GRAPHING TRIG FUNCTIONS

This is a quick note to reinforce my position on graphing trigonometry functions by using charts to calculate your 5 significant points (especially for Hira's upcoming quiz).  Drawing actual graphs take hours, so I’ll skip that part today and just focus on creating the charts of values.

If you want to take shortcuts and just wing it -- good luck.  But if you are looking for an A, I STRONGLY recommend doing the chart of values and plotting the graph from the data.  Even if you make a little mistake on the graph itself, the teacher has the chart to witness that you actually know what you are doing.  I’ll take partial points over no points any day.

So here are the charts and what to do with them.
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Take any trig equation in the form

    A (trig function) B ( X + C ) + D

Using the basic chart for the trig function named, multiply Y values by A and add D.  Divide X by B and subtract C.

Notice that adding D to Y uses the SAME SIGN as the equation, while subtracting C from X uses the OPPOSITE SIGN.  Remember my “Sandi Rule” -- any time you use a number that’s attached to a variable, CHANGE THE SIGN.

 

 TIPS:  

¶  Leave sufficient room in your chart to do the calculations.  Working with fractions, where you might have to find a common denominator, can spread out a lot.  Don’t skimp on your working space.

§  Notice that the COSECANT sits where the SINE DOESN'T and the SECANT sits where the COSINE DOESN'T.  ASYMPTOTES appear in the cosecant and secant where the sine and cosine are zero.
§§  The same does NOT hold for Tangent and its reciprocal Cotangent.

∆  Remember that B must be the coefficient of the binomial (X + C).  If you are given (3X + 9), don't forget to factor out the 3 from BOTH 3X and 9.

Using the charts, you can focus attention on the arithmetic to avoid calculation mistakes.  Then transferring the data to the coordinate plane allows you to concentrate on the visual aspects of the graph.  Make it pretty.  Use colors to draw your final graph.  Have fun!!  This is the artistic portion of math.

INDUCTIVE AND DEDUCTIVE REASONING

Simple logic was a favorite topic in my Philosophy degree, so my enthusiasm for the Geometry chapters that deal with reasoning can be easily explained.  The roots that are developed through the Geometry curriculum can be used in many other contexts: debate, essay writing, solving problems, and even convincing your parents to extending curfew.  In Geometry class right about now, if logic precedes introduction of two-column proofs, we should be thinking about inductive and deductive reasoning.

INDUCTIVE REASONING goes from the specific to the general, from observed details to global conjectures.  It’s what a scientist would use to develop a hypothesis after collecting data, discovering a pattern, and assuming that the pattern will continue.  It’s also the thinking that is expected on the last couple of questions in any data set on the ACT Science Reasoning section.

A determining factor of inductive reasoning is that conjectures based on an infinite universe can never be proven.  We can only DISPROVE them.  Let’s say a statistician collects a huge amount of data about  monkeys in a maze.  The data shows that every time a monkey consumed carrots before running the maze, the elapsed time was shorter than without eating carrots.  This pattern continued with tens of thousands of monkeys.  The scientist hypothesizes that eating carrots will always improve a monkey’s speed in running a maze.  It would be impossible to test every monkey, so we could never prove the hypothesis is universally true.  But find just one monkey whose performance did NOT improve and the hypothesis is disproven.

The same principle of finding a counterexample will be helpful later when we get to indirect proof.

DEDUCTIVE REASONING (general to specific) starts with a global statement, properties, definitions, postulates, or theorems and applies them to form conclusions in specific situations.  It assumes that what is true of the large group is true of all of the members of the group.  This is the form of reasoning we’ll be using in two-column proof in Geometry.

Within deductive reasoning, we find two important principles: the laws of DETACHMENT and SYLLOGISM.

                 DETACHMENT follows my proposal that mathematicians aren’t poets engaging in metaphors.  When mathematicians mean detach, they say detach.  So the Law of Detachment says that something can be removed or detached.

If P implies Q
and P is true (given)
then Q is also true.

Q can be detached from the original statement and concluded to be true. This is a primary construction in Geometry  proofs.

1.  If a polygon has three sides, then it is a triangle.
2.  This polygon has three sides.
3.  Therefore, it is a triangle.

                  SYLLOGISM:  If you learned the transitive property in Algebra, you already know about syllogisms.

The placement of P, Q, and R are important in syllogisms.  R is the conclusion you want to come to and is considered the MAJOR term.  P is called the MINOR term, and Q is the MIDDLE term.  As “transitive” implies, we’re moving over or across the Q to get P to relate to R.



VALIDITY AND SOUNDNESS

CAUTION!  When we’re talking about logical steps to conclusions, we need to distinguish between validity and soundness.  If an argument is VALID, it follows the syllogistic steps already listed.  Validity is determined by the form of the argument and says nothing about truth, either of the premises or the conclusion.  A conclusion can be valid but unsound.

If the premises are not true, even if they follow appropriate steps, then the conclusion is not sound.  In order for an argument to be SOUND, the proper logical steps must be followed and premises must also be true. 

Here’s an example of a valid argument which is NOT sound.

    All my pets are mammals.
    Lizzie the Lizard is my pet.
    Therefore, Lizzie the Lizard is a mammal.  (Valid, but false: a lizard is not a mammal. The conclusion is not sound because the first premise is not true.)

Some students try to determine the validity of a conclusion by deciding if it and the premises are true.  This is a slippery slope in logic.  A better approach is to put premises into the “P implies Q” format and look at the steps used to arrive at the conclusion.

