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Original Site Title: Appetizers and Lessons for Mathematics and
Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept
Development with How-TOs Français: 26
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Equations:
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Welcome: Site content may develop
critical thinking, improve reading and writing, and build
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Teachers: This December 2011, 5-phase framework
offers a context for mathematics & logic instruction. Phases 1 to
3 focus on skills with actual or potential value for
adult & daily life. College-oriented phases 5 & 4 focus on
calculus & preparation for it. Phases 1 to 4 may also serve
trades & professions not dependent on calculus.
Site Review: Math resources ... span ... arithmetic, logic,
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and how-tos .... provide a good foundation for high school and
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Home < Algebra Starter Lessons < 3 Solving Linear Equations < Step 1 Stick diagram and fractions << Skill Development Notes
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Introduction
Skill Development Notes
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Most students understand the use of stick diagrams in solving linear
equations. One recent student could not see the equivalent between
the stick diagrams and equations, but he could use the diagrams to
solve. More examples might have helped. Not all is certain in
mathematics education.
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A vertical pair of equal-length sticks (line segments) with the
second below the first is use to represent an equation in which the
length of a line segment is the unknown. A sequence of operations
easily seen and understood may shorten or lengthen or multiply each
stick by the same amount, so that a sequence of stick pairs or
equation results. The aim is obtain a pair of equal length sticks
with one has the sought for length and the other have a length given
by a number. That solves the equation. The solution method here
employ subtraction, multiplication and division to shift from one
pair of equi-length sticks to another pair in order to solve for or
isolate the unknown. Some students will not catch on to the idea that
operations should lead from one pair of equi-length sticks to
another.
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A three column table summarizing the operations appears in each
example and is followed a check of the solution obtained. By
checking, students know at the end of their calculation whether or
not a mistake has been made in obtaining a solution. The first column
in the table presents a sequence of stick diagrams. The third column
gives the corresponding equation. The middle describes the operation
in going from one pair of sticks to another (or one equation to
another). Ideally the description is written in a way that it
describe the operation on a pair of sticks and the corresponding
equations well. The filling in of the table introduces the notion
that what is done to one stick or one side of an equation has to be
done to the other side as well to maintain equality of the lengths
each stick or equation represents.
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The equations here and in the exercises may be solved without using
the stick diagrams. Some students may see the stick diagrams and
decide not to use them. But their use introduces the notion that what
is done to one stick or one side of an equation has to be done to the
other side as well to maintain equality of the lengths each stick or
equation represents. Ask students who see the stick diagrams and
decide not to use them to have patience and to take the time to
demonstrate they mastered stick diagram usage. Implicit in their
usage are all the rules of algebra for solving equations. Those rules
will be formally given later with reference and illustration by the
stick diagram method of solving equations.
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Students should be required to check that the solution they obtained
satisfies the original equation, and be told explicitly if the right
hand side does not equal the left hand side for your solution that
they have to look for the error (or if time is short, acknowledge
their solution is wrong). Finding that the the right hand side does
not equal the left hand side and saying nothing, or worse claiming to
have done the problem points to a lack of comprehension.
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The examples here involve only unit and simple fractions ¼ ½ ¾ mainly
because they easily inserted typed on the keyboard. Other fraction,
proper and improper, and mixed numbers appear in the exercises.
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The stick diagram method here employs only subtraction, division and
replication of segment lengths. Magnification and reduction of
diagrams is also useful to fit them in the width of a column. Example
equations are chosen so that all coefficient and terms in the stick
diagram method remain non-negative. The objective of the stick
diagram method is not to solve all linear equation, but to lead
students to solving linear equations by operations on equations by
themselves without any geometric representation by stick diagrams.
Using parallel arrows would be a method to extend the representation
to include positive and negative coefficients in equations,
preferably selected to have non-negative solutions.
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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.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
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
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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Home < Algebra Starter Lessons < 3 Solving Linear Equations < Step 1 Stick diagram and fractions << Skill Development Notes
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