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Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

30 pages en Francais || Parents - Help Your Child or Teen Learn
Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch.

Are you a careful reader, writer and thinker? Five logic chapters lead to greater precision and comprehension in reading and writing at home, in school, at work and in mathematics.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

Early High School Algebra

What is a Variable? - this entertaining oral & geometric view may be before and besides more formal definitions - is the view mathematically correct?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

Early High School Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.
- Coordinates - Use them not only for locating points but also for rotating and translating in the plane.
- What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. Many human-made objects are similar by design.
- 7 Complex Numbers Appetizer. What is or where is the square root of -1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of -1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
- Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails

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www.whyslopes.com >> Algebra Starter Lessons >> 10 Examples of Algebraic Reasoning >> 2 Fraction Operations Physical Development Next: [3 Inequalities Algebraically.] Previous: [1 Decimals Modular and Remainder Arithmetic.]   [1] [2][3] [4] [5]

Operations with Fractions

The operations here are derived from physical meaning of what it means to tale a fraction of an object.

Multiplication of Fractions

Let m, n, p and q denote whole numbers - 5, 7, 11 and 13 if you like.

We assume one n-th of $m \times n = n \times m$ objects is m. We further assume p n-ths of IT is p times [one n-th of IT], provided the latter can be calculated. The latter can be calculated when IT is a multiple of n.


\begin{eqnarray*} \\ \frac pn \times \frac mq & =& p \times \left[ \frac 1n \mbox{ of } \frac mq \right] \\ & =& p \times \left[ \frac 1n \mbox{ of } \frac {n \times m}{n \times q} \right] \\ & =& p \times \left[ \frac { m}{n \times q} \right] \\ & = &\frac {p \times m}{n \times q} \end{eqnarray*}

Observe how raising terms gives a multiple of n. The result gives multiply the numerators, multiply the denominators rule for fraction multiplication.

    The rule can be made more efficient, applied indirectly, by showing how to find common factors to cancel before products in the numerator and denominator of the result are calculated.

Observe the multiplication of fractions

\[ \frac pn \times \frac mq = \frac mq \times \frac pn \]

commutes because the multiplication of whole numbers commutes.

Reciprocal of Fractions

When p and q are nonzero whole numbers, the reciprocal of their ratio

\[\frac pq \]


\[\frac qp \]

Clearly, the reciprocal of a reciprocal is the original fraction. Further the product of a fraction and its reciprocal

\[\frac pq \times \frac qp = \frac {p \times q}{q \times p} = \frac {p \times q}{p \times q} = 1\]

The question what fractional multiple of $\frac pq $ gives 1 has the answer $\frac qp$ = the reciprocal of the fraction. If

\[\frac pq \times A = 1\]

then one-qth of boths sides must be equal. That gives

\[ \frac 1q \times A = \frac 1p \]

Now q times both sides must be equal. Then gives

\[ A =\frac qp \]

Thus A is the reciprocal.


\[ A \times \frac pq = 1\]


\[\frac pq \times A = 1\]

Therefore answer to the question of what fractional multiple of $\frac pq$ gives 1 is given by the reciprocal $A = \frac qp$

    In measuring with units and subunits, the question of how times $\frac pq $ units goes into 1 unit has the answer: $A = \frac qp$

Division of Whole Numbers - Forwards and Backwards

Let N =17, d= 5 and r = 2 below if you like.

Long division of a whole number N by a divisor d, another whole numbers, implies there is a natural number $q \ge 0$ and a remainder r $ 0 \le r \lt d$ such that

\[ N =q \times d + r = q \times d + r\times 1 \]

Thus d goes into N, q whole times with a remainder of r. That is only answer that can be given before multiplication by improper fractions or mixed numbers is understood. The case q = 0 arises when $N \lt d$.


\[ 1= \frac 1d \times d \]


\begin{eqnarray*} N &=& q \times d + r \frac 1d \times d \\ & =& q \times d + \frac rd \times d \\ & = & \left[q +\frac rd\right] \times d \end{eqnarray*} Thus with a knowledge of fractions and mixed numbers, we may say, d objects goes into N objects, $q +\frac rd$ times exactly. That result appears in some but not all primary mathematics work booklets.

