Mathematics 116
Notes and Reflections.
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The first year course mathematics consists of the
following five topics
- Natural numbers and negative integers: the
four basic operations and the order of operations (including
exponents), preparation for algebra.
- Rational numbers: fractions, decimals,
percentage and ratio.
- Measurement: angles, segments, perimeters and
areas
- Geometry: transformations (translation,
rotations, reflections), construction and properties of triangles and
quadrilaterals.
- Statistics: information from graphs and
diagrams, forming tables of statistical data, drawing line, bar and
circle graphs.
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| The above description comes from a page describing
mathematics 116, 216 and 314. I am not sure of its origin. |
Students in the first year of high school may come with a weak to
non-existence command of the times table (addition table too) and with a weak
to non-existence fraction sense and abilities. The most important
service of first year mathematics in high school is to consolidate fraction
sense and skills. See Solving
Linear Equations with Stick Diagrams if your students have a weak
command of fractions or if you want to develop algebraic thinking skills.
The ability to follow a multi-step process in a repeatable
and reproducible manner, modulo some accidents, is a sign that the students
master further multi-step operations in and outside of arithmetic. That is the
skill or intelligence we seek. Start emphasizing in it in arithmetic.
Calculators betray students by allowing them to skip a first example of a
multi-step process in which accuracy is demanded at each and every
step. The last topic, statistics, should be exploited as much as
possible to develop and reinforce fraction skills and sense.
See the more recent lesson plans Secondary
I - fractions & allied concepts (decimals, percentages).
Previous Thoughts
- Logic and Study Skills: Cover (?) logic in a
mathematics-free manner - see chains of reason and implication rules at this
site, and emphasize that the step in a decimal method and in solving a
problem are chains of reason giving a story or path to follow. Point
out that multi-step methods need to be followed with care as an error in one
step leaves all that follows unjustified, if not wrong. Point that
conclusions in arithmetic and in logic should be independent of the
performer or reader, repeatable and reproducible. Emphasis the skill
and patience to follow a multi-step methods precisely in any discipline is
points to that skill and intelligence in most other disciplines.
Reference: What
to do in School & Why and How
to Study Mathematic & Why, and first steps in logic: 1.
Introduction, 2.
Implication Rules 3.
Chains of Reason 4.
Induction (Longer Chains) 5
Knowledge Islands. The last points to the division of rule- and
pattern-based knowledge into separate components, and how different
entry-point may be equivalent or not.
- Geometry - Concrete and Hands-On: Students need to recognize
(name) and draw circles, rectangles, triangles and further kinds of
triangles and quadrilaterals. Students need to master drawing and
measuring instruments. In particular, some student under measure the
lengths or distance because they measure from the end of a ruler and
not from the zero point on the ruler. The simple isolated exercise of having
students measure a distance or a line segment will catch that error before
it affects further work. Students may use dividers and protractors to
duplicate triangles via the SSS, SAS and ASA angle triangle construction
methods. That lies the context for the assumption and extrapolation
that two triangles will be isometric (congruent) if and only if a subset of
the measures of corresponding sides and angles satisfy the SSS, SAS or
ASA condition. The latter represent minimal conditions for isometry.
We further explicityly assume that isometric triangles can be shifted
(translated), rotated and/or reflected, so that one triangle coincides with
another by one of these rigid body motions. In place of dividers to compare
or duplicate lengths of sides, rulers can be used as well. That leads to the
construction or duplication of triangles using the measures of angles and
sides, the data required by the by SSS, SAS or ASA methods. But
if the measure of angles and sides to be employed in this method do not come
from another triangle, one previously drawn, the measures or data may be
inconsistent with the triangle construction method, and they cannot be
applied. How to bisect angles and line segments via triangle based
geometric construction can be illustrated and explained in terms of the SSS,
ASA and SAS methods and isometry rules. The bisection method for line
segments provides a means to construct a perpendicular from a point off a
line to the line, and the possibility of reflecting the point across the
line. Students see and recognize reflections, translations and rotation in a
coordinate free manner, and understand them well enough to apply them to a
line segment or polygon. The geometry
area of this site contains some details but not all besides a further
development of ideas before coordinates. Motivation Query: How does
one provide a context for these exercise? The fact these underlying
exercises are easily done and repeated may lead to skills and confidence and
fun - success for students. The greater context may lie in the message that
mastery of the geometric concepts is part of a longer path to the
mathematics needed in practice - technical drawing, surveying, trigonometry,
carpentry, construction, electrician, engineering and science.
