Math: The Less-Easy Stuff

Where do hard problems come from? There are 3 main sources of high-difficulty problems: Problems that require many steps Problems that combine multiple concepts in a single problem Problems that require knowledge of more advanced concepts (trig, logarithms, complex numbers, etc.) Wrong answer choices often reflect common mistakes Where do hard problems come from?

Strategy Review Look at the problem If you think you can solve it quickly and accurately using conventional math, do so If you don’t know how to solve it, or it looks like the problem will take more than a minute to solve, use your “cheats”: Make-a-target Plug-and-chug Remember that all questions have equal value Remember that only the subject tests use the wrong-answer penalty as of March 2016 Strategy Review

Make-a-target Choose numbers that are: For example: Appropriate to the problem Easy to work with in the context of the problem (small, easy to divide, etc) For example: X 3 Even integer 2 Prime number less than 20 11 Make-a-target

You can make a target when: The question describes a relationship between numbers without giving any specific numbers There are variables in the answer choices Make-a-target

On these multiple-choice tests, one of the answers has to be right. The answers are usually in ascending or descending order, so start with C. If C isn’t right, you’ll know if you need to go up or down. Plug-and-chug

If the question asks for the smallest possible value…start with the smallest choice You can Plug-and-chug when there are numbers in the answer choices Plug-and-chug

Advice for student-produced responses (“grid-ins”) Bubble in your answer AND write it at the top. The answer to student-produced responses will never be negative, so if you get to the end of a problem and you have a negative number, or a number that doesn’t fit into the grid (like 10,000), go back and check your work. Your answer can include a fraction but you should always write fractions as improper fractions or decimals. If the answer is 7/28, you don’t need to reduce it to 1/4. If the answer is 45/30, you will need to reduce the fraction (or convert to the decimal form), in order to fit the answer into the available space. If your answer to a student-produced response features a repeating decimal, write as many  of the repeating decimals as will fit in the grid. If you’re gridding a decimal number that’s less than one, you should not include the zero to the left of the decimal. Some of the student-produced response questions will have more than one possible answer. Double check that you’re actually answering the question that has been asked. Advice for student-produced responses (“grid-ins”)

Tips for high miss rate question types Based on diagnostic and midterm results

Stats: mean, median, standard deviation Definition review: Mean = Average = Sum of terms/number of terms Median = Middle number when terms are placed in order Standard Deviation = How spread out the numbers are Mean and Median will probably need to be calculated; Standard Deviation will probably not – you’ll be asked about general trends If you do need to calculate SD: Step 1: Find the mean. Step 2: For each data point, find the square of its distance to the mean. Step 3: Sum the values from Step 2. Step 4: Divide by the number of data points. Step 5: Take the square root. Stats: mean, median, standard deviation

Two variables are given in terms of a third variable, called the parameter: x = 12a y = -2a2 + 12a Test questions will likely involve substitution or finding ordered pairs: y = -x2/72 + x a = 0, (0,0); a = 1, (12, 10); a = 2, (24, 16) Parametric functions

Don’t be intimidated if you have to manipulate complex numbers Treat them like any other polynomial for purposes of adding, FOIL-ing, etc. Remember that there are only four relevant, cyclical identities: i0 = 1 i1 = i i2 = −1 i3 = −i Complex numbers

Nested and recursive functions Do not overthink these! No matter how complex these get, just follow the rules and work the steps one at a time. For both: For every input in the domain, you get an output in the range The output of one function can act as the input of another Scratch work is vital For nested functions: Work from the inside out For recursive functions: Look for patterns formed by consecutive terms in the sequence Nested and recursive functions

matrices

Calculation-wise, these are actually really easy Calculation-wise, these are actually really easy. At most, you have to multiply factions or percents. The difficulty lies almost always in figuring out how many favorable and total outcomes exist in any given situation: Drawing 2 aces in a row from a standard deck: 4/52 X 3/51 because you remove an ace from the deck on the first draw Getting heads 3 times in 4 flips: ½ X ½ X ½ X ½ = 1/16 4 different combintions: HHHT, HHTH, HTHH, THHH = 4/16 = 1/4 probability

Trig and trig functions

Trig and trig functions

Trig and trig functions