his folder has a tree like structure. The child, same level and parent level webpages for this webpage follow..
More Links:
D
What to do in School & Why
E.How to Study Mathematics
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
Mathematics 586-116
A Secondary I Course
This is a high school mathematics course taught in Quebec. The Quebec
government document in this pdf
file presents the old course objectives. An abridged and
paraphrased and annotated version of the objectives follow
Compare and contrast with the course content indicated by the current reform -
see the brick offline.
My comments and reflections on mathematics 116 are preceded by (W:) in order to distinguish between them from the MEQ objectives as is or paraphrased.
Relative Importance of Objectives
1. number and operation sense.52%
2. apply knowledge of geometric figures.36%
3. representations of statistical data.12%
The overall aim to help students acquire the numerical skills needed by algebra - prerequisites.
| W: In speaking of earlier studies, what students are expected to know, I
have inserted the word should. When students lack prerequisites for
a topic, essential skills and concepts should be developed and
verified to provide a solid base for further studies at the start of
studies or in the course of studies.
The first first year course mathematics 116 appears to be a year of fraction skills and sense consolidation with an introduction to algebra. students are expected to have been acquired most of the needed skills and concepts in elementary. The site lesson plans for Secondary I mathematics- fractions & allied concepts (decimals, percentages) provide incomplete but strong support for mathematics 116. See what is useful.
|
Objective 1:
Master arithmetic skills needed for algebra
W: That and the consolidation of fraction sense and skills below suggests calling Secondary I, the year of fractions, in whyslope lessons plans.
The first step involves facilitating the transition to algebra ensures that arithmetic skills have been mastered. The primary emphasis is on students' understanding of the equal sign. To this end, give students the opportunity to apply the rules for understanding and indicating the order of operations, and help them realize that the equal sign does not mean "do this," or "do that," but rather "the expression on the right has the same value as the expression on the left and vice versa."
W: Yes, that is a good emphasis. It worth repeating. Say the equal sign = means the full expression on either side of it have the same value in stead of saying they are same. That takes in account our use of equal signs between equivalent fractions and between arithmetic or algebraic expressions that have the same value in the circumstances at hand. See the proper use of equal sign in other parts of this site
W: Improper use of notation in mathematics with regrets is not penalized on final examinations. That MEQ practice has leads some instructors to say notation need not be required in course work before final examinations. The result poor or non-communication of proper use of notation points undermines instruction in Quebec secondary schools and CEGEPs.
W: Students need to write what they mean in a standard manner. Insist upon and use proper notation
Objective 2:
Check and develop number and operation sense
MEQ: Number sense should be incorporated into all aspects of teaching and
learning about numbers, be they natural numbers, integers, decimals or
fractions. ...
MEQ: Students begin to develop number sense by working with the natural numbers,
subsequently enriching their experience through exposure to fractions and
decimals.
W: Working with natural numbers would include mastery of the 10 or 12 times table. Students who do not know and for whom the calculation of products of numbers 1 to 12 is not an automatic process have to do more mental arithmetic or more mental steps before they can master fractions. The lack of emphasis on time tables in primary undermines fraction skills and sense.
MEQ: Well-developed number sense makes it possible to anticipate the results of numerical operations and determine the degree of accuracy required in a given situation, ...
W: The following paragraphs provide more explanation of number sense. The ability to anticipate the results of numerical computations is fine, but it not as important at the ability to do arithmetic operations in a repeatable and reproducible and thus verifiaible manner, efficiently. The latter requires practice. Here again if efficiency lacks due to lack of numerical experience drill, students will have arithmetic difficulties in dealing with fractions and algebra. That is implicit or explicit in the following paragraphs.
Objective 2.1:
master calculations with natural numbers
MEQ: In the elementary school, students should develop an understanding of the four operations and the ability to perform these operations on natural numbers, both mentally and in writing.
W: Here again mental arithmetic should include the ability to form the sums and products of all pairs of numbers from 1 to 10 or 1 to 12.
MEQ: Verify or develop knowledge of: prime factors of the power of a number, of the order of operations (+,*,-, ./.)and of the rules of writing associated with indicating the order of operations.
MEQ: Develop & verify the ability to work with exponents, with chains of operations and equalities; and with (evaluation) problems involving several operations.
MEQ: Develop & verify the following arithmetic skills and concepts important to algebra
- Understand and apply the properties of the operations and of equality.
- Write a natural number as a product of prime factors.
W: Add to this the ability to calculate the least common multiple and greatest common divisor of a pair of whole numbers. Add to this the ability to use a calculator to rapidly obtain the prime factorization of a number. Include here mastery of rules for recognizing multiples of 2, 3, 5 and 10, if not 9 and 11. See the site area Number Theory.for how (and more) if you like
- Evaluate an Expression. Calculate the value of a sequence of operations on
natural numbers in accordance with the order of operations. The chain may
include one or two sets of parentheses.
