How To Study Math

How to Study Math Most bright math students don't really have trouble understanding complicated math; but they don't know how to learn it well, to retain it, to apply it to new situations. They underestimate the time they should spend studying and don't have good study methods. I read an article online once that talked about succeeding in chemistry in college. The author stated "The number one reason students fail chemistry is that they do not do enough problems". He states this over and over about 6 times in the short article. That stuck with me and is true for any math based course: math, physics, chemistry, etc. Some students feel if they've done a couple problems, they are done studying. I say you need to do so many problems that there is nothing that can be thrown at you that you haven't seen before. Do every problem you can get your hands on. This also reduces any anxiety during a test. You know you've prepared well, so you are more likely to try the tough problems and not fear them. AND you actually know quite a bit of stuff since you studied well and that knowledge will help you figure out new stretch type problems. My list of math studying advice: 1. You can never do too many problems. 2. Do several problems of each type until you are VERY comfortable with a technique, then do a couple more so it is cemented in your brain and becomes automatic. 3. Always do the homework that is assigned. If you have any difficulties, do MORE than what was assigned. Keep your homework organized. You need to study the homework before tests. 4. Check your answers - ALWAYS. If you have answers in the book, check them. If the teacher provides an answer key in the classroom or online, check them. If there is a classmate who does quality work, check answers with them. 5. Make a list of the types of problems you are responsible for. Every kind and their variations. Then write down an example of each and the steps involved. 6. Make a list of math techniques that you can apply to problems in the chapter you are studying. This is your bag of tricks. Memorize them! If you are working a new problem (or are taking a test) and get stuck, go through your list and see if any of those techniques might apply. 7. Keep all your quizzes and tests and be sure to go back through them to make corrections and learn from them. 8. Write yourself little notes as you learn little important things, little tricks, things to remember. 9. Find outside resources when available. Use all your resources. If there is stuff online from your teacher, be sure to use it. If there are outside review books, get several and use them. If the teacher has old practice tests, do them. You can never do enough problems. If you have an older sibling or a parent that can do this stuff, by all means take their assistance. 10. Take notes in class. Put the date on your notes. Have your friends take notes and date them also. Helps when comparing. Write down everything that the teacher puts on the board or overhead. The more material you have to work with, the more you can study. You need material to study from. Sometimes the more difficult types of problems are modeled for you in class. Study these. When you are stuck on a homework question, check your class notes. Sometimes problems done in class are previews of test questions. 11. READ your textbook. Often there are example problems in the textbook to help you with homework. And, the examples may give you a hint on the types of problems that might be on the test. 12. Study and do homework with a study partner. Find someone you can work well with. Do homework simultaneously - checking each other's work. Talk about different methods. Be sure you discuss the little details. This will help you remember. 13. Spend about 2 hours studying per section in a math book. Another rule of thumb is to spend 2 hours studying for every 1 hour in class. It might take you less time if you are efficient, but don't be afraid to

