www.whyslopes.com || Fit Browser Window

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

Return to Page Top


www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 7 Slopes and Velocity Next: [Chapter 8. Slope Interpretation.] Previous: [Chapter 6. Slopes and Vertical Shifts.]   [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]

Chapter 7. Slopes and Velocity

Distance Versus Time

Signed Distance along a Road

Suppose in traveling along a road, position at time t is given say by d = f(t). The coordinate d gives a signed distance to the origin or point of reference. Assume positions on one side of this origin have a positive d-coordinate and the positions on the other side of this origin have a negative d coordinate. The absolute value or magnitude of d, that is |d|, gives the unsigned distance to the origin or point of reference. The coordinate d will just be called the distance or signed distance hereafter.

Constant Speed and Velocity Example

Henry Snail walks along a path away from its origin for four hours at a rate of 6 kilometers per hour. He started at 1 p.m., eight kilometers from the origin.

Problem:   Graph his distance d to the origin versus the time t since he began.

Solution. His speed s = 6 kilometers per hour = 6 ·[km/hour]· Thus he travels

6 km in 1 hour and 12 km in 2 hours,
3 km in [1/2] hour and 18 km in 3 hours,
0.1 km in [1/60] hour and 120 km in 20 hours.

Harry travels a distance Dd = s Dt in time Dt. This information allows us to form the following table,
elapsed
time t
distance
D
d traveled
distance
d
to origin
point (t,d)
on graph
0hr 0km 8km A = (0 hr, 8 km )
1hr 6km 14km B = (1 hr, 14 km )
2hr 12km 20km C = (2 hr, 20 km )
3hr 18km 26km D = (3 hr, 26 km )
4hr 24km 34km E = (4 hr, 34 km )
and then to graph his travel distance d versus t.

Here the slope m has the role of speed or velocity. That is,

m
=
speed s of travel
=
rate of changeof distance with respect to time
=
6 ·
km
hour
 =  
rise
run
·

everywhere on the graph. Observe that 

d
=
6 ·
km
hour
 + 8 km. 

First Varying Velocity Example

Problem:   Graph the distance d to the origin of a path versus time t for the following journey of Harry Snail.

  1. At two o'clock in the afternoon, he is 50 km west of the origin, he travels further west at 100 km/hr. He drives at this speed for 1[1/2] hours.
  2. At half past three in the afternoon, he stops for one-half hour.
  3. He then drives eastward at 75 km/hr for the next two hours and then stops for another 2 hours.
No other information is available. Also find the slope for each portion of the journey.

Solution.

The trip has five segments. Comments on each segment or portion follow.

1. Before his trip begins, he could be stationary, that is, not moving. This possibility, a suspicion which cannot be confirmed, is represented by the horizontal dashed line. The slope of this speculative dashed line is

m = 0 km
2hr
= 0
So his slope or speed m is 0 or 0 km/hr, as you like. The dashed line in the above diagram could have and probably should have been left out. 

Footnote: When in doubt leave out, is a rule to follow in solutions of problems. Or, when in doubt say so, to show what is certain and what is not. Your credibility is at stake. Indicating precisely where you are guessing in a solution, identifies a question to be answered later by yourself or your instructor. And in marking assignments or tests, I would be less severe with mistakes explicitly identified as guesses than I would be with guesses deceptively presented as sure knowledge. Caution: Not all instructors will have this opinion.

2. The first described portion of the trip starts at the point A = (2 hrs,50km). He reaches the point B = ([3½] hr, 200km) after traveling at 100 kilometer per hour for one and a half hours. The slope of this portion of the trip m = 100[ km/hr] = 100 km per hour.

3. The second described portion of the trip lasts for one half hour. By remaining stopped (stationary) for [1/2] hour, his (t,d) coordinate changes from B = (3[1/2]hr,200km) to C = (4hr,200km). The slope or speed m here is again zero.

4. By traveling at 75 kilometers per hour back towards the origin for two hours, his position coordinates (t,d) change from C = (4hr,200km) to D = (6hr,50km). The slope

m = rise
run
= -75 km
hr
= -75 km
hr

5. Finally, he does not move for 2 hours. This gives the last portion of the graph with d = 50km and slope m = 0.

Changing Units (Digression)

The measure or unit of slope, speed and velocity in the last example is given by the ratio [ km/min] or km per minute, or km per one sixtieth of an hour. But other units (or ratio of units) are possible. The question becomes: what units do you like for the expression of these quantities. It is possible to change units of time and distance.