- Generate and use equivalent numerical expressions in evaluation of
expressions, using properties of Natural Numbers.
W: The area view of distributive law might help here.
W: For fun and comprehension, Ask Students to read backwards and aloud the decimal notation for whole numbers in groups of 3 (US tradition) and groups of 6 (UK tradition). According to the first (US) tradition, 56,789,777,521,314 becomes 314 units plus 521 thousands plus 777 millions, plus 789 billions plus 56 trillions while according to the second (UK) tradition it becomes 521 thousand and 314 units plus 789 thousand and 777 millions plus 56 (UK) billions. This reading aloud of large and larger whole numbers with 15 to 30 decimal places in an amusing fashion points students to a non-standard mastery of place value notation. Be sure to identify the method required for the final examination. This fun with place value may aid in the review and practice of written methods (no calculators) for comparison, addition, subtraction, multiplication and long division of decimal fractions via decimal "column" or place-value methods. Reading aloud multi-digit decimal representation backwards and forwards provides comic relief and informs at the same time.
Objective 2.2
Operations on integers
MEQ: In the elementary school program, the students learn how to use negative numbers to represent concrete situations. This also allows them to compare (positive and negative) numbers.
MEQ: Develop or verify the following abilities
- compare or order integers.
- perform the following operations on integers: addition, subtraction, multiplication, division and exponentiation (exponents should be limited to the positive integers). Use law of signs.
- Calculate the value of an expression involving a chain of operations performed on integers, following the order of operations. The chain may include one or two sets of parentheses.
W: The comparison of non-negative numbers (before and even after knowledge of negative numbers) is based on magnitude. Here a positive number M is greater than a positive number N when and only when the difference M-N is positive. The extension of the greater than comparison to the integers or signed numbers M and N is based on the sign of the difference M-N. So -5 is greater than -12 as -5 - (-12) = 7 > 0. Here -5 is 7 units more positive than -12 or seven units to the right of -12. Explain that comparison of pairs of non-negative numbers uses magnitude comparison and more positive comparison interchangeable while the comparison of integers in general relies on the position or more positive than concept. The Number Theory. section of this site shows how the properties of inequalities follow from the more positive than concept. See too the site essay rename the greater than sign (but do not rename in your classes).
Objective 2.3
Operation on Fractions
MEQ: In this program according to the next objective, the term fraction to indicates a rational number in the form a/b, where a and b are integers and b is not equal to zero.
W: In this current objective, are a and b restricted to Natural Numbers with b >0?
MEQ: In elementary school, students should learn how to read and write natural numbers up to one million and decimal numbers to the hundredth place. In writing these numbers, students must follow the principles of the positional system of numeration. Students also learn to express frequently used fractions in different forms (a/b, %, decimal notation).
MEQ: :Develop and verify the ability to use the decimal positional system of numeration to read and write a number and to compare numbers.
MEQ: Since rational numbers can be written in decimal notation, in fractional notation (a/b), and in scientific notation, and it is essential that students understand that the same number may be expressed in different forms. That understanding or connection is the primary goal of having them carry out transformations.
W: Presentation and development of scientific notation becomes vacuous if students do not have fraction sense and skills. Scientific notation itself has no value. It needs to be discuss in the context of relative and absolute error in measurements and in the results of calculations: addition, multiplication, subtraction and division. Scientific notation here appears to be an odd end.
MEQ: The ability to convert a number from one notation to another and a true understanding of the numeration system will enable students to use the symbols =, ... ,> and < and to compare numbers expressed in different forms.
W: (Repeated comment) :The comparison of non-negative numbers (before and even after knowledge of negative numbers) is based on magnitude. Here a positive number M is greater than a positive number N when and only when the difference M-N is positive. The extension of the greater than comparison to the integers or signed numbers M and N is based on the sign of the difference M-N. So -5 is greater than -12 as -5 - (-12) = 7 > 0. Here -5 is 7 units more positive than -12 or seven units to the right of -12. Explain that comparison of pairs of non-negative numbers uses magnitude comparison and more positive comparison interchangeable while the comparison of integers in general relies on the position or more positive than concept. The Number Theory. section of this site shows how the properties of inequalities follow from the more positive than concept. See too the site essay rename the greater than sign (but do not rename in your classes).
MEQ: The use of negative exponents to express powers of 10 need not lead into a study of the theory of exponents;
W: But the prime decomposition of whole numbers involves exponents. Here the product of prime decompositions of factors in a product leads to the prime decomposition of the product - that may be easily understood above.
students will be able to understand them if they come upon them when looking for a pattern. Develop number and operation sense in an activity in which they must mentally compare numbers such as 3.2 and -3.3 or 0.3 and 0,3 or 0.4 and 0.35.