This example is not even valid because the steps are wrong, but the conclusion could be either true or false:

P, Q, and R statements might go like this:
    All dogs are mammals.
    All cats are mammals.
    All dogs are cats.  (Not Valid because the steps are wrong, and not Sound even though both premises are true, and obviously false.)

Looking only at the truth of the conclusion to determine its soundness can lead to mistakes.
    All mammals are born alive.
    All elephants are born alive.
    Therefore, all elephants are mammals.  (The conclusion is true, but the logic is not valid because it does not follow the Law of Syllogism, and the conclusion is not sound because the first premise is not true.  The platypus is not a mammal but offspring are born alive.)

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SUMMARY FOR YOUR LOGIC STUDY CARD:

Inductive Reasoning: specific to general

Deductive Reasoning: general to specific
     Detachment: If A, then B.  A.  Therefore, B.
     Syllogism:  If A, then B.  If B, then C.  Therefore, If A, then C.

Validity:  Are the Laws of Detachment and Syllogism followed?

Soundness:  Are the premises valid AND true?

TRUTH TABLES

In arithmetic and algebra, we are  working largely with numbers.  When Geometry comes into play, a lot of the work involves words to convey theorems and postulates.  “If a triangle has three congruent angles, then it is an equiangular triangle.”  Try writing THAT with X and Y!!  So now we need a way to work with these statements and to know when a statement is true.

You’ve already learned some things about an “if-then” conditional statement. Remember converse, inverse, and contrapositive?  For truth tables, we’ll call the first part of the conditional statement "P" and the second part "Q."  I can’t think of a specific reason for selecting these letters, but they are the ones other mathematicians use, so why not? 

For a conditional statement like “If two angles form a straight line, then the angles are supplementary,”  P will be the “two angles form a straight line” part and Q will be “the angles are supplementary” part.  Notice that "if" and "then" are detached.  They always stay in the same order, but we'll move the P and Q parts around.

The original statement can be summarized as “If P, then Q.”   For further brevity, we can write

and read it as “P implies Q.”

If P is true, then not-P ( -P) is false.  And if Q is true, then not-Q (-Q) is false.
-----------------------------------------------
Now for the TRUTH TABLES.  There are four ways to judge the conditional statement.  The first part (P) can be true and the second part (Q) either true or false.  Or the first part can be false while the second part is either true or false.  The truth table looks like this:

Notice that the statement is only false under one scenario:  when the first part is true but the second part is not.  That may seem counterintuitive, so let’s look at it closely.


Let’s think of a more personal conditional statement “If you miss only one question on the quiz, I will give you an A.”  If you live up to your side and I give you the A, I’ve followed through on my promise, so the first line is true.  But if you miss only one question and I renege, then Q is false and the original implication must be false also. We can all agree to the first two lines of the truth table.

But when we consider the last two lines, it’s not as obvious why the statement is still true even when P is not.  If you miss more than one question, it doesn’t matter what grade I give you, I haven’t broken my promise, so the implication has NOT been proven false and is therefore still true. 

You already learned the CONVERSE, INVERSE, and CONTRAPOSITIVE of the conditional statement.  Here are the corresponding truth tables.

CONVERSE:  Q implies P.....

INVERSE:  not P implies not Q...

CONTRAPOSITIVE: not Q implies not P...

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Here’s the TRICK to accurately filling out the truth table on the next quiz.  Look at the first part of the combination. 
•If it is true and the second part is also true, the conditional statement is TRUE. 
*If the first part is true and the other part is false, the conditional statement is FALSE. 
*If the first part is false, the truth statement is TRUE, no matter what.

In other words, the only time the conditional statement is FALSE is when the first part is true but the second is false.

You probably won't find "truth tables" outside of this unit in geometry, but you'll find them useful in unexpected ways throughout life.  Need to evaluate the speeches of political candidates or sales pitches and advertisements?  Well, if the first part of the statement isn't true in the first place, then anything in the second part can be false and still not be a technical lie.  Understand truth tables and you won't be fooled in choosing your vote or buying a product or service.  This is the logic part of math and it's a valuable lesson.  In fact, LOGIC may be a course you could take in college, especially if you major in premed or prelaw.

BASICS OF STATISTICS for ADVANCED ALGEBRA STUDENTS

Some of the concepts in your first Statistics test are already part of your mathematical skill set.  You already know, for example, how to calculate average or mean, median, and mode.  These are measures of central tendency.  They tell typical values like (respectively) the score that everyone would have gotten on that test if all points earned had been evenly distributed among the students, the middle of the scores where half of the students got a higher grade and half a lower one, or the most frequent score received.

You probably understand the concept of range - the spread between two values, the difference between the highest score on a test and the lowest.  You may even have drawn box and whiskers diagrams and frequency graphs as long ago as elementary school.

In higher level Algebra courses, we add a few details to make statistical measures and displays even more useful.

Some useful SYMBOLS

 (Pronounced "mew") This Greek letter stands for the POPULATION MEAN.

 (Called "ex bar")  This symbol represents the SAMPLE MEAN.


(Small sigma from the Greek alphabet)  Sigma is the symbol for for POPULATION STANDARD DEVIATION.

The small ess is used to signify the SAMPLE POPULATION STANDARD DEVIATION.

Using the MEDIAN

Median is used to construct a BOX and WHISKERS PLOT.

The middlemost element in a collection tells us where 50% of the subjects or items are higher and 50% lower.  The middle element (median) of the lower half and the middle element (median) of the upper half are the first and third quartiles (Q1 and Q3).

