefore all integers are also rational numbers. Rational numbers either have no decimal (4/1), or have a terminating or repeating decimal (1/4 = .25, 1/3 = .3333).

Try this SAT math question for practice!

Equivalent fractions are fractions that simplify to the same form. For example, 6/8, and ¾. In a fraction, when you multiply the numerator and the denominator by the same non-zero integer, it’s like multiplying the entire fraction by 1. The value of the fraction doesn’t change.

To simplify fractions, you need to understand the LCM, lowest common multiple, and the LCD, least common denominator. The LCD is the LCM (the smallest number that is also a multiple of both integers). For example, if we were to solve: 2/3 + ¼  – 1/2. The LCD is going to be the smallest number that 3, 4, and 2 divide into evenly. Since 2 is already a factor of 4, we can choose 12. Since 3, 4, and 2 will all evenly divide into 12.

When you multiply fractions, you can multiply the numerators straight across, then the denominators straight across. Before you multiply, you can also cancel out numbers across the numerators and denominators. For example: 1/5 * 5/6 * 12/2 * 1/3 = ?. One way to start if by crossing out the 5 in the denominator of the first fraction, and the 5 in the numerator of the second fraction. That gives us:

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SAT Practice: How to Find the Area of a Polygon on the SAT

By posted September 19, 2011

Polygons can be a confusing concept for SAT students with even a strong grasp of Geometry because they don’t appear as often as triangles, circles, and other geometric shapes. You’re likely to find a polygon question towards the end of a SAT Math section as it’s considered a more challenging concept. Let’s review a few basic properties and then learn how to calculate the area for one!

Find out how you can Collaborate with your peers to reinforce your knowledge and theirs on Grockit today!

A polygon is a many-sided closed figure whose sides are straight lines. A regular polygon has sides of equal length.

Every time you add a side to a polygon, you add 180 degrees to the sum of its interior angles. That is why a triangle has a sum of 180 (3 sides), a square has a sum of 360 (4 sides), and so on.

The area of a square is side x side. The area of a rectangle is length x width. But what about for shapes with more than 4 sides?

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Multiple Choice on the SAT: How to Use the Answer Choices

By posted September 6, 2011

Contrary to what you may think, the answer choices on the SAT are not designed to trap or trick you! We may describe them as “tricky” or “tempting” but seeing the answer choices can actually be a huge asset on Test Day. After all, we know that 1 of the 5 must be correct! Unlike grid-ins, we can often utilize the answer choices in problem solving questions to help us find the solution and get better SAT scores overall. Here are 5 ways you can make the answer choices work for you on your SAT Test!

For Writing question, remember that “No Error” is a strong option. For Identifying Sentence Error test questions, parts of a sentence are underlined. For this SAT question, Choice (E) is “No Error.” There is not always going to be an error. In fact, about 1/4th of the time, “No Error” is correct! Trust your instincts. If a sentence “sounds” okay, and you’ve checked the other four choices and found no grammatical mistake, the SAT alternative is choice E.

Eliminate (-) or (+) choices in Sentence Completions using your prediction. On test day, don’t simply read the sentence and plug in the answer choices, re-reading the SAT sentence five times. Identify the keywords that relate to the blank and write in your OWN word for the blanks, or at least predict whether the blank is a (+) or (-) word. Ace these tricky SAT questions by then eliminating answer choices that contain words with the opposite word charge!

Find out how Grockit can predict your performance on the SAT

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SAT Math Practice: Data Interpretation

By posted June 30, 2011

Interpreting data on the SAT may come in many forms: charts, graphs, tables, or extrapolating information from a reading passage. Mastering all the different ways to interpret data will be an important part of scoring well on the SAT. Only by getting enough test prep, including spending some time in Grockit’s interactive games with trained instructors, will ensure you get the score you want. Make sure to remember the following tips and strategies when faced with a data interpretation question. Then, use them to solve the sample problem below.

Find out how Grockit can predict your performance on the SAT.

Tips and Strategies for data interpretation questions:

  1. Read the passage before looking at any questions. Reading the question(s) first may mislead you when you’re examining the passage. Although it may seem like you’re targeting your reading, the chances you’ll make a mistake are much greater if you read the question(s) first.
  2. Carefully examine any chart, graph or table. Make sure you know what information the chart, graph or table is relaying. Carefully examine the title and what the x- and y-axes represent. And try to analyze the information in the chart, graph or table.
  3. Don’t be afraid to take notes and/or write out the math. You should never try to compute a data interpretation question without writing down the figures. Although it may take a few seconds longer, the chances you’ll correctly answer the question are much greater if you write out the math.
  4. Look for patterns. If numbers and figures come easily to you, look to see if you can see a pattern in the data. But even if you think you know the answer because there’s a pattern, do the math just to be sure.

