Helpful Hints for Taking Math Tests

Helpful Hints for Taking Math Tests

The night before the test get plenty of sleep. Eat well prior to the test (both protein and carbohydrates.) Take your I.D., three of four pencils, your watch (and if allowed, a calculator with freshly installed batteries.) Arrive at the test site a little early. If your find yourself getting anxious, walk briskly for a few moments to get your heart rate up just a little (this takes the edge off any adrenaline jitters.) FOR PAPER and PENCIL TESTS There is usually a percentage of incorrect answers that are deducted from you total score, so you must be very wise in guessing. Your strategy is to maximize your score in the time permitted, without incurring penalties. On each section, you will be told how many minutes you have for that section. Note what time you start, and figure the time at which you have to complete that section. For example, if you start a section at 8:37 and you have 25 minutes, time will be called at 9:02. Now subtract 3 minutes from that time. Write that time down so you will know when you need to stop working problems and start filling in the answer sheet or grid. In our example, that would be 8:59. As you go through the problems: Work in the test book. Use any available blank space for figuring, as needed. Circle your answer choice Also write the letter of the answer choice next to the problem number. You will be filling in the answers on the blanks or grid during the last three minutes, so do not put anything on the answer sheet as you are working problems. Do the problems in three waves: First wave: Do the ones you know how to do and can do quickly. Star any problems that you think you know how to do, but will take more time. Circle any problems that you don’t know how to do. If it is not a multiple-choice problem, guess. There is no penalty for guessing on these short answer types. Second wave: Return to the problems that you starred. These are the ones that you’re fairly sure you can figure out, but that need more time. Typical of this level of problem would be the ones in which you must try all possible answers to eliminate the incorrect ones. Third wave: Spend what time you have remaining (not including your last three minutes) working on the problems. Up to now, you have maximized your points given the time constraint. On multiple-choice problems, if you can eliminate at least one answer as incorrect, guess. If you cannot eliminate at least one of the choices, it will be better not to guess. If there is no penalty for incorrect responses, then guess freely. When you have used up all your time but the last three minutes, stop working. Use the last three minutes to fill in your answer sheet, and check it at least once to be sure you have marked the correct response for each problem. You may be able to check them all twice. The PSAT and SAT usually have the questions roughly in order of difficulty. If you have 30 questions, you can be pretty sure that by number 18 or so, you will not find but a few problems that you can do very quickly and easily. Keep this in mind as you progress through the problems. If there are 30 questions and you find that number 27 is a "snap," you may be jumping to conclusions and/or not really understanding what the problem is asking. Be very wary of "obvious" answers more than half way through the section. It would be better to mark it with a star and come back to it. Put down your pencil and relax until the next section. You have done your best. Try these techniques on the following group of problems. FOR COMPUTER-BASED TESTS For tests taken on a computer, you need to find out if you will be able to skip problems and then go back to them. If you cannot go back after skipping problems, there usually is no penalty for guessing. (Ask about penalties for incorrect responses.)Just work each problem as quickly as you can, being careful to not spend more than 3 minutes on any single problem. Rather than use a lot of time on one problem, it will be better to guess and go on, so that you can at least try all the problems. If you can skip and go back, ask about penalties for incorrect responses. Then number a sheet of paper 1, 2, 3,…., 25, or however many problems are in this section. This will be how you record the ones you need to go back to. You will be entering your answer choices on the problems as you go, so there is no need to save time at the end for recording answers. On each section, you will be told how many minutes you have for that section. Note what time you start, and figure the time at which you have to complete that section. Write that time down so you will know when you need to stop. For example, if you have 30 minutes and you are starting at 10:07, you will have to stop at 10:37. Now take half of the allowed time and figure that from the starting time. In our example, half of 30 minutes is 15 minutes, so write down 10:07 +0 :15 = 10:22. We’ll call this you half-time. By this time you will want to have at least looked at most of the problems. As each problem is presented, decide if you know how to work it and can do it quickly. If so, then do it and enter your answer. If you think you can do it, but it will take some time, star its number on your list. You will want to come back to this one. When you find a problem that you have no clue about, circle its number on your list and go on. Continue through the problems, working the easy ones and marking the medium and impossible ones. Try to at least every problem read by half time. At half-time, if you have not finished looking at all the problems, continue as before, doing the easy ones, and marking the harder ones. When you have seen all the problems, start back and work on the starred, medium-difficulty ones. If you get all of these done and have time remaining, then try the circled, impossible ones. If there is a penalty for incorrect answers, you should not guess on these unless you can eliminate at least one of the choices. If you can rule out at least one choice, then it will be safe to guess. If there is no penalty for incorrect responses, then guess freely.

Geometry, Vectors and Transformations

Geometry, Vectors and Transformations

Geometry is the study of shapes of various sort. The simplest shape is the point . (It's quite difficult to explain what a point is, it is basically just a position, for instance, the very end of your nose is a point). Another simple shape is a straight line . A straight line is just the simplest shape joining two points together. A plane is a more complicated shape, it is a flat sheet, like a piece of paper or a wall. There are more complicated shapes, called solids , like a cube or a sphere. Here are some pictures of these things.

