(Math, Physics, Chemistry tutor at The Edge Learning Center)

In one of my previous articles, I talked about how to use a scientific calculator to perform algebraic manipulation. Please check it here before reading this article.

In part 2, I am going to show you how to apply this method in the SAT and ACT tests. I will use Casio fx-82ES Emulator to demonstrate how to use this technology to quickly solve algebraic questions. Don’t worry if you use a scientific calculator that is not Casio fx-82ES because every scientific calculator has a similar layout.

Before we use the trick, it is essential for us to use the substitution function in our calculators. We should first substitute 100 into the variable x before we do the following questions:

Let’s take a look at our first example:

[Example 1: SAT College Board Practice Test 6 Section 4 Q1]

After we make x becomes 100, we need to type the expression in the calculator:

To input x in your calculator, key in ‘alpha’ and ‘)’ in order. The output of the expression is 49803. The digits representing the constant term is ‘03’, which refers to 3, so the answer must be A.

[Example 2: ACT 2008-2009 Q6]

Similarly, we key in the expression to our calculator:

Now, let’s take a look at a more advanced example.

[Example 3: ACT 2005-2006 Q15]

To interpret the result ‘20894’, we need to apply the adjustment rule that I mentioned in my previous article:

Let’s check your understanding by answering the following questions!

[ACT 2010-2011 Q32]

Hint: You can already pick the correct solution after you figure out what is the constant term.

[SAT College Board Practice Test 3 Section 4 Q33 (free response)]

Suggestion: Find the values of a,b and c using the adjustment rule.

Blogs written: Winning a Nobel Prize by Solving ‘1+1’, Using a normal calculator to do Algebraic Expansion, 5 common mistakes made by IGCSE Math students, A guide to use online tools to perform mathematical modeling

About The Edge

The Edge Learning Center is Hong Kong’s premier Test Preparation, Academic Tutoring, and Admissions Consulting services provider. Founded in 2008, The Edge has helped thousands of students improve their ACT and SAT scores as well as their IB and AP grades. The AC team has just finished off another successful period in which students gained acceptance to schools such as Columbia, Yale, UChicago, UPenn, Oxford, and more! Check out our latest Admissions Results!

The post Using a normal calculator to do Algebraic Expansion (Part 2) appeared first on The Edge.

Many universities require students to submit two or three SAT Subject Test scores in addition to their ACT/SAT scores. Those students who plan to study STEM in the future will often choose to take the Math Level II test. The score range of this test is 200 to 800. Many top universities will only accept a score of 800 on this test. Math II is one of the most competitive subjects amongst the many offered by College Board: it is taken by the most number of students; more than a fifth of the students receive a “perfect” score of 800; and the average score is close to 700. Students who take the test for the first time find it overwhelming: not only does the test cover a wide range of topics, but it also asks them in unconventional ways. Even students who have done exemplarily well in their Math classes can find this test challenging at times. To help you achieve the higher score that you desire, here are some tips to follow when you take this test.

1. Time Management

Just like many of the standardized tests out there, the SAT Math II requires you to work out many questions in a limited amount of time. You will be given 1 hour to finish 50 questions; this works out to 72 seconds, or 1 minute 12 seconds, per question. However, you should not focus on the “per question” time frame. The test gets progressively harder, and this means that the time spent on the last 15 to 20 questions can be significantly longer than that spent on the first half of the test. You must move quickly at the beginning while maintaining your accuracy. You do not want to read a question blankly just to read it again. Underline the key information and consider its significance. Identify what exactly are you tasked to find. Apply all the values given in the question. If you end up omitting a given value from the question, ask yourself: what am I missing? The test makers know exactly what kinds of careless mistakes student often make, and they put them as trap answers. This means picking an answer choice without utilizing all the information given will most likely be incorrect (there are, of course, exceptions, but they are rare).

In other words: be vigilant, and spend appropriate amount of time based on the position of the question.

2. Choose Your Method Wisely

There are always different ways to solve a problem. In order to minimize the amount of time spent on a question, choosing an appropriate and effective method is extremely important. While many questions can be solved algebraically, there are times when graphing the problem using your graphic display calculator (GDC) will yield a result faster. Furthermore, because the entire test is multiple-choice, trying out the answers (back-solving) or making up values (picking numbers) can be more efficient than solving for the solution.

You can read more about the basic concepts of back-solving and picking numbers from James, one of our awesome Test Prep teachers here at The Edge. Below are few additional details to pay attention to for Math II:

i. The answer choices are often rounded solutions, not exact. This means back-solving will not guarantee the exact result. In that case, choose the answer choice that gives you the closest value. Here is an example:

A. -1.00
B. -0.52
C. 0.00
D. 0.52
E. 0.67

If we try each answer choice, we would get:
A. cos⁡(-1.00)=tan⁡(-1.00) → 0.54030…=-1.55740…
B. cos⁡(-0.52)=tan⁡(-0.52) → 0.86781…=-0.57256…
C. cos⁡(0.00)=tan⁡(0.00) → 1=0
D. cos⁡(0.52)=tan⁡(0.52) → 0.86781…=0.57256…
E. cos⁡(0.67)=tan⁡(0.67) → 0.78382…=79225…

We can see that none of the equation is actually correct. However, answer choice E gives us the closest result, making it the correct solution.

ii. If there is more than one unknown in the question, make sure to pick numbers that follow all of the given conditions. Let’s take a look at the following example:

In this question, we are working with three different unknowns: a, x, and y. The first two equations give us the conditions. Thus, we need to work with values that satisfy these conditions. In this case, we will let a=2. Using a calculator, we can then find:

Giving us:

Using these values and plugging them into the answer choices, we get that:
2x+y=2∙1.5849…+2.3219…=5.4918…
Which is the correct answer.

iii. Be mindful of the values we pick, particularly when we are working with trigonometric equations. Use the proper value (radian vs. degree) according to the question. If you prefer working with one over the other, make sure to convert your answer at the end accordingly. Also, avoid special angles like multiples of 30°, 45°, 60°, or 90° ( in radians, respectively). These angles have special properties, and they may lead to unique results that are often used as trap answers.

