Mathematics 586-116
The immediate objectives below come from the Quebec government course objectives online in this pdf file They provide a skills and concept checklist for the pre-reform 116 course. For the sake of clarity and precision, compare and contrast with the new content objectives in the current education reform. There should be an overlap.
My comments and reflections on the old mathematics 116 are preceded by (W:) in order to distinguish between them from the MEQ objectives
| Intermediate Objectives:
Objectives that specify the scope of a terminal objective, intermediate objectives might also be described as "reference. objectives." They are not intended as a series of steps to be completed one after the other. Such a process would give a very fragmented picture of teaching and learning. Rather, intermediate objectives are:
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There are no intermediate objectives for the first objective 1 as support for it is embedded as part of the other objectives. The following quote says clearly how. Here reading the intermediate objectives, I have clarified some and given my view on others.
MEQ Objective 1:
Mathematics 116 takes a two-step approach to preparing students for the study of algebra: the first step involves facilitating the transition to algebra by ensuring that the necessary arithmetic skills have been mastered; the second step enables students to grasp certain differences between arithmetic and algebra. In step one the primary emphasis is on expanding the students' understanding of the equal sign. To this end, certain aspects of Terminal Objectives 2.1 and 2.2 not only give students the opportunity to apply the rules for writing the order of operations, but also 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." Equalities can be created through concrete application of the commutative, associative and distributive properties, which are used extensively in algebra. Step two comes into play most particularly in the study of numbers (Terminal Objective 2.1). Students can learn how to use algebraic language by working with all the properties and rules that can be easily generalized. For example, after discovering the rule for converting one number into another, students learn to express the rule by moving gradually from descriptive language to symbolic language, which takes into account the basic rules of algebraic language. The goal is to have students realize that if one relationship can be generalized by a rule, this rule can be applied to other numbers. Terminal Objective 3.5 provides opportunities for carrying out this algebraic activity in a geometric context. The objectives that come under this general objective are integrated with the terminal (w: other) objectives on numbers and measurement. They are marked with an asterisk. |
Objective 2:
Check and develop number and operation sense
MEQ Develop number sense by working with the natural numbers, subsequently enriching ... experience through exposure to
fractions and decimals.Well-developed number sense makes it possible to anticipate the results of numerical operations, determine the degree of accuracy required in a given situation, and judge the appropriateness of a calculation done mechanically by means of an algorithm or a calculator.
Intermediate Objective 2.1:
operations with natural numbers - Understand and apply the properties of the operations and of equality.
- To associate the power of a given natural
number with its exponential notation and vice versa.
W: That may mean recognize a4 is shorthand for the four factor product a*a*a*a in which all the factors have the same value a where a is a number or an expression.
- Write a natural number as a product of prime factors.
W: The first intermediate objective is used here in this prime factorization or decomposition of natural numbers. Add 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
- Calculate the value of a chain of operations on natural numbers in
accordance with the order of operations. The chain may include one or two
sets of parentheses. (w: Evaluate simple to more complicated
expressions in a repeatable and reproducible manner.
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* To express a given natural number, using one or more
operations and following the rules of writing
associated with the order of operations. - * To generate equivalent numerical
expressions, each containing one or more arithmetic operations, using, for
example, the properties of the operations and of the equivalence relation.
W: 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.
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To translate a chain of operations on natural numbers into a verbal problem.
W: Does mean translate the verbal expression "double the number and add three" into an algebraic, symbolic or literal expression of the form 2n+3
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* To explain in their own words the rule relating a number and its rank in a sequence.
W: Translate 3n+5 into a verbal expression
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* To express in symbolic language the rule relating a number and its rank in a sequence.
W: for instance the expressions 2n, 2n+1 and 3n generate sequences of even, odd and multiples of 3
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* To use the rule relating a number and its rank in a sequence to find either the number occupying a certain rank in the sequence or the rank of a number belonging to the sequence.
w: Not sure what this means.
W: for instance the expressions 2n, 2n+1 and 3n generate sequences of even, odd and multiples of 3. Answer the questions what is the fifth odd number. Is 13 the 4th, 5th or 6th odd number?
Intermediate Objective 2.2
Operations on integers
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: Evaluate simple to more complicated expressions in a repeatable and reproducible manner.
Intermediate Objectives 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.
- Read a rational number expressed in decimal notation.
W: Rational numbers whose 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
MEQ: Students who have attained Terminal Objective 2.3 of this program will be able to use the positional system of numeration to read and write a number and to compare numbers. All 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. Helping students make this connection is the primary goal of having them carry out transformations. The ability to convert a number from one notation to another and atrue understanding of the numeration system will enable students to use the symbols =, », <, and > to compare numbers expressed in different forms.Intermediate Objective 2.4
Solve problems involving fractions or rational numbers,
- 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 appropriate. 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: Evaluate simple and then
more complicated expressions involving fractions in a repeatable and
reproducible, and thus verifiable manner.)
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.
Intermediate Objective 3.1
Create figures by means of isometric transformations
- 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.
W: 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.
Some Quotes from the document:
connections that will enable them to solve problems ...... for triangles and quadrilaterals. Students learn to analyze 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, measure the angles, segments and surfaces, explore the transformations of these figures, and make the
Despite the emphasis on triangles and quadrilaterals, the teacher should ensure that students continue to work with the other geometric shapes they have studied.
Students ... will be able 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. By carrying out the steps involved in constructing an image, the students learn about the fundamental concepts of parallelism, perpendicularity and angles. ... They also learn that numerous polygons can be created from the same figure by carrying out a single transformation or a series of similar transformations on
it.Intermediate Objectives 3.2
Master straight lines or angles,
State Rules & Properties to Justify Conclusions,
- 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.
W: Check that students know how to measure lengths with a ruler. Some rulers measure from their endpoints. Others do not. Not knowing about this difference between rulers can lead students to false results.
- 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.
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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. |
Appendix
MEQ: Statements Associated with Themes Covered in Mathematics 568-116 (Geometry)
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page 34, MEQ: Terminal Objective 3.3
Solve problems involving triangles
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 (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 further course (or lesson) on geometry.
- Construct the altitudes, medians, and perpendicular bisectors of a triangle. This ability comes from knowledge of how to draw a perpendicular from a point to a line, how to bisect angles and lines segments
- Express the relationships between the various types of triangles. This may mean classification: a triangle may be both scalene and right angled. A triangle may isoceles 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. The information may be written or drawn.
- To justify an assertion in solving a problem involving triangles. That is, I think, students should say when they are using a property of a triangle or type of triangle when arriving at conclusions.
MEQ, Prior Knowledge: 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.
W: Verify or review prior knowledge (?) before proceeding.
W: 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)
W: The site area on Euclidean Geometry written for older students explains these constructions
MEQ: Intermediate Objectives 3.4
work with convex quadrilaterals
- 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.
Intermediate Objectives 3.5
calculate perimeter or the area of certain polygons,
- 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. There-in the lies the start of the backwards or indirect use of formulas.
- 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)
Objective 4
work with statistical data
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.
MEQ: Statistics can help students ... to develop the ability to think critically.
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:. 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
Intermediate Objectives 4.1
Interpret tables and graphs
- To interpret the information in a table.
- To interpret the information in various graphs.
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: 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 practice estimating and calculating the arithmetic mean of a set of data.
Intermediate Objective 4.2:
present information about a situation by means of a table or graph
- 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 MEQ says the following about math 116 in the Math 216 objectives. Italics are mine.
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: Thus working with statistical data is seen as a means to reinforce and develop fraction sense and skills.
Quebec English Mathematics Education
A farce is a farce is a farce.
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