Last day to CRAM for the ACT! Let's look at Math.
Math equations that you will need to use.
SOH-CAH-TOA
Pythagorean Theorem
a^2 + b^2 = c^2
distance = Pythagorean Theorem
= √ [(X1 - X2)^2 + (Y1 -Y2)^2]
slope = rise over run
= ∆ y/ ∆ x
= (Y1 - Y2) / (X1 - X2)
Mean = Average = Sum / Number
Median is the middlemost
Mode is the most frequent
Midpoint = (the average x, the average y)
= [(X1 + X2)/2, (Y1 + Y2)/2]
All AREA formulas:
Parallelogram....bh
Triangle.....1/2 bh
Trapezoid (often forgotten, frequently needed).....1/2 (b1 + b2)h
Rhombus and Square (special ones using diagonals).....1/2 (d1)(d2)
Circle.....π r^2
General VOLUME formula:
Flat-top prisms and cylinders……area of the base times height
Pointed-top cones and pyramids..(area of the base times height) / 3
Probability
Successes / Total Possible
PROCESSES that will come in handy.
1) Set up ratios and proportions
for SIMILAR TRIANGLES
for RATES
for CONGRUENT FRACTIONS
2) Write out equations and substitute values
3) Use all the data provided.
4) Draw diagrams.
5) Use the calculator to visualize by graphing and to check calculations.
6) Compare fractional values by finding a common denominator.
Don't
get hung up on a single question. Remember that the score is an
accumulation of points, so missing 1 is not as bad as not finishing all
60. Plan to spend the least amount of time on the first 30, an average
of 1 minute or less on 31 through 45, and invest the remaining time
efficiently on the final 15. This last group will probably contain the
most challenging problems, but be sure to recognize the 2 or 3 that are
super simple; they are your reward for getting that far within the
allotted time.
GOOD LUCK!!!
Showing posts with label MATH. Show all posts
Showing posts with label MATH. Show all posts
Friday, February 8, 2019
Wednesday, November 17, 2010
THE ACT SUGGESTS DEPRESSINGLY LOW BENCHMARKS FOR COLLEGE READINESS
ACT Inc suggests "benchmark standards" which are intended to indicate a student's readiness of college. Using these criteria, the Chicago Tribune reported recently on how Illinois high schools measure up. In a nutshell, the information was not positive. The standards themselves are unbelievably low, leaving the impression that kids don't have to know much in order to "succeed" in college, and results of the ACT component on the Prairie State Achievement Exam cast a shadow on any school that hopes to prepare more than 75% of their students for at least a 50-50 chance of earning a respectable grade in freshmen level college courses.
Not wishing to be the bearer of only bad news, I’ll start by saying that entry level scores on college entrance exams, like the ACT component of the Prairie State Achievement Exam (PSAE), are simply a starting point. There are many reasons a student may not achieve his or her best score if the test is taken “cold.” Some of these reasons are academically founded, but others hinge on knowledge of the test’s format, structure, and expectations.
That said, let’s think about the recent Chicago Tribune article highlighting Illinois statewide statistics on student readiness for college. (Friday, November 12, 2010, section 1, pp1+) Tutoring Resources’ primary service area includes North Cook, Lake, and McHenry counties. For the most part, our students are among the highest scoring in ACT’s recent survey of “college readiness benchmarks.” The creators of the popular entrance exam estimate that meeting the standard indicates a student has a 50 percent chance of earning a B or higher and a 75 percent chance of earning a C of higher in typical freshman courses. (In the spirit of full disclosure, I am of the opinion that a C in a college course is not acceptable, especially in the core courses required of a freshman. As a parent responsible for tuition and other fees, I would be less than pleased by only a 50% chance that my pecuniary investment would have a reasonable return.)
Benchmark scores established by ACT Inc for purposes of comparison are stunningly low in my opinion: English 18, Math 22, Reading 21, and Science Reasoning 24. I’m bewildered that English and Math are the lowest standards, while Science Reasoning is set at well above the average composite score nationwide. An average of the four benchmarks is lower than a national average composite score and just slightly above the state composite average. Yet, at Tutoring Resources' highest ranking high school, only 51.6% of the students met all four benchmarks.
The ACT researchers do not imply that there is blame to lay on our high schools, our teachers, or even our children. I would agree with this restraint and could, if asked, provide proof that our schools are doing better than these statistics might suggest if used in the wrong context. I DO believe that awareness of the criteria used by test developers, knowledge of the testing format, experience with actual test materials, and other nonacademic issues can have a significant impact on the scores each student can achieve and should strive to attain.
It is my experience that almost every freshman will be required to take a composition course through the college’s English department and those professors have every right, in my opinion, to expect students from our high schools to know when to use a semicolon, how to punctuate between independent sentences, that its’ is not a word, the difference between their/there/they’re, and other common grammar rules tested on the ACT English section. To achieve an 18, the student needs to answer only about 54% of the questions correctly. To propose that a student who knows only one-half of the standard grammar concepts will succeed without tremendous effort is wishful thinking.
Similarly in Math, to earn a score of 22, a student needs to find correct solutions for 32 to 35 of the 60 questions (53% to 58%). Considering as many as 5% of the questions might involve Trigonometry and/or higher level concepts not covered in some high school courses, but understanding the statistical probability that careful guessing can be rewarded with as much as a 20% success rate, setting a “benchmark” so low is anticipating that our college-bound students will be able to sustain an enormously challenging effort level in college math classes.
My tutors and I take every ACT that is made public and are constantly evaluating the concepts tested and looking for patterns of study which will help our students. So I tend to be positive about the test itself and the publishers and creators. I’m sure that researchers at ACT are cognizant of the limited ability of a single test to predict college success. Perhaps these “benchmarks” are intended to acknowledge the other factors which play into a student’s achievements at the college level: motivation, maturity, and personality. But I’m not willing to send my kids off to distant parts with only a 50% chance of succeeding.
I actually like the ACT test, not so much in terms of college admission (which is a fact of life not within my control), but as preparation for college itself. I expect my students to arrive on campus with a significant understanding of English rhetoric and grammar and with a thorough comprehension of high school level Math concepts. I want them to be prepared to focus on style and content in their first English course, not where to put a comma. I want them to obviate the need for remedial math courses, at full tuition for no credit, before qualifying to enroll in 2 semesters of required college level math. While I cringe at the low scores suggested as adequate, the ACT test itself and the English and Math concepts included in the testing are without doubt important elements for reviewing what we learned in high school and should take into the world after graduation.
Every student can use preparation for the ACT test as a foundation for success in college. Review the necessary concepts; the ACT authors test common mistakes which should be avoided in a composition class. Create study guides for use in college classes because elementary arithmetic really is necessary when solving a complex calculus problem. Raise those individual section scores to the highest possible level to demonstrate the student’s true potential and maybe even qualify for a great scholarship, but most importantly, to reinforce the foundation for academic prowess in post-secondary education. Treat the ACT as a handy checklist for packing to go off to college:
√ toothpaste
√ pillow
√ computer
√ Tutoring Resources’ “25 Grammar Rules”
√ a note card of those pesky math equations
By using the ACT test as the "benchmark" for what curricular concepts need to be firmly in place before starting college, a student is using the required entrance exam to its fullest potential and greatly increasing the 50% probability of earning outstanding grades in the ubiquitous Freshman English Composition and College Algebra classes.
Not wishing to be the bearer of only bad news, I’ll start by saying that entry level scores on college entrance exams, like the ACT component of the Prairie State Achievement Exam (PSAE), are simply a starting point. There are many reasons a student may not achieve his or her best score if the test is taken “cold.” Some of these reasons are academically founded, but others hinge on knowledge of the test’s format, structure, and expectations.
