YOU are better than YOU think. Show yourself  how:  

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Mathematics for Ages 10-13 say

Awkwardness with an idea or skill often signals difficulty with previous ones. It may indicate at least one earlier skill has been missed or forgotten. When an awkwardness is felt or seen, learners should go or be taken back to practice the missing skills, more precisely the ones just before them to restore confidence and build skills, so that the learner can go further. 

Can your charge do the following?  Where not there is room for instruction - learning takes takes and patience, yours included. 

Arithmetic

1. Read and write numbers 1 to 100,000 or 1,000,000?
2. Explain place value in decimals from 1 to 100,000 or 1,000,000?
3. Compare the size of numbers 1 to 100,000 or 1,000,000?
4. Compare and order proper and improper fractions using the least common denominator or comparison of integer parts, as appropriate.
5. Read and write improper fractions.
6. Can you charge produce the 10 times table on demand?
7. Multiply 2 to 4 digit numbers by 2 digit numbers?\
8. Divide 1 to 5 digit numbers by 1 or 2 digit numbers and find the remainder using the long division algorithm?
9. Use the rules for recognizing multiples of 2, 3, 5, 9, 10 and 11?
10. Rewrite fractions as percentages or decimals, finite or repeating.
11. Express a decimal as a percentage, and vice-versa.
12. Recognize through its prime factorization, when a fraction will a finite decimal expansion..
13. Round decimals to the nearest tenth, hundredth or thousandth.
14. Explain the use of decimals to one, two, three or four decimal places.
15. Compare and order decimals.
16. Multiply and divide decimals by whole numbers or decimals.
17. Convert decimals into percents or fractions.
18. Solve for an unknown given equations with whole number coefficients.
19. Express an infinite, repeating decimal expansion as a fraction?
20. Express a fraction as a percentage, and vice-versa.
21. Express a fraction as finite or repeating decimal ?
22. Does your charge know that a fraction has a finite decimal expansion when and only when the denominator is equal to a product of 2s and 5s with no other primes in the prime decomposition/factorization of the denominator?
23. Understand powers, that is exponents,  in arithmetic?
24. Give the prime decomposition of a whole number?
25. Recognize multiples of 2, 3, 5, 10 and 11 with the aid of rules for this recognition?
26. Find the greatest common multiple and least common divisors using the prime decompositions for whole numbers in question? 
27. Can your child simplify square roots using factorization into squares or primes?
28. Can you charge use the greatest common divisor for a pair of whole numbers to compute their least common multiple? 
29. show in simple examples why fractions resulting from simplification or introduction of higher terms are  equivalent?
30. Powers of Ten: Can your child write 10, 100, 1000, 10000, etc as powers of ten?
31. Scientific Notation: Can your child write a decimal as the product of a power of ten with a number between 1 and 10?  Can your child use scientific notation to estimate the size of products and ratios of numbers written in scientific notation.
32. Can your child write a number given in Scientific Notation as a decimal?
33. Signed Numbers: Identify where signed numbers appear (position along a line, thermometers, negative assets or debts). Say how to add and subtract signed numbers.  Say how to multiply and divide whole numbers. State the law of signs.
34. Given the first term in a arithmetic sum and an additive constant, compute the further terms, one at a time, and one after another.
35. Find the sum of a finite arithmetic sum. Justify the formula by writing the finite sequence forward and backwards (Gauss's method).
36. Given the first term in a geometric sum and an multiplier, compute the further terms, one at a time, and one after another.
37. Find the sum of a finite geometric sum by means of a formula. Justification reserved to a future lesson on mathematical induction.

Extra/Enriched Arithmetic

Can your charge

1. Use Euclid Algorithm to find the greatest common divisor for a pair of numbers? Euclid Algorithm for this provides the quickest way to simplify  fractions - reduce to lower terms. This method (not commonly taught) provides the quickest way to simplify fractions and their products, and to find the least common multiple multiple of a pair of numbers or the least common denominator.
2. Explain why column methods for addition, subtraction and multiplication of decimals work? 
3. Show how the numbers appearing in decimal long division, or the work needed by it, imply a number is equal to a remainder plus quotient times remainder? A similar reasoning applies to the polynomial long division method to be met later.
4. Visualize the addition, multiplication, division and subtraction of lengths where the lengths are whole number or  fractional multiples, proper or not,  of a unit length?
5. Explain how the former gives a geometric viewpoint and motivation for arithmetic?
6. Use and justify the compound interest formula
7. Link annuity and mortgage payment/evaluation calculations to a geometric sums and sequence of payments or investments with a constant interest rate or rate of return.