A Thought: The above objectives (first step) may first be developed and verified with non-negative numerators and denominators.. Working with a mix of negative and non-negative ratios may come as a second step. See next objective.
Intermediate Objectives
- Read a rational number expressed in decimal notation.
W: Rational numbers whose denominators prime factorizations equal a product of 2s and fives, and no other primes, have finite decimal expansions. All other rationals have infinite decimal expansions - recurring. See the site area on Number Theory.
- Write a rational number in decimal notation. (Decimal notation: form
of writing that uses the base 10 positional system of numeration.)
- Order rational numbers when expressed in decimal notation.
- Write a decimal number in expanded form, and vice versa.
W: Here is a reinforcement of place value.
- Round off a rational number expressed in decimal notation (the order of
magnitude will either be given, or determined by the context).
- Convert a rational number from decimal notation to scientific notation,
and vice versa ("Decimal number," as used here, refers to the set
of decimal numbers (rational numbers that can be written as fractions whose
denominator is a power of 10). This ensures that, for a given objective,
students do not have to deal with numbers in which the period is not zero.
- Convert a decimal number from decimal notation to fractional notation
(a/b)
. - Convert a rational number from fractional notation (a/b) to decimal
notation.
- Convert a decimal number into a percentage, and vice versa.
- Convert a rational number from fractional notation (a/b) into a percentage
Note: All rational numbers a/b where a and b have no common prime factors and b is given by a product of twos and fives (2s and 5s) can be expressed or are equivalent to a decimal fraction, that is a fraction where denominator is a power of 10. All other fractions a/b where a and b have no common prime factors (so they are relatively prime), have infinite decimal expansions with period at most b. By long division and the aid of a calculator, students can be shown how to obtain the decimal expansion and how to recognize the period. The converse operation of converting a infinite decimal expansion of period q into a fraction is appears to be optional or the business of the second or third year of high school mathematics in Quebec.
Note: Comparison of fractions a/b and c/d may begin by converting both to a common denominator, the least, the product ab or another. The use of the product bd leads to, justifies and gives a context for the so-called cross-multiplication rule for comparison of fractions where a, b, c and d are non-negative. Here b and d are non-zero as well. Conversion to the common denominator bd also leads to, justifies and gives a context for a cross-multiplication rule for elimination of denominators for any equality a/b = c/d
Info with Comic Relief: Ask students to read aloud the decimal notation for decimal representation of decimal fractions in groups of 3 (US tradition) and groups of 6 (UK tradition). For example, according to the first (US) tradition, 56,789,777521,314.456,892,45 becomes 456 thousandths, 892 millionths, plus 450 billionths plus 314 units plus 521 thousands plus 777 millions, plus 789 billions plus 56 trillions - that is reading away from the decimal point on both sides. You may wish to have students re-read this from least significant to most significant contribution (or from greatest to small). And according to the second (UK) tradition the decimal fraction might be 450 thousand billionths, 456 thousand and 892 millionths plus 521 thousand and 314 units plus 789 thousand and 777 millions plus 56 (UK) billions. This reading aloud of large and larger whole numbers with 15 to 30 decimal places before and after the decimal point again in an amusing fashion points students to a non-standard mastery of place value notation. Be sure to identify the method required for the final examination. This fun with place value may aid in the review and practice of written methods (no calculators) for comparison, addition, subtraction, multiplication and long division of decimal fractions via decimal "column" or place-value methods.
Note: Scientific Notation is usually associated with significant digits. One error convention for positional notation is that the error in the last digit shown should be at most one half a unit. Significant digits are normally associated with relative error in a number. The study of the effect of numerical error in computations is empirical - see what happens, see if more or less digits are used to represent the numbers in a computation. The ratio of the magnitude of the change in a result to the magnitude of a small change in a number used in the results computation gives an indication of the results stability, and the possibilities for error control in the calculation. Exact calculation in error control are very difficult to do. Empirical methods are therefore used.
Objective 2.4
More operations on fractions or rational numbers,
MEQ: In elementary school, students are introduced to decimal numbers. They explore operations on numbers expressed in the form a/b, using concrete material and working with very precise restrictions. Division by a rational number is not taken up.
MEQ: In this program the term fraction to indicates a rational number in the form a/b, where a and b are integers and b is not equal to zero.
Develop and verify arithmetic algorithms involving decimal numbers or fractions.
W: I hope this means make sure that students can use column methods for addition, subtraction and multiplication; make sure students can do long division; and make sure students know how to simplify, multiply and add/subtract fractions efficiently, and divide too. Yes it does. See the following.
MEQ: With decimal numbers, students must be able to generalize the rules they have learned in order to add, subtract and multiply and must also know division and exponentiation.
MEQ: Further, develop and verify mastery of all the operations performed on fractions The aim is to equip students to deal with the fractions they encounter, regardless of the context. The problems may involve percentages, negative rational numbers, fractions, fractions greater than one, and the less common lowest-terms fractions.
