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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 >> Volume 2 Three Skills For Algebra >> Postscript-Unifying Theme - A Fourth Skill For Algebra Next: [ Chapter 14. Forward and Backward Use of a Formula.] Previous: [Chapter 13. Second Reading Guide.]   [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] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

# A Fourth Skill For Algebra

Volume 2, Three Skills for Algebra

### Direct and Indirect Use of Formulas, or Forwards and Backward Use of Equations

Every formula met in mathematics, accounting, science, technology etc may be used directly and indirectly, that is forwards and backwards.

The simple message that the forward and backward use of formulas (direct and indirect use) is part of high school mathematics and beyond names a required skill and allows us to recognize, identify and thus emphasize the most frequent pattern in high school mathematics and beyond.

This message needs to be given explicitly and early in secondary mathematics. Otherwise the underlying skill become part of the hidden, or silent and unspoken, agenda in mathematics courses.

Teachers: Consider combining the www.purplemath.com a two page lesson on solving literal equtions with the message above and the examples and exercises indicated below. The page banner above was Forward and Backward use of equations but it now reflects the purplemath lesson, Solving Literal Equations.

### First Site Example

Direct and Indirect Use of the Rectangle Area Computation Formula

Chapter 10 in discussing Direct use of A =WL assumes W and L are given. Indirect use assumes A and one of W and L is given, and leads to the calculation or formulas W = A/L or L = A/W. The explanation of those formulas is a step towards algebraic reasoning - the direct and indirect or forward and backward use of formulas.

More Examples: Formulas for perimeters and areas of squares, circles, triangles, rectangles etc can be used forwards and backwards. Finding the value of a proportionality constant k say in an equation y = k x represents an indirect or backwards use of an equation, a pre-requisite to further forward and backward use of the equation y = kx. The calculation of parameters a and b in y = ax + b (or y = mx +b) represents another backward use of a formula or equation. Quebec students in secondary III have met the forward and backward use of the Pythogorean equation c2=a2+b2 where c is the length of the hypotenuse and the two numbers a and b are the lengths of the other two sides (legs) of a right triangle.

To Do: : Post some details and exercises here to further illustrate and emphasize the forward and backward use of common formulas.

### Going Further (More on Substitution)

Chapter 10 before the forward and backward use of a formula goes further in showing how to describe a the calculation of a box V = H(WL) and show how to employ substitution (a new concept for students) to go between this formula and V = HA where A = WL. Details are given in the chapter. The details may be easier to grasp if numerical examples are added to this exposition.

Seeing how a box volume formula V = hA and V = h (WL) can be transformed into each other illustrates and may introduce the notion of equivalent expressions. The law applied here is A = WL is a geometric law rather than an algebraic law (like the distributive law). None, the idea that an expression represents a number or quantity and that there may be more than one ways to compute the number or quantity is key to the notion of equivalence. Students thus see how substitution in formulas leads to new formulas, how arithmetic patterns may be used to use formulas directly and indirectly, and how algebraic solutions may be more general or powerful than arithmetic solutions.

### Algebraic Exercises:

1. Find a formula for the area of square in terms of its perimeter (easy)
2. Find a formula for the area of circle in terms of its perimeter (easy)
3. Find a formula for the perimeter of square in terms of its areas (harder)
4. Find a formula for the perimeter of circle in terms of its areas (harder)

See www.purplemath.com two page lesson on solving literal equtions for hints or to learn more.

The exercises could be easier after reading the first sections of Chapter 15 and Chapter 14 here in Three Skills for Algebra. The chapter 15 material may be easier.

The first sections in Chapter 15, >Solving Linear Equations derives an algebraic formula for the solution of equations of the form ax + b = c, and so emphasize the use of algebraic shorthand reasoning to imply solutions for many problems of a given form at once. All the foregoing emphasizes the power of algebra, or the shorthand way of writing and reasoning with letters in place of numbers. That being said, numerical experience is still required with formulas and their graphs, otherwise the connection between numbers and algebra may too weak.

### A Deeper Site Examplefor now or later or never.

Chapter 14 introduces the direct and indirect use of the compound interest formula A = P(1+i)n.

Chapter 14 presents algebraic and arithmetic solutions that may be used to check the calculator skills of students while developing the algebraic way of writing and reasoning. In the compound interest formula A = P(1+i)n three of the four amounts A, P and i and n are assumed known, and the problem is calculate or find a formula for the missing fourth. The use of this formula is indirect when the left hand side quantity A is given or known, and the task is to find the value of the principal P, the interest rate i or the number of compounding periods n. Add to chapter 14 coverage, numerical confirmation that the algebraic solution works. The algebraic solutions for the indirect use of formulas involve substitution and assumes the pattern (AB)/B = A. Coverage of Chapter 14 is recommended as part of the next topic: exponents and radicals.

www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Postscript-Unifying Theme - A Fourth Skill For Algebra Next: [ Chapter 14. Forward and Backward Use of a Formula.] Previous: [Chapter 13. Second Reading Guide.]   [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] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

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.