Logic (Rule-Based Thought)

Reference: Chapters 1 to 7 in Volume 2, Three Skills for Algebra. Logic may be introduced in a math-free way.

1. Follow  steps in decimal  methods for addition, multiplication, subtraction and long-division, one at a time and one after another, with the knowledge that an error in one step makes all the rest wrong. See that results are repeatable and reproducible, independent of the person following the steps, in the absence of approximations.
2. Learn how to apply implication rules IF A THEN B, one at a time and one after another, to arrive at conclusions, one at a time or one after another, all with the knowledge that an error in one step makes all the rest wrong. Learn that results are repeatable and reproducible, independent of the person following the steps, in the absence of approximations. Recognize when an implication rule IF A THEN B does not apply, and when it is obeyed or disobeyed. The implication rule is not disobeyed when it does not apply. The implication rule may be obeyed or disobeyed when it applies or should apply.
3. Master the difference between one- and two-way implication rules.  See how implication based knowledge may divide into islands or bodies of knowledge.
4. Xtra: See the statement A or NOT A, the law of the excluded middle, as a consistency requirement for a collection of implication rules. Conjecture: If the collection is consistent, the law of exclude middle holds and may be used without harm. If the collection is inconsistent, using the law of the excluded middle preserves the inconsistency and may make it more obvious.

Algebra

1. Describing Numbers, Amounts and Quantities: Does your charge know how to talk about or describe numbers and quantities without writing letters or symbols on paper?  
2. Describing Calculations and Using Formulas: Does your charge know how to describe calculations with words or formulas, and to do those calculations given numbers to use in them?  
3. Replacement Principle: Does you charge understand that a symbol or expression in mathematics may be replaced by another when the other is known to give the same result.?  
4. Describing when different calculations will give the same result: Does your charge understand that the axioms for real numbers, properties of real numbers or rules for algebra use letters and symbols to describe when different calculations give the same result.?  
5. Explain the rules for recognizing  multiples of 2, 3, 5, 10 and 11 using arithmetic modulo these numbers. Hint:  ten and all its powers
   
       = 0 mod 2 or 5 or 10,
       = 1 mod 3 or 9
       = -1, mod 11.
   
6. solve linear equations and inequalities in one variable (or quantity).
7. graph lines y - mx + b by locating two points on this line. Can you charge  identify x- and y-intercepts?
8. Use axioms for real number to say when different algebraic or arithmetic expressions will yield the same results.

Sets:

Can your charge:

1. Recognize sets, empty or not.
2. Count the number of objects in a set
3. Understand no duplicates in a set
4. Find union, intersection, symmetric difference of sets
5. The three "safe" rules for set formation:  (1) subsets from existing set by specifying a property; (2) Cross-products of two sets; (3) Set of all subsets (Power Set Formation).
   
6. Modern Math Assumption: There is a set of Real numbers. Each real number may be represented by point on a real line, a signed finite or infinite decimal expansion. Some real numbers, those with a finite or repeating decimal expansions, may be represented by  signed fractions.
   

Measurement

1. Use rulers, scales, clocks and protractors for measurement. 
2. Explain what is a variable
3. Measure to the nearest eighth of an inch or millimeter
4. Define and Use units of length
5. Define and use units of weight or mass
6. Estimate areas of irregular polygons by partitioning - covering with squares or triangles.
7. Estimate and measure capacity (volume)
8. Estimate volume by counting cubes.
9. Estimate time to the nearest minute using an analogue clock
10. Find elapsed time
11. Figure elapsed time between AM and PM
12. Measure temperature and convert from one scale to another.
13. Add and subtract  intervals of time like 5 hours, 30 minutes, 15 seconds. Carries and borrows may appear as in decimal arithmetic, all associated with the conversion of hours into minutes and minutes into second, or vice-versa, during computation.
14. Multiply  intervals of time like 3 days, 5 hours, 30 minutes by whole numbers with carries first to aid conversion of minutes in hours  when more than 60, and second to aid conversion of hours into days when more than 24.