Multiplying by the number 1 does not affect the value of an expression, but the number 1 can be written in many ways. Some, not all, are helpful. These different ways can sometimes help in changing the units used for the expression of a quantity. A few slope or velocity based examples follow. Note that [60min/1hr] = 1 implies the following

m1
=
- 4
3
km
min
= - 4
3
km
min
×1
=
- 4
3
km
min
× 60min
hr
=
- 4
3
· 60
1
km
min
min
hr
=
4
1
· 20
1
km
hr
=
-80 km
hr
   
m2
=
8
3
km
min
× 60min
hr
=
160 km
hr
m3
=
0
m4
=
- 1
2
km
min
× 60min
hr
= -30 km
hr
m5
=
- 3
2
km
min
× 60min
hr
= -90 km
hr
In summary, multiplying by 1 = [60min/hr] does not change the speed or velocity m, but it does help to change the units from minutes to hours. Note to do the reverse change, multiply by
1 = hr
60min
instead. Note that care must be taken in selecting the ratio of units whose value is 1. A faulty choice will introduce more units instead of permitting some to cancel.

Remark.   An alternate method is to substitute 1 min = [1/60] hr into an expression. For example

m2
=
8
3
km
min
=
8
3
 
km
1
60
hr
=
8
3
· 60
1
· km
hr
= 160 km
hr
as before. Substitution replaces one unit by an expression for it in terms of another. With this method, there is little or no hazard of introducing units that don't cancel, but the algebra or arithmetic requires a little more thought. The choice of unit conversion method depends on where you would like to do the work or reasoning.

 

Motion with the Same Velocity

Two small problems follow, to further examine the situation where two graphs have the same slope everywhere. The slope in distance versus time graph is given by speed or velocity.

Information for the first problem. Paul starts at 3 kilometers north of his home. From there, for two hours, he walks northward at 5 kilometers per hour. Then he stops for a one hour rest. Next for one hour, he travels southward at 6 kilometers per hour.

First Problem: How far is Paul from his starting point?

Information for the second problem. John starts at a distance d0 north of Paul's home. He matches Paul's speed and movements. From his starting point, like Paul, he walks for two hours northward at 5 kilometers per hour. Then he stops for a one hour rest. Next for one hour, he travels southward at 6 kilometers per hour. Second Problem: How far is John from his starting point?

Solution to both problems.    A graph of Paul's motion is easily drawn with the help of the following table:


Time
Paul¢s
Position
0
3 km
2 hr
13 km
3 hr
13 km
4 hr
7 km
The graph follows.

In this graph

A = (0 hr,3 km),
   
B = (2hr,13 km),
C = (3 hr,13 km),
and
D = (4 hr,7 km).

Paul thus travels 7 km-3 km = 4 km from his starting point.

A graph of both Paul's and John's respective motions can be obtained from the next table.

Time
Paul¢s
John¢s
Position
Position
0
3 km
d0
2 hr
13 km
d0+10 km
3 hr
13 km
d0+10 km
4 hr
7 km
d0+4 km

The distance of John from his starting point is also 4 km. The scale has been omitted from the vertical axis in the following graph since d0 is unknown.  

This graph shows the case d0 > 3 km. How would the graph change if d0 was £ 3 km?

Questions

  1. What happens when two different graphs have the same slope over the same interval?
  2. Suppose you see two cars, one following the other and matching its speed exactly.

    (a) What would happen to the distance between the two cars?
    (b) Would the two cars travel the same distance in any given period?
    (c) How far would the following car travel if both started and stopped at the same time as the followed car?
    (d) How would you make the distance between the two cars change?

  3. How would the graphs of the motions of the two cars be related if the motion of one always matched the motion of the other, but with a thirty second delay? In the same length of time, how far would each travel? Hint: How is the graph of d = f(t) related to the graph of d = f(t-a)?
  4. Two cars travel along a straight road with the first car traveling twice as fast as the second, but still within the legal speed limit. Suppose the distance traveled by the first car is d1 = f(t). Find a formula for the distance traveled by the second car d2. Hint: First consider the case where both cars start moving at the same time.

www.whyslopes.com >> Volume 3 Why Slopes - A Calculus Intro Etc >> Chapter 7 Slopes and Velocity Next: [Chapter 8. Slope Interpretation.] Previous: [Chapter 6. Slopes and Vertical Shifts.]   [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]

Return to Page Top

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.


Return to Page Top