Math Strategies Tutorial What to Trust There are certain things in most standardized exams that you can trust. One of them is grids. If something is on a grid, you can trust that the figure is accurate, and you can use the grid lines to help you solve the problem. The general rule of thumb is that grids never lie. If the standardized test makers want to trick you, they will not put it on a grid. The next thing you can believe in is charts. Charts don't lie either. Whenever something is in the format of a chart, you can believe what it says. No traps here. It simply wouldn't make sense for a standardized exam to provide you with a faulty chart. Graphs also tend to be accurately drawn. If you see a graph on an exam, there's typically no reason to doubt it. Otherwise, why would they give it to you? The goal of the test designers is not to deceive you unjustly. After all, they've built their reputation on constructing fair and reasonable exams. The only exception to these rules is when a note appears on the standardized exam telling you not to trust a certain piece of information. If this is the case, don't trust it, no matter what it is. Total Distortion Sometimes its smart to go a bit wild and take a diagram and redraw it in an intentionally distorted manner, exaggerating certain features so that it is not to scale. This might seem crazy, but it can help you see the image in a new way and arrive a solution that might otherwise be obscured by the way in which the diagram is depicted. Once in a while the creators of standardized exams will try to trick you by providing you with a diagram that is labeled "Figure not drawn to scale." Then they try and confuse you by putting in a 45 degree angle that actually looks like a 20 degree angle, and making some of the lines appear longer or shorter than they actually are. If this is the case, it pays to play around with the diagram. You can change it in any way you want as long as you conform to any of the given data. The more you play with it, without breaking any of the conditions set forth, the greater your chances of coming across the solution. Less is More There are many ways to skin a cat, as the saying goes, but the best way is always the simplest. Don't make work for yourself. Before you dive into solving a problem, look for shortcuts. Try to think of the easiest possible way to get at the answer. There are many advantages to this. You might spend a little time up front thinking about shortcuts, but this might save you a lot of time on the backend. Also, the simpler your calculations, the less chance of making a mistake. Remember, there are often many ways to solve a problem and the first thing that comes to mind isn't always the most efficient. Create Visual Representations Visual representations can help in a wide variety of math questions, from those dealing with spatial relationships to those which define abstract groups or situations. If you can think of a way to take the abstract and represent it visually, it can be a great problem solving tool. Whenever you encounter a geometry question and there's no diagram or other visual aids, you should always draw your own. Creating a diagram can make it much easier to solve the problem. This said, many students are intimidated by the act of drawing. You don't have to be an artist or a draftsperson to create useful diagrams. All you need to do is to be able to create a visual model of the problem in your head and then put it down on paper. The drawing can be crude and messy and ugly, as long as you can understand it, and it helps you to work through the problem. Sometimes only a few lines forming angles is enough to get the job done. Keep in mind that most of the questions on standardized exams can be answered without ever creating a diagram, but we strongly encourage you to get in the habit of creating drawings whenever it seems helpful. First of all, putting something down visually enables you to clearly think through the problem and see solutions you otherwise might miss. Secondly, the act of creating a visual representation reduces the likelihood that you'll make a mistake. It's hard keeping track of everything in your head. The more you have down on paper, the less the chance of an error. Seeing is Believing Seeing is believing, and when it comes to looking at diagrams that appear on standardized exams, you should trust your eyes. Unless a diagram clearly states that the figure is "not drawn to scale" or has some other warning, you can count on what you see as being accurately drawn to scale. In other words, if one angle appears to be acute, then it is acute. And if a line looks longer than another line, it is longer. Whatever you see on the diagram is accurate, so go with your visual sense as much as possible. It can save time and help you work through the problem. Redrawing Diagrams There are times when diagrams are not drawn to scale. If this is the case, you should invest the time to redraw it to scale, and then use your eyes to help you solve the problem. You'd be amazed at what a difference this makes. It's much easier to solve problems when they look as they are. If they aren't drawn to scale, it can be quite confusing and misleading. People are visual, and we tend to think with our eyes, so the more you can help yourself in this regard, the higher your chances of answering correctly. In other words, don't be in a rush. Take the time to redraw anything which isn't to scale. Remember, if something is not drawn to scale, it is an invitation for errors. For instance, two line segments that appear to be the same length might actually be different lengths, or lines that look perpendicular may not be perpendicular, or an angle that seems to be obtuse may not be obtuse. You can't really tell anything until you recreate the diagram to scale. Only when you do this can the diagram actually help you to solve the problem. This is why it's so important to invest the time up front in redrawing it. One thing to keep in mind is that redrawing a diagram to scale is no easy task. You simply cannot trust what you see, so you have to go on facts. If two lines are supposed to be perpendicular but they don't look perpendicular on the diagram, then you have to draw them as