YOU are better than YOU think. Show yourself how:
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Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence in
work |
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Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
Mathematics In Biology I
Numbers and Percentages below are fictitious.
This assignment explores two different ways of describing growth of populations, one using doubling time and the another using annual growth rates. Doing the following questions will show you how the two different descriptions or models can be interchanged - allow you to switch between doubling times (or halving times) and annual growth or decay rates.
1. The Beluga whale population in the St. Laurent Rivers is decreasing at a rate of 2% per year. So after t = m years, the number left is
N(t) = N0 (1-0.02)m = N0 (0.98)m
where N0 denotes (is, represents) the initial population.
(a) Evaluate the factor (0.98)m for m = 0, 5, 10, 15, 20,
25, 30, 35 and 40 years.with the aid of a calculator.
(b) How years m will it take for the factor to be close to 0.5? How many
years m (the same value) will it take for the population to decrease to half its
initial value N0? Take N0 = 400 if you wish.
(c) Solve 0.5 = (0.98)m for m using the algebraic and
computational property that ln (ax ) x ln(a) for a > 0 and x any
real number. Here ln(x) = natural logarithm of x and ax may be
computed using your calculator. Parts (a) and (b) gives the numerical method
for solving this problem.
2. (a) For several years, the Blue whale population off an Antarctic is growing at 2.5% per year. At this rate of growth, a population of 1000 would increase as follows.
| m = no. of years | N(m) = population count |
| 0 | 1000 |
| 1 | 1025 = N(0)* 1.025 |
| 2 | ____ = N(1)* 1.025 |
| 3 | ____ = N(2)* 1.025 |
| 4 | ____ = N(3)* 1.025 |
| 5 | ____ = N(4)* 1.025 |
| 6 | ____ = N(5)* 1.025 |
Fill in the blank population numbers to the nearest whole number to estimate the population, one year after another.
(b) Evaluate the formula N(m) = A*(1+i)m for m = 0, 1, 2, 3, 4, 5 and 6, assuming i =2.5% = 0.025 and A = 1000. The results should agree with those computed one year at a time, and one year after another in part (a).
(c) Find the number of years m for which the factor (1+i)m has a value equal to 2. Using the numerical or algebraic methods followed earlier in question 1. The algebraic method is better - shows greater mathematical maturity.
(d) Let n satisfy (1+i)n = 2. Compute N(p)= A*(1+i)p for p = 0, n, 2n, 3n, 4n, assuming A = 1000, and i = 0.02= 2.5%. Do you need to know the value of i if you are given m.
(e) Let n satisfy (1+i)n = 2. Compute A*2m/n for m = 0, 1, 2, 3, 4, 5, 6 with A = 1000 again. Use your calculator.
(f) Let n satisfy (1+i)n = 2. Compute A*2m/nfor m = 0, n, 2n, 3n, 4n, assuming A = 1000, and i = 0.02= 2.5%.
3. The population of ponies on a isolated island doubles every four years for a decade or two. During that period the population numbers N(t) = 300*2m/4when t = m years. Show algebra implies N(t+1) = 2¼ N(t) regardless of the value of t.
(a) Find a number i so that 21/4 =
1+i.
(b) Compute the values of (1+i)m and 2m/4for
m=0,1, 2, 3, 4 and 5.
(c) If 21/4 = 1+i, simplify
A*(1+i)m - A*2m/4
4. The population of seagulls on a isolated island halves every four years for a decade due to a harsh environment change. During that period the population numbers N(t) = 300*(½)m/4when t = m years. Show algebra implies N(t+1) = (½)¼ N(t) regardless of the value of t.
(a) Find a number i so that (½)1/4 =
1+i.
(b) Compute the values of (1+i)m and (½)m/4for
m=0,1, 2, 3, 4 and 5.
(c) If (½)1/4 = 1+i, simplify
A*(1+i)m - A*(½)m/4
www.whyslopes.com
Algebra, Odds & Ends,
1. Hints for Exams 2A. Exact Arithmetic 2B. Fractions Briefly 3. What is a Variable? 4.. Square Roots 5. Straight Lines 6. Problem Solving Methods 7. Trig and Complex No. 8. Complex Applet 9. History of No.s 10. ln(x) and exp(x) 13. Rename the > Sign 14. Problems: Quadratics 15. Problems: Algebra Test 16. Problems: Linear Eqns I 17. Problems: Linear Eqns II 18. Problem Solving Hints 19. Functions & Sets 20. Independent Variables 21. Why Logic 22. Why Math 23. The 15 Times Table 24. The 20 Times Table 25. Algebra Formulas 26. On Learning Maths 27. Maths in Biology 28. Navigation +Time 29 Quibble-What is Algebra 30. Logic in Maths
Odd and Ends, Essays
Constant Retirement Rate Road Safety 3 Strikes Law in California. Math HOW-TOs 9 Steps in Maths
Study With Others: twiddla.com has developed a collaborative whiteboard with audio & text chat that allows students, tutors & teachers to explore & scribble on blank pages and copies of webpages together, If scribbling is awkward with one browser, try another.
In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.