Category Archives: math

Learning the unit circle for the ACT

unit circleOn a handful of ACT math tests, I’ve seen a unit circle question which requires outside knowledge of the coordinates around the circle. I had heard that there is a “hand trick” out there that students can use to recall those coordinates as you climb up and back down around the circle. Today I found a great blog post from “Mrs. B”. Check it out!


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What is “data analysis” according to GRE math?

The data analysis questions on the GRE Quantitative Analysis (i.e. Math) section require skills such as “describe past and present trends, predict future events with some certainty…” (The Official Guide to the GRE).    What that means for the test-taker is that questions will contain frequency distributions, bar graphs, circle graphs, histograms, scatterplots, and time plots.

histogram This is an example of a histogram.  The Y axis will often have probability or, in this case, frequency.  The percentages you see in the picture are frequencies that simply refer to how often the event in the X axis occurred.  In this case, we are told that out of a total of 25 families, the children were chosen for something.  There were six families who had 3 children and one can see on this histogram that the probability of one of those being chosen is 6/25 or 24%.

(Look at the x value of 3 and follow with your finger to the top where it would match about 24% on the Y axis.)  This is the kind of histogram interpretation that would be involved in an actual question.


Other data analysis questions may not involve a chart, but just be represented as a word problem.  For example, when given daily temperatures for 5 days in July, one would have to find the mean, median, mode and range.  A follow-up question might involve changing the given temperatures and then figuring the new quantities.

5 days in July: 90, 88, 85, 85, 87

Mean = (90+88+85+87)/5 = 87

Median = the number in the middle when put in order=  87

Mode = the most frequent number= 85

Range = the difference between the highest and lowest quantity= 5

If each day was 5 degrees cooler, what would the new mean, median, mode and range be?  (82, 82, 80, 5)


Hopefully this quick summary has been helpful.

Best of luck!  Tanya Panossian-Lesser



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Geometry on the GRE

This is a second post about the quantitative reasoning sections on the GRE.  In the previous post, I gave an arithmetic example.  Now I’m following up with a “medium” geometry question in that same “quantitative comparison” style, then a “hard” question in a multi-choice style.


from the Official Guide to the GRE



In the figure above, the diameter of the circle is 10.

Quantity A

The area of quadrilateral ABCD

Quantity B


Is Quantity A higher than B?  Or vice versa?  Are they the same?  Or is there too little information?

When reviewing our knowledge of quadrilaterals, we remember that this shape is a kite.  Kites’ properties include the following: the diagonals multiplied together equals twice the area of the figure.  This would be a good lead if we had information on what a BD chord (not shown) length would be… but we don’t!  So the answer will be that there is too little information.

Perhaps it’s been too long since you’ve thought about all the different quadrilaterals and their properties.  It is a great idea to review those topics.  Here’s one post about kites, for example.


There is also the multiple choice format in geometry.  The following is an example of a “hard” question.


from the Official Guide to the GRE



Parallelogram OPQR lies in the xy-plane, as shown in the figure above.  The coordinates of point P are (2, 4) and the coordinates of point Q are (8,6).  What are the coordinates of point R?

A. (3, 2)

B. (3,3)

C. (4,4)

D.  (5, 2)

E. (6, 2)

What we know about parallelograms is that their sides are parallel, and therefore have the same slope.  We are given the endpoints of PQ, so we should find the slope using the slope formula.  That requires you take the difference of the Ys and put them over the difference of the Xs.  This “rise over run” would be found with 6-4 over 8-2, or 2 over 6.  This is the same slope for OR, since it’s parallel to PQ.  So if we rise 2 and run 6 from point O, which is (0,0), then we will match answer E.

PWN the SAT’s Mike McClenathan’s review about lines is here and is super-helpful!


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Thinking about graduate school? Here’s what to expect from GRE Math

scantronThe college grads that I talk to in the field of education have plans to go on to graduate school.  A few that still have to take the GRE yet are nervous, “by far”, they say, about the math.    What will be on it?  How out of practice am I???

The range of topics on the GRE math includes the categories of arithmetic (do you know how to work with fractions?), algebra (can you solve an inequality?), geometry (do you know how to work with similar figures?), and data analysis (can you interepret a distribution chart?).  The two thirty-five minute math sections, (aka “quantitative reasoning”), have twenty questions each.  (This doesn’t include the unscored section which may or may not be math.)  The paper-based test has five more questions and five more minutes per math section.

Here are some question samples from the ETS Official Guide.

1.  A student made a conjecture that for any integer “n”, the integer 4n +3 is a prime number.  Which of the following values of “n” could be used to disprove the student’s conjecture?  (Select all answers that apply.)

A. 1

B. 3

C. 4

D. 6

E. 7

In this case, the answers are “B” and “D” because when either 3 or 6 substitute for “n”, the result is not a prime number.  (Reminder: a prime number is only divisible by 1 and itself, so if you had put 3 in for “n”, you would have had 15, which is not prime since it can be divided by 3 and 5.)

