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Appetizers and Lessons for Mathematics and Reason
by A. Selby, Ph. D.   Feedback & Questions

 20 pages in French: Algèbre  
 Définition d'une variable  
La raison basée sur les  règles et modelés

www.whyslopes.com  >> Odds & Ends  > >   1. Hints for Exams     [ Up ] [ Next ]

1. Taking Tests and Examinations:

On tests and examinations, start writing after you have read all problems. Then while you are doing one, your mind may have ideas for doing the others. Learning takes time and effort. No one but you can provide this for your education.

Proper notation is must. Or, improper notation means you do not how to express your ideas and solutions correctly, and so (ouch) deserve to lose marks. Avoid that if you can.

For questions in mathematics, science and technology that require written answers and you to show work, you must give (state, write) any and all formulas used in your solution and specify the values of the numbers or quantities that appear in them, you should give name of each formula (if it has one) and you should give names of any rules or physical principles used to obtain equations. Diagrams may be used by themselves or besides words to explain your shorthand notation for geometric or physical quantities. Try to to give enough information so that your answer makes sense by itself to someone who has not seen the question. 

Who will get a better mark?

  • Student A in answering the question what is the area of a rectangle that is 8 cm wide by 5 cm high writes 
      
    50 

     (instead of 40) alone and by itself without further explanation of how this number was obtained. He gets 0.
  • Student B in answering the same question writes

    50 square cm  = 50 cm2

    He might gets 0.5 marks out of four as the proper units are included.
  • Student C in answering the same question writes

    40  

    He gets 1 mark out of four as units do not appear.  If I say I am give you 10 units of something, you do not what what you will get until the unit is specified.. Answers to mathematics should be full and precise - exact.
  • Student D draws the rectangle, marks the lengths of width and height on it, write the formula, replaces letters by lengths, and then gets the wrong answer 50 cm2.  His mark is  3 or 3.5  out of four even though he gave the same answer as student B.  But the work done records the reasoning fully step by step, so the marker can observe and see evidence of what the student understands.  So credit is given. 
    height 
    H = 5 cm

    Area  A = ?
    width W = 8 cm


    Area A =  W H

                 =  (8 cm)(5 cm)

                 =   50 cm2    // Answer.

     

If Students D had put 40 instead of 50, his answer would have been worth 4 out of 4 (or more).

The show work safety net: By drawing diagrams, stating the formulas that you use , giving the values used in them, showing the substitution, and then doing the arithmetic you are demonstrating know-how and giving yourself a better chance of earning marks You may get part to full marks if the formula is correct and the arithmetic is not, or if the formula is wrong, while the arithmetic is correct. And in all this, do not abuse the equal sign.

Abuse of Equal Signs: Expressions of the form a = b = c means and demands a = b and a = c. For instance in computing the average of numbers 1, 2, 3 and 4, it is wrong to write

1+ 2 + 3 + 4 = 10 /4 = 2.5 as 1 + 2 + 3 + 4 = 10

while 1+ 2 + 3 + 4 = 10 is correct, but 1+ 2 + 3 + 4 is not equal to 10/4
The latter is the literal meaning of 1+ 2 + 3 + 4 = 10 /4 

You could and should write (1 + 2 + 3 + 4)/4 = 10/4 = 2.5

Note:   When you put an equal sign between two expressions,  

     first  expression     =  second expression 

you should mean that the value of the first expression in full  has the same value as the second expression in full.     Thus  the equation   1+ 2 + 3 + 4 = 10 /4   are incorrect as the expression 1+ 2 + 3 + 4 does not have the same value as the full expression 10/4.    That is the common standard you need to follow.  (If you someone teaches you differently, send them to this page).

See the webpage Proper Use of Equal Sign to learn more.

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Professor Whyslopes:

  • Site value lies in the difference between its ideas and yours.  

  • If one site explanation is not to your liking, try another. Each one is different.