That said, let’s think about the recent Chicago Tribune article highlighting Illinois statewide statistics on student readiness for college. (Friday, November 12, 2010, section 1, pp1+) Tutoring Resources’ primary service area includes North Cook, Lake, and McHenry counties. For the most part, our students are among the highest scoring in ACT’s recent survey of “college readiness benchmarks.” The creators of the popular entrance exam estimate that meeting the standard indicates a student has a 50 percent chance of earning a B or higher and a 75 percent chance of earning a C of higher in typical freshman courses. (In the spirit of full disclosure, I am of the opinion that a C in a college course is not acceptable, especially in the core courses required of a freshman. As a parent responsible for tuition and other fees, I would be less than pleased by only a 50% chance that my pecuniary investment would have a reasonable return.)
Benchmark scores established by ACT Inc for purposes of comparison are stunningly low in my opinion: English 18, Math 22, Reading 21, and Science Reasoning 24. I’m bewildered that English and Math are the lowest standards, while Science Reasoning is set at well above the average composite score nationwide. An average of the four benchmarks is lower than a national average composite score and just slightly above the state composite average. Yet, at Tutoring Resources' highest ranking high school, only 51.6% of the students met all four benchmarks.
The ACT researchers do not imply that there is blame to lay on our high schools, our teachers, or even our children. I would agree with this restraint and could, if asked, provide proof that our schools are doing better than these statistics might suggest if used in the wrong context. I DO believe that awareness of the criteria used by test developers, knowledge of the testing format, experience with actual test materials, and other nonacademic issues can have a significant impact on the scores each student can achieve and should strive to attain.
It is my experience that almost every freshman will be required to take a composition course through the college’s English department and those professors have every right, in my opinion, to expect students from our high schools to know when to use a semicolon, how to punctuate between independent sentences, that its’ is not a word, the difference between their/there/they’re, and other common grammar rules tested on the ACT English section. To achieve an 18, the student needs to answer only about 54% of the questions correctly. To propose that a student who knows only one-half of the standard grammar concepts will succeed without tremendous effort is wishful thinking.
Similarly in Math, to earn a score of 22, a student needs to find correct solutions for 32 to 35 of the 60 questions (53% to 58%). Considering as many as 5% of the questions might involve Trigonometry and/or higher level concepts not covered in some high school courses, but understanding the statistical probability that careful guessing can be rewarded with as much as a 20% success rate, setting a “benchmark” so low is anticipating that our college-bound students will be able to sustain an enormously challenging effort level in college math classes.
My tutors and I take every ACT that is made public and are constantly evaluating the concepts tested and looking for patterns of study which will help our students. So I tend to be positive about the test itself and the publishers and creators. I’m sure that researchers at ACT are cognizant of the limited ability of a single test to predict college success. Perhaps these “benchmarks” are intended to acknowledge the other factors which play into a student’s achievements at the college level: motivation, maturity, and personality. But I’m not willing to send my kids off to distant parts with only a 50% chance of succeeding.
I actually like the ACT test, not so much in terms of college admission (which is a fact of life not within my control), but as preparation for college itself. I expect my students to arrive on campus with a significant understanding of English rhetoric and grammar and with a thorough comprehension of high school level Math concepts. I want them to be prepared to focus on style and content in their first English course, not where to put a comma. I want them to obviate the need for remedial math courses, at full tuition for no credit, before qualifying to enroll in 2 semesters of required college level math. While I cringe at the low scores suggested as adequate, the ACT test itself and the English and Math concepts included in the testing are without doubt important elements for reviewing what we learned in high school and should take into the world after graduation.
Every student can use preparation for the ACT test as a foundation for success in college. Review the necessary concepts; the ACT authors test common mistakes which should be avoided in a composition class. Create study guides for use in college classes because elementary arithmetic really is necessary when solving a complex calculus problem. Raise those individual section scores to the highest possible level to demonstrate the student’s true potential and maybe even qualify for a great scholarship, but most importantly, to reinforce the foundation for academic prowess in post-secondary education. Treat the ACT as a handy checklist for packing to go off to college:
√ toothpaste
√ pillow
√ computer
√ Tutoring Resources’ “25 Grammar Rules”
√ a note card of those pesky math equations
By using the ACT test as the "benchmark" for what curricular concepts need to be firmly in place before starting college, a student is using the required entrance exam to its fullest potential and greatly increasing the 50% probability of earning outstanding grades in the ubiquitous Freshman English Composition and College Algebra classes.
Monday, August 9, 2010
NUMERACY: A FIRST STEP TOWARD PRESCHOOL MATHEMATICS
This is a special article on the blog, just for parents of all those youngsters I met at Northbrook Days and who I hope will become Tutoring Resources students in 8 years or so from now. Usually I’m totally focused on the preteen to adult ages. I like working with kids old enough to question a teacher, even challenge the teacher’s knowledge.
But Math education starts long before Middle School and parents are the first and most influential teachers that every child can have. So here are a few things to do before students get into Preschool.
NUMERACY: This is the internal understanding of numbers. It involves number order, recognition of number symbols, and the one-to-one relationship of counting. Using the Tutoring Resources Math Cards distributed at our booth, select a few of the positive numbers, say 1 through 5 for the youngest kids and up to 10 as a child progresses. Lay them out on the table in numeric order.
Activity #1: (for ages as young as 2) Point to each number in order and say it’s name. Show the child how to point to the letters as you say their names. Ask the child to repeat the numbers after you’ve said them, starting with one at a time and progressing to the whole list at once. Eventually, have the student name the cards without help.
Activity #2: Using the same numbers that have been practiced in Activity #1, ask the child to point out a certain number, first with numbers laid out in order and then with cards randomly spread on the table.
Activity #3: Using manipulatives (navy beans, candy pieces, dots....something large enough that the child can pick it up without much discomfort), demonstrate the concept of each number by laying one item on the #1 card, 2 items on the #2 card, etc. As a second step, put the manipulatives in one central location (like a dish) and demonstrate that each number is the previous number with one more added. Help the child “count” with you, identify the number verbally and pick out the card with that number on it. Evenutally, have the child lay out the right amount of manipulatives to correspond with a random number that you pick from your deck.
Activity #4: Start “counting” stuff.....lots of stuff. Count the number of chairs at the dinner table, the number of plates or forks while setting the table, the number of shoes in daddy’s closet, the number of winter coats in the front hall, the number of toy cars or stuffed animals or Barbie doll dresses. Count, count, count until it becomes “natural to count” 1-2-3... These are the “natural counting numbers” that the student will use in kindergarten and first grade and which are the first group of “Real Numbers” as described in Pre-algebra. Imagine! Your 3 year old is already learning one of the Pre-algebra concepts!!
OTHER RESOURCES: The library has numerous number books with themes that may be stimulating to your child -- animals, characters, etc. There are puzzles that reinforce the shapes of numbers; I just found one at the one-dollar store.
Repetition and exposure are the keys to teaching NUMERACY, the vitally important first step in the understanding and love of math. It is my goal that your child have FUN with numbers and develop a sense of accomplishment and competence that will spur him or her to success in Math class. To receive a packet containing the Tutoring Resources Math Cards, dice, dots, and sample games for first grade through sixth, contact your local Tutoring Resources center in Barrington or Northbrook. Visit the website at
www.tutoring-resources.com for contact information.
But Math education starts long before Middle School and parents are the first and most influential teachers that every child can have. So here are a few things to do before students get into Preschool.
NUMERACY: This is the internal understanding of numbers. It involves number order, recognition of number symbols, and the one-to-one relationship of counting. Using the Tutoring Resources Math Cards distributed at our booth, select a few of the positive numbers, say 1 through 5 for the youngest kids and up to 10 as a child progresses. Lay them out on the table in numeric order.