W: Lowest term here I assume means a fraction of the form a/b where a and b are relatively prime. Conversion to lowest terms is a simplification which I might stress when fractions results as the answer to a problem - For improper fractions a/b where a has a larger magnitude than b, conversion by long division to a mix number of the form C + (d/b) first may speed reduction to lowest terms. We could show students the calculator based shortcut where C = the integer part of the decimal provided by the calculator based calculation of a/b provided the calculator shows a non-zero decimal fraction after the decimal point. But the one button use of calculators to simply a/b may reduce or lessen the fraction sense of students.
MEQ: Operations on negative rational numbers offer an excellent opportunity for students to apply their knowledge of integers. Let students practice using rational numbers in learning activities relating to geometry or statistics.
W: Good idea the latter: Examples would be welcome. In statistics, average might involve positive and negative heights, assets and debts of people, etc.
MEQ Intermediate Objectives 2.4
- Perform the following operations on decimal numbers: addition,
subtraction, multiplication, division and exponentiation (exponents should
be limited to the positive integers).
W: Hopefully, this includes column methods for arithmetic operations, so that student can obtain results in a repeatable and reproducible, and thus verifiable manner.
- Calculate the value of a chain of operations on decimal numbers. The chain
may include one or two sets of parentheses.
W: If the numbers are decimal, and many are present, use of a calculator is appropiate. That being said, the foregoing comment on calculations without a calculator (excepts the students pencil and paper) is advised, I hope.
- Convert a fraction into an equivalent fraction.
W: Here a chance to emphasize simplification and the use of prime factorizations.
- Compare fractions.
W: Do this first with like denominators. Then convert fractions being compared to a common denominator. The cross product for comparision of fractions a/b and c/d is a consequent of converting both to fractions over the product bd of their denominators b and d. Here bd is not necessarily the least common denominator.
- Perform the following operations on fractions: addition, subtraction,
multiplication, division and exponentiation (exponents should be limited to
the positive integers).
W: Make sure operations here are efficient.
- To calculate the value of a chain of operations on fractions. The chain
may include one or two sets of parentheses.
W: Have this done by hand in the first instant. Calculators are useful but students need efficient hand-on, pencil and paper experience with fractions for the sake of algebra and trig, and senior high school mathematics.
Note: The site area Solving
Linear Equations with Stick Diagrams (URL?) re-enforces skills
with equalities and fractions - those given by ratios a/b where a and b are
non-negative. See too the site area on Fractions, Etc.
Objective 3:
apply, develop and verify knowledge of geometry or geometric figures.
MEQ: By the end of the elementary school students usually can associate several geometric shapes with their respective names.
MEQ: They begin by observing solids among the objects found in their environment. They learn the names for them and have a mental picture that they associate with each name. They then consider the sides of these solids that they have learned to name and distinguish; they have thus developed a general understanding of geometric shapes.
MEQ: The teacher's role at this point is to help the students learn to associate (or characterize) a figure with a set of properties.
W: I suspect this includes saying a circle or its perimeter consists of all points in the plane equi-distant from a centre, saying a triangle has three angles and three sides, saying a quadrilateral has four angles and four sides (sides whose interiors may not intersect), and so on.
MEQ: It is, among other things, by studying the geometry of transformations that students establish a system of relationships that enables them to make deductions to ascertain the validity of a statement. Numerous exploration activities enabling them to manipulate, construct, measure, compare, ...
W: I suspect by movements that is transformation including translations, rotations and flipping (reflections)) students may see how to make isometric copies of a polygon or polygonal region in the plane, and other regions too, regular or not Triangles and quadrilaterals are examples of polygons. The foregoing movement are the so-called rigid body movements. We may visualize the region, polygonal or not, in question by drawing it or tracing it on paper and moving the paper - that process involves not only the polygon but also its interior.
W: The approved MEQ textbooks for English language instruction in secondary II to IV, if not V, emphasis the properties of rigid body motions - that is translations, rotations and reflections. S
MEQ: This program provides for the establishment of such a system for triangles and quadrilaterals.
MEQ: Students learn to analyze (describe, recognize?) these polygons by discovering the properties of their angles, sides, altitudes and diagonals. They learn to construct polygons on the basis of specific data, assimilate the vocabulary pertaining to these figures (what vocabulary?), measure the angles, segments and surfaces, explore the transformations of these figures, and make the connections that will enable them to solve problems according to a geometric model of their own creation. Despite the emphasis on triangles and quadrilaterals, the teacher should ensure that students continue to work with the other geometric shapes they have studied.
Objective 3.1
Create figures by means of isometric transformations
Note: isometric transformation = rigid body motion (err, rigid region motion since we working with the plane) = a translation, rotation or reflection movement one at a time or in some combination, one after another.
