Step 2 Algebraic solutions for one unknown
1 Proper Equal Sign Usage
2 Three Examples
3 Four Examples
4 Four Examples Fractional Coefficients
5 Algebraic Solutions - Introduction
6 Algebraic Solution Example
Notes
This folder in lessons 2 to 4 offers examples of solving linear equations
with whole number and fractional coefficients. Solution checks are
usually includedd - emphasize when a check fails that the error or
errors may be found between the start of the solution and the end of the
check.
Lesson 4 reproduces the first part of Volume 2, Three Skills for Algebra,
Chapter 15. In this lesson, equations of the form ax +c = d are repeated
solved to lead students to see (we hope)the derivation of the solution
formula x = (d-c)/a as an algebraic shorthand description of the
numerical solutions. An algebraic check is included. Then solution
formula is employed to solve many numerical examples. Lesson 4 aims to
provide a step by step path into the algebraic way of reasoning with
letters and symbols, with the understanding that they are place holders
for numbers. Following in the footsteps of Lesson 4, Lesson 5 derives
algebraic solution of the more general equation ax + b = cx +d.
Before solving linears numerically, students should be familar formulas
for perimeters, areas and volumes. These formulas describes many possible
calculation, all at once, not all of which have to be done. After lessons
4 and/or 5, students may see the further power of algebra to solve many
like or similar problems at once.
In numerical examples or exercises involving the solution of linear
equations in one unknown with integral and fractional coefficients,
students should see again the domino effects of errors, should do and
record steps in an observable and verifiable or correctable manner,
should check results and be aware that when a check fails thatthe error
or errors may be found between the start of the solution and the end of
the check. Solutions should further develop and reinforce calculation
skills with fractions, but only after student success in solving linear
equations with integral coefficients that have integral solutions,
non-negative in the first instance.
Note: The literal or algebraic solutions of equations is introduced
further in a small example in Chapter 10 and in many examples in Chapter
14 of site Volume 2, Three Skills for Algebra. Chapter 14 in particular
emphasizes the forward and backward use of a formula - the compound
interest formula, while contrast arithmetic (a.k.a numerical) and
algebraic (a.k.a literal) solutions. The careful forward and backward use
of rules and formulas, proportionality relations included, represents a
unifying thread for mathematical and scientific subjects in secondary and
college.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
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The Logic of Injustice:
How Texas sent
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by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
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May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
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Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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