Quartiles divide the data into quarters, so at Q1, 1/4 of the scores are lower, while at Q3, 1/4 are higher.  The difference between Q1 and Q3 is the interquartile range, IQR.  It describes the location of the middle 50% of items in the group and is less influenced by extreme values than some other measures.  In fact, we use the IQR to determine whether values are too far from the central tendency to be worthy of consideration.

Say we want to know how many students are likely to attend the basketball game this weekend.  We’ve kept count of the number of attendees at the past 15 games and the interquartile range is 100.  But last month, there was one game that was attended by 700 fans!  That statistic is  so high, it could throw off measures of central tendency like the mean, so how do we tell for sure if we should count it as the highest value element or something else?

Identifying OUTLIERS

Using the IQR, we can calculate the limits beyond which certain values are too extreme to be of importance.  We call these outliers.

The limit of “reasonable” element values is 1.5 times the IQR either above Q3 or below Q1.

In our basketball game example, we can arbitrarily say that 100 is our mean.  the reasonable upper limit would be 1.5 X 100, or 150 above the Q3 count of 200 (another arbitrary example for demonstration sake), or 350 fans.  That 700 game was definitely an outlier.

On the box and whiskers plot, outliers are marked, maybe by an X but OUTSIDE the whiskers.

The measures forming the box plot (excluding any outliers) are sometimes called the “5 Number Summary.”

Using the MEAN

We’ve been calculating the arithmetic mean (average) since grammar school.  Almost every 5th grader will tell you that to figure out the average, “add ’um all up and divide by the number of ‘um.”  The use of the mean in statistics is yet another example of my guiding principle: you learned everything there is to know in math before you entered middle school.  From then on, we just use those things in new ways. 

So let’s use the mean to figure out how things are distributed around that central tendancy.  Let’s first calculate the VARIANCE by finding the distance of each value from the mean...

 then squaring it to remove the implication of more than or less than the average...

Now divide by the number of items and we have an average of the squares...

In these equations, I've used the symbols for SAMPLE MEAN.  To calculate the POPULATION MEAN, use mu.  Some texts distinguish the Sample Mean from the Population Mean by using n-1 as the denominator in the former.  I've never made that distinction, but check with your text to see what your class requires.

Either Variance equation used gives us a measure of how spread out the items are, but squaring the differences has provided a number that isn’t the same “weight” as the individual items.  To find the STANDARD DEVIATION, we need to undue those squares...

Use this equation for standard deviation when you are able to survey every member of the population.  If a smaller sample is used, errors could effect the standard deviation.  Experience has suggested a modest alteration in the equation in order to gain a more reliable result:

Use this equation for standard deviation when you only tested a sample of the entire population.  This minor alteration in the denominator was suggested after statisticians found that it reduces some of the inaccuracy created when a small sample is used to extrapolate to a full population.

Using Standard Deviation with a Normal Bell-shaped Curve

Given a normal curve (that bell-shaped graph that puts the mean in the middle along with median and mode and tapers off at both ends), we can predict that roughly 68% of the values will fall within one standard deviation of the mean, 95% within two standard deviations, and almost every value (99%) within 3 standard deviations.

Are you wondering where the percentages came from?  So did I.  It seems that statisticians for many, many years have collected and analyzed data and actually “discovered” that for things that approach a normal curve (like heights of adult males) the 68-95-99 standards are the areas under the curve at points very close to 1-2-3 standard deviations from the mean.  I’m just taking their word for it because I am not willing to do as much arithmetic as it would take to verify the data.  I’m accepting the percents as a postulate.


Using a Z score

When I want to compare two different surveys, like a student’s results on two separate Math tests, it would be handy to have a single measure of comparison.  Here’s where the Z-score comes in handy.  It tells how each score deviates from the mean of that particular test.

A Z-score of -1 is one standard deviation below the mean, while +2 means the grade was 2 standard deviations above.  To calculate the Z-score, find the difference between the score and the mean, and divide by the standard deviation...

These 5 equations are probably more than will be covered in the first Statistics test.  Much of what comes later in the course also relies on equations, so I suggest that you start a note card now, before too much information gets out of control.  

Coming up, we’ll look at correlation coefficient, confidence intervals, and probability indexes.  Go figure!

PARTIAL FRACTIONS (Revisited)

WHY DO WE STUDY PARTIAL FRACTIONS IN ALGEBRA AND PRECALCULUS?

This one is for Emma.  A "strange" question appeared on her Precalc review worksheet.  It didn't appear to be related to the transformation of graphs concept that was the focus of the homework.  The problem looked something like this:
         Solve for A and B:                    x             =       A    +    B   
                                       x^2 + 3x + 2      (x + 1)  (x + 2)       I wondered how many of the students didn't even know that this is a partial fractions question.  This is an honors class, but I don't recall the topic being covered in previous courses.  For the benefit of everyone in the class, here's the lesson we shared in our tutorial session.  (It's a repeat of the blog entry from June, 2013)

In the early days of Algebra study, we covered “collect like terms” so that the process would be a simple step once we got to FOIL (also called “double distributive” in some schools).  The same thinking encourages us to learn other basic algebraic steps before using them in more complicated mathematics.  In Calculus, we may need to decompose rational expressions before applying the rules of integration.  By doing the initial algebraic work of decomposition now, higher level concepts will be easier to learn and understand.

There are several situations which may be presented in Partial Fraction exercises.  We’ll start with the simple and work toward the more complex.

