What's curious about this problem?
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What's curious about the problem?Nearly all the problems on the SAT should be approached with this strategy, for it will likely point you to the actual strategy that gets you through the problem most efficiently. For example, here is the first problem on the no-calculator section of a practice test, so it's easy and can be done in the traditional way, but we'll do it differently, noticing we can easily transform one expression to look like the other:  • Use the information in charts and tables effectively. • Pay close attention to scale and read carefully. Don’t assume that the horizontal and vertical axes have the same scale. • Read how the axes are labeled. When you need to interpret the slope, putting the vertical units over the horizontal units will usually make this easy. • Two-way tables usually require you to fill parts of them in. Often totals are left out, and you’ll need to compute these. Only take the time to fill in the parts you need if you can determine this. Example 1 (practice test 6) Calculator ok We are given that this is a linear function. From the table we see that for a positive change of two units in x, we have a positive change of 6 units in y. Therefore an x change by +1 gives a y change by +3. From x=2 to x=3 is an x change of +1, so the corresponding y-value will be 4+3=7. Choice B is correct.  Example 2 (practice test 6) Calculator ok This is a two-way table. The problem is only interested in vanilla, so we only look at the Vanilla column. Since we want the fraction of those who chose hot fudge topping, we need the total who chose vanilla: 8+5=13. So the fraction is just: 8/13. Choice D is correct.  • Draw or extend lines in a diagram and label the givens and unknowns. • Whenever a circle is involved, get its radius! • Pay attention to whether or not the figure is drawn to scale. If “not drawn to scale” is not stated beneath the figure, then the figure is drawn to scale! In this case you can interpret the drawing according to how it looks. For example, If an angle appears to be 90 degrees, then it is 90 degrees; if a point appears to be the midpoint, then it is the midpoint. • If the diagram is not drawn to scale and looks disproportionate, it may be a good idea to redraw it to make interpreting it easier. For example, if an angle in a triangle is said to be 90°, but it looks acute, redraw the triangle so that angle looks like a right angle. • If no diagram is given, always draw one! You will most likely work faster with a picture to look at. Example 1 (practice test 6) Calculator ok For this problem, no diagram is given. So draw a number line, and plot the point -4 and two points labeled x, each with 3 units on either side of the point -4.   The two values of x are: -7 and -1. Plugging in, we see choice A works.
Strategy #16 Know important triangle facts, in addition to what’s in the reference table.12/30/2021
• Equilateral triangles: all angles measure 60° (or π/3 radians) and all sides have the same length.
• Isosceles triangles: two sides are the same length. Base angles are congruent. The altitude bisects the vertex and the base. • The longest side of a triangle is opposite its largest angle; the shortest side is opposite the smallest angle. • The sum of two sides of a triangle must be greater than the third side. The difference of two sides of a triangle must be smaller than the third side. • 30°-60°-90° triangles have side-length ratios of 1:2:\(\sqrt{3}\), and 45°-45°-90° triangles have side-length ratios of 1:1:\(\sqrt{2}\). These facts are extremely important to know because they occur frequently. They are found on the tests reference page, but it’s better to have them memorized. • cos x = sin (90° - x) where x is one of the two acute angles of a right triangle, and (90° - x) is the other acute angle. This is called the co-function identity. Using it will save time. The identity is a result of the bullet below. • The two acute angles of a right triangle are complementary. Example 1 (practice test 6) Calculator ok 
The two labeled angles are complements of each other (32°+58°=90°). And since the two acute angles of a right triangle must be complementary, \(\angle A\cong \angle E \), and since, BC/AB = sin 32°, DF/EF = sin 32° as well.
The way this usually comes up is when we know, let’s say, the total distance, and this distance is broken into two unknown parts. We need expressions for the two parts, but we don’t want to introduce a second variable. Rather, we want to express one of the variables in terms of the other variable. We will label one of the parts as X. Then, we will label the other part as Total Distance – X.  Example 1 (Practice Test 7) Calculator ok  The two parts in this problem are the volume of pieces of fruit and the volume of syrup. So, the sum of the parts equals the whole is:
fruit + syrup = volume of can. Since we want fruit turn this equation around to solve for fruit: fruit = volume of can — syrup. Remember that the formula for the volume of a cylinder is given in the reference section at the beginning of each math section: area of base x height of cylinder = volume. volume of fruit = (75 sq cm)(10 cm) — 110 cubic cm = 640 cubic cm. Choice C is correct. Strategy #14 Know how to work with ratios, rates, proportions, percents and probabilities12/17/2021
• The ratio of a to b is written as: \(a/b\) or as: a:b:c if there are three values.
• A proportion is an equality between two or more ratios such as: \(\frac{a}{b}= \frac{c}{b}\). • Similar Triangles have sides whose ratios form a proportion • Percent means, how much out of 100. It is a ratio and can be found by dividing the PART we are interested in by the WHOLE, then multiplying by 100: \[\frac{part}{whole}\times 100\] • The probability of an event is the ratio between the number of favorable outcomes (the part you are interested) to the number of all possible outcomes (the whole). If we want percent probability, then we’d multiply by 100: \[\frac{favorable\ outcomes}{all\ possible\ outcomes}\times 100\] • To convert a decimal to a percent – move the decimal to the right two places. • To convert a percent to a decimal – move the decimal to the left two places. • To calculate the percent increase or decrease from the old value to the new value: \[\frac{new\ value}{old\ value}\times 100\] • Know the formulas for density and speed. These are both ratios that express rates of change: the amount of change in the numerator for each unit change in the denominator.