3. Rely on Your GDC

The College Board recommends the students to use a GDC (graphing calculator) when taking the Math Level II test. You can find the list of approved calculators here. Don’t even think about taking this test without at least a scientific calculator; approximately 20% of the questions require a calculator, and another 20% that will take too long if solved by hand.

Having a graphing calculator can significantly cut down on the amount of time needed to work out certain questions. However, this tool is only effective if used correctly. You can read about my tips on using a GDC here.

It is important that you are familiar with the functions of your graphing calculator for the test. The last thing you want is to spend 5 minutes finding the “log” function on your calculator. Here are some steps to take that will help reduce the time spent on a question:

i. Check whether you are in Degree or Radian mode, especially when dealing with trigonometric equations. Be able to change mode on the fly.

ii. If you want to see something from the graph, make sure you have the right window size. The question will likely provide a hint (or even the exact values) about where your answer should fall within. Set the window first to avoid graphing twice.

iii.  Know where to find some uncommon symbols like i (imaginary numbers), ! (factorial), nCr(combination), nPr(permutation),  (summation), or |x| (absolute value). There is no guarantee that you will need to use all of these symbols, but you need to be ready.

4. Focus on the Basics

There comes a point when you read the question and have no idea what is going on. You can’t even decide what method to use to solve the problem. So what do you do?

This is when you must focus on the information given, apply whatever knowledge you have learned, and try to proceed to as far as you can. As I mentioned at the beginning, the Math II test requires students to know a wide range of topics (Number and operations; Algebra and functions; Geometry and measurement; Data analysis, statistics, and probability), and often time the question can be presented in an unfamiliar way. However, the College Board will never require the students to apply knowledge or concepts that are beyond the scope of high school level mathematics: instead, a question masks the basic knowledge in a complex setting. Here is an example involving multi-variable function:

If f:(x,y)→ (x+2y,y) for every pair (x,y) in the plane, for what points (x,y) is it true that (x,y)→(x,y)?
A. The set of points (x,y) such that x=0
B. The set of points (x,y) such that y=0
C. The set of points (x,y) such that y=1
D. (0,0) only
E. (-1,1) only

At first, students may see that this is a function question because of the notation used. However, they may not understand how to proceed as the function is defined as coordinates instead of operations. But we don’t need to understand the mechanics behind this multi-input/multi-output function (which is actually a transformation) in order to find the answer. Focus on what we need to achieve: the input needs to be the same as the output. If we then separate the two coordinates and look at them individually, we get:

x=x+2y→0=2y→y=0
y=y

The first statement tells us that y must be 0, which means we can eliminate A, C, and E. Neither statement tells us anything about x, so choice D, which says x must also be 0, has a limitation that is not supported by our calculation. Therefore, the logical solution would be B.

In the end, break down an unfamiliar question into smaller parts. Apply your knowledge and see where it can take you.

5. Know When to Give Up

One misconception about the Math II test is that in order to get a “perfect” 800, you must get every question right. That is not the case at all. In fact, out of the 50 questions, students often need to get 43 or more correct answers in order to receive a score of 800. This means that it is strategically viable to skip questions that are too difficult or too lengthy to solve. This raw score (the amount of questions you correctly answered) does not get reported to the university. In other words, there is no difference between getting all 50 questions correct, or only getting 43 questions right but skipping 7.

And skip you should. The SAT Subject Tests all have a “penalty” system: a fraction of a point is subtracted from a wrong answer, while no point is deducted for an unanswered question. The amount deducted depends on the number of choices given in the question: 1/4 point for a five-choice question, 1/3 for a four-choice question, and 1/2 for a three-choice question. Since all questions on the Math II are five-choice questions, an incorrect answer will result in a quarter point penalty. If you are able to eliminate two or three choices, then you should take the chance and give it a guess. Otherwise, play safe and skip.

About The Edge

The Edge Learning Center is Hong Kong’s premier Test Preparation, Academic Tutoring, and Admissions Consulting services provider. Founded in 2008, The Edge has helped thousands of students improve their ACT and SAT scores as well as their IB and AP grades. The AC team has just finished off another successful period in which students gained early acceptance to schools such as Columbia, MIT, UChicago, and more! Check out the rest of our 2018-9 Admissions Results!

The post 5 Tips on Tackling the SAT Subject Test: Math Level II appeared first on The Edge.

Test Preparation (ACT/SAT/SSAT), Math, Physics

Take a test that involves mathematics today, and the chances are that you will need a calculator. And not just a four-function calculator; many tests now demand a graphic display calculator (GDC). GDC has come a long way since its commercial debut in 1985. Nowadays, the new calculators have so many new tricks up their sleeves that there may be things that you don’t even know they can do.