Simple rational expression:

               2x + 3       
          (x^2 - 7x +10)        Notice that the degree of the numerator is smaller than the degree of the denominator.  This is an important restriction that must be met.  (see below: “What if the Numerator is a higher degree than the Denominator?”)

    STEP 1 - factor the denominator

              2x + 3  
            (x-2)(x-5)

    STEP 2 - using the factors from the denominator, list separate fractions to form an equation. (We don’t know the numerators yet, so let’s just keep them blank until Step 3).

              2x + 3        ?          ?  
            (x-2)(x-5)= (x-2) + (x-5)

    STEP 3 - Substitute letters for the unknown numerators.

              2x + 3         A          B 
            (x-2)(x-5) = (x-2) + (x-5)

    STEP 4 - Expand the right side to have a common denominator.

              2x + 3        A(x-5) + B(x-2)
            (x-2)(x-5) =   (x-2)      (x-5)

    STEP 5 - Since denominators on both sides of the equation are the same, set the numerators equal and distribute A and B.
   
              2x + 3    =  Ax - A5 + Bx - 2B

    STEP 6 - Collect like variable terms.

              2x + 3   = (A + B)x - 5A - 2B

    STEP 7 - Create a system of equations by setting coefficients of the variables equal.

            2  = A + B
            3  = -5A -2B            The reason I prefer this ‘systems’ method to some others is that I can use the matrix function on my calculator to get answers with only a few key strokes and no manual arithmetic.
    STEP 8 - Solve the system.

            A = -7/3
            B = 13/3

    STEP 9 - Seems like a lot of steps so far, but this is the last one.  Substitute A and B values into the equation from Step 3.

              2x + 3         -7/3      13/3 
            (x-2)(x-5) =  (x-2) +   (x-5)


WHAT ABOUT MORE COMPLEX DENOMINATORS?             
   
    More factors?  Include more fractions in Step 3....
   
                 2x + 3              A         B         C  
            (x-2)(x-5)(x-1) = (x-2) + (x-5) + (x-1)

    .... and continue with Steps 4 - 9  (SOLUTION:  A = -7/3, B= 13/12, C= 5/4)

    Repeated binomial factors?  Consider the highest degree of the repeated binomial and EVERY SMALLER DEGREE to set up the equation in Step 3....

                2x + 3             A         B           C    
            (x-2)(x-5)^2   = (x-2) + (x-5) + (x-5)^2

    ....and continue with Steps 4 - 9.  (SOLUTION:  A= 7/9, B= -7/9, C=13/3)

A quadratic factor?  If the denominator is quadratic, make the numerator of the fraction a binomial...

          2x + 3             A        Bx + C   
    (x-2)(x^2 - 5)   = (x-2) + (x^2 - 5)

    ....and continue with Steps 4 - 9.  (SOLUTION: A=-7, B=7, C=16)

    What if the Numerator is a higher degree than the Denominator? 
AHA!!  Another example of how simple algebra concepts are eventually used again as steps within more complex algorithms!!  To satisfy the requirement that Partial Fractions can only be constructed if the Numerator is a lower degree than the Denominator, we need to break up the rational expression even more by LONG DIVISION.  The Partial Fraction is constructed from the REMAINDER, expressed as a fraction with the divisor as the Denominator.

VECTORS - What is a UNIT VECTOR?

By vector definition, a UNIT VECTOR is one in which the Magnitude equals 1.  (To review how to calculate Magnitude, refer to thenormalgenius.blogspot.com article "Vectors: What is Magnitude?," published on 8/19/13.) 

In working with vectors, it is sometimes simpler to use, or the equation calls for, a unit vector in standard position, i.e., originating from (0, 0).  Transforming a given vector into a friendly unit vector is easily envisioned by relating it to common algebra and geometry.

Think of a vector as the hypotenuse of a right triangle.  Draw sides parallel to the x axis and y axis.

If you start with a right triangle (5,12,13 for example) you would need to multiply the hypotenuse by its reciprocal (the multiplicative inverse - one of the properties of real numbers) in order to get it back to 1 (the multiplicative identity).  Dividing each of the legs by 13 (the hypotenuse) would give you the coordinates that would create a special right triangle with hypotenuse of 1 (5/13, 12/13, 1).

Try it with any right triangle:

         3,4,5 transformed to a right triangle with hypotenuse of
         length 1 would be   3/5, 4/5, 1.

         x, x√3, 2x gives you sides of length (x / 2x) or 1/2
         and (x√3 / 2x) or √3 / 2.

         These numbers sound a little familiar, don’t they?  I’m thinking Trigonometry -- Sine and Cosine of 30° and 60° angles.  And they coincide with the 30°- 60° - 90° right triangle that has sides equal to our example and the hypotenuse terminating on the circumference of a UNIT circle.


So when the study of vectors comes up and you are asked to determine whether a vector is a UNIT VECTOR, if the magnitude is 1, you know it is.  For scalar or vector components, you will want to change a vector INTO a UNIT VECTOR: divide x and y by the magnitude.  In the not-so-distant future, as part of the study of vectors in Precalculus, you'll discover equations which contain terms that look like this...

and you'll recognize the unit vectors.  A discussion of components will highlight these terms in an upcoming blog article. 

