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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.

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.
- 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.
- 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 >> More Algebra >> 3 Quadratics Geometrically >> 9 quadratics physical and further context Next: [10 quadratic exercises.] Previous: [8 quadratics - backward use of various formulas.]   [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11] [12]

9. Quadratics: Applications in Geometry Physics Etc

 Problems may come from several sources.

  • Solving Systems of Equations - one quadratic, one linear.
  • Examples from Physics.
  • Constant Velocity Motion
  • Quadratic in Time implies Constant Acceleration
  • Constant speed and constant acceleration motion (enriched topic)
  • Examples from Economics (do, but view with suspicion)

Mastery of quadratics is needed for calculus and beyond in science, engineering, mathematics and other quantitative disciplines based on calculus (or special functions such as logarithms and exponentials.)

Remark: Applications in economics of quadratics exist, and you may meet them,  but those I have met seem more unreal, contrived, or artificial than the physic applications.

Remark: The quadratic formula may be used to solve  ax2+bx + c = 0 directly. Or, factoring by inspection and factoring by completing the square and using the difference of two squares can be use to say ax2+bx + c = a(x-r)(x-s) for some real numbers r and s

Problem Type: Intersection of a line and a parabola.

The intersection is found by solving a systems of Equations - one quadratic, one linear.

The intersection of a line  y = Ax + B and parabola y = ax2+bx + c may be found by solving Ax +B = ax2+bx + c.  The latter yields ax2+(b-B)x + (c-B) = 0 which can be solved for x by the most convenient you see, say by inspection, by completing the square or by the quadratic formula. The latter quadratic in x may have two, one or no solutions. For each x solving the quadratic, there is a y = Ax+B to be computed in order to obtain the coordinates (x,y) of an intersection point.

Example: Find the intersection of the

straight line y = 3x-3 and
the quadratic y = 3x2-6x+3

At the intersection points, if any, the right hand sides of the equations must give the same value for y. Thus comparison of the two sides gives the equation

3x-3 = 3x2-6x+3

linear on one side and quadratic on the left.  Add -3x + 3 to both sides

  3x  -3 = 3x2 - 6x + 3
-3x + 3 =       -3x  + 3   +
         0 = 3x2 - 9x + 6

That implies that the first coordinate of any intersection point must satisfy the quadratic equation:

 0 = 3x2 - 9x + 6

The latter can be solved with the aid of the quadratic formula or by factorization. The latter route may give the least amount of work: Let us try it.

0 = 3x2 - 9x + 6
   = 3(x2 - 3x + 2)   - take out common factor 3
   = 3(x-2)(x-1)  since 2 has two possible factorizations
                          2 = (2)(1) and 2 =(-2)(-1)
                         Here we are fortunate that -2 - 1 = 3.
                          That gives the factorization.

Now 0 = 3(x-2)(x-1) suggests x = 1 and x = 2  provide the x-coordinates of the intersection points.  Let's compute the y-coordinates for each x-value and verify that the two expressions y = 3x-3 and y = 3x2-6x+3 give the same values for y.

x 1 2
3x -3 3 -3 = 0 3(2)-3 =  6-3
= 3
3x2-6x+3 3-6+3 = 0 3(2)2-6(2)+3 =
12-12+3 =


(x,y) = (1, 0) gives one
point of intersection, and
(x,y) = (2, 3) also gives an intersection point.

Remark: In my scratch work, I made an error in the evaluation of  at x=2 and had to reconsider my derivation of the solution and  after a short delay, saw my error. Knowledge of how does not guarantee calculations are error-free, but practice may help you and I correct more quickly  from errors or inconsistencies in our solutions.

Remark: The above problem did not ask us to graph the straight line and quadratic in the region about their intersection points. However, a graph follows. 

Problem Type: Projectile Motion

Let t denote time. Let y denote height. Then quadratics  y = at2+bt+c  may be used to describe or approximate the height of thrown or  free-falling projectiles such as bullets, rocks and balls when air resistance is negligible or neglected. For such projectiles, equations of the form x = pt+q may describe the projective movement in a horizontal direction. 

If we express t in terms of x, we see that time t is given by a linear expression in x. That expression can be use to eliminate t in y = at2+bt+c to obtain a quadratic relation y = Ax2+Bx+C between the y and x coordinates of the projectile. So we conclude, the projectile follows a quadratic path in the xy plane. In the foregoing, the upper case letters A, B and C The letters A, B and C depend on the coefficients p, q, a, b and c. The do not have the same meaning or same value as the lower case letters a, b and c unless x = t in a unit-free description of the physical situation.