Even though the tool is getting more and more powerful, it is still up to the user to get the most out of it. Today I will give you 5 tips that will help you do just that.

I) Pay Attention to the OoO (Order of Operations)

Remember back in grade school when you learned about the order of operations? Let’s review that idea very quickly:

(Photo source)

When you do simply arithmetic, you must follow this order of operations. Of course, calculators follow the same order. What you need to pay close attention is the “Please”. Many students get the simple calculations incorrect because they don’t realize the importance of the parentheses. Parentheses are essential when your calculation involves negative values or complicated fractions. Case in point:

The careless mistake here would be to put the following in the calculator:

The eagle-eyed students will notice that the answer is incorrect because the square is applied incorrectly. Instead, the right way to calculate this is:

Notice how the parentheses around -2 affect the result of the power.

Another common mistake shows up when working with fractions. Let’s take a look at the example here:

Those who are not familiar with their calculators, thus not being able to input the expression as a fraction, may try to input the following:

That is an accurate interpretation, knowing that fraction is equivalent to division, but they will get an error from the calculator in this case. The problem? In a fraction, you need to calculate the numerator and denominator first before you apply division. Thus, the correct way to input this into the calculator is:

In conclusion, get into the habit of putting parentheses around negative numbers and numerator/denominator of a fraction. This will save you from unnecessary, careless mistakes.

II) R U RAD or Do You Deg?

No, it’s not about whether you are cool or watering something. In mathematics, there are two ways to measure the size of an angle: in radians or degrees. For those who are unfamiliar with radians, you can learn more about this concept here. In a very simplified manner, the difference between degree and radian is the measuring perspective. Measuring in degree is like a person standing in the middle of a circle and turning. Measuring in radian, on the other hand, is like walking around the circle with a radius of 1 unit.

(photo source)

This idea of radians vs. degrees becomes very important when you are dealing with trigonometric functions (sine, cosine, tangent). If your calculator is in the wrong mode, you will get the incorrect answer. Here is an example:

Find cos 60°

If your calculator is in degree mode, the answer will be a simple 0.5. However, if you accidentally leave your calculator in radian mode and punch in the same thing, the answer will now become -0.9524129804…

Make sure you know how to change the angle measurement in your calculator and make a conscious effort to adjust when you encounter a question involving trigonometric functions.

III) Solve With the G, Not With Your Hand

And now we get to the most important part of a GDC: graphing. Why spend all that money on a fancy machine if you are not going to use it?

One common misconception about graphing is that you can only do it when you are given some kind of . While that is the most common usage, which will tell you the features and behaviors of a function, it is also possible to use graphing to solving equations. Geometrically speaking, a graph represents all the possible pairs of  and  that satisfy the function. If you consider the two sides of an equation as two different functions, you can graph them separately and look for the intersection. Khan Academy has some easy-to-follow videos that explain how this idea works.

One thing to keep in mind: while graphing the solution can help simplify the more complicated equations, it’s also a double-edged sword that can complicate a simple question. Use good judgment before throwing everything at your GDC.

IV) [Window] Size Matters

See that window on that GDC of yours? You can adjust the viewing real estate. To master the art of solving an equation by graphing, you need to know where to find the solution. Photographers use different sized lenses to find what they want to capture, and, similarly, you need to adjust the viewing window to find the correct solution.

Finding the right barn requires different lenses (Photo source)

Changing the window size is equivalent to adjusting the domain and range (or the  and values, respectively). Unfortunately, your calculator is not smart enough to automatically change those values to give you the best view. Instead, you need to rely on your mathematical knowledge to narrow down the dimension. Consider information that may have been provided in the question. Can  be negative? Does the question already specify the domain? Are you expecting the answer to be large or small? Put all those ideas into your consideration before randomly adjusting the window size.

V) Machine with an imagination

In one of my previous blog, I talked about the concept of complex numbers, an increasingly popular topic being tested in both SAT and ACT. It’s a good idea to learn how to operate with by hand, but it’s also important to remember that a modern GDC has the ability to calculate expressions involving imaginary numbers. Simple operations like addition, subtraction, multiplication, division, and exponent can be done by using the designated on your calculator. Do not input , however, as your calculator will treat that as an undefined value instead of converting it to .

Just because it’s imagined doesn’t mean it non-existent (Photo source)

When you use your calculator next time, remember these tips!

Sources:

The Flight That Made The Calculator And Changed The World

http://americanhistory.si.edu/collections/search/object/nmah_599945

https://mashable.com/2014/04/25/graphing-calculator-tricks/#8YWJhxSg8Pq5

https://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/

https://www.khanacademy.org/math/algebra2/advanced-functions/solving-equations-by-graphing/v/estimating-a-solution-to-nonlinear-system-with-calculator

https://theedge.com.hk/decomplexifying-complex-numbers/

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About The Edge

The Edge Learning Center is Hong Kong’s premier Test Preparation, Academic Tutoring, and Admissions Consulting services provider. Founded in 2008, The Edge has helped thousands of students improve their ACT and SAT scores as well as their IB and AP grades. The AC team has just finished off another successful period in which student gained acceptance to schools such as Stanford, University of Chicago, Brown, and more! Check out the rest of our 2018 Admissions Results!

The post 5 Tips on Using Your GDC (Graphic Display Calculator) During a Test appeared first on The Edge.

