Oral Rules for Arithmetic
Each rule or pattern in below has value in its own right. But the earlier
ones lead to the last two. Moreover, these rules or patterns may
developed and understood verbally before any algebraic description of
properties of numbers, whole to real. The rules wordily given and
explained reflect and extend the common know-how in ways that may have
take-home value.
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Counts are independent of the order in which elements of a
(finite) set are counted. In practice, people may count
twice in the hope of detecting mistakes. Recounting is required when
earlier counts disagree.
Set View: [i]If f is a one-to_one map of the set of whole numbers 1
to m onto a set S, and g f is a one-to_one map of the set of whole
numbers 1 to n then m = n. [ii] Each count of the elements of a
finite S is given by a f is a one-to_one map f of the set of whole
numbers 1 to m onto a set S. [iii] Therefore all counts m of the set
are equal to as single number, which we denote by #[S].
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Counts may be obtained by forming and adding non-overlapping
subcounts in any order. In practice, addition of whole
numbers may be introduced as a counting shortcut. People may add
twice in the hope of detecting mistakes. Disagreement between
additions will require a recount - here another addition. Counting by
addition is the practiced in elections where polling stations report
their totals for inclusion in larger totals.
Set View: [i]If a finite set S is a sum of subcounts of disjoint sets
Bk where k ranges from 1 to m, then \[ \#[S] =
\sum_{k=1}^m \#[B_k] \]
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In arithmetic, sums of whole numbers, fractions and decimals
may also be obtained by forming and adding non-overlapping
subtotals. If one looks carefully enough, this practice
or principles is another consequence of the ability to count wholes
by addition.
Set View: [i]By expressing whole numbers, fractions and decimals over
a common denominator, raising terms as need, each of them can
expressed as count of a unit numerator fraction - a unitary fraction.
Thus we are counting multiples of the latter. The previous rule
applies. [ii] If a set of real numbers S is the union of disjoint
sets $B_k$ where k ranges over whole numbers 1 to m then \[\sum_{a
\in S} a = \sum_{k=1}^m \left(\sum_{a \in B_k} a\right) \]
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In arithmetic with signed numbers (integers and then
rationals, sums integers and rationals may be obtained by forming and
adding non-overlapping subtotals. Note when students use
both signed coordinates and arrows to represent movements on a map or
plan, the observation that the head to tail addition of arrows in
sequence may be done by subtotalling and that addition is commutative
informally implies most of this property.
Set View: If a set of real numbers S is the union of disjoint sets
$B_k$ where k ranges over whole numbers 1 to m then \[\sum_{a \in S}
a = \sum_{k=1}^m \left(\sum_{a \in B_k} a\right) \]
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In Arithmetic, Products of numbers may also be obtained
forming and multiplying non-overlapping subproducts. This
rule find application when students are shown how to group like
primes in the the prime factorization of whole numbers. It has
also has application in the discussion of decimals - the optional
explanation of how or why decimal methods for multiplication
work. Extension: This rule also
applies to division as division by a number is replaced by or
identified with multiplication by its reciprocal in the case of
fractions and a multiplicative inverse.
Set View: If a set of real numbers S is the union of disjoint sets
$B_k$ where k ranges over whole numbers 1 to m then \[\prod_{a \in S}
a = \prod_{k=1}^m \left(\prod_{a \in B_k} a\right) \]
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In arithmetic, products of nonzero numbers are nonzero, but
if one or more factors is zero, the associated product is
zero. This rule may answer questions about whether or not a
product is zero. This rules may be used to speed the calculation of a
product in which one of the factors, one given by the value of an
expression, happens to have the value zero. The rule may be
implied by observations about how and why the product of two nonzero
counts or two nonzero lengths cannot be zero.
Set View: If a set of real numbers S has no zero elements, then
\[\prod_{a \in S} a \ne 0 \]
The first two of the last three rules or patterns imply that sums and
products of terms and factors may calculated and grouped in
different ways even before algebra begins.
The distributive property of arithmetic with real numbers is introduced
elsewhere with the aid of geometry and coupled with a column methods for
calculating products of sums.
Set View: Let A and B be set of real numbers then
\[ [\sum_{a \in A} a
\times [\sum_{a \in B} b] =\sum_{[a,b] \in A \times B} ab = [\sum_{a \in
A}[ [\sum_{a \in B} ab] \]
The latter justifies column methods for expanding products of
polynomials.
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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
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Others are welcome to refine or exceed it. Please do.
Secondary
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How Texas sent
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May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
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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,
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Is your child able to add, subtract and multiply amounts
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
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Algebra
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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
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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
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if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
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calculus and more generally in the first year of college. Bon
Appetite.
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