MEQ: In elementary school, students are recommended or encouraged (or should have been) to observe the movement of objects, especially movement that modifies the position of objects without changing their shape or size. By using tracing paper, folding techniques, a mirror, or dot or graph paper, students (should have) learnt to identify the image of a figure obtained by translation, reflection or rotation, to draw this image according to the instructions provided and to describe the geometric transformation undergone by a figure.
Develop and verify the ability to use geometry instruments to accurately construct the image of a figure on the basis of instructions calling for a translation, a rotation or a reflection.
MEQ: By carrying out the steps involved in constructing an image, the students learn about the fundamental concepts of parallelism, perpendicularity and angles. Numerous polygons can be created from the same figure by carrying out a single transformation or a series of similar transformations on it.
Set the students a variety of tasks, sot that they acquire a more precise understanding of concepts, and develop techniques which they can apply when they study the properties of triangles and quadrilaterals.
MEQ: Activities consistent with objectives:
- analyze a construction
- make observations (the right angles remain, the segment and its image form an angle...),
- observe special cases (the axes of symmetry, rotations of 180o, the formation of quadrilaterals, reciprocal transformations...)
- test hypotheses (is there another translation that would yield the same image?...)
Note: A translation is needed of analyze and what it means to observe formation of quadrilaterals. Is a reciprocal transformation, the inverse transformation to a translation, rotation or reflection?
MEQ: The formal study of properties and their applications is undertaken in a subsequent program
W: in other words, the aim is to provide students with geometrical experience..
MEQ: For all objectives in geometry, "construct" means draw, using a ruler, compass, set square or protractor.
|
W: Students may arrive from elementary
school with the inability to properly use a ruler to measure. Check
that students can measure a known length between two points with the aid
of ruler where placing the end of the ruler on one of the point leads to
error - a too small value for the length. See diagram below. |
MEQ: Intermediate objectives 3.1
- Construct the image of a figure under a translation.
- Construct the image of a figure under a rotation.
- Construct the image of a figure under a reflection.
- Construct the axis or axes of symmetry of an angle (bisector), of a segment (median) or of a polygon.
Objective 3.2
Master straight lines or angles,
State Rules & Properties to Justify Conclusions,
MEQ: In elementary school, student should develop a general understanding of geometric shapes. Straight lines and their segments are treated as components of polygons. Straight, acute and obtuse angles and parallel and perpendicular straight lines are introduced with a view to equipping students to describe and classify polygons.
MEQ: Through construction activities, students should meet and assimilate the concepts associated with parallel lines, perpendicular lines and angles plus the appropriate terminology and symbols.
MEQ: Students should be taught how to make simple deductions such as determining the size of an angle from statements (say definitions and properties given the appendix) rather than by measuring the angle. Although they may take an intuitive approach to finding a measure given indirectly, no matter how it is found, develop and verify students ability to support their reasoning with the definitions or properties (those in the appendix and ?)
MEQ: Intermediate Objectives 3.1
- Identify parallel and perpendicular line segments
- Construct a straight line parallel or perpendicular to another straight line, in accordance with certain requirements.
- Measure an angle in various figures.
- Construct angles with the same vertex, using a protractor.
- Determine the measure of an angle from a statement (flat, straight or full angle; perpendicular lines; complement, supplement or bisector of an angle; angles with the same vertex).
- Justify assertions (show their reasoning) in solving a problem involving angles with the same vertex.
page 55. Appendix
In geometry, the students analyze (describe and recognize?) triangles and quadrilaterals - meaning (?) that they met their properties. Through hands-on activities, they increase their grasp of a number of concepts and improve their skills. They should become familiar with the definitions and some of the properties of the figures they are studying. They will use these definitions and properties to calculate missing measurements and justify any assertions used in solving (missing measurement and ? type) problems involving angles with the same vertex, triangles, and quadrilaterals.
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MEQ: Terminal Objective 3.3
Solve problems involving triangles
In elementary school. students should have learnt to distinguish among various triangles on the basis of their angles and sides. As a result, students should be able to draw and describe
- a right triangle,
- an equilateral triangle
- and an isosceles triangle.
Develop and verify student ability to recognize, describe and draw
- scalene,
- scalene right,
- isosceles
- isosceles right, equilateral
Should be able to identify the characteristics and properties of these triangles so that they can solve for missing measures (when feasible)
From knowledge of
how to draw a perpendicular from a point to a line, and
how to bisect angles and lines segments,
develop and verify student ability to construct any given triangle, as well
as the altitudes, medians and perpendicular bisectors for a triangle.
The foregoing definitions and verification of these definition, directly or
through activities or problems that require altitudes, medians and
perpendicular bisectors. The foregoing also employs triangle construction
methods: Side-Side-Side, Side-Angle-Side and Angle-Side-Angle.