(Test Preparation (ACT/SAT/SSAT) tutor at The Edge Learning Center)

What are ‘counting principles’? The fundamentals of counting principles appear in some form or another in the IB, AP and A-Levels maths curricula, and are common topics tested in the maths section of the ACT and SAT, so make sure you are prepared to deal with them!

Counting Principles

Think about a time when you went to Ocean Park with friends, and took a ride on the Mine Train. Maybe you had to figure out which order your friends should all queue up in order to sit with the people they wanted. For this challenge, you would need counting principles! It could even be used to figure out the different flavors of ice-cream you’d use to make the best sundaes or cocktails afterward.

(Photo: portlandsonic.com)

Whenever we’re considering the different number of ways something can occur, or if we’re trying to arrange items in a particular order, we are applying the fundamentals of counting principles. These topics are also the two main question types that appear in almost all maths courses, and solving questions on this topic is an important process.

With these steps, it can be relatively straightforward:

1. Counting Principle is the method by which we calculate the total number of different ways a series of events can occur. This is always the product of the number of different options at each stage.

Let’s look at an example of this to see how best to apply this principle:

(from ACT 65D, April 2008 paper)

This is a common example of a question that appeared in an actual ACT paper in 2008. In this case, there are 4 events that will occur, and in order to solve this question we need to:

• First, calculate how many different ways each of the four event can occur

Event	Numbers of Options
Choose a sandwich	4
Choose a soup	2
Choose a salad	2
Choose a drink	2

• Then, we can calculate the total number of possible outcomes by multiplying the number of options at each stage.

Total possible outcomes = product of how many different way each selection can be made

Therefore, total number of ways these selections can be made is 4 x 2 x 2 x 2 = 32 possible ways.

Hence, the correct answer is K.

Now, let’s consider the second type of question, where we are asked to consider events where a series of specific objects are drawn from a much larger pool.

2. Let us consider a class of 20 people, out of which we are interested in appointing 4 people for positions of responsibility in the school debate team. Assuming all members of the group are of equal competence and are all capable of carrying out the requirements of the positions effectively; we will need one treasurer, one secretary, one president and finally one vice-president. How many ways can this be done?

We can approach this question in a similar method to the previous question.

• First, let’s work out how many different ways we could pick a person for each position. It is important to note here that whether we pick the treasurer first or the president first will not actually affect the final answer:

Job title	Number of Options
Treasurer	20
Secretary	19 (as we have already picked 1 person to be the treasurer)
Vice President	18
President	17

• Therefore, the total number of possibilities for assigning 4 people out of 20 to these positions of responsibility is 20 x 19 x 18 x 17 = 116,280

These topics are sometimes described using the notation nPr and nCr, meaning ‘Permutation’ and ‘Combination’ respectively. For those of us having to survive IB Maths, combinations and permutations (only for HL) come up in both non-calculator and calculator papers. So, let’s have a look at how counting principles fits into the topic of permutations and combinations.

Permutations and Combinations

Permutations and combinations are the various different possible ways we can arrange or select an item or r items out of a sample size of n. You can think about these using our lovely Sets and Venn diagram terminology. If you have a set of n elements and you pick r elements to form a subset, the possible options for this subset are the ‘combinations’. With combinations, the order in which the elements are chosen does not matter, so ABC = CBA =BCA etc. If permutations are being considered, then the order of the elements does create different options, so ABC does not equal CBA etc.

Permutations   [order matters]

Combinations  [order doesn’t matter]

Your scientific calculator will always be able to calculate nPr and nCr for you automatically, but here we’ve shown you how they are actually calculated, for people who are curious!

So let’s think about the previous Debate Team question using these two ideas.

• To begin with, we have to ask ourselves if we need to permute or combine for this question? If a person is picked to be the Treasurer, is this the same as being picked or the President? Definitely not! So where the selection of a person is made in the process will make a big difference. If the order in which people are assigned to a role is important, this will be a permutation question!

Here, we can substitute 20 = n and 4 = r values for nPr. Then we have two ways we can solve this. We can either:

• Use a calculator and plug in  

Or, if we are feeling particularly assiduous we can always work it out manually by using the formula (given in the IB Data Booklet).

Notice how 16! cancels out from both the numerator and the denominator.

One great application of this is Mr. Potato Head. If you forget which is which, I like to use Potato Head to help me remember. The pieces you choose to put on are your combination of pieces, (the items you have used). The order in which you put them on is you permutation (how you have used them). Therefore, you are combining the parts to make one permutation!

In the ACT there will be no requirement for calculating permutations and combinations manually, as we always have our trusty GDC or scientific calculator which will come to our rescue. In the SAT they usually reserve combinations and permutations for the calculator section of the maths test. The IB gods, however, can ask students to manually calculate combinations in both the non-calculator and calculator papers at SL, and both combinations and permutations can appear in HL. Fortunately, they do provide both of these equations in the data booklets.

Here you can practice these questions for some fun scenarios.

On a final note, remember that whilst this can be a pain to study for a Maths exam, and we can only make so many sundaes before we become fat… we can use these ideas to think about any series of events, from predicting our opponents’ hand of cards in games like poker, through to how best to seat people at a table for a party!

Good luck permutatin’ out there, maths-fans!

Related Blog: Approach to SAT Math

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About The Edge

The Edge Learning Center is Hong Kong’s premier Test Preparation, Academic Tutoring, and Admissions Consulting services provider. Founded in 2008, The Edge has helped thousands of students improve their ACT and SAT scores as well as their IB and AP grades. The AC team has just finished off another successful EA/ED period in which student gained early acceptance to schools such as Stanford, University of Chicago, Brown, and more! Check out the rest of our 2018 EA/ED Admissions Results!

