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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.

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.

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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.
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- 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

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 6 Fractions and Ratios >> Fraction Operations by Raising Terms - A Simple Innovation Next: [1 What is a fraction.]   [1][2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

Explaining and Justifying Fraction Operations by Raising Terms

Making Fraction Operations Easier to Master

Primary level workbooks and lessons show how raising terms leads to equivalent fractions and to explanations of how to add and subtract fractions and even mixed numbers involving different denominators. But raising terms can also make fractions comparison, multiplication and division easy to understand and explain.

In mathematics, we value the thought-based explanation or development of operations over giving them by rote. The explanation provides greater comprehension that some, if not all students, may appreciate in building their skills and knowledge. Mathematics when taught at a higher level is deeper when built on explanation of why methods work.

Comparison Example or Model

The question of which is more for a pair of fractions $\frac{11}4$ and $\frac{15}4$ with like denominators is easily answered as 15 is 4 more than 11. In contrast, the question of which is more for a pair of fractions $\frac{8}3$ and $\frac{5}2$ with nlike denominators is also easily answered by raising terms, so that both have a like denominator, smallest or not. Here 6 is a common denominator. So raising terms gives $\frac{8}3 = \frac{2 \times 8}{2\times 3} = \frac{16}6$ while $\frac{5}2 = \frac{3 \times 5}{3\times 3} = \frac{15}6$ So the difference, smaller than I thought is,$\frac{1}6$ and the first fraction $\frac{8}3$ is $\frac{1}6$ more than the second fraction $\frac{5}2$

The common cross-multiplication method of comparing fractions with unlike denominators follows from the model by using the product of the unlike denominators as a common denominator. In constrast, a smaller or least common denominator could be used. The latter would lead to smaller numbers but whether or not the latter would lead to a gain in efficiency in the comparision of fractions with small denominators and numerators is a question, not resolved here.

Multiplication by raising terms

Product of fractions may be computed by forming the product the numerators and denominators, respectively, to obtain the numerator and denominator of the product. Now a fifth of a multply of five, for example 15, is easily. For 15, a fifth is 3 and 4-fifths is twice as much, that is 12. For example

$\frac{4}{5}$ of $15 \times \frac{1}{7}$ is $12 \times \frac{1}{7}$ or $\frac{12}{7} = 1 \frac{5}{7}$

To take a fifth of a number or fraction that is not a a multiply of 5, we may raise terms. A whole number example follows

$\frac{1}{5}$ of $11$ is $\frac{1}{5}$ of $\frac{5 \times 11}{5}$ or $\frac{11}{5}$

Now 2-fifths would be twice as much.

$\frac{2}{5}$ of $11$ is 2 times as much or $\frac{2 \times 11}{5}= \frac{22}{5}$

In calculuting, one or more fifths of a fraction - the multiplicand say, we may raise terms in the latter so the numerator is a multiple of 5. For example,

$\frac{3}{7} = \frac{5 \times 3 }{5 \times 7 }$

by raising terms is equivalent to a fraction with a numerator equal a multiple of 5. Thus one fifth of the latter is $\frac{1}{5} \mbox{ of } \frac{3}{7} = \frac{1 \times 3 }{5 \times 7 }$ and so fourth fifths would be four time more: $\frac{4}{5} \mbox{ of } \frac{3}{7} = \frac{4 \times 3 }{5 \times 7 }$ Thus raising terms implies the multiply the tops, multiply the bottoms rule for multiplying factions: $\frac{4}{5} \times \frac{3}{7} = \frac{4 \times 3 }{5 \times 7 }$

Division by raising terms

The whole number 4 goes into 11, 2 wholes with a remainder of 3. The remainder is $\frac{3}{4}$ of 4. Thus the whole number 4 goes into 11, exactly

$2 + \frac{3}{4}$

times. That is,

$11 \div 4 = 2 + \frac{3}{4}$

Now 2 is $8 \times \frac{1}{4} = \frac{8}{4}$

Therefore

$11 \div 4 = \frac{8}{4} + \frac{3}{4} = \frac{11}{4}$

exactly.

Likewise 4 units of measure or counting goes into 11 units of measure or counting,

$11 \div 4 = \frac{11}{4}$

exactly.

Likewise $\frac{4}{5} = 4 \times \frac{1}{5}$ goes into $\frac{11}{5} = 11 \times \frac{1}{5}$ exactly

$11 \div 4 = \frac{11}{4}$

times. So in dividing fractions with common denomominators, the rule appears to be to ignore the common denominators: $\frac{11}{5}\div \frac{4}{5} = \frac{11}{4}$

As further example,

$\frac{11}{99}\div \frac{4}{99} = \frac{11}{4}$

The foregoing covers the like-denominator case. In general, we may raise terms to obtain a like denominator, as follows.

$\frac{8}{3}\div \frac{5}{7} = \frac{8 \times 7}{3 \times 7}\div \frac{3 \times 5}{3 \times 7} = \frac{8 \times 7}{3 \times 5}$

from the like denominator case. The result can be written in reciprocal mutliplication form:

$\frac{8}{3}\div \frac{5}{7} = \frac{8}{3}\times \frac{7}{5}$

where division by a fraction $\frac{5}{7}$ is obtained by multiplying by its reciprocal $\frac{7}{5}$

Five Operations

In summary, the five operations of addition, subtraction, comparison, multiplication and division with fractions may be introduced and justified by raising terms.

This algebraic example shows algebraically how to fraction operations of addition, subtraction, comparison, multiplication and division may be justified. The latter provides readings for older students of algebra, and a lesson plan, an algebraic template, for tutors and teachers to transform into numerial examples and exercises. Actual lessons for students should use numbers in place of letters.

Fraction Operations Efficiently

Two webpages

introduce this topic.

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 6 Fractions and Ratios >> Fraction Operations by Raising Terms - A Simple Innovation Next: [1 What is a fraction.]   [1][2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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.
... 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.