- Decimal Notation: Place Value of Digits, individually and in groups
of three or six, for Decimal Representation of Natural Numbers. Column
Methods for addition, subtraction, multiplication and long division. The Number
Theory area of this site offers some justification for column methods.
- Proper Notation and Use of the Equal Sign: In column methods
for addition, subtraction and multiplication, digits have to written in the
proper place, else their meaning is not clear. Part of learning mathematics
is learning how to use notation properly. Improper use points to
misunderstanding and leads to difficulties or ambiguities, so what is
written differs from what was meant, and may be read or interpreted in a
third way. Now the equal sign in mathematics says that one full expression
on the left of the sign has the same value as the other on the
right. Here an expression may consist of several terms and
factors involving additions, multiplications, subtractions, division, roots
and "function" evaluation or it may consist of a single term or
number or quantity. Now 4 = 3 +1 and 10 = 6+ 4 = 6 + (3
+1) is correct, but is it wrong to write 6 + 4 = 3+1
as the equal sign says the full expression on its left has the same
value as the full expression on the right.
Improper notation in mathematics points to and leads to
difficulties. If a student solves an equations ¾ =
(¼)x by writing =3 besides the x, the student knows the answer, but
is misusing the equal sign. The equal sign requires the full expression on
its immediate left has the same value as the full expression on the
right. Proper notation allows you to record, develop and report
arithmetic and algebraic ideas in writing instead of in your
head. If you cannot write mathematics properly, your mathematics
education is at risk. For a series of expressions separated by equal
signs, each full expression must have the same value as the next full
expression in the series. Anything else points to incomprehension of the
equal sign and the associated concept of equality - having the same value.
- Addition, Subtraction, Multiplication, Comparison and Long Division of
Natural Numbers
N = {0, 1, 2, 3, 4 ... }= unsigned numbers using their decimal
representation. This assumes a knowledge and memory without the aid of a
calculator of the sums and products of all pairs of natural numbers from 0
to 10 - from 0 to 12 to 20 would be better. Comparison is based on
place value in decimal notation. Students need drill and practice so that
the foregoing becomes fast and automatic without a calculator.
Reliance on a calculator may hide some weaknesses and prevent the
development of arithmetic skill reflexes needed for doing arithmetic
with whole numbers and fraction exactly in later algebraic
reasoning and problem solving. Gifted students may visit the Number
Theory area of this website for an justification of decimal methods for
arithmetic operations of addition, subtraction, multiplication and long
division.
- Exponent Notation and the Prime Decomposition of Whole Numbers
W = {1, 2, 3, 4 ... }= {0, 1, 2, 3, 4 ... }\{0} = N \{0}
Rapid Prime Decomposition of whole number 1 to 121 is required. That
skill may be based on recognition of multiples of 2, 3, 5, 7 and 11
due to the square root method for rapid factorization in the Number
Theory area - calculators may be used here. The area contains more than
what is required for this topic. The tree diagrams used to develop the
prime decomposition of numbers should be regarded as shorthand for a
sequence of equalities. Least common multiples may be found and introduce
via a list method. Prime decompositions may be used to obtain greatest
common divisors and least common multiples.
- Fractions: Addition, Subtraction, Multiplication, Comparison,
Simplification, Equivalence and Division of Fractions A/B where
the numerator A is a Natural number and the numerator B is member of the set
of Whole Number W = {1, 2, 3, 4 ... } Division may be left in some places to
the second year of high school. The cross multiplication rule for comparison
of fractions A/B and C/D is a consequence of converting both fractions
to the common denominator BD. Least common denominators for the
efficient addition and subtraction of fraction may be found with the aid of
primer decomposition (exponents appear here) and in list method.
Teachers, if not students, may see site fraction
pages for an development of fraction sense and operations based on
operations with line segments.