The post Counting Principles, Combinations and Permutations appeared first on The Edge.

(Test Preparation (ACT/SAT/SSAT), English Literature, and GRE tutor at The Edge Learning Center)

As someone who teaches test preparation at The Edge, I’ve seen students make many, many mistakes on the Math sections of the ACT and the SAT. I’ve become somewhat of a catalog for these errors, dangling the most memorable in front of my students in the form of protracted cautionary tales: “you don’t want to be like the guy who made a subtraction error in the middle of a probability question, rendering the remainder of his work totally useless, do you?” or “make sure that these trig formulas are etched into the folds of your brain, or you might taste some of the misery of this one crestfallen student who put his faith in the arcane power of COH TAH SOA.”

Read more from Levi in his previous blog “Case Study: How does a college use standardized test data?”

The type of problem that I want to talk about today, however, doesn’t have any of these silly anecdotes as a preface, doesn’t lend itself well to any sort of jokes. It’s also by far the most issue I see when I’m reviewing mock tests with students. They look at the question they missed and say the phrase I’ve heard about a thousand times now: “I don’t even know how to start.” These students might even know the fundamental information required to solve the problem, but the chicanery inherent to standardized testing can sometimes make it difficult to know how to apply this info.

With this issue in mind, we’re going to take a look at a math problem from an old SAT, a problem that used to decimate my students. We’ll break down some useful steps for securing that initial “foothold” into a difficult problem, helping you to develop a more organized and useful reaction to the challenges that appear on standardized testing.

Step Number 1: Read the problem as carefully as you can, identifying what areas of math you will need to pull from

As intimidating as this problem might appear at first glance, make sure you understand precisely what it’s asking. In this case, we’re trying to find out how many sides this covered polygon has, and we’ve been given a few pieces of information: the polygon has equal sides and the two labeled angles, x and y, add up to 80. Based on the information given, it’s clear that we need to know about the angle properties of polygons. Let’s move on to the next step!

Step Number 2: List the pieces of mathematical knowledge that might be relevant

Right now, you might still be looking at the question quizzically, wondering what to do. This is why we should start cataloguing what we know about the angle properties of polygons: as we wade through what we know, we will run into facts and formulas that are relevant to us. Additionally, if you cannot recall the upcoming facts, you should review the angle properties of polygons!

Fact 1: All of the angles in a triangle add up to 180º

Fact 2: All of the angles in a quadrilateral add up to 360º

Fact 3: The sum of the interior angles of a regular polygon is (n–2)(180º), where n represents the number of sides of the polygon

Step Number 3: Attack the problem with the information that you’ve listed

Looking at the problem, there’s no visible application for Fact 1. There is no visible triangle in the problem, so we should discard it.

There is, however, an ugly-looking quadrilateral created by that irritating blank piece of paper! We have an in! Since we know that all of the angles in a quadrilateral add up to 360º (thanks Fact 2!) AND that x and y add up to 80, we can determine those interior angles that we see.

Now, we’re not done yet. We now know the degree measure of each interior angle, but how can we use this information to find out the number of sides that this polygon has? Let’s look at Fact 3 and see if we can use it.

(n–2)(180º), where n represents the number of sides of the polygon

It may not seem immediately evident that we can use the formula, but with a little bit of mathematical trickery, this equation serves as the express ticket to the end of the problem. If we divide the sum of the interior angles of a regular polygon by n, we will find the value of one of its interior angles, and guess what? We already know that an interior angle of this polygon is equal to 140º. Let’s do some math:

And that’s it. This stupid polygon hiding cowardly under a sheet of paper has 9 sides, and you used nothing other than facts you already knew to uncover its shameful truth.

It’s easy to get intimidated by the harder math problems on standardized tests, but it’s also reassuring to know that they’ll never throw integral calculus or linear algebra at you; as long as you keep your cool and make sure to consciously index the knowledge you have in your head, any problem will crumble at your fingertips.

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About The Edge

The Edge Learning Center is Hong Kong’s premier Test Preparation, Academic Tutoring, and Admissions Consulting services provider. Founded in 2008, The Edge has helped thousands of students improve their ACT and SAT scores as well as their IB and AP grades. The AC team has just finished off a very successful year in which 84.62% of their clients were accepted into one of their top 3 schools and an astounding 48.15% of their Ivy Plus* applicants were accepted. (The general acceptance rate was only 7.61% last year) Check out the rest of our 2017 Admissions Results!

*Ivy Plus: All Ivy League Schools + Stanford & MIT

The post Approach to SAT Math appeared first on The Edge.

(Test Preparation (ACT/SAT/SSAT), Math and Physics tutor at The Edge Learning Center)

Disclaimer: before you read further, the word “decomplexifying” is not a real word. It just sounds better than “simplifying” when put in front of “complex numbers.” So please do not use this word in your English papers.

One of the more advanced topics that are popping up in ACT and the new SAT is complex numbers. While the name suggests that it is a complicated concept, both tests only require students to know the basic operations. However, many students might not have encountered this concept in school before, and tackling this on their own can be a daunting task. Today I would like to give you some tips on how to work with this number system for the exam, and introduce some interesting facts should you find the topic fascinating.

>> Read more on Leo’s previous math blog “Fun! Facts about Factorials!”