Now, keep these tips and strategies in mind as you examine the following example:

What is the average (arithmetic mean) height, in inches, of the 5 students’ mothers?

  1. 60
  2. 62.5
  3. 65
  4. 70
  5. 72

After you examine the graph carefully, you should notice that the y-axis has the height of each student’s mother in inches. That means you will only refer to the information on the y-axis, as the x-axis (height of father) is not relevant to this question. With this in mind, we see that two mothers are 60 inches tall, one is 65 inches tall, and the other two are 70 inches tall. To find the average, we need to add all their heights together and then divide by 5, the total number of mothers. Let’s do the math:

60 + 60 + 65 + 70 + 70 = 325 (all heights together)

325 ÷ 5 = 65 (average height—or arithmetic mean—of the mothers)

Now that we’ve done the math, we can see that C is the right answer!

(There’s another, faster way to solve this problem. If you’re good at noticing patterns, you would’ve seen that the average of the two mothers at 60 inches tall and the other two mothers at 70 inches tall would’ve been 65 inches. This is the same height as the other remaining mother, meaning the average height for all the mothers would be 65. Looking for patterns like this can save you time when completing data interpretation questions, but if you’re stuck, just do the math to get the correct answer.)

Data interpretation questions on the SAT require you to read and examine any charts, graphs or tables closely. Getting enough test prep, which you can get in Grockit’s interactive games, will ensure you’re ready for any data interpretation questions that come your way on test day.

Grockit can target your study plan to improve your SAT score

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Elementary Probability

By posted May 31, 2010

Probability questions on the SAT basically test if you know one concept – the definition of probability.

Probability = Number of times a certain event might occur/Total number of events that might occur

So if a question tells you that there are 120 dorm rooms and 24 of them are painted yellow, the probability that a freshman is placed in a yellow dorm room is 24/120 = 1/5 because there are 24 yellow rooms that the freshman could be placed in, out of a total of 120 rooms.

Reverse probability

To make things difficult, the question might ask you a reverse probability question.  For example, if there are 120 dorm rooms and Amy has a  chance of 1/6 being placed in a blue room, then how many blue rooms are there?  Using the formula above, if there is a one in six chance of getting a blue room, that means that the number of blue rooms out of the total number of rooms is 1/6.  Since there are 120 rooms in total, then 1/6 = blue rooms / 120.  This works out to 20 blue rooms.

Probability of an event not happening

Another type of question is one like this.  Suppose there are only blue and red marbles in a bag.  Let the number of blue marbles be b and the number of red marbles be r. If the probability of drawing a blue marble is 5/7, what is the value of b/r?  The only number you have to work with is 5/7.  This number represents the odds of picking a blue marble, meaning that 5/7 = number of blue marbles / total number of marbles.  You can thus let there be 5 blue marbles and 7 marbles in total.  This means that there are 2 red marbles, so the value b/r is 5/2.

This type of question is essentially testing if you know how to find the probability of an event not happening.  If the probability of getting a blue marble is 5/7, that means that the probability of not getting a blue marble (i.e. getting a red marble) is 1 – 5/7 or 2/7.

Probability of multiple events

The most difficult type of probability question on the SAT generally involves 2 dice.  Remember the question tells you that Joe rolls a pair of dice and forms a fraction x/y where x represents the number rolled on the first die and y represents the number rolled on the second die?  It then asks for the probability that this fraction equals 1.  To work out this problem, you need to know the total number of dice rolls that could occur.  Each dice can roll 6 different numbers, which means that with 2 dice, there are 6 x 6 = 36 possible combinations.  Of those 36 combinations, there are 6 ways of forming a fraction that equals 1.  If a 1 and 1, or a 2 and 2, or a 3 and 3, or a 4 and 4, or a 5 and 5 or a 6 and 6 is rolled.  In each of these cases, the fraction will equal 1.

So since there are 6 cases that give us the value we need, out of a possible 36, the probability is 6/36 = 1/6

Don’t forget that the probability of an event can never exceed 1.  If the probability of an event is 1, that means that it is certain to happen.  If the probability of an event is 0, that means that there is no chance of it ever happening.  Probability values never go higher than 1 or lower than 0, so if you find yourself with such an answer, double check to see what went wrong.

Check out Grockit for more probability practice!

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Number Lines

By posted April 29, 2010

There are typically two types of number line problems:

  • questions that show you an actual number line that require you to estimate and know properties of positive/negative numbers and decimals
  • and questions that require you to translate a word problem into a number line.