 In practice, you may meet

  1. Vertical projectile motion - the position of a falling object subject to the constant pull of gravity at or near the earth's surface can be described using quadratics expressions y = at2+bt + c with time t in place of horizontal coordinate x as the independent variable.  The direct use of this equation is to calculate coordinate y given the value of time t. One indirect use of this equation gives the value of y and asks for the value or possible values of t.  You will need to solve a quadratic equation for t and if there are two numerical solutions, decide which one is required or selected by the information at hand. Further indirect uses of the formula may give you values of y and t, clearly or not, and ask you find the values of the coefficients a, b and c, before using y = at2+bt + c directly, or indirectly again.
  2. Projectile Motion in the Plane:  Here  y = at2+bt + c and x = Bt + C describes a falling body in the vertical xy plane near the earths surface. You may be asked to analyze these equations forwards and backwards.
  3. Free Sliding Object on a slanted plane. y = at2+bt + c and x = At2+ Bt + C but simplifications may follow, will follow, from using a slanted coordinate system with x or y coordinate in the plane.  The equations for situation B or C may reappear. This might be an enriched problem in a senior high school physics course.
Problems of type B: A cannon ball leaves the mouth of a cannon with an initial horizontal velocity of 800 meters per second in a direction (say the x-direction) and an initial vertical velocity of 600 meters per second (say the y-direction).  (i) How will the ball be when the ball is 4000 meters horizontally from the initial position.  (ii) When and where will the ball hit the ground? (iii) What is the maximum height of the ball?  Assume the position of the ball as it leaves the cannon mouth is almost ground level, say y = 2 meters.

Solution: We assume air resistance is negligible for this cannon ball projectile, that it flies in a vertical plane with upward direction is positive. Then the x and y coordinates of the projectile are given by two formulas from physics, namely

x = x(t) = x0 + vxt and

y = y(t) = y0 + vyt - ½gt2

where t = elapse time since the projectile was in its initial position, where g = 9.8 meters per second square = acceleration of a free-falling object due to gravity at the earth's surface, where (x0 , y0) give the initial position of the projectile; and where  (vx , vy) = the initial velocity of the projectile. The latter means vx =  initial horizontal velocity  and  vy = initial vertical velocity

Substitution (Use) of data in equations.

We will take the initial position   x0 = 0 and use vx =  initial horizontal velocity = 800 meters per second. So

x = x(t) = x0 + vxt  = 0 +

800 m

t =

800 m


Now (½)(9.8) = 4.7 gives

y = y(t) = y0 + vyt - ½gt2 = 2 m +

600 m

 t  - 4.7   m

(i) Now we want to find the value of y when x = 4000 meter. The latter condition implies

800 m


= 4000m

t  =      sec 
800 m
4000 m

 Therefore t = 5 seconds when x = 4000 meters.  That implies the value of y is given by the formula

y = y(t) = 2 m +

600 m

 t  - 4.7   m

evaluated at t = 5 seconds.   That yields

 y = [2  + (600)(5)  - 4.7 (5)2] m = [3002 - 4.7 (25)] m

    = 2884.5 meters when x = 4000 meters. 

That completes the solution to part (i).

In part (ii), the question is when will y(t) = 0 after t = 0. That requires the solution of the equation

0 = y = 2 m +

600 m

 t  - 4.7   m

or equivalently

0 = 2 +


] - 4.7[   t
] 2

The positive solution T+ =  t/sec of this equation follows from the quadratic with the aid of a calculator. That completes part (ii).  Calculate the negative solution T- as well for use in part (iii).

Exercise: Compute T+ ,  T- and then the t-coordinate h below of the maximum height with the aid of a calculator.

For part (iii), the high point of the trajectory

 y = 2 m +

600 m

 t  - 4.7   m

occurs on the axis of symmetry t = h = -b/2a. The latter can be computed directly.  The latter can be compute directly, or you can use symmetry to observe h = (T+ + T-) is half-way between the two zeroes of y = y(t).   From the value t = h, the maximum value of y = y(t) can be obtained, again with the aid of a calculator.

Remark 1: In mathematics courses, I would advise students to, try to delay or postpone the use of calculators and hence the appearance of approximate calculations in a solution as much as possible. The objective is to obtain an exact solution - one in which there are no approximations. 

Remark 2:  The description of projectile motion is   provides a war-like qualitative idea of the flight of projectiles.   Suffice it to say, I do not like the connection of mathematics to the arts of war, past and present. Mathematics skills and concepts have been driven by various motivations in consumer life, business, construction, science (planetary movements included), technology and war.  Projectile motion provides the application of quadratics most easily visualized and so most useful for the development of mathematical skills - ouch.

www.whyslopes.com >> More Algebra >> 3 Quadratics Geometrically >> 9 quadratics physical and further context Next: [10 quadratic exercises.] Previous: [8 quadratics - backward use of various formulas.]   [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11] [12]

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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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