There is also a comparative section.  One must compare the quantities expressed for Quantities A and B, then decide if one is greater than the other, or if they are equal, or if not enough info is given.

Here’s an example:

1.  Quantity A           (3 raised to exponent -1) divided by (4 raised to exponent -1)

Quantity B            4 divided by 3


The answer would be that the quantities are equal since after you calculate 3 raised to the -1 and 4 raised to the -1, you have the reciprocals.  1/3 divided 1/4 can be rewritten as 1/3 multiplied by 4.  That equals 4/3.


Those two examples represent problems in arithmetic.  In a future post I’ll share some examples of other math problems.


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SAT questions: How to Tell Which Are Easy, Medium or Hard

The guessing penalty on the SAT lowers your total points by 1/4 point for each wrong answer. The questions are designed so that statistically more students will get the hard ones wrong. So how do we know which are easy, medium or hard?
The answer is simple for some sections, and more complicated for others. (It would be helpful to have a practice test in front of you as you read this, by the way.) Starting with the math section, one can simply skip the last questions of the section to avoid the hard questions. Remember, those complicated, time-consuming questions are worth the same point value as the easy, faster one. So if you have 20 questions, and you are not aiming for a 700-800 on the section, you can skip the last two questions entirely.
On math sections with grid-ins, there are two things to know:
1- the multi-choice questions have their own order of difficulty, just like the other math section, and then the grid-ins start over with easy to medium to hard
2-in general, the grid-ins tend to be less complicated- so don’t be intimidated!

The shortest Writing multiple-choice section – the one with only one type of question- also has a simple order from easiest to hardest. However, the writing sections with a variety of grammar/improving writing types of questions are not in order as a section as a whole, but just the sentence corrections are. (These questions offer a sentence and one has to decide if it needs a correction.)

The Critical Reading section, with its varieties of questions, also has a more complicated order of difficulty. The Sentence Completions have their own order, so if this is not your strong point, skip the last one or two. Then the passage questions are not in an order of difficulty, but it’s worth it to know that some are trickier than others. For example, the single paragraph passages are short- but often have harder questions. The same goes for the dual passage questions; if these are difficult, at least answer the questions that pertain to just one passage.  And as always, “except” questions take longer since one usually has to rule out all the true answers first.
As I mentioned in my description of the math section, skipping questions is a good idea for those who are not aiming for a 700-800 on a section. Need to practice ASAP? Try the test offered online on the college board website; it’s administered and graded for free.

The next three SATs will be on November 3rd,  December 1st, and January 26th.

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Getting Ready for Mid-terms: Online Lessons in Math from Pre-Algebra to Calc

Mid-terms week is coming up at many high schools, and even if your teacher isn’t giving a true “mid-term” due to the Hurricane Sandy’s disruption, many are still giving a major test. Take charge by going online and re-learning concepts from a new perspective.  Making your brain process complicated information in different ways is a solid way to retain that information.

Here are some for the self-directed learner:

a) Plan for at least a 60 minute session.  It may be nearly impossible to go on the internet and *not* do your daily “quickly check” of some favorite entertaining sites.  If you do that, set a timer to 5 minutes and stick to it, bookmarking sites you want to finish later.  Experiment with studying in the morning, afternoon and evening to find out when your brain is most focused.   Exercise first for ten-twenty minutes; it gets your brain running on full blast.

b)  After a study session, family members or friends can help by asking what topics were covered.  Get more out of the conversation by asking what concepts were covered differently than they were in school.

Ready to begin?  Here are two noteworthy sites:


Getting past the annoying title to this page, “Math for Morons Like Us”, I would recommend a visit to this site.  It turns out that it was certainly not created by morons; I found their Algebra, Geometry and Algebra II key concepts to be taught clearly and concisely.  But this is not meant for the struggling math student- it would need more of a variety of presentation.  Nor is it for those who are looking for more advanced coursework.  But for those who have had success learning “by the book” so far, this website should definitely be bookmarked.  The quizzes at the end of each chapter are helpful as well.

Example page:

The trig. ratios, sinecosine, and tangent are based on properties of right triangles.  The function values depend on the measure of the angle.  The functions are outlined below.

sine x = (side opposite x)/hypotenuseright triangle

cosine x = (side adjacent x)/hypotenuse

tangent x = (side opposite x)/(side adjacent x)


And now, the rock star of online learning… (drumroll)……  The Khan Academy


It’s a fair guess that anyone who has already explored the web searching for math help has heard of this site.  It’s simply the most comprehensive online school of mathematics there is, as far as I know, and remarkably, free.  After a few years of development, the creators have added SAT math, the natural sciences, and some Social Studies as well.  Pretty impressive, right?  And I haven’t even mentioned the best part: these are video presentations by articulate instructors at the chalkboard who are using helpful visual tools such as underlining, colored chalk, and even subtitles.

Want to give it a try?  A topic that frequently challenges students to the point of exasperation is “word problems”.  Here’s link to that lesson at the Khan Academy:

Working with online math brings the opportunity to learn lessons in new ways.  It can be lot more fun than opening the same textbook you’ve been opening for months.

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