- A Step into Algebra: the site area Solving
Linear Equations with stick diagrams uses operations on sticks or line
segments to introduce the idea of finding the length x of a line segment
from an equation represented by equality of lengths of a line segment formed
by multiples of the line segment of length x plus another line
segment. The physical addition, subtraction, duplication or
multiplication and division of line segments introduces algebra (solving for
an unknown) in a concrete manner while maintaining and reinforcing fraction
sense and operations. All operations on pairs of sticks or line segments
representing the sides of the initial equation correspond to a sequence of
manipulations and the sticks. In the first instance, students are required
to present the sequence of stick diagrams and equations in parallel in
tabular form with the aim of leading students to dispense with the stick
diagrams after they absorbed a geometric or physical understanding of the
solution method. Teacher may be to be firm with students who make the
transition too quickly and who do not want to provide proof, a written
record, of the ability to work with the stick diagrams and equations in
parallel. On the other hand, if a student is able to do everything
without stick diagrams, have him or her aid the others or assigned him or
her some enriched material or rest and recreation. The site area Solving
Linear Equations with stick diagrams has further ideas which can be used
to develop and refine algebraic skills in the first and further years of
secondary school. Here in solving equations, students are following a
multi-step procedure in which the results, their answers, can be checked. If
the check fails, students have to redo the procedure or find the error in
their implementation of it. This develop and reinforces the ability to do
arithmetic and algebra in a repeatable and reproducible fashion.
- Another Step into Algebra: Seeing the shorthand description of a
numerical pattern As preparation or further warm-up for the
comprehension of the algebraically described commutative, distributive and
associative laws of Natural Numbers (and fractions too), a review of
fraction addition may be combined with the algebraic description of
the numerical patterns of addition,
| namely (i) |
R
N |
+ |
T
M |
= |
RM+NT
NM |
| and (ii) |
R
DE |
+ |
T
EF |
= |
RF+DT
DEF |
The first pattern represents the use of the product of the denominators
to obtain a common denominators while the second pattern represents the use
of a lower common denominator when the denominators have a common divisor or
factor E. One can show by example that the greatest common
divisor leads to the best choice of E, the one that requires the least
amount of the right hand side. As a step towards algebraic reasoning,
give numerical example or two of the first example (i) in which M and N are
relatively prime, so that MN = the least common denominator, and tell the
students which numbers are playing the role of R, N, T and M. Then
test them on their ability to identify the numbers playing the roles of
these letters or actors or placeholder for numbers in the formulas,
identities or addition laws (i) and (ii). The exercise or aim here is for
students to recognize that a shorthand formula (i) describes a numerical
pattern with letters that have or correspond to different numbers in
different examples. This step done as part of a lessons or a review of
fractions.
Note: These formulas can be justified via the conversion of the
fractions in the sum to a common denominator, and then applying the rules
for the addition of fractions with like denominators, that is the
rule
Some students might follow the development but the development should not
be imposed on all- or you could refer interested parties to the simpler fraction
pages or the more advanced Number Theory
page what is a fraction
at this site, and discuss details outside of class.
- Geometry: Exact and Approximate Evaluation of Formulas:
Measurement of angles, line segment lengths and perimeters of geometric
shapes, and the calculation of areas from formulas provided opportunities
for calculations with whole numbers, fractions and exact or approximate
decimal measures. There in lies an opportunity to discuss significant digits
(error of less than half a unit in the last digit) and to discuss
alternative interval estimates or location of numbers and quantities. The
effect of error in measurement on further results can be mentioned. In long
calculations done with a calculator, students should be instructed to enter
all significant digits of the original data and for the sake of accuracy,
not to round intermediate results. With the aid of a calculator, the
consequences of small variations in the original measurements or data for
the evaluation of a formula for perimeter, area or some other can be
explored to estimate the error or variation in the results due to
inaccuracies in the data. What is the effect of not using all the
significant digits in the measurement or data is a question to consider? The
foregoing may lead to a first numerical comprehension of error control and
continuity in calculations. A mix of exact calculations (no calculator) and
approximate calculations (with calculators) is advised. Showing
how to carry units through calculations is an exercise in arithmetic and
algebra.