perpendicular lines. The same goes for angles: if an angle is marked 75 degrees, but it looks more like it's 30 degrees, then you have to draw it as 75 degrees. And if two line segments appear to be the same length, but one is clearly marked as larger than the other, then you have to redraw the lines according to their designated length. If you trust the numbers and not what you see on the diagram, your recreation will be accurately scaled. The only caveat is that it takes a lot of effort and time to recreate a diagram to scale. If you can see the solution (using the numbers and not relying the accuracy of the diagram), then it's better to forego the task of recreating the drawing to scale. That will only slow you down unnecessarily. Missing Lines Some diagrams are missing lines. Yes, the creators of standardized exams are tricksters, and they'll resort to all types of clever schemes in order to fool novice test tasters. Of course, these types of tricks are not unjust. They are always done in a way which is both fair and reasonable. If you find yourself staring a diagram and get that sinking feeling that some critical piece of information is missing, well maybe you're right. Try adding a line here or there. Fool around with it, and see if something jumps out at you. You'd be amazed at how a single line can change your whole perception of the problem and bring a solution into view. Shaded Regions You'll come across diagrams where certain regions are shaded and others are left unshaded. If the question asks you to find the area of the shaded region, a simple method is to first find the area of the entire figure, then subtract the area of the unshaded region. This will leave you with the area of the shaded part. Sometimes you'll be able to compute the area of the shaded region without resorting to this technique, but often this is not the case. That said, if the question happens to ask for the area of the unshaded region, just reverse the process, subtracting the area of the shaded part from the area of the entire figure. Either way, it's the same technique. Units Units are everything. If you answer the question correctly, but don't check to make sure the units are right, then you haven't answered the question correctly. The creators of standardized exams tend to ask you for an answer in specific units, such as meters and centimeters, or feet and inches, or seconds, minutes, hours, etc. If you do all the work but don't convert your answer to the proper units, you'll get it wrong. Many standardized exams even try to trick you by labeling the diagrams with different units from what's requested for the answer. Also, there may be wrong answers corresponding to the units on the diagram. A good way to avoid making a mistake is to circle the units in which the answer must be in. This will help remind you to convert to those units at the end of your computation. It's too easy to forget in your haste to move on to the next question. Bring a Calculator If a standardized exam allows you to use a calculator, it's foolish not to bring one. And don't make the mistake of bringing a brand new calculator. You should be well acquainted with your calculator and all of its functions. Practice using it many times before you use it on the exam. Don't Overuse Your Calculator Watch out that you do not become totally dependent on your calculator and feel that you must use it on everything. Sometimes it's faster to compute the numbers in your head than trying to type them into the calculator. Also, calculators are prone to mistakes. It's easy to mistype a number. A good idea is to first compute rough estimate in your head and write it down on paper. Then when using your calculator, you can see instantly if you're way off base. Also, thinking things through without a calculator can help you arrive at solutions you otherwise might not see. Lists If you see the words "how many," you can be pretty certain that it's time to make a list. Lists help you to see all the possibilities. It's hard to keep everything in your head, and writing the items down allows you to keep from leaving things out and counting certain items more than once. As you create the list, look for patterns. If you see a pattern developing, you might be able to accurately compute the remaining items without completing the entire list. Even when the question doesn't ask "how many," a list may be in order. It really depends on the question. If you find yourself trying to track a large number of possible items, a list may be the best solution. Symbols When taking a standardized exam, you may come across a variety of strange symbols, some of which you've never seen before. Don't freak out. It isn't that bad. Whenever you see a strange symbol, rest assured that the definition of that symbol is close at hand. The question will always tell you exactly what that symbol means. It's smart to think of the symbol as a variable. It can mean whatever the creators of the exam want it to mean. Once they tell you its meaning, just use it like you would any other variable. Don't be afraid to plug it into equations or use it as part of your computations in any way that makes sense. Just because it looks different than x or y, doesn't mean it functions any differently. You may also notice that some symbols crops up again and again in other questions. Typically, the first time you see a symbol in a question, it involves only numbers. The second and third times it appears, it usually becomes more complicated. A good way to practice is to create your own symbols and use them in equations. You'll quickly see that there's nothing mysterious about these symbols. Once you get the hang of them, you won't panic when they appear in a question. Multiple Equations Sometimes you'll encounter more than one equation. This can be disconcerting. But it's not hard to deal with. Whenever a question involves two equations, you should either add them together or subtract them. As a rule of thumb, when there are three or more equations, always add them together. The one thing you should keep in mind is that if the question doesn't ask you to actually solve the equations, then don't do it. It's a waste of time. Only do what the question asks you to do. Less is more!