Activity #1: (for ages as young as 2) Point to each number in order and say it’s name. Show the child how to point to the letters as you say their names. Ask the child to repeat the numbers after you’ve said them, starting with one at a time and progressing to the whole list at once. Eventually, have the student name the cards without help.
Activity #2: Using the same numbers that have been practiced in Activity #1, ask the child to point out a certain number, first with numbers laid out in order and then with cards randomly spread on the table.
Activity #3: Using manipulatives (navy beans, candy pieces, dots....something large enough that the child can pick it up without much discomfort), demonstrate the concept of each number by laying one item on the #1 card, 2 items on the #2 card, etc. As a second step, put the manipulatives in one central location (like a dish) and demonstrate that each number is the previous number with one more added. Help the child “count” with you, identify the number verbally and pick out the card with that number on it. Evenutally, have the child lay out the right amount of manipulatives to correspond with a random number that you pick from your deck.
Activity #4: Start “counting” stuff.....lots of stuff. Count the number of chairs at the dinner table, the number of plates or forks while setting the table, the number of shoes in daddy’s closet, the number of winter coats in the front hall, the number of toy cars or stuffed animals or Barbie doll dresses. Count, count, count until it becomes “natural to count” 1-2-3... These are the “natural counting numbers” that the student will use in kindergarten and first grade and which are the first group of “Real Numbers” as described in Pre-algebra. Imagine! Your 3 year old is already learning one of the Pre-algebra concepts!!
OTHER RESOURCES: The library has numerous number books with themes that may be stimulating to your child -- animals, characters, etc. There are puzzles that reinforce the shapes of numbers; I just found one at the one-dollar store.
Repetition and exposure are the keys to teaching NUMERACY, the vitally important first step in the understanding and love of math. It is my goal that your child have FUN with numbers and develop a sense of accomplishment and competence that will spur him or her to success in Math class. To receive a packet containing the Tutoring Resources Math Cards, dice, dots, and sample games for first grade through sixth, contact your local Tutoring Resources center in Barrington or Northbrook. Visit the website at
www.tutoring-resources.com for contact information.
Sunday, August 8, 2010
GET READY FOR THE NEXT LEVEL OF MATH
With the Fall semester right around the corner, conscientious students are starting to recognize that familiar “butterfly” feeling. What will my classes be like? What will I be expected to learn? to do? to know already?
It’s that last question that you can control. Be prepared to move forward in your next math class by reviewing what you already know. Here’s a handy list of concepts for the most frequently required math classes.
ELEMENTARY ARITHMETIC:
-- Start practicing adding, subtracting, multiplying, and dividing.
-- Practice multiplication tables and number families.
PRE-ALGEBRA:
-- Practice those computation skills so adding, subtracting, multiplying, and dividing are quick and easy for you. This will give you the mental space to learn the first concepts that will take you from arithmetic to mathematics.
ALGEBRA I:
-- Review last year’s work in Pre-Algebra if you still have it.
-- Ask the local library if there is a copy of the Algebra textbook on the shelf. It would probably be a reference copy, so you won’t be able to check it out. Make a photocopy of the Table of Contents and leaf through the first chapter which is frequently a review of past material.
-- Be prepared for these concepts:
COLLECT LIKE TERMS
SOLVE EQUATIONS WITH ONE VARIABLE
PROPERTIES OF REAL NUMBERS (Commutative, Associative, Distributive, Identify, Inverse, Reflexive, Transitive, Substitution)
GEOMETRY:
Do you want the good news first or the bad news? The good news is that there is very little in the average Geometry curriculum that you don’t already know. You’ve been learning a little Geometry every year since Kindergarten, which is the bad news. Some of the concepts you learned maybe 9 years ago and they may be buried under a lot of other stuff that you’ve learned in the interim. Start digging!
-- SHAPES -- square, rectangle, rhombus, parallelogram, trapezoid, circle, triangle. Know how to tell what each shape is (properties) and the equation for finding the area.
-- AREAS -- yes, I know I’ve mentioned that already, but it is a basic concept that causes trouble for many students. Know how to calculate areas and perimeters, too.
-- VOCABULARY -- The names of shapes is an example of vocabulary that you will be expected to already know. Add point-line-plane to the list, as well as diagonal, radius, diameter, arc, area, perimeter, volume (and related equations), Pythagorean Theorem, isosceles, equilateral, equiangular, skew, obtuse, acute, complementary, and supplementary. There may be more terms that you’ve already learned. Can you add to the list?
-- ALGEBRA -- “WHAT? I have to know last year’s Algebra in order to succeed in this year’s Geometry?” Yes. You’ll be asked to solve Geometry problems by using one-variable equations (like area) and systems of equations too. There may even be a few quadratic equations, so review factoring.
The biggest hint for success in your Geometry class is to learn the postulates and theorems well and as they are introduced.
ALGEBRA II:
-- Review your Algebra I notes from 2 years ago. If you haven’t developed the practice of filing notes from each class, resolve NOW to archive old class material now that you see how helpful it can be in future years.
-- Be prepared to factor quadratic equations and FOIL binomials.
-- Review the real number properties that allow you to solve equations.
TRIGONOMETRY:
Triangles are paramount shapes in Trig, so review everything you've learned about them. Pythagorean Theorem will have new applications and SohCahToa with be the basis for learning 3 more side relationships. A little review of circles will be helpful in the first couple of weeks when you're introduced to the Unit Circle.
A huge hint for success in Trig is to learn the Unit Circle when told to do so. You can expect to be tested on it in a stringent, timed format. After enough time to forget what you've memorized, you'll be expected to actually USE the information, so try to find patterns in the angles and the sine, cosine, and tangent values. This strategy will help you access the information in later problems.
PRECALCULUS:
Factoring is a stumbling block for even some high ability students. Practice it until you could "factor in your sleep." Review graphs of second and third degree equations, square roots, absolute value, and rational equations. A quick review of your old Algebra II work can help you to recall concepts you learned previously. Much of Precalc will be an exercise in deeper analysis of these same concepts.
IN GENERAL:
Check out older blogs for more detailed tips for each class.
GOOD LUCK! Not everyone is as excited about the beginning of a new semester as I am, but I know for sure that success in the first few weeks of school can lay the groundwork for a productive course and a good grade. My goal is for YOU to be the first student in the class to answer a question correctly. I want the teacher to go home that night believing that YOU are the best student ever. My experience shows that this can become a self-fulfilling prophesy that can enhance your classroom and exam performance.
It’s that last question that you can control. Be prepared to move forward in your next math class by reviewing what you already know. Here’s a handy list of concepts for the most frequently required math classes.
ELEMENTARY ARITHMETIC:
-- Start practicing adding, subtracting, multiplying, and dividing.
-- Practice multiplication tables and number families.
PRE-ALGEBRA:
-- Practice those computation skills so adding, subtracting, multiplying, and dividing are quick and easy for you. This will give you the mental space to learn the first concepts that will take you from arithmetic to mathematics.
ALGEBRA I:
-- Review last year’s work in Pre-Algebra if you still have it.
-- Ask the local library if there is a copy of the Algebra textbook on the shelf. It would probably be a reference copy, so you won’t be able to check it out. Make a photocopy of the Table of Contents and leaf through the first chapter which is frequently a review of past material.
-- Be prepared for these concepts:
COLLECT LIKE TERMS
SOLVE EQUATIONS WITH ONE VARIABLE
PROPERTIES OF REAL NUMBERS (Commutative, Associative, Distributive, Identify, Inverse, Reflexive, Transitive, Substitution)
GEOMETRY:
Do you want the good news first or the bad news? The good news is that there is very little in the average Geometry curriculum that you don’t already know. You’ve been learning a little Geometry every year since Kindergarten, which is the bad news. Some of the concepts you learned maybe 9 years ago and they may be buried under a lot of other stuff that you’ve learned in the interim. Start digging!