The  works like a  …

If a question asks you to calculate , we know from basic algebra that we can combine like terms. Thus, we would combine the terms with  and add the two constants together to get . Turns out the  in a complex number works very much like a  in an algebraic expression: you can combine them. Hence adding and subtracting complex numbers are as easy as combining like terms. Multiplication works almost the same: we can apply the FOIL method when we multiply two complex numbers together, like below:

The only thing we have to remember is…

 needs to be changed

A complex number is defined as , where both  and  are real numbers. We are not allowed to put any other terms in there. So when we end up with the term , we need to rewrite it. Because  is defined as, we have:

As a matter of fact, we can apply higher integer powers to  and find a pattern:

As we continue, we can see that when we are given a positive integer power of , we can rewrite it in terms of power of  (and if you remember your rules of exponents, this means we multiply the power with 4), and whatever is left will tell us the final form of the number. In another words, these are the rules you need to remember for evaluating , being a positive integer:

1. if  is a multiple of 4, then 
2. if dividing  by 4 and the remainder is 1, then 
3. if dividing  by 4 and the remainder is 2, then 
4. if dividing  by 4 and the remainder is 3, then 

Dividing requires rationalization

If you get a question where the imaginary number is being placed in the denominator of a fraction, we need to recall from our rules of radicals that roots are not allowed in the denominator. Because , which makes it a radical, we need to rationalize the denominator to make it into a proper form. For example, if we are given:

We need to multiply both the numerator and the denominator by  to achieve a in the denominator and get rid of the . Thus, we get:

Recalling from the previous part that when we expand,  will become -1, we will end up with the answer:

In complex number, we have a special name for the value we multiply to rationalize: a complex conjugate. The idea is as simple as changing the sign for the number with  attached to it. The proper definition is as follows:

Let  be defined as a complex number such that , where  and  are real numbers.

Then the conjugate of , denoted either  or , is defined as:

With this knowledge in your arsenal, you will be able to tackle any questions from the SAT or ACT that involve complex numbers. However, if you are interested in more advanced idea about this number system, here are some additional facts.

It’s not all “imaginary”…

While the name sounds like it’s “made up”, we can actually visualize complex numbers.

Don’t make the “imaginary friends” pun when you work with complex number, courtesy of xkcd.com

The first thing we should see is that complex numbers are made up of two components. If we think of them as the “” and “” of the coordinate plane, we are able to create a plane that has “real” and “imaginary” as the two axes. This plane is called the “complex plane”, and it is here that we visualize complex numbers. Instead of being a point on a plane, however, complex number is shown as a vector instead. That’s because we would end up operating with these entities, and we simply cannot “add” or “multiply” two points. Once we have this plane in place, there are many ways we can visualize some of the more advanced mathematical concepts. For example, proving the Fundamental Theorem of Algebra becomes a task of drawing when we apply this concept.

It’s more than just 

            Using the “” and “” of the coordinate plane as a foundation of visualizing complex number is really an application of the more general “Cartesian Plane” structure. However, there are other ways to represent a complex number. One common way to write a complex number is using the “Polar Coordinate.”

            Polar coordinate uses the angle formed between a line and the length of that line to represent the end of the point. Since we mentioned earlier that complex numbers are drawn as vector in the complex plane, we can easily convert the “” and “” coordinate into polar coordinates by creating a simple right triangle and follow some trigonometric operations.

Here is a picture of how polar coordinate works. Having trouble understanding this? No worry, let Ron the Cartesian Bear help you out.

So why bother with representing the same value in a different way? One reason is the ease of calculation using this different form. Turns out that finding the product, quotient, power, and roots of complex numbers in this form is as simple as adding, subtracting, multiplying , and dividing the angles, respectively (the latter two, which involves power and roots, follow a very important formula called the De Moirve’s Theorem).

Here is a picture to help you remember the difference between Polar and Cartesian: Polar is round, Cartesian is boxy.

And we are only scratching the skin of complex numbers. Come join me and learn more about this amazing number system that is bigger than the real numbers you are used to working with. And before we go, let’s see if you are as smart as Homer and understand the joke here:

Need help with your ACT/SAT? Check out our courses here or call us for more information on private lessons/additional course offerings.

Check out The Edge’s other ACT/SAT Blogs

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About The Edge

The Edge Learning Center is Hong Kong’s premier Test Preparation, Academic Tutoring, and Admissions Consulting services provider. Founded in 2008, The Edge has helped thousands of students improve their ACT and SAT scores as well as their IB and AP grades. The AC team has just finished off a very successful year in which 84.62% of their clients were accepted into one of their top 3 schools and an astounding 48.15% of their Ivy Plus* applicants were accepted. (The general acceptance rate was only 7.61% last year) Check out the rest of our 2017 Admissions Results!

*Ivy Plus: All Ivy League Schools + Stanford & MIT

The post Decomplexifying Complex Numbers appeared first on The Edge.

(Test Preparation (ACT/SAT/SSAT), English Literature, English Builder tutor at The Edge Learning Center)

For many students, the mathematics sections of standardized tests are like a skinny guy at a sumo competition: no big deal and nothing to really worry about. As YouTube teaches us (watch here), though, you can’t trust a skinny sumo wrestler. Here are six things you should know when approaching the math sections of the SAT, ACT, SSAT, and more.

