Formulas:
- Algebra:
- Difference of squares: \(a^2 - b^2 = (a + b)(a - b)\)
- Sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
- Difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
- Geometry:
- Area of a trapezoid: \(A = \frac{1}{2}(a+b)h\), where \(a\) and \(b\) are the parallel sides, and \(h\) is the height.
- Law of Sines: \(\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}\)
- Law of Cosines: \(c^2 = a^2 + b^2 - 2ab\cos(C)\)
Tricks and Strategies:
- Plugging in Values:
- If the question asks for a relationship (like "which of the following must be true?"), pick easy numbers for the variables and test the options.
- Backsolve from the Answers:
- Start with the middle option (C) and use the given answers to work backward. Adjust based on whether you need a larger or smaller answer.
- Draw and Label:
- For geometry questions, always draw a figure and label all given values. Even if a figure is provided, draw it again. Your version might be larger and easier to work with.
- Estimation:
- Instead of calculating the exact value, estimate to make calculations faster. This is especially useful when the answer choices are widely spaced apart.
- Cross-Multiplication for Inequalities:
- When dealing with fractions in inequalities, cross-multiply but remember: if you multiply or divide by a negative number, reverse the inequality sign.
- Square of a Binomial:
- Recognize the pattern: \((a+b)^2 = a^2 + 2ab + b^2\)
- Symmetry in Number Properties:
- In many questions, if a particular positive value works, its negative counterpart will too (e.g., if 3 is a solution, so is -3).
- Odd/Even Properties:
- Remember:
- Even ± Even = Even
- Odd ± Odd = Even
- Even × Even = Even
- Odd × Odd = Odd
- Even × Odd = Even
- Remember:
- Quick Multiplication:
- For numbers close to 100: To multiply \(97 \times 98\), think of how much each is away from 100. \(97\) is 3 away and \(98\) is 2 away. \(97 - 2 = 95\) (this is the first part of the answer). \(3 \times 2 = 6\) (this is the second part of the answer). So, \(97 \times 98 = 9506\).
- Factorization for Simplification:
- Recognizing common factors can help simplify seemingly complex expressions.
- Combining Like Terms Quickly:
- Instead of doing long addition or subtraction, group like terms together to simplify calculations.
- Watch Out for Zero:
- Remember that \(0\) is an integer, even, and neither positive nor negative. Many students overlook properties of zero in haste.
- Special Triangles:
- Familiarize yourself with the properties of 30-60-90 and 45-45-90 triangles.
- Remember the Answer Could Be in the Question:
- Sometimes, especially in word problems, re-reading the question can provide clues or even the solution to the problem.
By practicing these tricks and integrating them into your problem-solving routine, you can significantly speed up your performance on the SAT math section. Remember, though, that understanding the underlying concepts is crucial – shortcuts are just tools to help you apply that understanding more efficiently.
- FOIL Method for Binomial Multiplication:
- \( (a+b)(c+d) = ac + ad + bc + bd \)
- Remembering the pattern can save time instead of distributing each term individually.
- Difference Between Squares:
- Quickly factor expressions like \( a^2 - b^2 \) as \( (a+b)(a-b) \).
- Using Answer Choices to Your Advantage:
- In some problems, you can plug answer choices into the problem to see which one works, starting from the middle value.
- Shortcut for Percentages:
- To find 15% of a number, find 10% (move the decimal one place to the left) and add it to 5% (half of 10%).
- Quickly Squaring Numbers Ending in 5:
- For numbers like \( 25^2 \), you take the "2" (from 25), multiply it by "2+1=3" to get 6, and append 25 at the end. So, \( 25^2 = 625 \).
- Rapid Multiplication with 9's:
- For 9 times any digit (e.g., 9×4): Subtract 1 from the digit to get the first number and subtract that from 9 to get the second number. So, \( 9 \times 4 = 36 \).
- Avoiding Common Trap Answers:
- The SAT often includes answers that are derived from common mistakes. If your answer seems too obvious or was too easy to find, double-check your work.
- Quickly Determining Divisibility:
- For 3: If the sum of the digits is divisible by 3.
- For 4: If the number formed by the last two digits is divisible by 4.
- For 6: If it's divisible by both 2 and 3.
- For 8: If the number formed by the last three digits is divisible by 8.
- For 9: If the sum of the digits is divisible by 9.
- Patterns in Powers of 2:
- Recognizing that powers of 2 double (2, 4, 8, 16, 32, 64, ...) can help in problems where exponential growth or division is involved.
- Recognizing Common Ratios in Fractions:
- \( \frac{1}{5} \) is 20%, \( \frac{1}{4} \) is 25%, \( \frac{3}{4} \) is 75%, etc. Having these in mind can speed up percentage problems.
- Looking for Patterns or Symmetry:
- Sometimes sequences or geometric figures have patterns or symmetries that can be exploited to find shortcuts.
- Breaking Down Complex Fractions:
- For something like \( \frac{a/b}{c/d} \), remember it’s the same as multiplying by the reciprocal: \( \frac{a}{b} \times \frac{d}{c} \).
- Use the Choices to Estimate:
- If the answer choices are numbers that are far apart, you can often estimate or round numbers to make calculations faster.
- Using 0 and 1 for Variables:
- If a problem has an unknown and no constraints are given, try plugging in 0 or 1 as they often simplify calculations.
- Parallel Line Angles:
- Remember that alternate interior angles are equal, corresponding angles are equal, and co-interior angles sum up to 180°.
By mastering these tricks, along with the earlier ones mentioned, you can navigate through math problems more efficiently. However, always ensure a strong foundational understanding of the concepts behind each trick. Practice makes these tricks second nature and helps to identify when to use each one.