-- SHAPES -- square, rectangle, rhombus, parallelogram, trapezoid, circle, triangle. Know how to tell what each shape is (properties) and the equation for finding the area.
-- AREAS -- yes, I know I’ve mentioned that already, but it is a basic concept that causes trouble for many students. Know how to calculate areas and perimeters, too.
-- VOCABULARY -- The names of shapes is an example of vocabulary that you will be expected to already know. Add point-line-plane to the list, as well as diagonal, radius, diameter, arc, area, perimeter, volume (and related equations), Pythagorean Theorem, isosceles, equilateral, equiangular, skew, obtuse, acute, complementary, and supplementary. There may be more terms that you’ve already learned. Can you add to the list?
-- ALGEBRA -- “WHAT? I have to know last year’s Algebra in order to succeed in this year’s Geometry?” Yes. You’ll be asked to solve Geometry problems by using one-variable equations (like area) and systems of equations too. There may even be a few quadratic equations, so review factoring.
The biggest hint for success in your Geometry class is to learn the postulates and theorems well and as they are introduced.
ALGEBRA II:
-- Review your Algebra I notes from 2 years ago. If you haven’t developed the practice of filing notes from each class, resolve NOW to archive old class material now that you see how helpful it can be in future years.
-- Be prepared to factor quadratic equations and FOIL binomials.
-- Review the real number properties that allow you to solve equations.
TRIGONOMETRY:
Triangles are paramount shapes in Trig, so review everything you've learned about them. Pythagorean Theorem will have new applications and SohCahToa with be the basis for learning 3 more side relationships. A little review of circles will be helpful in the first couple of weeks when you're introduced to the Unit Circle.
A huge hint for success in Trig is to learn the Unit Circle when told to do so. You can expect to be tested on it in a stringent, timed format. After enough time to forget what you've memorized, you'll be expected to actually USE the information, so try to find patterns in the angles and the sine, cosine, and tangent values. This strategy will help you access the information in later problems.
PRECALCULUS:
Factoring is a stumbling block for even some high ability students. Practice it until you could "factor in your sleep." Review graphs of second and third degree equations, square roots, absolute value, and rational equations. A quick review of your old Algebra II work can help you to recall concepts you learned previously. Much of Precalc will be an exercise in deeper analysis of these same concepts.
IN GENERAL:
Check out older blogs for more detailed tips for each class.
GOOD LUCK! Not everyone is as excited about the beginning of a new semester as I am, but I know for sure that success in the first few weeks of school can lay the groundwork for a productive course and a good grade. My goal is for YOU to be the first student in the class to answer a question correctly. I want the teacher to go home that night believing that YOU are the best student ever. My experience shows that this can become a self-fulfilling prophesy that can enhance your classroom and exam performance.
Tuesday, May 4, 2010
FINAL EXAMS -- 3 WAYS TO PROCRASTINATE
College students will be returning home soon, signaling the approach of high school final exam time. Around this point in the semester, if you are like I was, you feel your brain is already full and there is little motivation to learn any more before summer break. So here are three strategies which will improve your final exam scores and simultaneously provide valid excuses to avoid studying until the last minute.
1. Since final exams are frequently cumulative, you will need to collect data from the entire semester or even the whole year. Do that now. Rummage through your closet, your backpack, under the bed. Find all old homework assignments, class notes, study guides, tests & quizzes (if you're allowed to take them home).
Create a separate "bin" for each subject. I use sweater boxes because they're wide enough to hold an 8.5 X 11 sheet of paper without folding and tall enough to collect a lot of stuff. When you run across old work, drop it into the appropriate bin.
2. I'm a visual, creative sort of person, especially when I'm avoiding doing any real work, so I like to "decorate" my bins. I recommend card stock since it holds up well. A different color for each bin can be stimulating, but I generally steer toward plain white so I can color the sign with magic markers which takes longer but looks like I'm preparing to study. I write the subject in fancy script and add things that remind me of the subject matter. Math, for example, lends itself to formulas written in elaborate calligraphy or to geometric diagrams illustrating concepts like "if sides, then angles." History suggests a time line or caricatures of historic figures. Science might involve terminology, molecular diagrams, or physics diagrams. English could include the novels discussed with reference to setting, characters, or actions. The options are limitless as long as they relate to the subjects covered during the semester. Check out textbooks for suggestions.
3. Ask the teacher for the Final Exam Study Guide. The one from last year is probably what will be used again this year, so it's in the classroom file somewhere. Start asking for it now so the teacher has a warning to locate it and have it printed. Since this will take at least a week or so, you won't have to actually work on it for a while but you've demonstrated to the one person who evaluates your grade that you are a conscientious, dedicated, hard-working, well-organized student (and that can't hurt your grade).
Keep reading ONLY if you really ARE that hard-working student who wants to start preparing for finals now. The activities described above are, in reality, the basis for study. By collecting old work, you have the chance to review what has been learned during the semester. The next step might be to create a list of the concepts that will probably appear on the final exam. To get ahead of the next blog, organize the collected material into topic, unit or chapter. You might even create a cover sheet that lists the concepts that were addressed in each unit or add the information to your decorative subject sign (which is your personal study guide in disguise).
By spending time designing your "sign" you are moving related concepts to the creative side of your brain where they will be more readily available for solving new problems. Keep adding things and periodically review what's on the card -- a strategy for getting details into long term memory.
Asking the teacher for the study guide really DOES create the impression that you are committed to earning a remarkable grade! You might become a self-fulfilling prophesy!!
Don't tell anyone that you're already studying for finals, though. Let everyone (except your folks) think you're just killing time until a cramming session the day before the exam.
1. Since final exams are frequently cumulative, you will need to collect data from the entire semester or even the whole year. Do that now. Rummage through your closet, your backpack, under the bed. Find all old homework assignments, class notes, study guides, tests & quizzes (if you're allowed to take them home).
Create a separate "bin" for each subject. I use sweater boxes because they're wide enough to hold an 8.5 X 11 sheet of paper without folding and tall enough to collect a lot of stuff. When you run across old work, drop it into the appropriate bin.
2. I'm a visual, creative sort of person, especially when I'm avoiding doing any real work, so I like to "decorate" my bins. I recommend card stock since it holds up well. A different color for each bin can be stimulating, but I generally steer toward plain white so I can color the sign with magic markers which takes longer but looks like I'm preparing to study. I write the subject in fancy script and add things that remind me of the subject matter. Math, for example, lends itself to formulas written in elaborate calligraphy or to geometric diagrams illustrating concepts like "if sides, then angles." History suggests a time line or caricatures of historic figures. Science might involve terminology, molecular diagrams, or physics diagrams. English could include the novels discussed with reference to setting, characters, or actions. The options are limitless as long as they relate to the subjects covered during the semester. Check out textbooks for suggestions.
3. Ask the teacher for the Final Exam Study Guide. The one from last year is probably what will be used again this year, so it's in the classroom file somewhere. Start asking for it now so the teacher has a warning to locate it and have it printed. Since this will take at least a week or so, you won't have to actually work on it for a while but you've demonstrated to the one person who evaluates your grade that you are a conscientious, dedicated, hard-working, well-organized student (and that can't hurt your grade).
Keep reading ONLY if you really ARE that hard-working student who wants to start preparing for finals now. The activities described above are, in reality, the basis for study. By collecting old work, you have the chance to review what has been learned during the semester. The next step might be to create a list of the concepts that will probably appear on the final exam. To get ahead of the next blog, organize the collected material into topic, unit or chapter. You might even create a cover sheet that lists the concepts that were addressed in each unit or add the information to your decorative subject sign (which is your personal study guide in disguise).