Read more on Steve’s previous blog “Reading Dead Redemption”

1. Know Your Situation

Each test is different. They all test slightly different topics in slightly different ways with slightly different rules. The ACT, for example, lets you use a calculator on the entire math portion, but the SAT only allows calculators on one math section, and the SSAT doesn’t allow calculators at all (the jerks!). Similarly, the SAT, warns in its official documentation that it tests standard deviation, but it has yet to use the formula. It’s only been testing the general concept, so memorizing and practicing the algebra isn’t the best use of prep time. Knowing what you can and cannot do lets you plan better. Learning how to use your overclocked TI-9000 to solve polynomial division is great until you get to your desk on test day and discover that it’s not allowed. Rules and guidelines for the ACT are here and for the SAT here.

2. Know how to use your calculator

Okay, so your calculator’s on the approved list (good old TI-85!), but if you don’t know how to use it properly, it’s about as useful as a chocolate teapot. Many calculators, even simple scientific models, have powers beyond what their poor users know. Often, students don’t realize that they can use their calculator to find sin^-1 or to multiply the dread matrices. Take time to get comfortable with how to operate your machine quickly and effectively. User manuals and guidebooks are freely available online from manufacturers, and the internet has plenty of hacks and tricks from calculator cowboys. HOWEVER! Make sure to leave your calculator in factory condition! Modified calculators or downloaded apps and programs may be considered cheating, which leads to trouble nobody wants.

3. Know how not to use your calculator

That said, even the greatest calculator ever built can’t do everything, even if you’re allowed to use it. At any rate, the meat computer in your head is far more powerful than the silicon one in your hand. Practicing without a calculator exercises your brain’s math skills (when did you last review your multiplication tables? Prime numbers?) and also helps you understand when using the calculator might actually slow you down. Being able to solve any and all problems with raw mathematic brawn is awesome, but it might be slow and it’s certainly exhausting, especially under test conditions. Knowing when not to use your calculator means recognizing when you don’t need to actually crunch the numbers. How? Well….

 4. Know that there are patterns

Finding multiples of ten is easy: if it ends in a zero, it’s a multiple of ten. Fives are pretty easy, too, and then so are twos. This is probably written so deeply in your brain that you can’t remember if you were taught it or just always kinda-sorta knew. Seeing these patterns means you don’t have to go through the whole process of actual calculation, which is where simple mistakes creep in. There are way more of these patterns than you might realize, too. Some of them are general math ideas, like how raising any number to the power of 2 gives up a positive number, and others are more related to the specific ways certain tests approach certain concepts (if only there were somewhere you could go for that kind of expertise…). Similarly, a lot of ideas in math are related to other ideas, so if we can’t exactly find the “right” method, we can often figure out another way using a slightly different area of math. We can, for example, use our knowledge of functions to find the solutions to a system of equations Practice with both general math and your specific test is one of the best ways to start seeing these kinds of patterns.

5. Watch for simple mistakes

Ah, simple mistakes, the enemy of math-magicians everywhere. We’ve all missed points in math class because we missed a negative sign or made a simple calculation error. Even the best of the best of the best of math students make these simple mistakes, but there are things we can do to help prevent them. The first thing is to go back to the top of this list and read it all again. Everything here can help prevent simple mistakes and so does writing down all of your work. None of the tests we’re talking about offers style points for not showing your work, and trying to do a bunch of mental arithmetic in the stressful, timed conditions of test day leads to more and more simple mistakes. Sure, you might catch the errors in time, but how many minutes does that cost, and how many minutes do you have? This brings us nicely to our last point….

6. Know that test prep math isn’t exactly the same as classroom math

The goal in the classroom is knowledge. The teacher wants you to understand the ideas in the lesson. The goal on test day is the score. This subtle difference can greatly influence how you approach to test preparation. Your emphasis should be on efficiency, and your study plan should look at the best way to get points based on individual strengths and weaknesses. Now if only there were somewhere to go for that kind of expertise…. The Edge Learning Center.

Read more about Calculator Etiquette and Other Tips

Need help with your ACT/SAT? Check out our courses here or call us for more information on private lessons/additional course offerings.

Causeway Bay: 2972 2555 / Mong Kok: 2783 7100

About The Edge

The Edge Learning Center is Hong Kong’s premier Test Preparation, Academic Tutoring, and Admissions Consulting services provider. Founded in 2008, The Edge has helped thousands of students improve their ACT and SAT scores as well as their IB and AP grades. The AC team has just finished off a very successful year in which 84.62% of their clients were accepted into one of their top 3 schools and an astounding 48.15% of their Ivy Plus* applicants were accepted. (The general acceptance rate was only 7.61% last year) Check out the rest of our 2017 Admissions Results!

*Ivy Plus: All Ivy League Schools + Stanford & MIT

The post Math Effect: 6 Things You Should Know About the ACT, the SAT, Calculators, Math, and You appeared first on The Edge.

(Test Preparation (ACT/SAT/SSAT), Math, Physics at The Edge Learning Center)

Students often first encounter the idea of factorial when they learn about probability, particularly combinatorics. They often see it again in binomial expansion or binomial distribution. If they go into advanced calculus, they would see it popping up in Taylor series and Maclaurin series. Most students will just remember that factorial is defined as “multiplying all the positive integers up to the particular number.” Today, I would like to share some facts about this exciting(!) operator in mathematics and certain rules to follow when working with it.