INDEX - THE NORMAL GENIUS ARTICLES

    CATEGORY   
DATE        TOPIC
----------------------------------------------------------------

    A BLOG INTRODUCTION

  3/19/10     THE NORMAL GENIUS

    ACT

  11/17/10   THE ACT SUGGESTS DEPRESSINGLY LOW BENCHMARKS FOR  COLLEGE READINESS
  7/3/10       4 MISCONCEPTIONS THAT MAY BE LIMITING YOUR ACT SCORE
  3/20/10     ACT LINK
  4/2/10       FOOD FOR THOUGHT (EATING YOUR WAY TO AND THROUGH THE ACT)
  3/29/10    QUICK TIPS FOR THE ACT
  7/27/10    SCHOLARSHIP AWARDS BASED ON YOUR ACT SCORE
  3/31/10    TIPS FOR ACT TEST DAY

    ACT - ENGLISH

  7/22/10    ACT: ENGLISH GRAMMAR
  3/19/10    IMPROVING ACT ENGLISH

    ACT - MATH

  3/24/10    ACT MATH CHECKLIST
  3/20/10    IMPROVING ACT MATH, PART ONE
  3/24/10    IMPROVING ACT MATH, PART 2
  3/22/10    REMEMBERING MATH FORMULAS

    ACT - READING

  3/25/10    IMPROVING ACT READING

    ACT - SCIENCE REASONING

  3/30/10    QUICK REVIEW OF ACT SCIENCE REASONING

    ACT - STUDY PLAN

  4/6/10      ACT - APRIL 10, 2010 (3 DAY STUDY PLAN)
  5/7/10      SELECTING A STUDY PLAN FOR THE ACT
  5/4/10      SOPHOMORES: GETTING READY TO STUDY FOR THE ACT
  3/26/10    START SLOWLY. WORK SMART. DO IT TODAY!
  5/31/11    STUDY FOR FINALS OR THE ACT?

    ALGEBRA

  7/21/13    QUICK STEPS TO COMPLETE THE SQUARE
  7/21/13    QUADRATIC EQUATIONS & THE PARABOLA (ALGEBRA II)
  7/18/10    PRE-SEMESTER REVIEW FOR ALGEBRA - PROPERTIES OF REAL NUMBERS
  2/8/11      LINEAR & ANGULAR VELOCITY
  7/21/13    ALGEBRA II: QUADRATIC EQUATIONS & THE PARABOLA
  7/21/13    QUICK STEPS TO COMPLETE THE SQUARE
  8/6/13      THE SUM OF TWO SQUARES - BREAK THE RULES
  8/6/13      IMAGINARY NUMBERS; TOOLS AND TRICKS  

    AP CALCULUS

  1/7/11      ORGANIZE YOUR RESOURCES (CALCULUS)
  3/31/11    SPECIAL STUDY FOR THE AP CALCULUS AB TEST
  4/27/11    AP CALCULUS AB
  5/1/11      WHAT THE TEACHERS ARE ASKING ABOUT AP CALC AB
  9/11/11    PREPARING FOR CALCULUS

    AP STATISTICS

  8/27/12    GETTING READY FOR AP STATISTICS - LEVELS OF MEASUREMENT
  8/18/12    GETTING READY FOR AP STATISTICS: VOCABULARY BASICS
  8/11/12    GETTING READY FOR AP STATISTICS: MEAN-MEDIAN-MODE
  8/4/12      GETTING READY FOR AP STATISTICS: COMMON GRAPHS

    CHEMISTRY

7/26/13    FIRST STEP IN CHEMISTRY - THE ATOM
8/19/13    BALANCING EQUATIONS

    COLLEGE

8/28/10    EARLY DECISION COLLEGE APPLICATION
10/4/10    PAYING FOR COLLEGE
6/2/11      A SPECTACULAR GPA IS NOT ENOUGH
1/22/12    PLANNING YOUR COLLEGE VISITS
1/9/12      PAYING FOR COLLEGE: SAVING MONEY ON TEXTBOOKS
4/1/10      EARN COLLEGE CREDIT WITH CLEP TESTS
7/25/10    ESSENTIAL STUDY MATERIALS FOR COLLEGE

    ELEMENTARY

1/17/12    TIPS FOR ELEMENTARY SCHOOL PARENTS: SHOULD WE HIRE A TUTOR?
8/9/10      NUMERACY: A FIRST STEP TOWARD PRESCHOOL MATHEMATICIANS
8/2/13      SINGAPORE MATH: USING BAR MODELS
8/9/13      SINGAPORE MATH: PARENTAL INVOLVEMENT
8/11/13    SINGAPORE MATH: AT-HOME RESOURCES

     EXAMS

5/4/10      FINAL EXAMS -- 3 WAYS TO PROCRASTINATE
5/27/10    FINALS - ACT - SUMMER!

    GENERAL MATH

8/8/10    GENERAL MATH    GET READY FOR THE NEXT LEVEL OF MATH

    GEOMETRY

1/4/11      THE AMBIGUOUS CASE (GEOMETRY & TRIG)
7/17/10    PREPARING FOR THE NEXT LEVEL OF MATH: GEOMETRY

    PRECALCULUS

3/30/11    COMPLEX NUMBERS..TO POLAR COORDINATES..TO RECTANGULAR COORDINATES
8/26/12    PARAMETRIC EQUATIONS: MATH ART
6/27/13    PARTIAL FRACTIONS - THE BASICS
6/26/13    VECTORS IN A NUTSHELL
6/27/13    PARTIAL FRACTIONS: the basics
8/19/13    VECTORS: WHAT IS MAGNITUDE
8/20/13    VECTORS: WHAT IS A UNIT VECTOR:

    PSAE

4/25/10    DO YOUR BEST ON THE PSAE
4/13/10    JUNIORS: 2 WEEKS TO THE PSAE STUDY PLAN
3/19/10    PRAIRIE STATE ACHIEVEMENT EXAM
3/31/10    SAMPLE QUESTIONS FOR THE PSAE (PRAIRIE STATE)
4/7/10      SENIORS! PSAE & GRADUATION

    PSAT

8/9/12    GETTING READY FOR THE PSAT

    SAT

2/17/11    TAKING THE SAT?