By spending time designing your "sign" you are moving related concepts to the creative side of your brain where they will be more readily available for solving new problems. Keep adding things and periodically review what's on the card -- a strategy for getting details into long term memory.
Asking the teacher for the study guide really DOES create the impression that you are committed to earning a remarkable grade! You might become a self-fulfilling prophesy!!
Don't tell anyone that you're already studying for finals, though. Let everyone (except your folks) think you're just killing time until a cramming session the day before the exam.
Tuesday, April 13, 2010
JUNIORS: 2 Weeks to the PSAE Study Plan
Fourteen days to the PSAE for Illinois high school juniors and time for a little "intensive study." If you haven't already, get a few copies of old ACT tests. The library or college department at your high school should have copies of the ACT preparation materials which include one sample. There are numerous prep manuals at bookstores. ACT publishes one and I also recommend the Barron's version because of it's similarity to real ACT questions and the clarity of answer explanations. Some online sources have samples too. Check out previous blogs for links.
YOUR 14 DAY PLAN...
1. Take a sample test, possibly over 4 days (one day for each section if you're under a time constraint) and keep track of how much time it takes to finish each section. Scoring your work will give you an indication of how much you need to study and timing will alert you to any need for pacing practice.
2 (One day) Choose the section in which you scored the lowest. Identify the concepts tested in each question you missed. Read the instructional material on that concept in the Barron's text or other answer key. Complete any sample questions available.
3. (The next day) Take another test on your #1 section. Score. Recognize improvement. Look over the rationale for the correct answer for any question you missed.
4. Repeat the process on the next lowest scoring section -- 2 days of review -- one day to study and the next day to retest and review. Repeat Step 4 until all sections have been studied.
If you used 4 days to take the original test, you've spent 12 days studying. Two more to go.
5. Take another full test. If pacing has been problematic, follow these guidelines:
ENGLISH (75 questions in 45 minutes): There are 5 articles, each with 15 questions. Try to finish all questions in an article in 9 minutes. Answer each question as you come to it. Don't skip any. Put answers directly on the scantron.
MATH (60 problems in 60 minutes): This may appear to give you 1 minute for each question, but the easier ones come at the beginning and the more difficult ones after #45. Try to spend just 30 seconds on most of the first 30 questions. From 31 to 45, plan on one minute each. From 46 through 60, you'll have about 2 minutes for each. If a problem requires too much time, skip it if you aren't looking for a perfect score. Either mark the scantron lightly so you know to return to the question, or default the question right away but lightly so you can change the answer if you have time to come back and work on it again.
READING (40 questions in 35 minutes): Each Reading passage has 10 questions. Plan to spend 8 minutes on each passage, leaving 3 minutes extra for the "extra hard" questions. Try answering direct questions first and inferential ones once you have an idea what the article is about. Mark answers directly on the test booklet until all 10 are complete and then transfer your selections to the scantron before moving to the next article.
SCIENCE REASONING (40 questions in 35 minutes): For pacing, plan to spend one less minute than the number of questions in the data set. This timing gives you 2 additional minutes; that's not much, so use it wisely. Answer questions in the order they are asked within the data set and mark your alternative choices directly on the scantron. BUT if you're not reaching for a 35 or 36 (nearly perfect or perfect score), hold an extremely difficult data set for last. Default the answers if you think you'll run out of time, but do so lightly enough so that you can erase if you get back to it and work out answers.
READY? SET? STUDY!
YOUR 14 DAY PLAN...
1. Take a sample test, possibly over 4 days (one day for each section if you're under a time constraint) and keep track of how much time it takes to finish each section. Scoring your work will give you an indication of how much you need to study and timing will alert you to any need for pacing practice.
2 (One day) Choose the section in which you scored the lowest. Identify the concepts tested in each question you missed. Read the instructional material on that concept in the Barron's text or other answer key. Complete any sample questions available.
3. (The next day) Take another test on your #1 section. Score. Recognize improvement. Look over the rationale for the correct answer for any question you missed.
4. Repeat the process on the next lowest scoring section -- 2 days of review -- one day to study and the next day to retest and review. Repeat Step 4 until all sections have been studied.
If you used 4 days to take the original test, you've spent 12 days studying. Two more to go.
5. Take another full test. If pacing has been problematic, follow these guidelines:
ENGLISH (75 questions in 45 minutes): There are 5 articles, each with 15 questions. Try to finish all questions in an article in 9 minutes. Answer each question as you come to it. Don't skip any. Put answers directly on the scantron.
MATH (60 problems in 60 minutes): This may appear to give you 1 minute for each question, but the easier ones come at the beginning and the more difficult ones after #45. Try to spend just 30 seconds on most of the first 30 questions. From 31 to 45, plan on one minute each. From 46 through 60, you'll have about 2 minutes for each. If a problem requires too much time, skip it if you aren't looking for a perfect score. Either mark the scantron lightly so you know to return to the question, or default the question right away but lightly so you can change the answer if you have time to come back and work on it again.
READING (40 questions in 35 minutes): Each Reading passage has 10 questions. Plan to spend 8 minutes on each passage, leaving 3 minutes extra for the "extra hard" questions. Try answering direct questions first and inferential ones once you have an idea what the article is about. Mark answers directly on the test booklet until all 10 are complete and then transfer your selections to the scantron before moving to the next article.
SCIENCE REASONING (40 questions in 35 minutes): For pacing, plan to spend one less minute than the number of questions in the data set. This timing gives you 2 additional minutes; that's not much, so use it wisely. Answer questions in the order they are asked within the data set and mark your alternative choices directly on the scantron. BUT if you're not reaching for a 35 or 36 (nearly perfect or perfect score), hold an extremely difficult data set for last. Default the answers if you think you'll run out of time, but do so lightly enough so that you can erase if you get back to it and work out answers.
READY? SET? STUDY!
Monday, March 29, 2010
QUICK TIPS FOR THE ACT
April 10....the next national administration of the ACT. Really too late to start studying, but a few quick tips could help raise that score by a point or two. For English and Math you might need 2 more correct answers to gain one more section score point; each Reading and Science Reasoning question should usually give you a point. Four more section score points are needed to raise the composite average by just one point.
ENGLISH: Recognize the punctuation questions relating to independent sentences. You always need a full sentence in front of a period, semi-colon, comma-conjunction, or colon. For the first 3 you also need a full sentence (not a prepositional phrase) after the punctuation. A colon can be followed by either a full sentence or a list or just more detail.
Recognize prepositions. They introduce phrases, not full sentences, and might need a comma at most.
SHORTER IS BETTER. At least consider the shortest alternative since it's the right answer more often than not.
MATH: Avoid silly mistakes, especially on the first 30 questions. And work problems for the entire hour; there are some easy questions in the last 10.
READING: Keep moving. Don't get "stuck" on one difficult question. Unless you're working toward a perfect score (which means you've been studying for weeks), you don't need to get every question right. Take the loss of one point rather than denying yourself the chance to gain 2 by finding easier questions further along in the section.
SCIENCE REASONING: Get the first two questions in every data set correct. These ask if you can read the data given and the answer should be "sure." If the last 2 questions are too hard, default them and move on to the next data set where the first few questions will be easier.
IN GENERAL: Leave nothing blank. Pick a default answer and use it to fill in bubbles for any question you can't answer. ONE DEFAULT ONLY. Pick the first, second, third, fourth, or fifth alternative and use only that one for default. Statistically you will be correct 20 - 25% of the time. Arbitrarily picking a different alternative every time could result in a success rate of 0%!
