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. Early High School Arithmetic
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. 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? Early High School GeometryMaps + 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 humanmade 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 TermsMaking Fraction Operations Easier to MasterPrimary 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 thoughtbased 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 ModelThe 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 crossmultiplication 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 termsProduct 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 4fifths 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 2fifths 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 termsThe 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 likedenominator 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 OperationsIn 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 EfficientlyTwo 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 headtotail
addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.
Pattern Based ReasonOnline Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule and patternbased 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 storytelling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge. Site Reviews1996  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; patternbased 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
crossproducts, 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 howtos 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. 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. 