By raising terms, we see that mixed number

\[ q +\frac rd = \frac {q \times d}d + \frac rd = \frac {qd+r}d = \frac Nd \]

is equivalent to the fraction $\frac Nd$ - improper when $N \ge d$. The foregoing leads us to write

\[ N \mbox{ objects} \div d \mbox{ objects} = \frac Nd \]

Thus the improper[?] fraction $\frac Nd$ gives the exact number of times d objects go into N objects. Long division or inspection allows the latter to be expressed as a mixed number $q +\frac rd$

Division with Like Denominators

Take m = 4 on first reading if you like.

Now suppose the object in question is an m-th of another object. That would imply

\[ N \mbox{ m-ths} \div d \mbox{ m-ths} = \frac Nd \]

Or, in fraction notation

\[ \frac Nm \div \frac dm = \frac Nd \]

Division with Unlike Denominators

Take A =3, B =5, C = 11 and D = 13 on on first reading if you like.

In the case of unlike denominators, we raise terms to transform the division question into a like denominator case:

\begin{eqnarray*} \frac AB \div \frac CD &=& \frac {A \times D}{B \times D} \div \frac{B \times C}{B \times D} \\ & =& \frac {A \times D} {B \times C} \end{eqnarray*}

Showing how to divide defines the operation. But

\[ \frac {A \times D} {B \times C} = \frac AB \times \frac DC \]

Hence division by a fraction $\frac CD$

\[\frac AB \div \frac CD = \frac AB \times \frac DC = \]

as well. So it has the same result as multiplication by the reciprocal $\frac DC$ - the latter serves as constant of proportionality $K$ for division by $\frac CD$. That is,

\begin{eqnarray*} \frac AB \div \frac CD &=& \frac AB \times \frac DC \\ &=& \frac DC \times \frac AB \\ &=& K \times \frac AB\end{eqnarray*}


\[K = \frac DC = 1 \div \frac CD \]

Addition, Comparision and Subtraction with Like Denominators

Take a =5, b =11 and m = 5 on on first reading if you like.


Given a pair of whole a and b the distributive law for counting says

\[ a \mbox{ objects } + b \mbox{ objects} = [a+b] \mbox{ objects} \]

Now if the object is given by an m-th of another, we likewise have

\[a \mbox{ m-ths } + b \mbox{m-ths} = [a+b] \mbox{ m-ths} \]

In fraction notation that gives the like denominator fraction addition rule:

\[ \frac am + \frac bm = \frac{a+b}m \]


If in the pair, the number a is more than b, we may write

a objects is more than b objects

Now if the object is given by an m-th of another, we have

a m-ths is more than b m-ths

In fraction notation we may write

\[ \frac am > \frac bm \]

where we read > as more than.


Now if in the pair, the number a is more than b, then by a distributive law for counting

\[ a \mbox{ objects } - b \mbox{ objects} = [a-b] \mbox{ objects} \]

Now if the object is given by an m-th of another, we have

\[ a \mbox{ m-ths } - b \mbox{ m-ths} = [a-b] \mbox{ m-ths} \]

In fraction notation that gives the like denominator fraction addition rule:

\[ \frac am - \frac bm = \frac{a-b}m \]

Addition, Comparision and Subtraction with unlike Denominators

Take A =4, B =6, C = 11 and D = 13 on on first reading if you like.

The sum, comparision and difference of two fractions $\frac AB$ and $\frac CD$ may done by raising terms [if need-be] to apply like denominators. We will study the raising term parts.

One common multiple of the denominators B and D is their product

\[M = B \times D = D \times B\]

Let M denote a common multiple of the denominators B and D. Then

\[ M = b \times B = d \times D\]

where \[b = M \div B = \frac MB\] is the number of times B goes into M, and \[d = M \div D = \frac MD\] is the number of times D goes into M. are proportional to M. The least common multiple of B and D gives the smallest values for b and d.