SIT UP! If you start to fade, take a second to sit up straight, stretch, breath deeply. Get the brain oxygenated again and dig back in!!
GOOD LUCK! And remember that if your score doesn't turn out to be what you hoped, there is also June, September, and October for 2011 graduates. With any of these test dates, you'll have results in time for early admission to most colleges.
ENGLISH: Recognize the punctuation questions relating to independent sentences. You always need a full sentence in front of a period, semi-colon, comma-conjunction, or colon. For the first 3 you also need a full sentence (not a prepositional phrase) after the punctuation. A colon can be followed by either a full sentence or a list or just more detail.
Recognize prepositions. They introduce phrases, not full sentences, and might need a comma at most.
SHORTER IS BETTER. At least consider the shortest alternative since it's the right answer more often than not.
MATH: Avoid silly mistakes, especially on the first 30 questions. And work problems for the entire hour; there are some easy questions in the last 10.
READING: Keep moving. Don't get "stuck" on one difficult question. Unless you're working toward a perfect score (which means you've been studying for weeks), you don't need to get every question right. Take the loss of one point rather than denying yourself the chance to gain 2 by finding easier questions further along in the section.
SCIENCE REASONING: Get the first two questions in every data set correct. These ask if you can read the data given and the answer should be "sure." If the last 2 questions are too hard, default them and move on to the next data set where the first few questions will be easier.
IN GENERAL: Leave nothing blank. Pick a default answer and use it to fill in bubbles for any question you can't answer. ONE DEFAULT ONLY. Pick the first, second, third, fourth, or fifth alternative and use only that one for default. Statistically you will be correct 20 - 25% of the time. Arbitrarily picking a different alternative every time could result in a success rate of 0%!
SIT UP! If you start to fade, take a second to sit up straight, stretch, breath deeply. Get the brain oxygenated again and dig back in!!
GOOD LUCK! And remember that if your score doesn't turn out to be what you hoped, there is also June, September, and October for 2011 graduates. With any of these test dates, you'll have results in time for early admission to most colleges.
Friday, March 26, 2010
START SLOWLY. WORK SMART. DO IT TODAY!
I know from experience how difficult it is to get moving on a task that seems as massive as working toward the score you deserve on the ACT. But this weekend of Spring Break is the perfect time to start.
Begin slowly: download a few questions from the ACT website, sign up for the question of the day while your there, get a copy of The Real ACT Prep Guide (about $25). Complete a small part of each section:
English -- answer the 15 questions in any of the 5 passages.
Math -- work 10 problems.
Reading -- answer the 10 questions in any of the 4 essays.
Science Reasoning -- pick one data set and answer all of the questions. Just for fun, keep track of the data sets as you do them and we can look later at a pattern that should tell which are the easiest and which are more challenging.
Work Smart: Stick to this daily plan and you'll gain knowledge of each of the sections. Don't time yourself yet. Concentrate on thinking about the problems you're working. It should take you only about half an hour to do this work but at the end of the week, you'll have completed 1.5 English sections, 1 Math, 2 Reading, and one whole Science Reasoning. That's a pretty good study week, especially since you may be on holiday!! CONGRATULATIONS!!
Begin slowly: download a few questions from the ACT website, sign up for the question of the day while your there, get a copy of The Real ACT Prep Guide (about $25). Complete a small part of each section:
English -- answer the 15 questions in any of the 5 passages.
Math -- work 10 problems.
Reading -- answer the 10 questions in any of the 4 essays.
Science Reasoning -- pick one data set and answer all of the questions. Just for fun, keep track of the data sets as you do them and we can look later at a pattern that should tell which are the easiest and which are more challenging.
Work Smart: Stick to this daily plan and you'll gain knowledge of each of the sections. Don't time yourself yet. Concentrate on thinking about the problems you're working. It should take you only about half an hour to do this work but at the end of the week, you'll have completed 1.5 English sections, 1 Math, 2 Reading, and one whole Science Reasoning. That's a pretty good study week, especially since you may be on holiday!! CONGRATULATIONS!!
Wednesday, March 24, 2010
ACT MATH CHECKLIST
Oh, yeah. There's a LOT of Math to know for the ACT and it is easy to skip over some important rule or formula. Here's a checklist of the concepts mentioned in previous blogs. Use it to refresh your memory and insure that you've covered each concept sufficiently. As you take practice tests, highlight the math facts that caused errors and review those again.
BASICS
___ fractions
___ percents
___ percent change
___ word problems
___ ratios
___ average
___ area
ALGEBRA
___ factoring
___ distance formula
___ midpoint formula
___ rules of exponents
___ graphs
___ functions
___ conic equations
___ logs
___ sequence and series
___ probability
___ variation
___ systems of linear equations
___ matrices
GEOMETRY
___ special right triangles
___ SohCahToa
___ parallel lines and transversals
___ properties of quadrilaterals
___ circles
___ sum of interior angles of a polygon
___ volume
___ similar triangles
___ relation between lines, areas, and volumes
TRIGONOMETRY
___ Pythagorean Identities
___ The Unit Circle
There. That wasn't so bad, now was it? Stay tuned to this blog site for analysis of recent ACT test questions and any changes that we discover.
BASICS
___ fractions
___ percents
___ percent change
___ word problems
___ ratios
___ average
___ area
ALGEBRA
___ factoring
___ distance formula
___ midpoint formula
___ rules of exponents
___ graphs
___ functions
___ conic equations
___ logs
___ sequence and series
___ probability
___ variation
___ systems of linear equations
___ matrices
GEOMETRY
___ special right triangles
___ SohCahToa
___ parallel lines and transversals
___ properties of quadrilaterals
___ circles
___ sum of interior angles of a polygon
___ volume
___ similar triangles
___ relation between lines, areas, and volumes
TRIGONOMETRY
___ Pythagorean Identities
___ The Unit Circle
There. That wasn't so bad, now was it? Stay tuned to this blog site for analysis of recent ACT test questions and any changes that we discover.
IMPROVING ACT MATH, PART 2
Now that the first 30 Math questions are perfect, and the next 15 are within sight of "perfect," it's time to buckle down on the last 15 questions and possibly fill in some of that Algebra II that may have appeared earlier on the test.
ALGREBRA II
- rules of exponents - when multiplying the same base, add the exponents; to raise an exponent to a higher power, multiply (and don't forget the coefficient if it's in a parenthesis with the variable); when dividing the same base, subtract the exponents; A NEGATIVE EXPONENT CHANGES THE POSITION OF THE BASE, NOT IT'S POSITIVE OR NEGATIVE VALUE (this is the most widely missed question relating to exponents).
- graphs - recognize linear, quadratic, and third degree graphs and translations of each. Know how to calculate or recognize slope, X intercepts (these are solutions -- any ordered pair satisfies the equation but only the roots solve it), Y intercepts, and points of intersection (where two graphs meet). Practice telling a story that fits a combination graph.
- functions - recognize odd and even functions from their exponents. Use the vertical line test to prove that an equation is a function. Use both the vertical line test and the horizontal line test to prove that the inverse is also a function (or that the original function is "one-to-one.") Practice substitution and compositions of functions.
- conic equations - circle (Did you realize that this is also the Pythagorean Theorem?), ellipse, hyperbola, and quadratic, too.
- logs - basic rules for expanding and simplifying.
- sequence and series - arithmetic and geometric. Arithmetic has a common difference (add the same thing every time to get the next term) and geometric has a common ratio (multiply by the same thing every time to get the next term).
- probability - successes divided by possibles. (Do you see a similarity with percent change?) Probability questions are appearing in the first 30 questions on a regular basis these days.
- variation - direct ( Y = kX) and inverse ( Y = k/X).
- systems of linear equations - substitution and elimination methods are used to solve for one variable at a time.