W: The site area on Euclidean Geometry written for older students explains these constructions
Develop and verify student comprehension of concepts and vocabulary associated with triangles by having them meet, describe and manipulate figures and solids, some of whose faces are triangles.
The properties studied should be conclusions (or discoveries) that the students arrive at while carrying out their activities. The students should not, however, be required to prove these properties. In some cases, students will be shown how to deduce properties through clear reasoning and from definitions or established properties.
My Opinion: Yes, allow students to discover the properties if time permits, but when the allotted time is up or nearly finish give a summary that consolidates the information. Find out what mix of (i) learning by discovery (the constructivist way) from hands-on experience and (ii) direct explanation supported by hands-on experience works with your students. At the end of the day, a subject oriented or final examination oriented approach requires skills and concepts to be checked and verified, with clear, authoritative, feedback to the student to say which ones are missing or in need of repair.
page 35, MEQ Objectives,3.3
- Construct a triangle given the measures of the three sides, the
measures of two angles and the adjacent side, or the measures of one angle
and two adjacent sides
W: The foregoing are known as the Side-Side-Side, Side-Angle-Side and Angle-Side-Angle triangle drawing or construction methods. These methods might also be known as copying or reconstruction methods. See the discussion of isometric triangles in site area on Euclidean Geometry.
- Construct the altitudes, medians, and perpendicular bisectors of a
triangle.
W: This ability comes from knowledge of how to draw a perpendicular from a point to a line, how to bisect angles and lines segments. Give cases where the altitude of a triangle hits the base of the triangle or the line through it.
- Express the relationships between the various types of triangles.
W: This may mean classification: a triangle may be both scalene and right angled. A triangle may isosceles and not equilateral.
- Determine the measure of an angle or a segment on the basis of a
information identifying types of triangles, or to the altitudes,
medians or perpendicular bisectors of triangles.
W: The information may be written or drawn.
- To justify an assertion in solving a problem involving triangles.
W: That is (?), , students should say when they are using a property of a triangle or type of triangle when arriving at conclusions.
page 36, MEQ: Objective 3.4
work with convex quadrilaterals
MEQ: In the elementary school program, the students learn to distinguish among the quadrilaterals. With the help of concepts such as parallelism and perpendicularity, they draw and describe the various quadrilaterals.
MEQ: Develop and verify students abilities to identify the properties of the sides, angles and diagonals of each type of quadrilateral (scalene trapezoid, isosceles trapezoid, right trapezoid, parallelogram, rhombus, rectangle, square).
MEQ: By identifying common properties, students establish certain relationships between different types of quadrilaterals -
W: how one type may also be a subset of another: a squarer is a rectangle. Are there any further relationships?
MEQ: Students should be able to apply information about a figure, its type and hence its properties, for its construction or for determining missing measures.
MEQ: Activities that enable students to use their knowledge of geometric transformations, straight lines, angles and triangles are recommended or encouraged. During exploration and manipulation activities and when presenting their discoveries, the students should learn new vocabulary, develop their understanding of concepts and improve their ability to construct figures and deduce information. See my opinion above.
MEQ: Intermediate Objectives 3.4
- Construct a quadrilateral, given sufficient data.
- Express the relationships between the various types of convex quadrilaterals - recognize that one type may also be another or not.
- Determine the measure of an angle or a segment in a convex quadrilaterals from information about its type.
- Use properties of each type of convex quadrilaterals to arrive at and explain conclusions or calculations.
Objective 3.5
calculate perimeter or the area of certain polygons,
MEQ: In elementary school, students should learn to estimate and measure the dimensions of an object, choose the most appropriate unit for expressing a given measure and establish relationships between the different units of length used in the metric system (SI). Students should learn that they can determine the perimeter of a polygon by adding the lengths of its sides. Area is studied by examining the idea of coverage. By the end of their elementary school, students should be able to take a simple example and quickly calculate the area using the numbers representing the dimensions of the figures. Students have learnt about the units of measure for area used in the international system, without, however, going into the relationships between them - how to convert one unit to another?
MEQ: Develop and verify student ability to establish relationships between the dimensions of a figure and its perimeter or area, and use these relationship forwards and backwards, or directly and indirectly.
MEQ: Students will then draw on the characteristics of polygons to
express these relationships and increase their understanding of variables. To
this end, they learn to distinguish between a variable and an abbreviation, and
become aware that the variable represents a number in the formulas for perimeter
and area that they will learn to generalize.
W: Notes on algebraFor triangles and quadrilaterals, the above paragraph is pointing to the use of letters as shorthand the area and perimeters, side lengths and where defined, heights. The use of these letters permits a shorthand description of how to compute perimeters and some areas from side lengths and heights. So student may see a shorthand formula or algebraic description of how some numbers and quantities, here lengths and heights, can be used to compute other numbers and quantities, here areas and perimeters.