Two gaps

  • The Old Algebra Gap:  Algebra  appears with too few words of explanation in high school and college mathematics.  Online Volumes 2 and 3 offer remedies.   Chapters 8 to 12 in Volume 2  put more words into the explanation and comprehension of algebra.  Chapter 14 in Volume 2 with its explicit discussion of the direct and indirect use a formulas identifies a unifying theme for mathematics and logic - all rules and patterns will be used forward and backwards. Chapters 2 to 6 and 12 to 18 in Volume 3 may further ease or avoid the very challenging use of algebra in the high level mathematics: calculus.    Calculus requires earlier high school mathematics at full strength: (i) This logically complete but long lesson on  complex numbers shows how to simplify the senior  high school exposition of circular trig functions upto to formulas in the plane  for vectors dot and cross-products.  The lesson provides the route that would have been taken in course design if the key element of the lesson, a December 2009 invention,  had been available in the 1950s.  For further algebra skill development. See the site coverage of fraction with units, proportionality,  ratios and rates, polynomials, quadratics functions  and straight line slopes and equations.
  • The Arithmetic Gap: An exact and efficient mastery of arithmetic with decimals and fractions is best (required)  for the high level  study of mathematics alone and in science, technology and business.   Pages here on arithmetic with decimals and integers,  on  fractions and solving linear equations with fractional operations on stick diagrams may help fill the gap.  That exact and efficient command should be obtained in the last years of primary school and the first years of secondary school.   

 Skill mastery in mathematics has to be seen to believed.  To that end,  learn or teach how-to write and draw the steps in mathematical figuring or  reasoning  clearly. Do not try to save space by doing a sequence of step in one place. Instead, do or record the steps in sequence on a separate lines to make each step obvious and verifiable.   

 

 

Odds & Ends

Group I

1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
3. What is a Variable?
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
8. Complex No. Applet
7. Trig and Complex No.
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
20. Independent Variables
21. Why Logic
22. Why Math
23. The 15 Times Table
24.  The  20 Times Table
25. Algebra Formulas
26. On Learning Maths
27. Biology - Growth & Decay
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths
31. Real Number Operations
Learn More

Group II 

Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths
Two  Gaps

[ Up ] [ Next ]

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For Senior High School  & Calculus Students

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Words  to clearly introduce algebra and variables have been missing in course design. For people who cannot do algebra, 
the missing words may explain or ease their difficulties.  Volume 2 ,Three Skills for Algebra,  in Chapters 8 to 14 & 18 etc, puts words before symbols to providing the missing words in a way that enrich the comprehension of all.  Those words form the middle part of a algebra (and logic) lessons aimed at helping or improving all of  high school mathematics and also calculus course design & delivery. 

For Avid Readers in School & Out - Online Books 
   1.  Elements of Reason. 1996 
1A. Pattern Based Reason  1995 
1B. Math Curriculum Notes 1996 
2. Three Skills for Algebra  1995 
3.Why Slopes & More.Math 1995
Tour their  forewords.   

Calculus Prep or Help: See Volumes 2 & 3, and this bigger Calculus Guide.  If your  calculus   questions is not answered here, submit it. Over time, that may complete the site development of calculus. 

For Parents: Speaking Skills, Reading & Writing,  Preparing for Science,  ends, values and methods for work and study,  parent- friendly maths skill development booklets for ages 4-14.

Mostly For High School

Intro to Solving Linear Equations 
- a different paths for junior and even senior high school students. Question for Tutors: When do you use and when you skip the stick diagram method here?

Fraction Skills,  thought-based  development, Ages 10 to 14 may need a tutor.  Students who have to understand in order to do may like the development in all or part. 

For Senior High School Mathematics & Calculus

5 wordy Logic Chapters
4 curious Algebra Chapters
Words before & besides symbols. A Key Algebra forward & backwards Chapter    

First Calculus Preview (1st intro)
Four Calculus Chapters  (2nd intro)
Intro to Complex Numbers (long)
Intro to Mathematical Induction (romantic & wordy at first)

Tutors & Instructors: These lessons introduce skills differently Would you recommend them? 

More Topics 

 1. Decimal Arithmetic  Reference!
2. Integers - Intro to Signed No.s
3.  Fractions - fully explained.
4.  Fractions  with Units  
5.   Number Theory, 
6.    Solving Linear Equations  
7  Formulas for- & backwards -  
8.  Proportionality, Back- & For-wards.   
9. Logic Chapters:   
10.  Euclidean-Geometry  
11.  Slopes & Equations of Straight Lines.  (Take I. See take II below)
12.  Why Study Slopes. 
13. Maps, Plans,  Similarity & Trig,  
  (Take II included here)
14.  Quadratics: Starter lessons
15.  Polynomials: Starter lessons 
16 Why Factor Polynomials:  
17   Functions - Forwards & Backwards.  
18.  Exponents, Radicals & logs.  
19. Complex Numbers before trig (new advance/ starter lesson)
20.  DC Electric Circuits Etc 
21. Real  Analysis 
22. The Olde Complex No, Trig
& Vector Section.
23. More Calculus Stuff
- written after Volumes 2 and 3.