Read more on the importance of math in life in Leo’s blog “A link between Math and Life”

You don’t need to yell

The exclamation mark notation was introduced by French mathematician Christian Kramp in 1808. The idea of multiplying consecutive whole numbers has been around for centuries; people had been using this method to find out the distinct way to order multiple objects (i.e. permutation). Having this notation, as Kramp personally suggested, makes writing down things much easier.

You might see it as Pi

The need of multiplying numbers is not unique to the consecutive whole numbers. As a matter of fact, just like the need to add certain terms together (which we use the Sigma notation, Σ, to denote such operation), it is more common to see the product notation Π. A typical Pi notation works just like a Sigma notation; instead of adding, you just multiply each term! Here is an example:

It gets huuuuge!

If you have a single-line display scientific calculator with only 2 digit displayable exponent, you would likely find that typing 70! will give you an error. It doesn’t mean the value cannot be calculated. 70! is bigger than 1 googol, which makes it impossible for such calculator to display. Here is a list of some of the bigger factorial values:

Your calculator/computer doesn’t just multiply

Mathematically speaking, calculating factorial is easy: you just multiply a bunch of numbers together. However, simple as it may seem, most computers don’t find the answer by just multiplying. The sheer value means there are certain limitations. Factorials are always integers because it’s the result of multiplying integers together. Modern computers covert our everyday numbers into binary, using only 0s and 1s, before they do any calculation. Older computers are limited to working with 32 binary digits, or bits, which translate to a maximum of 2,147,483,647 in decimal. For the newer computers with 64-bit architecture, it can store an integer in decimal value up to 9,223,372,036,854,775,807 . Which is a big number. But if you look at the table, you will see that a 32-bit computer can only calculate up to 12!, while 20! is the limit for a 64-bit computer. Beyond these boundaries, most common computers have to resort to approximation to provide an answer.

It only works for whole numbers… or does it?

Most students know that we can’t do 2.1! or (-3)! because factorials are limited to positive whole numbers. However, mathematicians have extended the application of factorial, called the Gamma function, into complex numbers (with the exception of negative integers). This application is often utilized in mathematical analysis and complex analysis.

Above is the graph of the Gamma function plotted for real values.

At this point, you would probably say, “yeah, that’s exciting(!) and all, but it’s not like I will ever need to apply any of that. Show me something I can use now!”

So here it is, certain rules and tricks that you want to remember about factorials.

0! = 1

As weird as it may sound, this is a fact that we must remember. There are several ways to explain why this must be true. In mathematics, the empty product rule (which is very different than the “null factor law.” I have written an application of the null factor law in my previous blog, which you can find here) states that when we are multiplying no product, the result is set as default to the multiplicative identity, which is 1. 0! Literally means we are not multiplying any number, so it falls into the empty product rule. Thus, the result should be 1.

Another way to explain 0! is to look at it like finding ways to order items. We learn from permutation that ! gives you the number of unique ways to order  items in distinct sequence. Now think about what 0! represents; it is kind of like “how many ways are you able to put nothing in order.” And the answer is… 1 way – by ordering nothing. More importantly, this definition fits into the model or permutation and combination.

A third way to look at this is to see how consecutive factorials relate to each other. For example, we know that:

but we can also rewrite it as:

Which relates the two consecutive numbers. If we continue to move our list down, we can see that:

nCr and nPr will always give you a whole number

This is a reminder to those who is given a non-calculator question that involves combination and permutation. If you need to calculate these values, the result will always be a non-zero whole number. Keep in mind that both of these functions are finding “number of ways” to perform something, so getting a fractional result would not make any sense.

You can’t “cancel”, “split”, or “combine” factorials

A common mistake that students make is doing something like:

Which is very tempting to do, because they look just like a fraction. However, if we expand the terms, we will see that:

which is different. We can think of the factorial as an operation that needs to precede multiplication and division, which limits how we can simplify terms.

Thus, we also need to remember that:

Simple counter examples will help us check that the operations above are invalid.

However, you can still manipulate factorials

One of the most important techniques regarding factorials is that we can rewrite a factorial as products, which can be rewritten again if we are being careful. For example:

While we are not splitting the 15 into smaller numbers, we are able to “peel” each layer off and create a different factorial. This means the following ways to rewrite a factorial are all valid:

It is also possible to “add layer” by making the factorial bigger, like this:

Thus, here is another way to make a factorial look different:

Now let us take a look at an example and see how we can actually make certain factorials look simpler.

or

The key is to work with the larger factorial and change it to a multiplication between individual term(s) and a smaller factorial.

And now you know more about this unique operator in mathematics. Next time when you encounter the exclamation mark in your math assignment, don’t fret! Remember the rules and tricks and you can shred them with ease.

Hopefully you won’t have this face anymore when you are given a factorial question.

Need help with your AP/IB/IGCSE Math? Check out our courses here or call us for more information on private lessons/additional course offerings.

Causeway Bay: 2972 2555 / Mong Kok: 2783 7100

About The Edge

The Edge Learning Center is Hong Kong’s premier Test Preparation, Academic Tutoring, and Admissions Consulting services provider. Founded in 2008, The Edge has helped thousands of students improve their ACT and SAT scores as well as their IB and AP grades. The AC team has just finished off a very successful year in which 84.62% of their clients were accepted into one of their top 3 schools and an astounding 48.15% of their Ivy Plus* applicants were accepted. (The general acceptance rate was only 7.61% last year) Check out the rest of our 2017 Admissions Results!

*Ivy Plus: All Ivy League Schools + Stanford & MIT

The post Fun! Facts about Factorials! appeared first on The Edge.