The use of letters as abbreviations for lengths and areas in polygons and circles provides an easier introduction to algebraic ways of writing and reasoning than the context-free phrase. Let x, q and r be numbers. The novice may react in an offended manner to this phrase and say give m the numbers. Yet less offense will be taken, if we say Let x, q and r be the lengths of three line segments or Let s be the number of units in the area of that circle. The geometric or physical significance of the letters turns them into placeholder or pronouns for numbers and quantities easily visualized. Again, it is easier for students to accept the height of a rectangle and to say it is h units, than it is for them to say let h be a number. With the use of letters to denote quantities or numbers, expression involving those letters become meaningful. They describe calculations that could be done. By using letters to denote lengths or non-negative numbers, the commutative law for multiplication represents the notion that two different ways to compute the area of a rectangle should provide the same result, the distributive law and the foil method represent two different ways to calculate the areas of a rectangle as a whole or as the union of subrectangles. The commutative law for addition represents the ideas that the order in which two line segments are placed or measured does not affect the overall length. The distributive law can also be associated with the notion that a change of units (change of currency) should not affect a sum. Geometric significance here provides a scaffolding for the introduction of algebra with positive or non-negative quantities. By algebra in the first instance, we mean the role of shorthand notation in denoting numbers and quantities, and beyond that in describing calculations and the equality of calculations. The simplest context for introducing algebra appears before or apart from the use of negative numbers and lengths and areas are non-negative. The site area Solving Linear Equations with Stick Diagrams may be used to introduce and re-enforce the skills and concepts in class. In junior high school mathematics, the question of what is a variable is answered by considering our ability to describe or talk about numbers and quantities without doing any arithmetic nor measurement, and without using letters. We can say a number or quantity is unknown or not, changing or not, constant or not, varying in one way but not another. A number or quantity that does not change, is constant in one sense may be called a constant. A number or quantity that varies on one way (over time, over distance or between examples) may be called a variable number or quantity or in brief a variable. Letters in mathematics may denote numbers and quantities, preferably named numbers and quantities. We should say a letter is a variable (respectively constant) when the number or quantity it represents or stands is a variable (respectively constant) in one way or another. |
MEQ: The algebraic aspect should not, however, be cause for shifting the
focus from the aims of this objective, which are to consolidate the concepts of
perimeter and area, to enable students to make a connection between the
characteristics of polygons and their measures, to use units of measure
properly, and to apply their knowledge to polygons that can be broken down into
component polygons.
| W: Additive Nature of Area.
Develop and verify the ability to compute area of a region by dividing it into submersions for which area calculations follows from formulas known to the students. Then summing the areas of the subregions yields the total area. The question of how much does it cost to paint walls in a home, or cover floors with carpets or linoleum. All the foregoing gives the addition method for calculation area. Imagine a rectangular area containing a half-circle. The additive property implies the region of the rectangle outside the circle has an area given by subtraction. The additive property of area can be used as indicated above to develop algebraic thinking skills. |
MEQ: Activities in which students have to apply the properties of polygons in calculating perimeter or area, compare polygons with the same perimeter or area and estimate the perimeter or area of a figure are recommended or encouraged. By studying the measures of polygons, students learn how to analyze them accurately.
Hopefully the latter is a call for students to calculate in a repeatable and reproducible manner.
MEQ: Intermediate Objectives 3.5
- Know when to calculate the area and when to calculate the
perimeter.
W: Think of cost of painting a floor area versus the cost of painting the walls of a room (doors ignored). Does that work?
- Calculate the perimeter of a triangle or a quadrilateral, given sufficient
data.
- Express the relationship between the perimeter and the sides of a triangle
or a quadrilateral, using shorthand notation for geometric quantities,
taking into account that different types of figures involve different
numbers and quantities
. - Calculate the length of one side of a triangle or a quadrilateral, given
the perimeter and sufficient data.
Note: Here formulas for perimeters are used in reverse.
- Convert a measure of length from one unit to another.
Note: If you take a length that equal to the sum of two or more sub-lengths, the change of units is given by a multiplication, and it can be applied in two different ways, namely to the total length or to the sublengths. The equality of two different ways to do the conversion leads to the distributive law. There-in lies the principle that the description and addition of lengths should be independent of the choice of units (Can someone give a better wording).
Note: The a geometric or physical assumption or concepts that results should be independent of the choice of unit length (or unit direction and length) give a context to imply (or reflect) the distributive laws for real numbers, vectors and complex numbers.
- Calculate the area of a triangle or a trapezoid, given sufficient
data.
Note: This objective also ties in with one of the steps intended to prepare students for the study of algebra.)
- Describe the relationship between the area and certain dimensions of a
triangle or a trapezoid using shorthand notation for geometric quantities.
- Convert a measure of surface area from one unit to another.
