Category Archives: Math

 Answers Don’t Crawl Out of The Rubble

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If you haven’t seen it yet, take a moment to watch this video which shows one of the more feel-good stories to come out of the horrific damage caused by the tornadoes in Oklahoma.

This woman goes into the storm with a plan. Granted, it’s not the best plan, but it’s a plan nonetheless. Then the storm comes and throws everything around and all that’s left is a pile of rubble. And somehow, under a pile of debris, appears the dog that was feared to be lost. It’s a tremendous story, but it’s not a model for you to take in your test prep.

Some students have what I call “Mad Scientist Disease”. They imagine that upon seeing numbers and equations they should immediately start performing calculations, scribbling notes so quickly that smoke rises from the paper as the sweat pours down the intensely furrowed brow. After several moments of this process they imagine that an answer will reveal itself as correct. They create rubble and hope that the thing they hope for will crawl out from under it.

Math problems are not best solved by hoping for the miraculous. Unlike the tornadoes in Oklahoma, students who go down this road are creating the disaster rather than dealing with a force beyond their control. Rather than jumping directly into calculations you should begin at the end. Figure out what it is that you need to solve for in order to find your answer and then figure out what information you need in order to get that intermediate information. Once you’ve plotted out your steps, THEN it’s safe to begin the process of doing calculations.

Planning your approach before the danger comes is the best way to minimize the danger of a natural disaster, and the best way to survive your test unscathed.

 

 Excuse Me, But Do You Speak Math?

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In three years of high school Spanish, I learned a lot of words. I am confident that if dropped into a Spanish-speaking country I could successfully order food, find the bathroom and greet the people I met in an awkward and excessively formal way. In my three years of law school I learned almost as many words, which seems surprising given that all of my classes were taught in English. The fact is that although theoretically in my native tongue, joinder and mandamus came off as foreign to me as cebolla and encantado once had been.

 

While sitting in a business meeting recently and struggling to keep up with acronyms, who was working in which “space” and whether or not the team had sufficient bandwidth to take on a project, I had a realization. Many areas of our lives have a different language that seems completely natural once we’ve familiarized ourselves with it, but can be extremely difficult to penetrate at first.

 

Math too has a language all its own. The string of words that appear in the question can seem impossibly far from the equation that you know is supposed to be elicited. Here are some quick tips for gaining some comfort with the words that compose math problems.

 

  1. Find the equal sign- If you’re untangling a word problem, it can be difficult to decide where to begin. Start in the middle. Any equation has two sides and separating those two sides is a word that is equivalent to the equal sign. It may be “is”, “equals”, “are”, “was”, or “will”. Focus on finding that word that means equals and start from there.
  2. Represent percents as x/100- One of the common language issues is translating percents into an equation form. Just remember that per means “out of” and cent means 100 (just like century or centipede).
  3. Take your time- Most of the mistakes made in translating from English to math are not gaps in knowledge, rather they’re careless errors. If you take your time you’ll certainly realize that 10 liters of a substance evaporating means to subtract or that separation into equally sized groups means division. Walk it through step-by-step.

 

Here’s an example problem for some practice:

 

A positive number y is multiplied by 3, and this product is divided by 2. If y percent of this product is 10, what is the value of y?

 

Focus on that key language “y percent of this product is 10”. The “is” in that phrase signifies the equal sign, to that’s where we’ll start. Something equals 10.

 

The next step is to unpack that something. We start with the y. Multiplying by 3 yields 3y. Then we divide that by 2.

 

“y Percent Of”

 

Remember the second tip. We can translate that “y percent of” language by simply substituting y/100:

 

 

At this point we have a common equation that you can surely solve. Remember to find the equal sign first and take your time and you’ll get your translating done with ease.

 Invisible Gorillas: Prime Factors

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In the last few blog posts we’ve explored “Invisible Gorillas”. As mentioned in the original post, Invisible Gorillas are patterns that recur on standardized tests that you can easily miss if you’re not looking for them. However, they’re simple to spot if you’re on the lookout for where they might show themselves.