• The slope is the rate of change for a graphed line of a linear equation. It's the ratio: \(\frac{change\ in\ y\ }{change\ in\ x\ }\) and it expresses how y changes for each unit change in x. The usual formula is: \(\frac{y\ _{2}-y\ _{1}}{x\ _{2}-x\ _{2}\ }\). • Average and Mean are the same thing in this test. • If the individual quantities are not known but their average is, you can let the average value stand in for each of the unknown quantities. Sometimes the average for a group of objects is given, and the average of a smaller group within that group is also given but we don’t know their individual values. Let the objects in the smaller group all equal their given average. This lets us not worry about individual values. Example: Six children have an average height of 62 inches. Five of those children have an average height of 60 inches. What is the height of the sixth child?     The height of the sixth child is 72 inches. • When you know the number of things, and you know their average, multiply these two values to get what the things add up to. From the example above, the heights of 6 children added together was: 
Pythagorean triples refer to right triangles in which the three sides are each nice, whole numbers. Know the following two and their multiples:
3-4-5 5-12-13 Whenever you are dealing with a right triangle look for Pythagorean triples (and their multiples), as they will save you time from working through the Pythagorean Theorem. The test uses them frequently. These come up very frequently, but can be hiding. Look for them when dealing with quadratic expressions and save time. x^2-y^2=(x+y)(x-y) x^2+2xy+y^2=(x+y)^2 x^2-2xy+y^2=(x-y)^2 Look for, and recognize, pieces of them that may be in different parts of the problem. Connect the dots. Example 1 (practice test 6) No calculator  We can recognize the difference of two squares, the first of the three expressions. p = +2 or -2. Choice A is correct.
• Addition and Subtraction: Cross out common quantities that are being added or subtracted on both sides of an equation or inequality.
• Multiplication and Division: Cross out common quantities that are being multiplied on each side of the equation if you can be certain they don’t equal zero. Quantities that are being divided on both sides of an equation can also be crossed out. In the case of inequalities be sure the quantity being multiplied or divided is not negative. • Rational Expression: If each side of an equation is a single rational expression (i.e. a fraction) with common denominators cross out the denominators. • Powers: If each side of an equation is a single power with common bases cross out common bases and set the exponents equal to each other. If the bases are not the same, often the problem is set up to make them equal (What’s curious? Strategy #2) From your algebra classes you are used to finding the value of a single variable. However, many questions on the SAT ask, instead, for the value of an expression and not necessarily for a single variable. Sometimes a question wants you to identify an equivalent expression to the one that is given.
Be able to translate directly from expressions in english to mathematical expressions, equations or inequalities. Knowing how to do this will make many problems much more simple.
Sum: is an answer to an addition problem. Difference: is an answer to a subtraction problem. Product: is an answer to a multiplication problem. Quotient: is an answer to a division problem. What or when: fill in with x (or some other letter that makes sense). Is, was or will be: replace with an = sign Is more than: replace with a > sign Is less than: replace with a < sign N is more than x: write, n > x N is less than x: write, n < x Is at least: use a greater than or equal to sign. Is no more than: use a less than or equal to sign Of: means multiply. Per, out of, for every, to every: means divide. Percent: means divide by 100 N percent: write n/100 N less than x: write x – n (reverse the order and subtract) N more than x: write n+x When you are working problems toward the end of a section, if choice A looks tantalizingly easy, be careful! It might be a lure choice and incorrect. Remember, the difficulty increases from the beginning to the end of each section.
Some questions ask, “Which of the following...?” Frequently, these problems cannot be answered without reading through the list of choices. They are often descriptive. When the choices must be read, start with choice D, and work up. The correct answer will almost always be found in choice C or D. Very seldom must you read into choices A or B.
Some examples:
When you are stumped, you can try substituting reasonable numbers into the unknowns. Though it takes more time, this can at least allow you to eliminate a couple of choices so you can guess more effectively. No penalty for wrong answers
There is always a fairly quick way through the problems. If you find you are doing lots of calculations, then you are most likely on the wrong approach. Be sure to use Strategy #1 above to find the correct approach. Oftentimes you can find the correct answer in the multiple choice sections by doing just enough work to pick out the correct response. Examples:
Example 1 (practice test 6) Calculator ok   This may be the most important strategy because most of the problems are designed to be solved in a simple way. Very often, there is a key or trick, that once recognized allows you to easily solve the problem. See the examples below. You definitely don’t want to get caught up in long calculations. It’s a sign you are going about it in the wrong way. Before jumping in and steaming ahead on a problem, look to see if there is anything that’s curious about it. Do you see any patterns or interesting relationships amongst any of the given values or expressions? Paying attention to what’s curious will help you find the best approach without wasting time. Here are a couple of examples.
Draw the triangle and observe that one leg has a length of 4 units and the hypotenuse, 5 units. What's curious? It's a 3-4-5 right triangle! The other leg is 3, so tan P = 3/4.
Before you automatically begin solving for x, look first at the expression on the left side of the given equation, and now look at the expression we want the value for. What's curious? Notice that 2x + 1 is just half of 4x + 2. So, 2x + 1 = 1/2 of 4. The answer is 2.
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David PorterDavid has tutored students from grade school through junior college in chemistry, physics and all levels of math for thirty years. 
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