The product common multiple $M = B \times D = D \times B$ gives $b=D$ and $d=B$

Raising terms gives two fractions

\begin{eqnarray*} \frac AB = \frac {A \times b}{b \times D} = \frac {A \times b}M \\ \frac CD = \frac {C \times d}{d \times D} = \frac {C \times d}M \end{eqnarray*} with like denominators to add, subtract or compare. The foregoing - see site lesson on inequalitiesproviding it is site to do - implies

  • $\frac AB$ is more than $\frac CD$ when and only when $ A \times b$ is more than $C \times d$

  • $\frac AB$ is equivalent to $\frac CD$ when and only when $ A \times b = C \times d$

  • $\frac AB$ is less than $\frac CD$ when and only when $ A \times b$ is less than $C \times d$ /

The foregoing also implies the addition formula

\begin{eqnarray*} \frac AB + \frac CD &=& \frac {A \times b + C \times d} M \\ &=& \frac {A \times [M \div B] + C \times [M \div D]} M \end{eqnarray*} and the subtraction formula addition formula

\begin{eqnarray*} \frac AB -\frac CD &=& \frac {A \times b - C \times d} M \\ &=& \frac {A \times [M \div B] - C \times [M \div D]} M \end{eqnarray*} When $M = B \times D = D \times B$ is the product of the denominators. Then the addition and subtraction formulas become

\begin{eqnarray*} \frac AB + \frac CD &=& \frac {A \times D + C \times B} {B \times D} \\ \frac AB - \frac CD &=& \frac {A \times D -C \times B} {B \times D} \\ \end{eqnarray*}

www.whyslopes.com >> Algebra Starter Lessons >> 10 Examples of Algebraic Reasoning >> 2 Fraction Operations Physical Development Next: [3 Inequalities Algebraically.] Previous: [1 Decimals Modular and Remainder Arithmetic.]   [1] [2][3] [4] [5]

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Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic?

Play with this [unsigned] Complex Number Java Applet to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

Site Reviews

1996 - Magellan, the McKinley Internet Directory:

Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it.

2000 - Waterboro Public Library, home schooling section:

CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as how (and why) to learn mathematics in school; pattern-based reason; finding a number; solving linear equations; painless theorem proving; algebra and beyond; and complex numbers, trigonometry, and vectors. Also section on helping your child learn ... . Lots more!

2001 - Math Forum News Letter 14,

... new sections on Complex Numbers and the Distributive Law for Complex Numbers offer a short way to reach and explain: trigonometry, the Pythagorean theorem,trig formulas for dot- and cross-products, the cosine law,a converse to the Pythagorean Theorem

2002 - NSDL Scout Report for Mathematics, Engineering, Technology -- Volume 1, Number 8

Math resources for both students and teachers are given on this site, spanning the general topics of arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos with clear descriptions of many important concepts provide a good foundation for high school and college level mathematics. There are sample problems that can help students prepare for exams, or teachers can make their own assignments based on the problems. Everything presented on the site is not only educational, but interesting as well. There is certainly plenty of material; however, it is somewhat poorly organized. This does not take away from the quality of the information, though.

2005 - The NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4, Number 4

... section Solving Linear Equations ... offers lesson ideas for teaching linear equations in high school or college. The approach uses stick diagrams to solve linear equations because they "provide a concrete or visual context for many of the rules or patterns for solving equations, a context that may develop equation solving skills and confidence." The idea is to build up student confidence in problem solving before presenting any formal algebraic statement of the rule and patterns for solving equations. ...

Senior High School Geometry

- Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

Calculus Starter Lessons

Why study slopes - this fall 1983 calculus appetizer shone in many classes at the start of calculus. It could also be given after the intro of slopes to introduce function maxima and minima at the ends of closed intervals.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.

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