 

I am not old. Granted, I didn’t get my first cell phone until I was in college. And yes, the first music I ever bought was on cassette tape. And sure, Google didn’t exist when I was born, but still. I am not being old. So why didn’t my teachers cover prime factorization when I was in school?

 

It’s not like the concept of prime factors is anything new. I would give you some historical evidence of this, but this newfangled Google thing gives me some trouble sometimes. But still, why wasn’t prime factorization more prevalent in schools when I was growing up?

 

For those of you as old as I am (or at least as disadvantaged as I was in not getting this information in school) prime factorization says that we can break any number down to smaller pieces—it’s prime bits—and make it easier to handle. We do that by first taking any factor pair and then seeing whether the factors are prime, or whether they are non-prime, in which case we repeat the process until we have all primes. Take the following example:

 

 

Our analysis shows us that 84=2*2*3*7. This in itself seems pretty simple, but making things simple is often the major step that you need to take in order to solve what looks like a complex problem. Take the following for example:

 

 is a factor of 540. What is the greatest possible value of x+y?

 

Although the form makes this look daunting, it’s simply an exercise in prime factoring. Watch what happens when we re-write the information in that form:

 

 

So, 540= 2*2*3*3*3*5. All that’s left to do is re-write that in a form that more closely resembles the form in which the problem was given.

 

 

Once the number has been broken into its prime factors, it’s plain to see that we can only squeeze two twos and three threes out of this number. There’s no different way we could break it down to get more twos or threes. Prime factoring figures out how many pieces of each type we have and allows us to re-assemble them in any way we see fit. So the greatest possible x is 2 and the greatest possible y is 3. Thus, 2+3 is equal to our answer of five.

 

Whether you’re old or young or somewhere in between, jump on board with prime factorization because taking numbers and breaking them into their component prime parts is a great way to solve problems and a recurring pattern on your test that you’ll be sure to recognize if you look for it!

 Invisible Gorillas: Answer Choices in Different Forms

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In the last few blog posts we’ve explored “Invisible Gorillas”. As mentioned in the original post, Invisible Gorillas are patterns that recur on standardized tests that you can easily miss if you’re not looking for them. However, they’re simple to spot if you’re on the lookout for where they might show themselves.

 

Today we’re looking at the form of the answer choices. It’s a frustrating experience to feel that you’ve done every step correctly in a math problem and to reach the end only to realize that your choice isn’t there. Self-doubt can quickly creep in leading to thoughts of “Where did I mess up?”, “Do I need to start over?” or “Whyyyyy is this test sooo harrrrrrrd?”

 

Avoid the whininess of that last question and realize that having the value you reached at the end of your work not line up with the values in the answer choices is a common experience, and it doesn’t mean that you did anything wrong. Often by slightly transforming the answer choices the testmaker can take a simple question and make it one capable of separating the unprepared student and the test-taking superstar (that’s about to be you!).

 

Here are three common transformations that you’re likely to see.

 

  1. The answer is a combination of variables- Perhaps there’s a fact pattern that’s set up to have you solve for x. After going through all the math you’re confident that you’ve correctly solved for x, but you don’t see the value that you got. Double check to make sure that the question doesn’t ask for some form of x, such as 2x or x+y. It’s hard to get the right answer when you’re answering the wrong question!

 

  1. The answer is a reduced fraction- This is perhaps the simplest one on the list, but the correct answer to the problem will be the fraction reduced to it’s simplest form. If your answer is 18/51 and you don’t see that in the choices, don’t worry. The testmaker’s version—6/17—is likely just a reduced form of the fraction that you didn’t initially see.

 

  1. Your answer has a radical at the bottom of a fraction- For some reason many students fail to see that the following is true:

 

 

If you’re left with a square root in the denominator of a fraction, simply multiply both the top and bottom of your fraction by the same square root in order to simplify. Remember, that by the definition of a square root when you have this:

 

 

You can re-write it as:

 

And cancelling will leave you with:

For example:

 

 

Keep these examples in mind as you practice and you’ll realize that some of those times when you start to worry that you’ve gotten the wrong answer are just situations where you haven’t recognized your correct answer in another form!