- Another Step in Algebra: Obtain Formulas for perimeters of
polygons in terms of letters denoting lengths of sides.
- An Oral Element of Algebra - words before and besides symbols. The
concept of a number or quantity that may vary or not between examples, over
time, or in one direction by not another, leads us to describe numbers and
quantities as variable or not, constant or not, in one sense or direction,
if not another. The site essay What
is a Variable to learn more. Explicitly talking about and describing
numbers and quantities without necessarily representing them by letters
represents a missing step in the conceptual development and exposition of
mathematics. The concepts of a number or quantity being variable, constant,
known, unknown, private, confidential or secret can be understood apart from
the use of the shorthand letters and symbols which may denote a number or
quantity alone or in an expression. Along side and/or following the
introduction of letters or symbols to denote numbers, amounts and
quantities, we we say a number or quantity that may vary between examples is
called a variable, along with the letter that represents it. See
chapters 8 and 9 in the online book Three Skills for Algebra to learn
more. There is more to mathematics than doing arithmetic.
- Another Step in Algebra, Direct Use of Formulas. The direct use of
formulas for perimeters and areas of rectangles and circles provides one use
of the shorthand service of letters or algebra in describing calculations.
The Greek letter p represents a constant, a
number that does not change. But other letters in representing lengths
and areas may vary between examples and so are variable. Here we say a
number or quantity that may vary between examples is called a variable,
along with the letter that represents it. See the site essay What
is a Variable to learn more. Evaluation of formulas using exact
and approximate calculator based exercises connects letters to numbers and
quantities which may vary between examples.
- Another Step in Algebra, Replacement of Expressions by Others with the
same value. There are two ways to calculate the volume of a box.
One may form the product of its dimensions or one may multiply a height by
the base area. The explanation why both ways give the same number employs
replacement of one expression by another of the same value, and may involve
some discussion or rationalization of the procedure, cumulating in the the
explicit adoption of the rule or pattern or Substitution Axiom that
two expression with or representing the same value may replace each in the
modification of formulas.. See chapter
10 the online book Three Skills for Algebra for more details of
the example. There is more to mathematics than doing arithmetic
- Another Step in Algebra, Indirect or Backward Use of Formulas, The
direct use of formulas for perimeters and areas of rectangles and circles
means a single quantity on the left hand side of equation can be computed
from the values of those on the others side. Students may shown how to
obtain arithmetic and algebraic solutions for the indirect use of formulas
in which numbers and quantities normally given need to be found. The
algebraic solution follows the numerical pattern in order to arrive at
formula for the missing number or quantity which holds for all numerical
instances of the general problem - how to compute a number or quantity if
the number appears in a formula for another quantity. This process of
comparing and contrasting arithmetic and algebraic ways for finding a
solution points students to the power of algebra in solving many problems at
once.. The model for this is illustrated via the direct
and indirect use of the compound interest formula (a topic not
necessarily in secondary I mathematics) but teachers should see the model
and emulate with the formulas met in secondary I.
- Adding and Subtracting Areas: Some areas can be computed from
formulas. Other areas may be seen as fractions of areas which can be
computed. Still other areas may be seen as the sum or difference of regions
whose areas can be computed. That leads to the calculation of areas via
adding and subtracting the areas of rectangles, triangles and circles.
Exercises can be given involving the cost of painting the walls of a home or
house or apartment in all or part, with or without the ceiling, the
cost of carpeting a room or two in an home - the exercise can be
personalized and made more relevant for students by posing it in the context
of their homes. The cost of a sails for a sailing boat might provide
further examples. The foregoing can be done numerically exactly in simpler
cases and approximately with the aid of a calculator in more complicated
cases. The additive property of area in this discussion of adding and
subtracting areas sets the stage for an explanation of the distributive law
in expanding (a+b)c or (a+b)(c+d) in the case of nonnegative numbers or
lengths a, b, c and d. The Number Theory
area develops this notion further.