    STUDY SKILLS

8/22/10     FIRST DAY OF CLASS: COLLECT CONTACT INFORMATION
9/6/10       THE DIFFERENCE BETWEEN LEARNING & STUDYING
9/17/10     USING TESTS & QUIZZES TO MAXIMIZE YOUR STUDY PLAN
12/31/10    WHAT MOTIVATES THIS STUDENT?
1/14/11      TEST STRESS: CHICAOG TRIBUNE ARTICLE
2/18/11      MATH 911
9/13/11      PREPARE TO STUDY FOR SEMESTER FINALS
6/7/12       PAINLESS SUMMER GEEK
7/2/12       DOG DAYS MATH
5/19/10     STAYING MOTIVATED TO STUDY
7/8/10       SUMMER DATE BOOK = PSEUDO ASSIGNMENT PLANNER

    TEST TAKING

5/4/10      FINAL EXAMS -- 3 WAYS TO PROCRASTINATE
5/27/10    FINALS - ACT - SUMMER!  

    TRIGONOMETRY

6/28/13    TRIG SUM & DIFFERENCE IDENTITIES

VECTORS: WHAT IS MAGNITUDE?

VECTORS:  WHAT IS MAGNITUDE?

In terms of vectors, magnitude is the length of a line joining the origin to an ordered pair on the coordinate plane.  Its symbol is a number with a double line in front and in back....  ||V||

This is nothing new but you may not recognize the vector definition as the hypotenuse of a right triangle, calculated with the Pythagorean Theorem.

Let’s look at Vector G, (5, 12).  Find the point on a graph and draw a line from (0,0) to the point.

  Draw lines from the
  point to the x axis and      
  from there to the origin
  creating a right triangle.
                 
  Label these lines.

  Using the Pythagorean Theorem, the Hypotenuse is 13.

Since the equation for MAGNITUDE is

, magnitude is nothing more than the hypotenuse of a right triangle.  The magnitude of 

V (5, 12) = ||13||.

When you need to use Magnitude in a vector situation like finding the Norm, direction, unit vector, or scalar components, think PYTHAGOREAN THEOREM and HYPOTENUSE.

CHEMISTRY: BALANCING EQUATIONS

There are 2 tricks to balancing equations.  First, spelling the technical word:
                       STOICHIOMETRY

Second, insuring that there are an equal number of atoms of each element on both sides of the equation, following the Law of Conservation of Mass.

The left side of a balanced equation is called the reactant(s).  The right side is called the product(s).

The number of atoms is determined by multiplying the coefficients and subscripts.

To balance a chemical equation, change only the coefficients.  This changes the number of elements or molecules used or produced without changing the element or molecular configuration itself.

Let’s balance an easy one...

We have 2 H’s on the left, so we need 2 on the right.

And now the 2 Cl’s on the right match the 2 on the left.  We’re in balance.

Here’s one with more steps:

Start with the most complicated configuration, maybe Pb3
on the right.   We need 3 Pb’s on the left.

 

That gives us 12 Cl’s on the left, so we need the same number on the right.

Now there are 12 Na’s on the right, so we need the same number of Na atoms on the left. 

TIP:  Because the PO4 on the right is held together with parentheses, keep the PO4 on the left as a single entity as well.

And counting up the PO4’s shows that we have 16 on both the left and the right.  We’re in balance.  Notice that we didn’t have to write the coefficient ‘1’ because it is implied.

TIP:  In the balancing act, it is easier to make the final adjustments if there is a reactant that stands alone, a single element.  Try to save these until last.

BALANCING with FRACTIONS

Sometimes when balancing an equation, you'll wind up with terms that can't be split up.  What if you have O2 on one side and O3 on the other?

Assume that O3 is the more complicated entry so we need to get 3 oxygens from the diatomic O2 on the left.  Think of fractions.  We need one and a half O2s or 3/2.

Even though this equation is technically "in balance," most teachers will want you to use only whole numbers as coefficients in chemical equations.  Think about what you would do in a simple Algebra equation to get rid of a fraction.  You'd multiply by a common denominator, so do the same here.  Multiply ALL terms on BOTH SIDES by the denominator 2 to yield

TIP:  The diatomic oxygen is so frequently involved in these fraction situations, that I try to "balance" the oxygens last.  

Although these examples seem simple, the concept itself is simple also, regardless of how complicated a balancing question may appear.  Just remember

MATTER CAN BE NEITHER CREATED NOR DESTROYED.

SINGAPORE MATH: AT-HOME RESOURCES

WARNING:  THERE IS MORE TO SINGAPORE MATH THAN THE CLASSROOM

Even critics of the Singapore Math program could not object to the proclaimed objectives of teaching students to the mastery level, using real world situations for problem solving, and encouraging communication in math.  After all, these same goals have been recommended by the National Council of Teachers of Math for decades.  Everyone interested in math wants our children to love math, not just tolerate it through 11 or 12 years of compulsory education.

