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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.

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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

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www.whyslopes.com >> Arithmetic and Number Theory Skills >> 7 Arithmetic and Fractions with Units

Notes

In my school days, mathematics courses were too pure to mention or show how to work with units in arithmetic and algebraic calculations. So calculations with units in chemistry and physic course appeared without sanction from my mathematics courses. But mathematics courses should support the use of units, and calculations with units may appear in the application of trigonometry and the development of calculus. This subsection of the fraction folder provides a remedy.

Arithmetic and algebra with units represents alternate title for this subject of fractions with units.

An operational command of calculations with units could be sufficient for further use in the representation of rates and proportionality constants and for further use in calculation in chemistry, physics and money matters. In this arithmetic with units of measurement, products and quotients of monomials may be formed and simplified with monomials that contain units to unlike powers. In contrast, sums and difference of monomials to have a physical context may only be formed and simplified only when the monomials are real multiples of each other. Inclusion of this topic will help later in examples of exponent addition and subtraction with monomials in one to several variables x, y, z, ... and in their products or quotients.

Unit of measurement are part of applied mathematics and external to pure mathematics. Yet calculations involving units of weight, mass, length, time, money (hand it over) and so on appear in the measurements and calculations of daily life alone or as part of calculations with rates and proportionality constants.

  1. Origin and Addition of Some Units: Measurements system appear in daily life, science and technology. They involve calculation with units. This lesson introduce standalone units, and indicates how to add subtract multiplies of them in a way that resembles the forward and backward use of a distributive law for counting and measuring.
  2. Units and Equal Signs: Mathematical practice requires the equality sign to have the "forward and backward" reflective property that a = b when and only when b = a. Here may read the equality sign to mean the same as or is equivalent to. Yet in daily life, the statement 3 apples + 4 oranges is, gives or equals 7 fruits, departs from this practice. We are not going change the habits of daily life, but should be aware of the difference here between the technical and common use of the word equals or the symbol = for it.
  3. Products of Units and Numbers: The first part of this lesson shows how to add subtract numerical multiplies of them in a way that resembles the forward and backward use of a distributive law for counting and measuring. Carries and borrows in addition and subtraction provide a unit-free form of conversion. In counting and arithmetic, conversions are further present in expressing counts or numbers in groups of ones, tens, hundreds and so on, and in adding, subtracting and multiplying such counts. The second part of this lesson talks about changing and converting the units used to measure or keep track of quantities. Your fortune may be measured in pennies. or in dollars? The third part show how to form products of quantities. The operations here are similar to work with monomials where the product of 4xy with 5 xy3z is 20 x2y4z. Instead of using letters x, y and z, we use measurement units and counting units as well, and could be more meaningful for students.
  4. Fractions With Units. This lesson introduces fractions with units and extends the concept of equivalent fractions to fractions with units. These fractions with units are analogous to ratios of monomials and their simplification:

        12 xy3
    8 x2y2

     =  

     3 y
    2  xz 

  5.  Simplification of Fractions with Units. Simplification of fractions with units is analogous to to simplification of ratios of monomials (fractions with monomials) in one to several variables.

       24 yw8
    9 w2y4z22

     =    8
     3

     w6
     y3z22

     
  6. Reciprocals, Division and Compound Fractions for Fractions with Units. This two part lesson shows how to (i) write the Reciprocals of Fractions with Units, and how to (ii) divide Fractions with Units, and how to (iii) evaluate the corresponding compound fractions where numerators and denominators are fractions with units. Equivalent fractions reappear here as part of the simplification of results. The operations here analogous to calculating reciprocals of ratios of monomials in variables x, y and z, etc; to dividing such ratios and to evaluating compound fractions with those ratios as numerators and denominators. three part lesson shows how to (i) write the Reciprocals of Fractions with Units, and how to (ii) divide Fractions with Units, and how to (iii) evaluate the corresponding compound fractions where numerators and denominators are fractions with units. Equivalent fractions reappear here as part of the simplification of results. The operations here anonymous to calculating reciprocals of ratios of monomials in variables x, y and z, etc; to dividing such ratios and to evaluating compound fractions with those ratios as numerators and denominators.
  7. 50167_Converting_or_Changing_Units.html">Conversion of Units in Fractions With Units: A physical quantity may be described in different units. Here we show how speed (it is written as a fraction with units) written in distance per hour may be expressed in distance per minute. The trick of multiplying by a fraction with value 1 (a fraction with units which is equivalent to one) appears here.

To Learn More about how about how fractions with units are treated and occur in daily life and physics, see the Units in Calculations pages in site Volume 3: [7 Velocity] [7 Varying Velocity Example] [7. Velocity Calculation] [7 Changing Units] [7 Same Velocity Motions] [10 Slopes without Units.] [10 Units & Slopes] [10 Units in Cost vs. Quantity] [10 How Units Appear] [10 Unit Elimination] [10 Partial Elimination][10 Interest & Units] [12 More on Units]

In general quantities in daily life involve basic units of measure, that is of time, length, mass (or weight) and of money too - an artificial concept. The units and their numerical multiples may be multiplied together form monomials - expressions equal to a real number times a product of units to nonnegative powers. Writing these monomials as numerators and denominators in fraction like expression (they look like fractions) gives fractions with units. This section on arithmetic and algebra with units shows how form and simplify expression for monomials form by products of units and fractions formed by taking monomials for numerators and denominators.

Advanced Students: The presentation here is informal, but it could be codified in a formal way. See the first or second chapter of Henri Cartan's work below for a model. . That codification coupled with invariance ideas leads in the physical sciences to similarity analysis and similarity requirements for solutions of equations.

End Notes for Teachers

  1. In applied mathematics calculations with proportionality, multiples of simple and compound units will serve as proportionality constants and rates with physical or social meaning. An operational command of calculations with units could be sufficient for further use in the representation of proportionality constants and for further use in calculation in chemistry, physics and money matters where units appear.
  2. Operations with monomials involving units and their quotients are similar to operations on monomials in variables x, y, z etc and their quotients. The latter too may represent formal operations on expressions that have no meaning for students other being marks on paper. In contrast, the previous note implies a role for polynomial like operations on units in calculations.
  3. In the modern axiomization of mathematics as seen in school and colleges, the codification is limited to pure numbers. Units are not discussed. But as a service to the physical and social sciences, money matters included, the algebraic role of units in their computations can be codified or modeled. That can be done informally. Henri Cartan's work Elementary theory of analytic functions of one or several complex variables, translation of THEORIE ELEMENTAIRE DES FONCTIONS ANALYTIC D;UNE OU PLUSIEURS VARIABLES COMPLEX, could be extended to provide a formal codification of operation with units.

www.whyslopes.com >> Arithmetic and Number Theory Skills >> 7 Arithmetic and Fractions with Units

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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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