- Algebraically Described Properties of Unsigned Numbers, Natural
Numbers and/or Fractions, Geometrically Illustrated and Implied. The
geometric context given below gives meaning or potential meaning to the
letters a, b and c, they denote lengths or the number of units in a length,
and its avoids phrase equivalent to Suppose or Let x be a number, a
phrase that may lead to the student objecting to the use of letters
and saying give me the number. The foregoing treatment takes a, b and
c to be lengths, a more meaningful proposition to students than saying
Let a, b and c be Natural, Rational or Real numbers greater than or equal to
zero
- The commutative law ab = ba for multiplication can be
illustrated by using the formula base-times-height to compute the area A
of a rectangle with sides of length a and b, in two different with way.
Here a rotation of 90 degrees interchanges the base and height and hence
the lengths a and b in computing the area.
- The commutative law a+ b = b +a of addition can be illustrated
by placing two line segments of length a and b, adjacent to each other,
so that they form a line segment of length c. Rotation of
180 degrees about the center of the line segment of length c will
interchange the order of the segments and give two ways a + b and b+a to
compute the length c of the combined segments. Hence a+b = b +a.
- The associative law (a+b)+c = a+(b+c) for addition can be illustrated
by computed the length of a long segment formed by three segments of
length a, b and c place side by side in a line. The total length can be
measured in two different ways from left to right or vice-versa.
- The distributive law a(b+c) = ab +ac represents two different ways to
compute the area of a rectangle of height a and base length b+c.
Computing the area directly gives the left hand side. Computing the area
by dividing the base into segments of length b and c gives the right
hand side. The foregoing builds on the addition and subtraction of areas
considered earlier. The Number
Theory area points to rectangle area generalization of the
distributive law for the computation of products of two factors, each of
which is given by a sum of terms. There are implications for the
justification of decimal methods for multiplication and methods for
multiplying polynomials, that need not be mentioned to students.
- The associative law (ab)c = a(bc) for multiplication comes from
computing the volume of a box (rectangular parallelepiped) in two
different ways. Rotating the box changes the height from c to a and the
base from a times b to b times c. Here we assume the volume is
independent of how computed. That implies the associative law.
The foregoing rules are connected to the use of the equal sign. They say
when two different calculation or expression will have the same value, and
so can be interchanged or replace each other. The discussion or
illustration of replacements is another issue that may be treated before or
after
The same properties of addition and multiplication for fractions can
be implied geometrically as well, and possibly developed at the same as
above. The lengths a, b and c above do not have to be whole numbers.
They could be rational or irrational numbers. A richer
and more complex route would be to derive the above laws for fractions
a, b and c from formulas for addition and multiplication involving the
natural or whole number numerator and denominators, and the above laws
restricted to whole or natural numbers.. The latter route would be for
gifted students only as an end of year extra. The question of what
happens when at least one of a, b and c is zero may be explored briefly in
class - a digression. The above algebraic shorthand description of the
properties for natural numbers and fractions can be stated again latter. The
online Chapter
18 in Three Skills for Algebra explores and illustrates further the
meaning of these properties, their shorthand description of the properties
of arithmetic and geometry.
Illustrations of the distributive law from a geometric viewpoint tell
students how to expand expressions like (a+b+c)(d+ e +f +g) by calculating
the area of a large calculated in two ways, directly from width times length
and indirectly from the sum of the area of 12 smaller subrectangles.
| a |
ad |
ae |
af |
ag |
| b |
bd |
be |
bf |
bg |
| c |
cd |
ce |
cf |
cg |
|
d |
e |
f |
g |
The foil method for expanding (a+b)(d+e) is a special case. While the
mathematician may think of the expansion of (a+b+c)(d+ e +f +g) when a
to g are real numbers as a sum of twelve terms ad, ae, ... cf, cg, the
student in the first instance may more grasped the concept by thinking of
the numbers a to g as lengths and seeing the expansion as one of two ways to
compute the area of a large rectangle. That provides a concrete example on
which the mathematical perspective may stem immediately for some and later
for others. The Number Theory area
shows how the distributive law or this geometric of it can be justified and
applied. Some of the applications may help in class.
