Mathematical Applications on the SAT
The College Board emphasizes that the Mathematics section on the new SAT is intended to test especially the mathematical knowledge that will be relevant for a broad range of careers—not only the mathy professions like accounting, statistics, or chemistry—as well as for the needs of daily life. Mathematics for the non-mathematicians, in other words.
This shift away from abstract or specialized mathematics towards mathematical applications is evident in the many questions that draw on real-world scenarios to frame their problems, from household finances and personal loans to off-the-cuff grocery-store calculations, simple estimates of productivity, and even computation of the volume of text messages sent and received (the last could be handy for figuring out your cellphone bill!). While there remain more “traditional” math problems testing, for example, parabolic graphs and systems of inequalities, these are now a shrinking portion of the material.
In a slightly more abstract way, the shift in emphasis can also be seen in the form that the questions take. Rarely do the mathematical problems that we face in the world come to us in the form of ready-made equations. On the contrary, we have to devise the mathematical equations ourselves by selecting the variables and operations that are appropriate for a particular situation. In most cases, these real-world equations will be embedded in a broader explanatory context that uses ordinary English in addition to, or instead of, the symbolic language of mathematics. This context shows us why we are using mathematics and what problem it will help us solve.
When problems that mix mathematics and ordinary language show up on tests, they are usually called “word problems.” In dealing with such problems, it can be as important to determine the significance of the information you’re given and the meaning of the results as to perform the algebraic or geometric calculations themselves. It’s not so much math for its own sake, as math for the sake of solving a concrete problem. My goal in this post is to provide a brief introduction to some of the forms that word problems take on the SAT, and how you can start coming to grips with them.
The simplest word problems (although not always the easiest) are those that are just a mixture of mathematical equations and English statements. Consider the following example:
You are given at first two mathematical equations and a chunk of text. In all word problems, the first thing you should do is read the text carefully. Treat the word problem as though it were a passage on the reading section: underline, circle or otherwise mark significant information, which will likely include explanation of variables and a statement of your task.
Here, we are first told what the three variables (b, c, x) given in the equations represent: the price per pound of beef is b, of chicken c, and the number of months after July 1st is x. We are then given the problem: “What was the price per pound of beef when it was equal to the price per pound of chicken?”
The next step in tackling these kinds of word problems is to translate the ordinary language into math. The SAT will always give you enough information on word problems to be able to change the English expressions into mathematical expressions. By substituting variables and operations for English expressions, we turn word problems into recognizable algebraic or geometric equations.
If we perform this substitution for our question, “What was the price per pound of beef when it was equal to the price per pound of chicken?”, we come up with, “What was b when b = c?” The relevance of the two equations given at the beginning of the problem should now pop out. We need to set them equal to each other, determine the value of x, and then plug the value of x back into the equation for b in order to find our answer. This is simple algebra that you can easily handle.
Of course, one of the difficulties of word problems is mastering the technique of translating English into math and vice versa. You will know much of it already from your math classes, but this is where practice with SAT sample questions will help you become familiar with the language of the test. Your tutor will also be able to give you some of the basic equivalences.
Charts and Tables
Some word problems will utilize charts or tables to visually represent information. Take this example:
It is useful to adopt a modified strategy for tackling word problems when you have a chart or table at hand. In these cases, do not begin by reading through the preliminary information and by trying to decipher the chart, but instead immediately determine the task. The reason for this is that it will be difficult to evaluate the significance of the information you are provided until you know what to look for. If you know what you need from the table, you can go directly to what is relevant for your purposes. We are told here that we need to find a “reasonable approximation” of the number of earthworms 5cm under the surface of earth in the entire plot.
Once you have determined your task, go back and read the text that you skipped while going to the task. It may contain important information that will help you understand the variables and presentation of the chart, and it is never safe to ignore it. This problem is a case in point. We learn that the students marked off 10 “randomly selected,” non-overlapping regions of the field measuring 1m x 1m each. Moreover, we learn that the entire plot is 10m x 10m. This information is vital, for, once we translate the English into math, it allows us to ascertain that the randomly selected regions account for only 10% of the size of the entire plot. (1m x 1m = 1m2; there are ten such plots, thus 10 x 1m2 = 10m2; whereas the plot is 10m x 10m = 100m2; finally, 100m2 /10m2 = 10.) Therefore, we cannot merely sum the values given in the chart to determine our answer, but must rather multiply that figure by 10 in order to find an approximation for the entire field. This little trick is what makes the problem challenging.
Finally, look at the chart itself, using the task to guide your eye. We are looking for a “reasonable approximation” of the number of earthworms 5cm under the surface of earth in the entire plot. We have already hinted at the answer above: since the preliminary information lets us know that the plots do not overlap and that altogether they are 10% of the total area, we need only find their sum and then multiple by 10. A rough approximation of the number of earthworms in the 10 plots is 1500. Multiplying by 10 we get 15000, which happens to be the correct answer, C.
Lastly, word problems can also incorporate graphs of all sorts. Here is an example:
Just as for charts and tables, do not start with the graph, but instead look for the task. In this case, our task is contained in problem #15, which asks what the C-intercept in the graphs represents.
From here, we can work our way back to the graph and the other information provided by the problem. Reading the labels for our graph proves to offer almost all of the information that we need to solve our problem. We see that C is the cost renting a boat for h hours. Answer choices B and C are immediately off the table, then, since they deal with numbers of boats and hours, not cost. A and D are better, but a moment of reflection will show that the intercept here must be a single value, not a relationship. A is the correct answer then. D is a potential pitfall for those who read the problem quickly and see the problem “What does C represent in the graph?” instead of “What does the C-intercept represent in the graph?”
Word problems can be challenging for a number of reasons, and each one demands a slightly different approach. But if you treat each one as a self-contained reading problem and master the art of translating English into math, you will also master word problems on the SAT mathematics sections.
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