And even the critics may not call enough attention to the need for out-of-school support if students are to achieve that mastery of basic arithmetic concepts.  The Singapore Math classroom curriculum stresses conceptual understanding to the delight of advocates, but procedural and skills proficiencies are often the under-reported secondary requirements that critics demand and that are an integral part of Singapore's education program. 

Unlike the American system in which “school” ends at 2:30 and is replaced by dance class, sports, and other extracurricular activities, in the country which inspired Singapore Math, nearly every student receives supplementary homework assignments, as well as additional rehearsal through math clubs and “after school tuition classes” like our tutoring services.  The reported 1995 (http://nces.ed.gov/pubs99/1999081.pdf) through 2007 (http://nces.ed.gov/timss/table07_1.asp) math success in Singapore may be more heavily dependent on the cultural significance placed on education than we realize.

With this possibility as our hypothesis, effective implementation of Singapore Math in American schools will require at least moderate changes to our priorities.  It will be necessary that after school activities devote more time and effort to the repetitive work -- those dreaded “math facts” like multiplication tables.

Practice in basic computation can be tedious for both parent and student.  Multiple worksheets with 25, 50, 100 problems are boring, to be generous.  A parent verbally making up sample problems is demanding.  The chances are that none of us is going to follow through long enough for our kids to gain the level of skill that we would like.  We need a helper.

Tutoring Resources uses a game we call “War on Integers” which is played like the familiar card game War.  Contact us at www.tutoring-resources.com for a free copy of instructions for playing with addition, subtraction, multiplication, or division.  The game encourages the student to practice in a fun way that doesn’t depend on age or grade level to “win” and promotes the much needed rehearsal.

The internet is another readily available resource.  There are many sites that provide colorful, entertaining games for students from kindergarten through sixth grade and even higher.  My personal favorite (because I played it with my own kids back in the day) is NUMBER MUNCHERS.  I have it on floppy disk (I did mention that it was “back in the day”) but for today’s electronically-literate youth, it’s available for iTunes, iPhone, and iPad Touch.

Other sites to explore include
www.wallofgame.com  (Number Munchers is available on this site.)

www.Sheppardssoftware.com  (A listing of several useful sites.)

www.IXL.com  (This one makes it easy to access appropriate games by grade level and even goes through Algebra.)

www.Cool.math-games.com (This site may not be as distracting for kids who are accustomed to websites as it was for me, but I would not recommend it for anyone with focus and concentration issues.)

As a general rule, games will require dexterity with either arrows or a mouse.  I found them all easier to play on a desktop than laptop computer.

Want more suggestions?  Just search the web for ‘math games’ and be prepared to spend considerable time checking out a myriad of sites that might appeal to the specific interests of your student -- and playing a few yourself.  I lost more than a few minutes on the pop-up www.ahhhhh.com, which is apparently a soft drink website ad, as mezmerizing as it is subliminal.

If our kids are going to get the promised rewards from Singapore Math, parents need to make it a point to provide the supplements that will fill in the gaps.  Luckily, plenty of resources are available to make our job easier. 

SINGAPORE MATH: PARENTAL INVOLVEMENT

So your school district is all in a lather over Singapore Math.  Some are dead set against the changes coming in the Fall and others are so enamored that they seem almost manic.  This is a common scenario:  two sides, both at the extremes.  Critics and advocates alike will implement the new methods, procedures, and texts like scientists conducting an experiment.  The classroom is the lab.  Results will be evaluated somewhere down the line and assessments made about the effectiveness of the program.  It has happened before with “new math,” Everyday Math, and many other attempts at improvement that have resulted in both success and failure. 

But YOU are the parent and what happens to, for, and with your child is a permanent fixture of his or her education with important future ramifications. To YOU this “experiment” MUST WORK.  Supporters of Singapore Math do not hesitate to stress the importance of parental support for successful implementation.  Some suggest that after-school tutorials and homework are integral factors in the reported success of elementary students in Singapore.  If you are among the group of the hesitant, unsure of your own math knowledge and ability to adjust to Singapore Math, this article will help you become a competent homework assistant.



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TIPS FOR PARENTS NEW TO SINGAPORE MATH

1.  Many of the “informational” publications about SM are lean on specifics.  You’ll hear that the Singapore Math program as amended for American Schools teaches mathematical concepts from concrete through pictorial to abstract, devoting more time to fewer topics and stressing “mastery.”  This limited information seems to me to be more of a sales pitch than useful data. 

It is up to the parent to drill down to details that will help each student to learn the math fundamentals.

For example, here is the sequence of study for the first week or two of each grade:
    Kindergarten:      Numbers to 5
    First Grade:         Numbers to 10
    Second:              Numbers to 1,000
    Third:                  Numbers to 10,000
    Fourth:                Place Value of Whole Numbers
    Fifth:                   Whole Numbers
    Sixth:                  Positive Numbers/Number Line

With this information in mind, you can help prepare your student for the first few days of school by reviewing the numbers that will be covered.

2.  To ensure that you are in the loop throughout the school year, be sure you are receiving the “School-to-Home Connections” newsletters from the teacher at the beginning of each chapter.  These include a summary of the upcoming lessons, vocabulary, and an activity to support learning at home.  The newsletters are available to the teacher as prewritten resources which can be printed for general distribution, emailed, or posted on the school’s homework website. (Sixth through eighth grade newsletters are called “Family Letters.”)