- Word Problems in Essentially One Unknown. The indirect use of
formulas described and the introduction to Solving
Linear Equations with stick diagrams gives students algebraic tools to
be used in problem solving, or translating word problems in to one or more
equations. Solving
Linear Equations with stick diagrams shows how to solve system of
equations in a few to several unknowns, systems that are essentially
one variable or one-unknown. Many word problems require students to
identify the key unknown and to express all others in terms of it in order
to obtain one equation in one unknown. That process bypasses the power of
algebra and makes word problems harder than need-be. Teaching students to
solve systems of equations which have essentially have one unknown would
provide an easier route and allows algebra to aid students instead of
confusing them.
- Statistics: Calculation of average, median and extremes test
student ability to follow and employ definitions. Instruction and
exercises for forming line, bar (histograms?) and circle graphs and diagrams
provide an opportunity for students to graphically represent fractions as is
or written as a percentage, and so may test or solidify fraction sense
and skills. Students should not only be able to construct graphs a and
diagrams, they should also be able to interpret them. Statistical data
may be collected for items or topics of interest to students. One
reason for the inclusion of statistics and graph interpretation is the
development of critical thinking skills, the ability to recognize the
limitations of statistics and graphs met in daily life - when are they
accurate, when are they misleading or not, and how the choice of scale and
location (y-intercept) influence graphs and lead to impressions of great or
small variation. Each statistic provides a window, a blinkered view of
a set of statistical data. The question of which statistic, the average or
median will give the best impression of salaries in a company or cost of
houses in an area, points to the limitations of statistics - the blinkered
view that statistic provide of data. In repeated measurement of a
single line segment, the average of a set of measurements may give a better
estimate of the true value of a coordinate or quantity - that points to
calibration methods and/or the scientific or technological use of statistics
(averages) for the sake of greater accuracy or less probable error. That may
be mentioned to students. But the presentation of statistics to develop
critical thinking skills with numbers and their interpretation is some what
absurd in classes where student command of arithmetic with and without
calculators does not lead to repeatable and reproducible results. The
prerequisite for critical thinking is the ability to follow multi-step
methods, one step at a time, and one step after another, with care because
of the knowledge that an error in one step leads to bad or incorrectly
justified results. If a student lack precision in reading and writing
mathematics, in doing calculations on paper, the development of critical
thinking skills via the study of statistics is hopeless.
- Natural Numbers and Negative Integers: Upto now
unsigned natural numbers and fractions have been considered. They can
provide some coordinates on a half-line, by introducing raised
prefixes + and - in front of unsigned numbers, coordinates for a line that
extends in both directions from a point chosen to be the origin of the
coordinate. Here positive sign prefixed numbers lie on one side (the right?)
while negative signed prefixed numbers lie on the other. Now 0 denotes
the origin and it can be given the positive or negative prefixes, or not. So
the origin is denote by three symbols. Finally, unsigned numbers
are taken to be same as positive prefixed unsigned numbers. NOW
OPERATIONS OF ADDITION, MULTIPLICATION, SUBTRACTION AND Division can be
defined - saying how to do computation defines it - in the first instance
for Natural Numbers and Negative Integers. . That can be done quickly.
Students may practice the operations. The properties of multiplication and
addition can then be presented as an extension of the algebraically
properties met earlier for unsigned numbers. The earlier practice and
understanding may lead to an understanding of the extension. The same
or similar discussion with rational number m/n where m is an integer
and n is a whole number can be left for later - the next year of mathematics
instruction. The set of integers Z consists of the set
theoretic union of the Natural Numbers {0, 1, 2, 3, 4 ... } and
negative whole numbers {-1,-2,-3,-4,-5, ...} or { -1,-2,-3,-4,-5,
...}. The unsigned natural numbers N = {0, 1, 2, 3, 4 ... } may be
identified with signed numbers: {+0,+1,+2,+3,+4,+5,
...}. The latter notation with the prefixed negative or positive or
negative sign raised is preferred for technical reasons in pure mathematics
but mathematics instruction would not suffer greatly if the raised prefixed
position was replaced by prefixes not raised to indicate positive and
negative numbers. Operations with integers may be familiar to students
from earlier schooling. What may be new here is the shorthand
algebraic description of the associative, commutative and distributive
laws for addition and multiplication of integers. A quick statement of
these laws and the assertion that laws or properties of numbers like
these for will be consulted whenever we need justify operations in
arithmetic or solving equations. The Number
Theory area indicates or echoes alternative approaches which teachers
may read for personal enrichment, but not give in class.