- matrices - setting up from a system of equations, adding, multiplying, and expanding through scalar multiplication. (Here's a perfect place to use the calculator to it's highest purpose!)
GEOMETRY
- good news - most of the geometry used is actually part of the elementary curriculum AND you won't have to do any proofs.
- special right triangles - Pythagorean Triplets, 30-60-90, and 45-45-90. Although you can always use the Pythagorean Theorem to find these relationships, the ACT and other college entrance exams are TIMED and knowing that 5-12-13 is a triplet can save precious minutes.
- SohCahToa- sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent.
- parallel lines and transversals - know which angles are congruent and which are supplementary.
- properties of quadrilaterals - parallogram, rectangle, rhombus, square, trapezoid, and kite.
- circles - all radii are equal. (This is the most valuable property of a circle and will usually figure into a solution if a circle is used in a question.) Central angle equals the arc, inscribed angle equals 1/2 the arc, and outside angle equals (the big arc minus the little arc) times 1/2. Lines tangent to the circle are perpendicular to the radius.
- sum of interior angles of a polygon - (the number of sides minus 2) times 180 and each angle in the regular polygon is that equation divided by the number of sides. In a later blog, we'll look at another way to get the same answer without the need to remember an equation.
- volume - (if the sides are perpendicular to the base) the volume is "the area of the base times the height." (if the shape is a pyramid or cone) divide the answer by 3. The equation for finding the volume of a sphere will probably be given in the question.
- similar triangles - all angles are congruent and sides are proportional -- set up congruent ratios.
- relation between lines, areas, and volumes - if the sides are in the ratio 2 to 3 (2/3 or 2:3) then the ratio of the areas is (2/3) SQUARED, and the ratio of the volumes is (2/3)CUBED. Lines are first degree, areas are second degree, and volumes are third degree.
TRIGONOMETRY
- good news - probably half of the questions that the ACT calls Trig are from the Geometry curriculum in many schools. Go back and review everything you know about triangles, especially those pesky word problems involving shadows and ladders leaning against buildings.
- Pythagorean Identities - yet a fourth application of the Pythagorean Theorem. It is no wonder that we frequently say it is the "most important equation in all of math"and luckily it's taught around the fourth grade. Sine^2 + Cosine^2 = 1.
- The Unit Circle - know your reference angles and what's positive in each quadrant. ACT trig questions will usually refer to the axes and angles in radians, so know how to convert from degrees if needed.
You might be amazed at all of the Math that you've learned through the years. There are probably only a very few concepts that you haven't studied, so preparing for the Math section is really refreshing your brain and bringing the useful information to the top where it can be accessed quickly on demand.
ALGREBRA II
- rules of exponents - when multiplying the same base, add the exponents; to raise an exponent to a higher power, multiply (and don't forget the coefficient if it's in a parenthesis with the variable); when dividing the same base, subtract the exponents; A NEGATIVE EXPONENT CHANGES THE POSITION OF THE BASE, NOT IT'S POSITIVE OR NEGATIVE VALUE (this is the most widely missed question relating to exponents).
- graphs - recognize linear, quadratic, and third degree graphs and translations of each. Know how to calculate or recognize slope, X intercepts (these are solutions -- any ordered pair satisfies the equation but only the roots solve it), Y intercepts, and points of intersection (where two graphs meet). Practice telling a story that fits a combination graph.
- functions - recognize odd and even functions from their exponents. Use the vertical line test to prove that an equation is a function. Use both the vertical line test and the horizontal line test to prove that the inverse is also a function (or that the original function is "one-to-one.") Practice substitution and compositions of functions.
- conic equations - circle (Did you realize that this is also the Pythagorean Theorem?), ellipse, hyperbola, and quadratic, too.
- logs - basic rules for expanding and simplifying.
- sequence and series - arithmetic and geometric. Arithmetic has a common difference (add the same thing every time to get the next term) and geometric has a common ratio (multiply by the same thing every time to get the next term).
- probability - successes divided by possibles. (Do you see a similarity with percent change?) Probability questions are appearing in the first 30 questions on a regular basis these days.
- variation - direct ( Y = kX) and inverse ( Y = k/X).
- systems of linear equations - substitution and elimination methods are used to solve for one variable at a time.
- matrices - setting up from a system of equations, adding, multiplying, and expanding through scalar multiplication. (Here's a perfect place to use the calculator to it's highest purpose!)
GEOMETRY
- good news - most of the geometry used is actually part of the elementary curriculum AND you won't have to do any proofs.
- special right triangles - Pythagorean Triplets, 30-60-90, and 45-45-90. Although you can always use the Pythagorean Theorem to find these relationships, the ACT and other college entrance exams are TIMED and knowing that 5-12-13 is a triplet can save precious minutes.
- SohCahToa- sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent.
- parallel lines and transversals - know which angles are congruent and which are supplementary.
- properties of quadrilaterals - parallogram, rectangle, rhombus, square, trapezoid, and kite.
- circles - all radii are equal. (This is the most valuable property of a circle and will usually figure into a solution if a circle is used in a question.) Central angle equals the arc, inscribed angle equals 1/2 the arc, and outside angle equals (the big arc minus the little arc) times 1/2. Lines tangent to the circle are perpendicular to the radius.
- sum of interior angles of a polygon - (the number of sides minus 2) times 180 and each angle in the regular polygon is that equation divided by the number of sides. In a later blog, we'll look at another way to get the same answer without the need to remember an equation.
- volume - (if the sides are perpendicular to the base) the volume is "the area of the base times the height." (if the shape is a pyramid or cone) divide the answer by 3. The equation for finding the volume of a sphere will probably be given in the question.
- similar triangles - all angles are congruent and sides are proportional -- set up congruent ratios.
- relation between lines, areas, and volumes - if the sides are in the ratio 2 to 3 (2/3 or 2:3) then the ratio of the areas is (2/3) SQUARED, and the ratio of the volumes is (2/3)CUBED. Lines are first degree, areas are second degree, and volumes are third degree.
TRIGONOMETRY
- good news - probably half of the questions that the ACT calls Trig are from the Geometry curriculum in many schools. Go back and review everything you know about triangles, especially those pesky word problems involving shadows and ladders leaning against buildings.
- Pythagorean Identities - yet a fourth application of the Pythagorean Theorem. It is no wonder that we frequently say it is the "most important equation in all of math"and luckily it's taught around the fourth grade. Sine^2 + Cosine^2 = 1.
- The Unit Circle - know your reference angles and what's positive in each quadrant. ACT trig questions will usually refer to the axes and angles in radians, so know how to convert from degrees if needed.
You might be amazed at all of the Math that you've learned through the years. There are probably only a very few concepts that you haven't studied, so preparing for the Math section is really refreshing your brain and bringing the useful information to the top where it can be accessed quickly on demand.
Monday, March 22, 2010
REMEMBERING MATH FORMULAS
Here's a little trick for remembering those pesky math formulas. Write them out on a notecard and tape it to the UPPER LEFT CORNER of the mirror you use every day to brush your teeth, style your hair, and admire your countenance. You'll find yourself glancing at it, but don't make it an effort. When looking for a particular formula in your brain, move your eyes UP AND LEFT. Is the notecard there?
If this subtle method does not work for you, if you don't "picture" the card when it isn't physically in front of you, try the next step. Add a little picture to the formula to give yourself another visual clue.
Still having trouble? Try "hearing" the music of a formula. You're probably familiar with the rhythm of (A squared plus B squared equals C squared). Pick a different formula each day and say it in your mind several times while you're doing those mirror tasks.
Whenever I'm working a problem, I first write down the applicable formula. That makes even tough problems nothing more than a matter of substitution and calculation.
WHAT FORMULAS SHOULD YOU KNOW FOR THE ACT?
