|
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. For students of reason in society, science and technology: Pattern Based Reason describes origins, benefits and limits of rule- and pattern-based thought and actions. Not all is certain. We may strive for objectivity, but not reach it. Postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theories and practices. Site's Best LessonsFor Logic
These online chapters may amuse while leading to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. For Arithmetic
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style. For Algebra
What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct? |
whyslopes.com >> Arithmetic and Number Theory Skills >> 8 Arithmetic with Signed Numbers 2 signed and unsigned numbers as coordinates. 3 signed coordinates for maps and planes. 4 signed coordinates for regions in space. 5 lengths and signs of numbers. 6 adding signed numbers. 7 negative and additive inverse. 8 multiplying signed numbers. 9 subtracting signed numbers. 10 dividing signed numbers. 11 What are real lengths and numbers. Folder Content: 10 pages. NotesArithmetic with signed numbers includes arithmetic with integers, rational numbers (signed fractions) and beyond them, real numbers. Lesson 1 has just become the last lesson 11. Writing is an iterative affair. In retrospect, being last appears to the best place for it. Lesson 2 signed and unsigned numbers as coordinates introduces signs as prefixes to provide coordinates along a full number number. This use of signs + and - as prefixes to numbers in service of providing coordinates for a full line provides an initial context and motivation for signed numbers. Optional Reading: Lesson 3 signed coordinates for maps and planes and Lesson 4 signed coordinates for regions in space describe the further use of signed numbers in pairs or triplets to locate points. Lesson 5 lengths and signs of numbers. Signed numbers have a sign prefixed to a unsigned part. The latter part may be called its magnitude, absolute value or length of the signed number. Lesson 6 adding signed numbers introduces methods for adding signed numbers - those with a common signs and those with diferent signs. To add two or several numbers with a common sign, use the slogan prefix the common sign to the sum of their lengths. To add two numbers with unlike signs, prefix the sign of the longest [or largest] to the difference, the longest length minus the shortest length. Lesson 6 describes these methods for adding with words and with algebra, and then gives many, many examples. Lesson 7 negative and additive inverse for each number IT, identifies its additive inverse - a second number which when added to IT gives a result of zero. People who think algebraically may think x in place of IT. Lesson 7 is preparation for lesson 9. The accompanying slogan for computing a negative or additive is simple: keep the length, but change the sign prefixed to it. In that change, a plus + becomes a minus -, and a minus becomes a plus. Lesson 8 multiplying signed numbers is based on the slogan, multiple the signs, multiple the lengths to compute the produce of two or more sign numbers. Twenty or so multiplication examples employing integers, proper and improper fractions, and symbols denoting real numbers are given.
Teachers:
The case of signed mixed numbers is not covered here, but they can be rewritten as signed improper fractions. How to multiply mixed numbers without this conversion must wait mastery of the distributive law and perhaps associated column multiplication methods to exploit. The multiply the lengths, multiply the signs slogan provides a prequel to and slogan and rule multiply the lengths, add the angles for multiplying complex numbers. Lesson 9 subtracting signed numbers shows how to subtract a number by adding its negative or additive inverse. Multiple examples are given. Those examples are followed by two interpretations of subtraction, the more-than interpretation [i], that the length of the difference between two numbers gives the number of units, one is more than another; and the geometric distance interpretation [ii], that the length of the difference of two numbers gives the distance between two. There-in lies a prequel to the discussion of length calculation along a coordinate line using absolute values. Lesson 10 dividing signed numbers presents slogan multiply the signs, divide the lengths to say how to divide signed numbers. Lesson 11 What are real lengths and numbers describes how numbers may describe length and position along straight lines - number lines. It easy to understand the associated use of whole numbers and fractions - proper and improper, but it may come as a surprise that there are points on straight lines whose distance to the origin is not a whole and/or fractional multiple of a unit length. Thus more numbers to describe lengths appear. Thus extra numbers - the irrationals - together with proper and improper fractions form the unsigned real numbers. The latter provide coordinates along a half-line. Remark: If a number is written without a sign, its sign is deemed to the plus sign. With that convention, the arithmetic operations described below can also be applied to expressions involving a mix of signed and unsigned numbers. A NuanceLocation of Signs: The use of signs + and - in the super-prefix position, examples +5 and -3,in the introduction of integers appeared in modern mathematics secondary and college education 1967-75 say. But the but was not used in practice with rational numbers given by unsigned fractions (a/b). With the latter, signs were employed as prefixed position but not in the superscript position. In the following lessons, signs appear in prefix position at normal or superscript hieght, or somewhere in between. Whether or not the symbols + and - serve as number signs or as the number operations - here addition and subtraction, or calculating a negative inverse - is usually well indicated by the context, with any ambiguity being harmless. For example -5 may indicate negative 5 - the number - or the calculation of negative inverse of 5, a calculation that has value negative the number. whyslopes.com >> Arithmetic and Number Theory Skills >> 8 Arithmetic with Signed Numbers |
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? 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; 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. ...
For Geometry
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
to scale. For Calculus
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. |