Note: Here is an opportunity to show that what is a fraction according to one choice of unit may be whole multiple of a unit with another choice. One can also illustrate here the product of fractions - see why four fifths of two thirds is four times two fifteenths.
- Calculate the perimeter or area of a polygon by transforming it or
breaking it down into triangles or trapezoids. :( This
objective also ties in with one of the steps intended to prepare students
for the study of algebra.\)
Note: The logic-algebra area of this site gives an algebraic explanation of the Pythagorean theorem and based on this type of computation and the foil method)
MEQ: Objective 4 Page 41,
work with statistical data
MEQ: Mathematics 568-116 completes the introduction to descriptive statistics by teaching students about the construction of tables and graphs.
MEQ: The goal here is to teach students how to work with data so that they are less dependent on the media's interpretation of tables and graphs. By organizing the collected data into tables or graphs in such a way as to illustrate the different aspects that they want to highlight, students will be better able to interpret the statistical representations they encounter in books, magazines and newspapers.
MEQ: Statistics can help students to better understand various situations or
events, and to develop the ability to think critically.
W: NotesCritical thinking skills begin with ability to apply multi-step methods, carefully one step at a time and one step after another, with repeatable and reproducible results. In other words, talking about developing critical thinking skills in the interpretation and presentation of data is a bit absurd when students can not do arithmetic well and precisely. Critical thinking in collecting data can be explored by pointing out how to bias or influence data (how to favor one reply over another, how the sampling technique may influence answers). So data collection can be misleading. Critical thinking in interpreting graphs can be explore by pointing or discovering in the construction of graphs how the choice of scale and how location on the vertical axis (not always zero) of the horizontal axis may visually amplify or reduce variability. So graphs can be misleading. |
MEQ: Statistics is a subject that allows for ... use of simulations, studies and surveys,. activities that encourage both the active participation of the students and teamwork. They may also provide an opportunity to carry out long-term projects.
W: suggestions for data collection activities would be welcome. Here are few. Number of television sets, siblings (brothers or sisters), number of people with computers, portable music players (current models), distribution and average age of owners of various items, and how to lie with statistics and graphical presentation of data. - topic suitable in all or part with critical thinking. See next topic
page 42, MEQ: Terminal Objective 4.1
Interpret tables and graphs
MEQ: In elementary school, students should learn how to use different tables and graphs relating to such subjects as temperature, marks, and favourite shows in order to obtain specific information (e.g., the highest, the lowest, the most common measure). Many activities allow students to practise estimating and calculating the arithmetic mean of a set of data.
Idea: For critical thinking, point out that different data sets with very different distributions can have the same mean. Taking the mean to replace a data set with a wide dispersion is akin to looking at the world with blinkers.
MEQ: Develop and verify the ability to draw conclusions from a table or graph. It is recommended that students be exposed to a wide variety of graphs. On the basis of their initial observation of a situation, students can identify a trend and draw conclusions. They learn that the way data are presented can have an impact on the overall analysis of a situation.
MEQ: Activities where students use of information presented in different forms (tables or graphs) and compare their interpretations with those of their classmates are recommended or encouraged.
page 43, MEQ: Intermediate Objectives 4.1
- To interpret the information in a table.
- To interpret the information in various graphs.
page 44, MEQ: Terminal Objective 4.2:
present information about a situation by means of a table or graph
MEQ: Elementary school helps students develop the ability to classify information. Students are given many opportunities to present statistical information relating to a variety of subject areas. Students learn to classify objects, geometric figures and various results in an appropriate way (increasing order, decreasing order, alphabetical order). These activities provide adequate preparation for constructing tables.
MEQ: Develop, practice and verify the ability to gather, compile and organize statistics, following the rules for each type of representation. The type of graph students are asked to construct will depend on the data studied. Students should use the order, scale or legend they need to highlight a given detail.
MEQ: Activities in which students learn to use statistical representations to communicate information or to compile data with a view to discovering a relationship or drawing a general conclusion are encouraged.
page 45, MEQ: Intermediate Objectives.
- Tabulate data.
- present data in a horizontal or vertical bar graph.
- Present data in a broken-line graph.
- Present data in a circle graph.
- Highlight a detail in a table or graph.
W: The circle graphs may be connected to fractions and percentages.
W: The MEQ says the following about math 116 in the Math 216 objectives.
MEQ: In Secondary I, the students saw that a percentage was one way of writing a rational number. They carried out transformations involving percentages and expressed percentages as fractions, most notably in statistics when drawing circle graphs.
W: So working with statistical data was seen as a means to reinforce and develop fraction sense and skills.
W: See and contrast the above with these lesson plans for secondary I. The latter are intended to support the old MEQ curriculums for 116 and further courses. The emphasis there on fraction sense and skills, and efficient mastery of arithmetic with whole numbers and fractions is needed for the old curriculum in mathematics 216 and for senior high school mathematics - course which still follow the old curriculum.