When a question gives you a drawing of a number line, always look out for which points are between -1 and 1.  Numbers whose absolute values are less than 1 are points B, C, D, E, F, G in the picture below.  An important property to note is that when these points are multiplied with another point that is greater than 1, it makes that number smaller.  Try it!  Suppose G is 0.5 and H is 1.5.  GxH is 0.75 which is smaller than H.

Another important property to note is what happens when you multiply two negative points together or when you multiply a negative and a positive point.  Two negatives make a positive, meaning BxC gives you a positive number – either E, F or G because those points are positive and less than 1.  A negative and a positive makes a negative.  So BxG would either be point C or D.  If you estimate what the values of those points are, you would be able to say with greater certainty too that BxG has to be point D.

Here’s another number line question where the line is given.

If there are 8 equal intervals between 0 and 1. What is the value of x?  I would start counting the number of points until I get to the one labeled √x and realize that it is the point 6/8 which is also 3/4 .  Since √x – 3/4 that means that x – 9/16.

The other type of question requires you to translate a word problem into a number line.  Let’s start off with an easy word problem.

The number m – 4 is how much less than the number m + 5?

I would draw a number line like the one above.  And let m equal a number.  Suppose m=0.  That means that m-4 = -4 and m+5=5.  If I circle -4 and 5 on the number line and count the spaces in between, I would realize that they are 9 units apart.

Here’s a longer question.  An important thing to note about “number lines” is that it generally refers to the x-axis.  So if points A and C are located on a number line such that AC=6 that means you can draw and x-y graph and put A and C as two points on the x-axis, 6 units apart.  If I then tell you that point E is also on the x-y plane and located so that AE=3, where could you put E?   You could put E on the x-axis between A and C or to the left of A.

That means that if A is at (1,0) and C at (7,0), then E could be at (-2, 0) or (4,0)

Or E could not be on the number line if you put E 3 units above A or 3 units below A.

The most difficult type of number line problem is the word problem.  Try sketching a map of the buildings and stores in the following problem: Highland High School lies exactly halfway between the East and West bridges of town. Piggy’s Pizza lies halfway between the high school and the East bridge. Paul’s sub shop lies somewhere between the high school and the West bridge. All buildings form a straight line from the East bridge to the West Bridge. If Paul’s is 8 miles from the West Bridge and Piggy’s is 13 miles from the East bridge, how far is it from Paul’s to Piggy’s?

Did you manage to get the figure above?  Since Highland HS is exactly halfway between the bridges that means that it must be 26 miles – 8 miles = 18 miles between the high school and Pual’s sub shop.  So Piggy’s to Paul’s = 13 miles + 18 miles = 31 miles.

If you liked the examples on the page, try a custom number line math SAT game on Grockit!

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Inequalities

By posted April 15, 2010

We can break SAT inequalities questions down into three types: those involving a word problem, those involving algebra and those involving absolute values. Let’s tackle the word problems first.

Word Problems

Do you remember this question on Grockit?

A pulley can handle no more than 800 lbs of weight. It is currently holding 4 steel frames that weigh 112 lbs each. Amanda wants to load as many bricks onto it as she can without it breaking. If x represents the total weight of bricks, in lbs, that she can add, which of the following inequalities could be used to determine possible values of x?

From the question, I have deduced the following information:
• I can have a MAXIMUM of 800 lbs
• I have 4 frames, each weighing 112 lbs. That means, all 4 frames weight 4*112 = 448 lbs
• In addition to the 4 frames, I want to load x lbs in bricks.
I can thus conclude that x + 448 lbs must not exceed 800 lbs or my pulley will break.
In mathematical notation, x + 4(112) < 800 Don’t get tricked if the answer is written as 800 > x + 4(112). This statement is exactly the same as the statement above. 800 is greater than x + 4(112) is the same as x + 4(112) is less than 800.

Algebraic Inequalities

Another type of inequality involves algebra.  Suppose y = 2x + 4 and x < 3 and you need to find an inequality involving y.

  1. Start with the inequality you have x < 3
  2. Look at the equation involving y.  There is a 2x in it.  Since x < 3, that means 2x < 6
  3. If 2x < 6, that means 2x + 4 < 6 + 4 = 10.  Thus 2x + 4 < 10
  4. But 2x + 4 is simply y, so we can conclude that y < 10

The trick is to make your inequality look like the equation.  Can you work the next two examples out on your own?

If 2y = 2x + 4 and x < 3, find an inequality involving y

Answer: Like before, we have 2x < 6 and 2x + 4 < 10.  But now we have 2y < 10, meaning y < 5

If y = -2x + 4 and x < 3, find an inequality involving y

Hint: Don’t forget, that when multiplying an inequality by a negative number, you have to switch the signs.