Remark: In pure mathematics, the development of Natural Numbers N
= {0, 1, 2, 3, 4 ... } may be followed by the development of Integers Z,
next by rational numbers (signed fractions) and finally by Real
Numbers. But in high school mathematics in place of theoretical
set-theoretic notions to actually construct or codify these numbers and
define operation on them, text books and teachers use a unit lengths and
coordinates or locations along a line or half-line to represent these
numbers, imply their existence, and to represent operations on them.
The high school or college development of complex numbers may go
further and represent these numbers by points in the plane to imply their
existence, and define operations on points in the plane using rectangular
and polar coordinates.
- Fractions and Negative Rationals. Develop and
verify three arithmetic operations (addition, subtraction, multiplication)
with signed fractions. Operations with rationals may be familiar to
students from earlier schooling. What may be new here is the shorthand
algebraic description of the associative, commutative and distributive
laws for addition and multiplication of integers. A quick statement of
these laws and the assertion that laws or properties of numbers like
these for will be consulted whenever we need justify operations in
arithmetic or solving equations.
Checklist
- Decimal Notation: Place Value of Digits, individually and in groups
of three or six, for Decimal Representation of Natural Numbers. Column
Methods for addition, subtraction, multiplication and long division.
- Number Theory: Primes, Composite Numbers, Exponents, Prime Number
Factorization, Factor Trees, lcm, gcd, use with fractions.
- Fractions: Fractions or ordered pairs of whole numbers numbers a
and b written in form a/b) Decimal Representation, Percentages.
Connection of ratios to fractions. Efficient ways to add, subtract,
multiply and simplify fractions.
- Measurement of angles, perimeters, line segment and areas with
rulers and protractors. Dependence on unit of length.
- Geometry: construction and properties of triangles, limitations of
construction methods, practice with ruler and compass in construction,
congruence or isometry of triangles, invariance of measures of a rigid body
under translation, rotation and reflection.
- Integers: Four Operations of addition, subtraction, division and
multiplication. Law of signs, Geometric or Displacement Significance of
Addition, Subtraction as adding the negative or additive inverse.
- Preparation for Algebra: Linear Equations and Stick Diagram,
How to Solve Essentially One variable systems, Solving Word Problems, How to
describe or talk about numbers, amounts and quantities, numbers or
quantities that vary in one way or direction but not another. The role
of letters as definite or indefinite pronouns (or pronumbers). Compare
problem solving in mathematics with solution of jigsaw puzzles -- there are
methods to speed the solution of jigsaw puzzles, but trial and error still
required.
- Statistics: Collecting Data. Interpreting and
Creating Diagrams, Charts and Tables of Data including line, bar and
circle graphs. Here is an opportunity to Check and emphasize fraction
sense and operations while covering statistics. Talk about critical
thinking, how a single number (average, median, range) gives an ideas about
data but does not describe it fully. Talk about faulty impression left
by different visual presentations of data. How to best present the data for
one end or another. The treatment of statistics is not
essential.
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Quebec English Mathematics Education
A farce is a farce is a farce.
Area Intro
Copy Right Matters
Curriculum Cuts
Intermediate Objectives
MEQ Objectives
116 Textbooks
116 Objectives
116 Check List
116 Suggestions
216 Objectives
216 Check List
216 Book Review
216 Nonsense or BullShit
216 Suggestions
314 Objectives
314 Check List
314 Suggestions
416 Objectives
416 Check List
416 Suggestions
436 Objectives
436 Checklist
436 Suggestions
436 Book Reviews
436 Nonsense in
514 Objectives
514 Suggestions
514 Book Reviews
536 Objectives
536 Suggestions
536 Book Reviews
Area pages represent an effort to follow and understand the objectives of the
1997-2005, the prior reform, and the
text books required and used 1997-2005. In retrospect, the objectives and texts
in question
are too incoherent, too full of nonsense, for rational comprehension and for
service as a base for the current reform. A farce is a farce,
is a farce
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