spend a lot of time studying math. You need to go over concepts - make a list of the techniques - and do a bunch of problems to practice, and list anything else you feel is important to remember. That could take 2 hours per section. Each section is a separate concept with techniques and variations. Give it its due time. The goal isn't to "get by" but to learn it well. 14. You need to analyze what it is that you need to know. MAKE LISTS. MAKE CHEAT SHEETS of all the important things in the chapter. This will help you study for the current chapter test and will help you review for the final later. 15. When studying for a test, you should be reviewing stuff you already learned - this is not the time to learn something for the first time. Be sure to keep up with class, doing and checking all homework as you go. Be sure you can do all the types of problems you are responsible for. Master the material. Then when you are at the end of the chapter, you are simply reviewing and bringing it fresh in your mind. 16. One good technique is to review by reworking homework problems from scratch. If you can get through them and get the right answers, that’s fantastic and good review. If you cannot, then you can look at your own homework sets to see how to do the problems. It is likely that test questions will be similar to homework problems. If you had a homework problem on a certain technique but you had trouble with it, don’t expect your teacher will be nice and happen to skip that topic on the test. You should assume that it WILL be on the test, and you should work to learn it. Consider yourself warned, if you had a homework problem on it. 17. Also look at homework problems near the problems you were assigned, especially word problems. 18. Go see the teacher if you have any questions. If you study 2 days before the test, then you can still ask questions on the day before the test. 19. Be sure to go back through class notes and chapter quizzes when you make your lists to study for the chapter. Don't leave anything out. 20. Once you've made your list, study it. Memorize techniques, go over the tricky problems enough so that you'll recognize them on a test and you'll remember what to do. 21. Find a practice test or make yourself a practice test and do it under test conditions. See how you do. 22. Another quote: "Studying is what you do to firmly fix new information in your memory. You study to learn and to improve your ability to recall what you need to know." I often ask students, "How will you remember this?" Find a way to get it into your brain so that you'll know it the next time you need it. 23. Never, ever, say you "looked it over". You need to DO IT, and do LOTS of it. If you are having difficulty in a math class: 1. ADD MORE MATH TIME INTO YOUR WEEK. You need time to do homework AND time to study math. Those are two separate time commitments. It is not enough to only do the homework if you hope to learn it well. 2. Be sure you take very detailed notes in class. 3. READ your math textbook. 4. Do all your homework problems and check answers (back of book or teacher). 5. Fix any homework problems you had trouble with. Fix all quizzes. 6. Rework EVERY homework problem. a. Do this the day you do the homework the first time when it’s fresh in your mind. b. Do this a day or two later to see if you still remember how. c. Do this before the test. 7. Be diligent to recognize when you do and when you do not fully understand how to work a problem. If you understand a general technique but still don’t get the correct answer, then analyze which steps are giving you trouble. 8. Ask teacher for help – in class or after class. If your teacher is not available, ask a different teacher. 9. Ask fellow students for help. Try to form a study group to do homework together. 10. Ask a parent for help (yes, some parents are actually good at math and can help you.)

11. Get a tutor. Terry Guay, 2008

Working math problems: • When doing a problem, you don't need to know how it will turn out when you start. Just start and see where it leads you. Remember to simplify (but not round) as you go; you may get to a point where you "see" what to do next by what form your intermediate work is taking. •

Sometimes you can work a problem from both ends and meet in the middle.

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When possible, do not round until the end of the problem. Work your calculator or your math on paper so that you keep it as exact as possible.

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Be very accurate. Double check your steps. When factoring, multiply it back to see if it matches the original.

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Be very careful of fractions. o Need common denominator to add or subtract, so you multiply top and bottom by same amount to adjust denominator.

x −1 2 x + 3 x − 1 3x 2 ( x + 3)( x − 1) − (3 x)(2) − = ⋅ − ⋅ = 3x x + 3 x + 3 3x 3x x + 3 3 x( x + 3) ( x 2 + 2 x − 3) − (6 x) x 2 − 4 x − 3 = = 3 x( x + 3) 3 x( x + 3) o When multiplying, just multiply across (remember 2 is really 2 )

x − 1 2 x − 1 2( x − 1) 2 x − 2 2⋅ = ⋅ = = 3 1 3 1⋅ 3 3 o To divide fractions, “copy change flip”

x − 1 2 x − 1 x ( x − 1)( x ) x 2 − x ÷ = ⋅ = = 3 x 3 2 3⋅ 2 6 •

Clever Trick:

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Things you can do with radicals:

(x – 4) = (–1)(4 – x)

2 ⋅ 8 = 16 = 4 50 = 25 ⋅ 2 = 5 2 3⋅ 3 =3

( 3)

2

=3

1

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To solve a LINEAR equation (all x’s are to the 1 power), put x’s on one side, numbers on the other. 2 x + 1 = 5x − 7 2 x − 5 x + 1 = −7 − 3 x + 1 = −7 − 3 x = −8 −8 8 x= = −3 3

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To solve an equation that is NOT LINEAR (some x is not to the 1 power, x2 or x-1 or 1/x or put everything on one side, zero on the other. 2 x + 1 = 5x 2 − 7

x ),

2 x − 5 x 2 + 1 = −7 2 x − 5x 2 + 8 = 0 − 5x 2 + 2x + 8 = 0 5x 2 − 2 x − 8 = 0 Then solve using appropriate techniques: Quadratic Techniques: Factor, use quadratic formula, or graph and find zeros. Radical Techniques: Isolate radical and square both sides Exponential and log Techniques: Isolate exponent or log; then change form to other form •