Answer: If x < 3 , then -2x > -6.  That means that -2x + 4 > -2 so we get y > -2

Absolute Value Problems

The last type of question tests your knowledge of absolute values.  Absolute values are denoted by two straight lines | |.  Absolute values make negative numbers positive.

So |10| = 10 and |-10| = 10 too.

Now that we’ve established what absolute values do, we can solve absolute value inequalities.  Here’s a Grockit question:

A theater company is auditioning for actors to portray the leading character in a new play. The company is looking for actors between the ages of 20 and 40 (inclusive). Which of the following inequalities can be used to determine whether an actor of age a is eligible to audition for the part?

  1. | a-10 | ≤ 40
  2. | a-20 | ≤ 40
  3. | a-30 | ≤ 20
  4. | a-30 | ≤ 10
  5. | a-35 | ≤ 5

The answer is 20 ≤ a ≤ 40. But how do we make that look like one of the inequalities above?

  1. Take the average of 20 and 40.  That’s 30.
  2. Subtract 30 from everything to get 20 – 10 ≤ a – 30 ≤ 40 – 30
  3. You get -10 ≤ a – 30 ≤ 10  Note that the left and right side of the inequality is the same number, except that the one of the left is negative
  4. Now we can say, | a – 30 | ≤ 10

So the answer is clearly choice D.

Can you work out the lower and upper bounds of the other choices?   I’ll do Choice A for you.

Choice A says | a – 10 | ≤ 40.

=> -40 ≤ a – 10 ≤ 40

=> -40 + 10 £ a – 10 + 10 £ 40 + 10

Thus Choice A is effectively saying -30 ≤ a ≤ 50, which is not the answer at all.

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Graphing Linear Equations

By posted January 27, 2010

To understand how to sketch linear equations, you must first understand what makes up a linear equation.  Any linear equation can be expressed in this format, y = mx + c.  Note that whenever I refer to “m” or “c”, or the slope and the y-intercept as we shall soon call them, y must be the subject of the equation.  For example, if I have the equation 2y + 3x = 4, m is not 3 and c is not 4!  You have to make y the subject:

2y + 3x = 4

=> 2y  = -3x + 4

=> y = -3/2 x + 2

This means that m = -3/2 and c = 2

Now that we’ve got that out of the way, we can move on to discuss the significance of m and c. The value of m is what you know as the slope or the gradient of the line.  This determines the direction and extent the line “tilts”.  If m is positive that means that the line slopes from the bottom left to the top right, like a checkmark.  Something like this: “ / ”

If m is negative, that means the line slopes from the bottom right to the top left, like this “ \ ”

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Looking at the diagram above, you should now be able to tell me that all three lines have a positive slope and that m > 0 because they all slope like /

But what else can you tell me about these lines?  Let’s look at the blue line and the green line.  the blue line has equation y = x+3 and the green line has equation y = x-1.  Do you notice that they are both parallel?  This is because they have the same slope, or the same value of m, which in this case is 1.  The takeaway here is that all parallel lines have the same slope!

Now let’s look at where these lines intersect or “cut” the y-axis (that’s the vertical axis).  The blue line cuts at y = 3, the red line cuts at y = – 1 and the green line also cuts at y = -1.  What this affects is the value of c.

c is what is known as the y-intercept.  Whatever the value of c is, that’s where the line will cut the y-axis.  The blue line cuts at y = 3, and accordingly the equation of the line y = x+ 3 has c = 3.

The red line has equation y = 2x – 1, implying that c = -1 and true enough, it cuts the y-axis at y = -1.

Just to make sure you understand what I mean by graphs with a negative slope, here is a picture of two parallel lines with a negative slope.  They are parallel meaning the slope is the same.  The only difference is that the blue line intersects the y-axis at y=3, meaning that c=3 for the blue line. (c=0 for the red line).

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One last important property to know about linear graphs is the relationship between perpendicular lines.  If there are two perpendicular lines with slopes m1 and m2, then the relationship is defined as such

m1 x m2 = -1

Take a look at the graphs below.  I have drawn one blue line with the equation y = -2x + 3.  Now, you already know that m = -2 for the blue line, meaning that any line that is perpendicular to it has to have what value of m?  ½?  That’s right!

To illustrate the point, I have drawn 3 graphs that are all parallel to it. See if you can work out which equation is which line.

equation 1: y = ½ x

equation 2: y = ½ x + 1

equation 3: y = ½ x – 3

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Did you get it right?  red = equation 1.  pink = equation 2.  green = equation 3.

The important thing to remember with linear graphs is what m and c means, and to always always make y the subject of the equation before you start doing anything else.

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