Level I Material: New Stuff
Time and Date Matters
 Level I Arithmetic. 
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
 Logic Chapters (leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of Maps, Plans,  Similarity & Trig,  to appear here).

For Instructors
- Education Essays   (opinions, possibilities, references) 
- Free Advice and Directions for teaching primary & high school maths will be given in online meeting place with voice & whiteboard.   
- Math & Logic  How-TOs 
1. Arithmetic
2. Algebra
3. More Algebra
4.  Beginner Geometry
5.  More Geometry
6. Calculus 
7. Show Work or Logic 
 These may be too dense for students.

Offering ideas to change education makes this site different.  Nothing ventured, nothing gained.  Site material is mathematically  correct, and where not, please report errors. The two level program POMME in the site entrance implies multiple paths for instruction. Supporting those paths in turn implies a clear destination  for site development and perhaps a new name.

 

 

www.whyslopes.com >>  Odds & Ends  >>  1. Hints for Exams     [ Up ] [ Next ]

Road Safety Message   Walk on a side walk. If that is not possible, try  not to  walk on a road with your back to the traffic.
Try to see what  trucks, cars, buses or bicycles are coming, so that you may step out of their way.  Put safety first. .

Support for Technical Mathematics from Number Theory to Calculus Prep
A. More Arithmetic a must for algebra etc D. Logic In Mathematics G. Algebra with Take Home Value I. Vectors & Functions
Decimal Lesson - Reference  
Counting & Addition   (8 lessons)
Comparison to Subtraction  (9 lessons)
Multiplication ( 11 lessons)
Long Division  (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites 
-Primes Factorization
-Greatest Common Divisors & Multiples. 
-Prime Factorization Aids  (Learn how to find factors quickly)
-Prime Factorization Examples 
-Counting & Generating. Factors. 
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
 Fractions (< 20 lessons)  Essential Skills & Concepts 
Ratios & Fractions (3 lessons):  Similarities & Differences   
Units in calculations Fractions  with Units
B.  Basic Algebra
Solving Linear Equations  
- in one unknown. Intro  with stick diagrams?
the normal way  & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of  word problems.
  - still in more unknowns:  Gaussian Elimination via substitution, by equality or comparison, by operations on equations. 
C. More Algebra
Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra  
What is a Variable.  The answer here  is a simple prequel to the modern mathematics viewpoint.
First, every rule & pattern U meet in math, logic & science will be used forwards and backwards.  Get a head start with this theme by reading  Chapter 14 in Three Skills for Algebra.  Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial  backward  use of the proportionality formula.
 Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.  
If you can not read or write precisely, you will have difficulty in following instructions.  One wordy remedy  is given by chapters 2 to 5  in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in  Euclidean-Geometry    In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.
E.  More Geometry
The Pythagorean Theorem. Chapter 17 from  in Three Skills for Algebra uses algebra and geometry   to show why the  Pythagorean equation  for right triangles holds. Its forward and backward use  is common exercise..  At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be  fractional multiples of a unit length. That geometrically implies a  need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with  Maps and Plans drawn to scale  give coordinate-dependent base  for senior high school development of similarity, trig, vectors and straight lines.   
Complex Numbers: This lesson on Complex Numbers  draws on Euclidean and Analytic geometry. Sbortcuts simplifiy  trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products. 

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This  complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry,  personal finance, and calculus. To learn more, if you study calculus,  see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters
  1. Geometric Sums Etc,
  2. Notation For Sums,
  3. Personal Money Maths and
  4. Some Finite Mathematics
identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry:   -  Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one.  I may combine them later.  -In my school days, slopes appeared year after year.   This Why  Slopes calculus preview on graphs of functions y = f(x) explains why.  Enjoy.
Quadratics and Polynomials: -  Operations on Polynomials: Meet a light and ultraquick geometric introduction to  multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials.    Compare Fractions  with Units. Enrichment: A Plus:  The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic. 
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials  high school mathematics has technical value as part of the senior high school mathematics preparation for calculus.  This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials,  their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.  		Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms.  Vectors and their properties appear in physics, its mathematical description and formulation. 
 - Functions - Forwards & Backwards.  Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics.  comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic. 

J. Pre-Calculus Skill Check
-  Arithmetic Skill Check.  In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.  
-  All the skills and concepts in  Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with  your other duties.  Good luck.

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