Pythagorean Theorem
Average
Rate of Change
RT = D
Slope
Midpoint
ALL Areas
ALL Volumes
Circumference
SOH-CAH-TOA
And if you think of others, add your suggestions to this blog through "comments."
If this subtle method does not work for you, if you don't "picture" the card when it isn't physically in front of you, try the next step. Add a little picture to the formula to give yourself another visual clue.
Still having trouble? Try "hearing" the music of a formula. You're probably familiar with the rhythm of (A squared plus B squared equals C squared). Pick a different formula each day and say it in your mind several times while you're doing those mirror tasks.
Whenever I'm working a problem, I first write down the applicable formula. That makes even tough problems nothing more than a matter of substitution and calculation.
WHAT FORMULAS SHOULD YOU KNOW FOR THE ACT?
Pythagorean Theorem
Average
Rate of Change
RT = D
Slope
Midpoint
ALL Areas
ALL Volumes
Circumference
SOH-CAH-TOA
And if you think of others, add your suggestions to this blog through "comments."
Saturday, March 20, 2010
IMPROVING ACT MATH, PART ONE
Let's face facts. Math is not the favorite subject for every student. The good news is that the ACT Math section is geared toward an average high school curriculum. If you've completed an Algebra II course, you have been exposed to all the knowledge necessary to achieve a 30 on the test. A Trigonometry course adds the possibility of even more points.
Concentrate on achieving a perfect score on the first 30 questions. You need to review a few elementary concepts, some lower level Algebra, and maybe a couple of concepts that are unexpected because they receive such little attention in school.
ELEMENTARY ISSUES
- fractions - What do yo need to add or subtract fractions? (a common denominator, then add or subtract the numerators) How do you multiply fractions? (multiply straight across the numerator and straight across the denominator) How do you divide by a fraction? (invert and multiply)
- percents - convert to decimals and, usually, multiply
- percent change - divide the amount of change by the original amount
- word problems - write an algebra sentence. "Of" means multiply. "Is" means equals. "And" means add. "Per" creates a fraction or ratio. I try to avoid mistakes with "less THAN" and "greater or more THAN" by putting them at the end of the expression from the start. (X is 3 less than Y) translates to
"X = Y - 3."
- ratios - Try naming the numerator and denominator so they don't get mixed up. If you're writing a proportion, be sure the numerators have the same label. Cross multiply only in proportions, otherwise multiply like fractions.
- average - (also called arithmetic mean or simply mean) add the items up and divide by the number of items.
- area - know the formulas for parallelogram, trapezoid, circle, and triangle. They will come up more than once. And while you're at it, know perimeter and circumference too.
- Pythagorean Theorem - "A squared plus B squared equals C squared" (aˆ2 + bˆ2 = cˆ2)
ALGEBRA BASICS
- distributive property - a common mistake in distributing is losing a sign, especially from the negative terms.
- FOIL - (sometimes taught as "double distributive" because two distinct terms are distributed to those of the second binomial) First, Outer, Inner, Last. The most common error is from squaring a binomial -- (x + 3)ˆ2 -- does NOT equal (xˆ2 + 3ˆ2). (x+3)ˆ2 = (x+3)(x+3). FOIL.
- factoring - be prepared to factor out commons to make a problem easier; factor 4 terms by grouping; factor quadratics and cubics. Recognizing special situations like the difference of 2 perfect squares can save time in problem completion.
- distance formula - d = √[(X -x)ˆ2 + (Y -y)ˆ2]. Did you know that this is actually the Pythagorean Theorem?
- midpoint formula - call upon your outstanding knowledge of "average." The midpoint is (the average X, the average Y), so add 'um up and divide by 2.
- slope - Rise over Run -- ∆Y/∆X -- (Y - y)/(X -x) -- Up and Over (On Saturday morning, you've got to GET UP before you can GO OVER to a friend's house.) Common problems include mixing up the numerator and denominator and missing the importance of the slope's direction. Use your knowledge of positive and negative slope to eliminate misleading alternative answers.
THE UNEXPECTED
- probability - the number of successes divided by the possible outcomes (Success/Possibles). This number is never more than one.
- variation - I start by setting up an equation for either direct or inverse variation and then plugging in values, making this a substitution problem. Direct variation (Y varies directly with X) --
Y = kX. Inverse variation (Y varies inversely with X) -- Y =k/X.
A flawless first half of the Math section will provide a firm foundation for achieving a score well above average and you're ready to focus on the last 30 problems. More (higher level) Math concepts will be in the next post. For now, celebrate your Math Genius who just earned an A+.
Concentrate on achieving a perfect score on the first 30 questions. You need to review a few elementary concepts, some lower level Algebra, and maybe a couple of concepts that are unexpected because they receive such little attention in school.
ELEMENTARY ISSUES
- fractions - What do yo need to add or subtract fractions? (a common denominator, then add or subtract the numerators) How do you multiply fractions? (multiply straight across the numerator and straight across the denominator) How do you divide by a fraction? (invert and multiply)
- percents - convert to decimals and, usually, multiply
- percent change - divide the amount of change by the original amount
- word problems - write an algebra sentence. "Of" means multiply. "Is" means equals. "And" means add. "Per" creates a fraction or ratio. I try to avoid mistakes with "less THAN" and "greater or more THAN" by putting them at the end of the expression from the start. (X is 3 less than Y) translates to
"X = Y - 3."
- ratios - Try naming the numerator and denominator so they don't get mixed up. If you're writing a proportion, be sure the numerators have the same label. Cross multiply only in proportions, otherwise multiply like fractions.
- average - (also called arithmetic mean or simply mean) add the items up and divide by the number of items.
- area - know the formulas for parallelogram, trapezoid, circle, and triangle. They will come up more than once. And while you're at it, know perimeter and circumference too.
- Pythagorean Theorem - "A squared plus B squared equals C squared" (aˆ2 + bˆ2 = cˆ2)
ALGEBRA BASICS
- distributive property - a common mistake in distributing is losing a sign, especially from the negative terms.
- FOIL - (sometimes taught as "double distributive" because two distinct terms are distributed to those of the second binomial) First, Outer, Inner, Last. The most common error is from squaring a binomial -- (x + 3)ˆ2 -- does NOT equal (xˆ2 + 3ˆ2). (x+3)ˆ2 = (x+3)(x+3). FOIL.
- factoring - be prepared to factor out commons to make a problem easier; factor 4 terms by grouping; factor quadratics and cubics. Recognizing special situations like the difference of 2 perfect squares can save time in problem completion.
- distance formula - d = √[(X -x)ˆ2 + (Y -y)ˆ2]. Did you know that this is actually the Pythagorean Theorem?
- midpoint formula - call upon your outstanding knowledge of "average." The midpoint is (the average X, the average Y), so add 'um up and divide by 2.
- slope - Rise over Run -- ∆Y/∆X -- (Y - y)/(X -x) -- Up and Over (On Saturday morning, you've got to GET UP before you can GO OVER to a friend's house.) Common problems include mixing up the numerator and denominator and missing the importance of the slope's direction. Use your knowledge of positive and negative slope to eliminate misleading alternative answers.
THE UNEXPECTED
- probability - the number of successes divided by the possible outcomes (Success/Possibles). This number is never more than one.
- variation - I start by setting up an equation for either direct or inverse variation and then plugging in values, making this a substitution problem. Direct variation (Y varies directly with X) --
Y = kX. Inverse variation (Y varies inversely with X) -- Y =k/X.
A flawless first half of the Math section will provide a firm foundation for achieving a score well above average and you're ready to focus on the last 30 problems. More (higher level) Math concepts will be in the next post. For now, celebrate your Math Genius who just earned an A+.
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