Once you have a problem solved, check the answer IN THE ORIGINAL EQUATION. o Even if you did the problem totally correct, some answers are wrong. o Watch for extraneous solutions, particularly if you squared somewhere in the problem. o Eliminate answers where you would create a zero in the denominator o Eliminate answers where you would be taking a square root of a negative amount o Eliminate answers where you would be taking a log of a zero or negative amount

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Squaring means “multiply by itself”. - 32 = - (3)(3) = - 9

only the “3” is squared

(-3)2 = (-3)(-3) = 9 (x+3)2 = (x+3)(x+3) = x2 + 6x + 9

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Factoring difference of squares x2 – 1 = (x + 1)(x – 1) x2 - 4 = (x + 2)(x – 2) 4x2 – 9y2 = (2x + 3y)(2x – 3y)

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Factoring with a like group….bring the group out front (reverse distributive property) 3(x-2) + 2x(x-2) = (x-2) (3 + 2x)

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Factoring by grouping. x3 + 3x2 + 2x + 6 (x2)(x+3) + 2(x+3) (x+3) (x2 + 2)

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Factor completely x3 – x x (x2 – 1) x (x + 1) (x – 1)

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Factor sum/difference of cubes (the SOAP formula) (a3 + b3) = (a

+

(a3 – b3) = (a

– Same

b) (a2 b) (a2

ab

+ b2)

+ ab

+ b2)

–

Opposite

Always Positive

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Factor trinomial (reverse FOIL) x2 + 11x + 30 = (x + 5) (x + 6) x2 + x – 30 = (x – 5) (x + 6) x2 – x – 30 = (x + 5) (x – 6) x2 – 11x + 30 = (x – 5) (x – 6)

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Factor trinomial when coefficient of x2 is not 1 “split the middle” 2 ax + bx + c = 0 2x2 - 9x - 5 = 0 find ac ac = 2(- 5) = - 10 find b b = -9 find two magic numbers whose product is ac and sum is b find two magic numbers whose product is -10 and sum is -9 the two magic numbers are -10 and 1 now split the middle term using the magic numbers 2x2 - 9x -5 =0 2 2x + 10x - 1x - 5 = 0 factor by grouping (you can rearrange the middle terms if it helps) 2x ( x + 5) -1 (x + 5) (x + 5) (2x – 1)

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Different forms of a fraction are equivalent; we prefer the negative sign to NOT be on the bottom. −3 =

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Different forms of a fraction are equivalent. 3⋅

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6 −6 6 = =− −2 2 2

x 3 x 3x 3 x 3 1 3x 1 = ⋅ = = ⋅ = ⋅x = ⋅ = ⋅ 3x 2 1 2 2 2 1 2 2 1 2

Simplifying fractions. You must have items that multiply before you cancel. 6 3⋅ 2 3 = = 8 4⋅2 4 2 x + 6 2( x + 3) x+3 = ⋅= = x+3 2 2 1

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Square roots of expressions, be careful. (this cannot be simplified)

x2 + 4 ≠ x + 2 ( x + 5) 2 =

(

x+5

)

2

= x+5

(be careful if you are solving for x, check for extraneous solutions)

•

Exponents

x 2 ⋅ x3 = x5 ( x ⋅ x) ⋅ ( x ⋅ x ⋅ x) = x 5 ( x3 )2 = x6 ( x 3 ) 2 = ( x 3 ) ⋅ ( x 3 ) = ( x ⋅ x ⋅ x) ⋅ ( x ⋅ x ⋅ x) = x 6 x5 x3 x5 x3 x3 x5 x3 x5

=

x⋅ x⋅ x⋅ x⋅x x⋅x = = x2 x⋅ x⋅ x 1

= x 5−3 = x 2

x⋅ x⋅ x 1 1 = = 2 x⋅ x⋅ x⋅ x⋅x x⋅x x 1 = x 3−5 = x −2 = 2 x x −2 1 −2 x = = 2 1 x 1 x2 = −2 x 1 =

2

x x2   = 2 y  y x0 = 1 x3 = x 3−3 = x 0 = 1 3 x 1 2

x = x 1 3

x =3 x