3.  Many of the “new techniques” used in SM are similar to systems used in the 1950s and 1960s.  That’s a little before the grammar school dates of most of today’s parents, but here is a tremendous opportunity to get gramma and grampa involved.  Ask them about “friendly numbers.”

I am of that era and remember breaking large numbers into workable groups. (In fact, I still use this system for mental math, which is a chore for me rather than a talent.) For example, studies have shown that most people can “picture” numbers up to 6 in patterns found on playing cards.  The number 7 becomes more difficult to envision, but it can be broken up into 5 and 2.  Adding 5 and 7 is the same as adding 5 plus 5 plus 2.  Adding 5s is a familiar pattern; 5 + 5 = 10.  Add another 2 to find the sum of 5  + 7.  In Singapore Math, this is called “adding with regrouping.” Learning the process is facilitated by using manipulatives.

Subtracting with regrouping is similar.   Seven can also be broken up into 3 and 4.  So 13 - 7 becomes 13 - 3 to get 10 and then subtracting 10 - 4.  Ten is the friendly number in this case.

For early elementary students, using manipulatives to find number patterns (called number families in some very old textbooks) is a common practice.**

**  Manipulative “dots” are available free of charge to fans of this blog.  Become a fan and send a request for a set of 100 Dots through the comments section that follows this article or email tutoring.resources@yahoo.com

4.  Singapore Math uses “place value mats” as early as first grade.  The classroom teacher should have a template which can be copied and distributed to families, or you can easily make your own by replicating the following pattern.  Columns should be long enough to accommodate 5 manipulative dots (breaking 10 up into friendly 5s).  Laminating the mat will ensure that it holds up to 6 years of single student use or as a sibling hand-me-down.  At the second grade level you’ll want headings for thousands, third grade ten thousands.

5.  ADDITION (OR SUBTRACTION) WITH RENAMING are the SM terms for “carry the one”  and “borrow” with which you might be more familiar.  Be sure to review the vocabulary from the School-to-Home newsletter and ask the teacher if the connection to the math you know is not immediately obvious.

6.  After manipulatives, SM might move to drawings to transfer tactile knowledge to the more conceptual/visual format.  You may want to practice setting up word problems using this system.  It is not complicated when you've mastered the routine steps for creating bar models.  Check out The Normal Genius blog, "Singapore Math: Using Bar Models," posted on August 2, 2013.

7.  Other techniques that SM uses to engage students and stimulate higher level thinking include a heavy use of word problems (called “practical applications” in some older textbooks and college courses) and classroom discussions.  The format frequently presents a story that would use some form of math to solve a problem, let the students work independently or in groups to “think it out,” followed by presentations of their solutions.  This practice has long been supported by the National Council of Teachers of Math which emphasizes both real life situation applications and the importance of communication in mathematics either through speaking or writing.  It encourages students to identify and recognize their own problem solving thinking, a metacognitive element of learning.

You can help your student prepare for this task by practicing explaining the steps to a solution for sample homework problems.  Your own ability to model the process could be a valuable asset for your student.

8.  Thinking through problems is only part of the SM formula.  Rehearsal of basic math facts is still required but will need to take place at home.  Ask the teacher for supplemental materials outside of the workbook exercises.  Try chanting the multiplication tables (like you did to learn them) while throwing a ball back and forth or bouncing it on the floor for students who need to move while learning.  Or chant to a particular beat for the musically inclined.  Use your imagination to design drills that meet the specific needs of your unique student.

9.  Not necessarily part of the Singapore Math program, but a useful tool for upper elementary students when word problems can become quite complicated, is a strategy I call “Is that your final answer?”  It sets the framework for what the answer will look like, set up at the beginning of the problem solving process and insuring that the student doesn’t stop too soon or go down a blind alley while working out a solution.  In either math symbols (4X = ___ ) or words (4 pencils cost ___ ), the practice is similar to checking the question after working through the math.

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So your school district is all in a lather over Singapore Math.  But you are calm, cool, and collected because you are prepared to help your student make the most of the program and maybe even gain confidence in your own math skills at the same time.

The SUM of Two Squares -- BREAK THE RULES!

Here’s something nobody else in your Algebra II class (maybe even the teacher) will know.  (Do not try to be the wise-cracker until you're ready to PROVE that this works!!)

Since first learning to factor, we've been told that "the rules” are... you

    -  CAN factor the DIFFERENCE OF TWO SQUARES (referred to by some texts as ‘DOTS’)  into conjugates.....

    x^2 - 16 = (x - 4)(x + 4)

    - CAN NOT factor the SUM OF TWO SQUARES.  This second admonition is true when the instructions say "factor completely across the integers" because, as you are about to see, the answers are NOT integers.

BUT You CAN factor the sum of two squares if you've studied imaginary numbers and know a simple little TRICK.

        You will still use conjugates, but there needs to be an i in the second term of each binomial.....

    x^2 + 16 = (x - 4i)(x + 4i)

Why does it work?  Because when these two binomials are FOILed, the middle terms (O and I) cancel each other out.  AND since i^2 is simply -1, the multiplication of the last terms gives you the task of subtracting a negative number, which is simplified to ADDING!!

Try factoring these sums of perfect squares and FOIL the binomials to verify that you actually CAN “break the old rules” when you know enough math!!

PROBLEM A:     p2 + 64

PROBLEM B:     121 + y2 (hint: use the commutative property first and attach the i to the square root of 121)

PROBLEM C:     x2 + 36 y2

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 And this concludes the demonstration of how a math nerd can break all the "rules" and get away with it, mathematically speaking!!