Geometry

1. Measure angles in the plane using a protractor?
2. Construct triangles with the three methods SSS, ASA, SAS?
3. Know hen these methods SSS, ASA, SAS fail for construction in the plane?
4. Understand the SSS, ASA, SAS congruency (isometry) assumption for triangles in the plane.
5. Does you charge understand that translation, rotation and flips (reflection) preserves the angles and sides (the measures) of triangles? So these actions lead to congruent (isometric) triangles - triangles with the same measures.
6. Understand the concept of parallel lines - lines in the plane not meeting?
7. Know that failure of ASA method on both sides of a transversal to two lines in the plane implies the two lines are parallel? (Most likely his or her teacher does not know? The point made here is an innovation for the exposition of mathematics.)
8. Draw circles, squares, rectangles and triangles, given their dimensions.
9. Parallelism and perpendicularity of lines and line segments
10. Define, draw and use angles including acute angles and obtuse angles (measure too?)
11. Recognize (name) the different types of triangles and quadrilaterals.
12. Identify and use translations, reflections and rotations.
13. Count the number of edges (sides) and vertices of triangles, rectangles, pentagons, hexagons or octagons,
14. Measure angles directly
15. Measure angles indirectly using the assumption that the of angles in triangle is 180 degrees.
16. Measure angles in a triangle indirectly by using the isometry or similarity of triangles.
17. State and Apply the Pythagorean Theorem
18. Construct angles, triangles, rectangles, circles using compass and straight edges.
19. Explain and use the SSS, SAS and ASA methods for triangle construction. Explain when they fail
20. Explain how the ASA method fails when the sum of interior angles to a transversal between two lines adds up to 180 degrees or two-right angles - the two lines are parallel.
21. Estimate and compute areas of rectangles, triangles and cricles by measuring, by covering with small squares (partitioning) or by formulas.
22. Estimate and compute areas of convex polygons by measuring, by covering with small squares (partitioning) or by formulas.
23. Relate similarity and proportionality to map reading and scale models. Explain how lengths, areas and volumes and angles are affected by map or model scales.
24. Draw 2D representations (projections) of 3D objects?
25. Use vectors to represent movements, one at a time or one after another on a map. Show how equivalent vectors or movements may be added  using rectangular coordinates.

Enriched Arithmetic, Algebra and Geometry - A base for complex numbers and trigonometry.

1. Addition and subtraction of displacements to the left or right along a line.
2. Use of signed numbers as rectangular coordinates.
3. Use of rectangular coordinates to define addition and subtraction of points or vectors in the plane.
4. Use of signed or unsigned numbers as polar coordinates in the plane
5. Use of polar coordinates to define multiplication of points in the plane.
6. Introduction of complex numbers
7. Suggested High School/College Assumption: There are sets of Real and Complex numbers. Each complex number may be represented by a point in the Cartesian Plane. Each such point has rectangular coordinates and polar coordinates that may be measured or determined from each other. Each real number may be represented by point on a horizontal axis, a signed finite or infinite decimal expansion. Some real numbers, those with a finite or repeating decimal expansions, may be represented by  signed fractions. (This assumption is for all students and teachers who do not specialize in pure mathematics, or before such specialization, for ease of exposition.
8. Axioms for complex numbers including the distributive law.
9. Consequences of the Distributive Law: law of sines, unit circle definition and properties of sine, cosine and tangent functions; proof of Pythagorean theorem as a  consequence of equality of two different ways to multiply.

Calculator Usage:

1. Convert to and from scientific notation
2. Use a calculator to speed-up the long-division process.  Here decimals will be truncated to integer parts.
3. Use a calculator to speed-up the application of Euclid's method in finding greatest common divisors, and then least common multiples or denominators. Here decimals will be truncated to integer parts.

Decimal arithmetic approximations are  not acceptable in the exact derivation of formulas. But the use of calculators may be accepted and encouraged after the mastery of arithmetic operations (+,-, times, division) without it. Calculators may be used in evaluating formulas. Students have to be taught when to use a calculator and when not. Arithmetic with small numbers should be by hand to keep skills alive and current.

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Maths for Ages 5+

Ages 5 or 6
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Ages 7 or 8
Ages 8 to 9
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Age 14
Where is it going

D What to do in School & Why  

E.How to Study Mathematics

To read, write and spell, your children need to learn and memorize the alphabet. Anything less would be absurd. That being said, learning and using mathematics demands that your children meet key skills and concepts, and not skip any. Where local schools do not provide the latter, you need to provide remedies.

Care and Precision: If your child  can learn to follow multi-step methods carefully and precisely in arithmetic, he or she may do so  in other subjects, as well. Get your child or teen, if you can, to sit down and study. Suggest he or she aim for skill and concept development and perfection for their own sake, not that of their teachers.

The will to learn is the key to success in school.  Parents do have to be educated to support or guide their children and teens. What matters more is support for the will to learn, for children and teens to be  told to try to learn and to ask teachers, their schools or classmates for help and more help, as needed. Teachers and parents need to push students, help them find the will to learn, teamwork helps.

The main reason and focus for high school mathematics is or should be preparation for calculus. That requires skill and knowledge perfection with fractions, algebra, geometry, trig and functions. Many high school programs do not provide this. Make sure alone or with help that your children and teens have a good command of fractions. 

 

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