Simple geometric figures

If you have a line and a plane, you can find the point where the line cuts through the plane. In fact, sometimes you can't find the intersection, because they don't meet and sometimes the line is inside the plane so they meet at every point on the line , but this doesn't happen in the cases we're interested in. We call this the intersection of the line and the plane. Here is a picture of what this looks like.

Intersection of a line and a plane

A vector is a mathematical way of representing a point. A vector is 3 numbers, usually called x, yand z. You can think of these numbers as how far you have to go in 3 different directions to get to a point. For instance, put one arm out pointing to the right, and the other pointing straight forward. I can now give you a vector and you'll be able to find the point I'm talking about. For instance, if I say x=3, y=1, z=5, you find the point by walking 3 metres in the direction of your right hand, then 1 metre in the direction of your left hand, and then getting a ladder and climbing up 5 metres. Here is a picture of a vector.

Picture of a vector and directions

Vectors are written as (x,y,z), for instance (1,2,3) means move 1 in the x-direction, 2 in the y-direction and 3 in the z-direction.

One confusing thing about vectors is that they are sometimes used to represent a point, and sometimes they are used to represent a direction. The vector (1,0,0) can mean both ``the point you get to if you move 1 unit in the x-direction from the starting point'', or it can mean ``move 1 unit in the x-direction from where you are now''.

Proof with pictures

                             Proof with pictures

The algebraic identities (a+b)c=ac+bc and (a+b)2=a2+2ab+b2 can be justified by pictures, as Figures 1 and 2 show. 

Figure 1		Figure 2

Arguments of this nature can be found in Euclid's The Elements (book II) . Like Euclid, we will assume throughout this discussion that a, b and c are non-negative. 

Inequalities can also be demonstrated by pictures. For example, the inequality (a+b)2≥a2+b2is shown by figure 2. 

Figure:3

In figure 3, two rectangles, each of area ab, fit inside the two squares of areas a2 and b2, showing that a2+b2≥2ab.

In the April 2000 issue of Mathematics Magazine , Claudia Alsina gives further examples of inequalities which can be proved by pictures.

Figures 4 and 5 demonstrate the inequality a2+b2+c2≥ab+bc+ca. 

In Figure 4, three squares of areas a2, b2 and c2 are shown, assuming (without loss of generality) that a≥b≥c.

Figure 4		Figure 5

In Figure 5, three rectangles of areas ab, bc and ca are fitted inside the three squares, showing that a2+b2+c2≥ab+bc+ca. 

Using this inequality, then from Figures 6 and 7 the inequality a3+b3+c3≥3abc can be demonstrated.

Figure 6		Figure 7

The two rectangles have the same base length a+b+c, but the rectangle in Figure 6 has height a2+b2+c2, which as we have seen is greater than the height ab+bc+ca of the rectangle in Figure 7. So the area of the rectangle in Figure 6 is greater than the area of the rectangle in Figure 7. 

The two rectangles are each divided into nine small rectangles, with areas as shown. The six green rectangles in Figure 6 have the same areas as the six green rectangles in Figure 7 (a2b, b2c, a2c, ab2, bc2, ac2). Comparing the remaining areas shows that a3+b3+c3≥3abc, as required.

The AM-GM inequality

The inequalities a2+b2≥2ab and a3+b3+c3≥3abc can be written in the form

x1+x22x1+x2+x33≥≥(x1x2)1/2(x1x2x3)1/3

where

x1≥0,x2≥0,x3≥0.

These are special cases of the important Arithmetic Mean - Geometric Mean inequality (the AM-GM inequality)

x1+x2+…xnn≥(x1x2…xn)(1/n),

where

xi≥0,1≥i≥n

When is an inequality an equality?

When an inequality is established, it is always important to know under what circumstances equality can occur.

A re-examination of Figure 3 (see Figure 8) shows that the inequality a2+b2≥2ab becomes an equality if and only if the blue region has zero area. 

Figure: 8

The blue region is a square of area (a−b)2, which is zero if and only if a=b.

Similarly, a re-examination of Figure 5 (see Figure 9) shows that the inequality

a2+b2+c2≥ab+bc+ca

is an equality if and only if the two blue regions have zero area.

Figure: 9

This occurs if and only if (a−b)2=(a−c)(b−c)=0,

i.e. if a=b=c

It follows that the inequality a3+b3+c3≥3abc is an equality if and only if a=b=c.

(Remember that throughout this discussion it was assumed that a≥0, b≥0 and c≥0. It can in fact be proved that a3+b3+c3≥3abc under the weaker assumption that a+b+c≥0, and that equality holds if and only if a+b+c=0. How? Just look hard at the factorisation

a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−bc−ca)