THE ONTARIO CURRICULUM: PROPOSED REVISIONS
OCTOBER 2005
Mathematics: Grade 12
Geometry and Algebra
Side-by-Side
The comparison charts comparing the original to the proposed revisions are intended as a guide to assist you with the review. In general:
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Grey cells indicate the expectation has been moved from this location
An expectation in the left column, and nothing to the right of it (or the word deleted) means the expectation has been deleted
An expectation in the right column, and nothing to the left of it, indicates the expectation is new
When expectations are written side by side, the left column shows the original expectation and the right column shows the proposed
revision for that expectation
In some cases, notes in bold italics have been added for clarification
Please Note: Original expectations that are not associated with a revised expectation have not necessarily been removed or deleted.
Some overall expectations have been incorporated into specific expectations. Some specific expectations may have been combined, or moved to
another section of the program.
Original
Revised
Course Code: MGA 4U Name: Geometry Revised: Geometry and Algebra
and Discrete Mathematics
Grade: 12
Program Area: Mathematics
Revised: The Geometry and Algebra of
Strand: New strand added.
Conics
Specific Expectations—Section: New section
added.
New: Overall Expectations
demonstrate an understanding of the properties of a
circle, represent circles in a variety of ways and
solve problems related to circles;
demonstrate an understanding of the properties of a
conics, represent conics in a variety of ways and
solve problems related to conics;
Specific Expectations—Section: New section
added.
Revised: Investigating the Circle
solve problems, through investigation using a variety
of tools (e.g., dynamic geometry software, paper
folding and algebra) involving lines described as a
locus of points (e.g., locus of points equidistant from
two points, locus of points equidistant from two
intersecting lines)
demonstrate an understanding of the circle as a locus
of points with rotational and reflective symmetries
that fix the centre of the circle;
demonstrate and apply an understanding of terms
related to the circle (e.g., circle, chord, tangent,
diameter, radius, inscribed angle, cyclic
quadrilateral);
demonstrate an understanding of radian measure by
converting angle measure between radians and
degrees;
solve numerical and geometric problems involving
circumference, area, length of arc and area of sector
of a circle; Sample problem: Show that for two
sectors of a general circle, the ratio of the
corresponding angles, the corresponding arc lengths
and the corresponding areas are the same.
solve numerical and geometric problems involving
the angle, chord, tangent and secant properties;
use translations, through investigation, to represent
circles algebraically (i.e., (x - a)² + (y - b)² = r²) and
use coordinate geometry to prove angle, chord,
Page 2 of 10
Draft of Proposed Senior Mathematics Curriculum
tangent and secant properties associated with a
circle; Sample problem: Given the equation of a
circle, compare the lengths of the two tangent
segments that are created from a given exterior
point.
Specific Expectations—Section: New section
added.
New: Investigating the Conics
represent (e.g., a diagram created by hand, a sketch
created using dynamic geometry software) a
described locus of points and determine, through
investigation, the properties of the locus (e.g., the
locus of points equidistant from two fixed points is
the right bisector of the line segment joining the two
fixed points);
determine the graph and equation to represent a
described conic locus using a variety of tools;
Sample problem: Use technology, paper folding and
algebra to explore the locus of points equal in
distance from a point and a circle.
identify, through investigation using the locus
definitions, the key symmetries of a conic section
(e.g., the mirrors which fix the foci or interchange
the foci, are symmetries of the graph, the mirror of
the parabola fixes the focus and reflects the directory
onto itself;
identify the standard forms for the equations of
conic sections in standard position
sketch the conic sections given the standard form of
the equation of a conic in standard position
solve problems involving the intersection of lines
and conics;
demonstrate, through investigation, an
understanding of the conic sections, including the
circle, and intersecting lines, as plane sections of a
three-dimensional cone. Sample problem: Explain
why an artist draws the top of a can as an ellipse.
pose and solve problems drawn from a variety of
applications of conics (e.g., the planets move in
elliptical orbits with the sun at one of the foci) and
justify the solutions; Sample Problem:
Strand: Geometry
Revised: The Geometry and Algebra of
Vectors
Section: Overall Expectations
Unchanged: Overall Expectations
perform operations with geometric and Cartesian perform operations with geometric and Cartesian
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Draft of Proposed Senior Mathematics Curriculum
vectors;
determine intersections of lines and planes in
three-space.
vectors, and solve related problems;
determine intersections of lines and planes in threespace, and solve related problems.
Specific Expectations—Section: Operating
with Geometric and Cartesian Vectors
(Expectation moved here; formerly GE1.01)
represent vectors as directed line segments;
Revised: Developing Vector Methods in TwoSpace
define and represent a vector, geometrically as a
directed line segment and algebraically using
vertical and horizontal components and give
examples of situations that can be represented by
vectors (e.g., displacement, velocity and forces)
determine the components of a geometric vector and
the projection of a geometric vector onto the
coordinate axes and on other vectors;
determine the components of a geometric vector and
the projection of a geometric vector onto the
coordinate axes and on other vectors;
add and subtract vectors represented in component
form and as directed line segments, using a variety
of tools (e.g., paper and pencil, graph paper,
technology) and interpret the results.
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: The Geometry and Algebra of Vectors,
Section: Developing Vector Methods in Two-Space
perform scalar multiplication on vectors represented
in component form and as directed line segments,
using a variety of tools (e.g., paper and pencil, graph
paper, technology) and interpret the results.
determine the dot product of two vectors represented
in component form and as directed line segments,
using a variety of tools (e.g., paper and pencil, graph
paper, technology) and interpret the results.
represent and apply lines in two-space using the
vector, parametric, symmetric, and scalar equations;
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: The Geometry and Algebra of Vectors,
Section: Developing Vector Methods in Two-Space
solve problems involving applications that can be
modelled using vectors in two-space (e.g., velocity
and force);
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: The Geometry and Algebra of Vectors,
Section: Developing Vector Methods in Two-Space
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: The Geometry and Algebra of Vectors,
Section: Applying Vector Methods in Three-Space
(Expectation moved here; formerly GE1.03)
determine the components of a geometric vector
and the projection of a geometric vector;
(Expectation moved here; formerly GE1.03)
determine the components of a geometric vector
and the projection of a geometric vector;
represent vectors as directed line segments;
perform the operations of addition, subtraction,
and scalar multiplication on geometric vectors;
(Expectation moved here; formerly GE1.05)
determine and interpret the dot product and cross
product of geometric vectors;
determine the components of a geometric vector
and the projection of a geometric vector;
model and solve problems involving velocity and
force;
determine and interpret the dot product and cross
product of geometric vectors;
represent Cartesian vectors in two-space and in
three-space as ordered pairs or ordered triples;
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Draft of Proposed Senior Mathematics Curriculum
perform the operations of addition, subtraction,
scalar multiplication, dot product, and cross
product on Cartesian vectors.
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: The Geometry and Algebra of Vectors,
Section: Applying Vector Methods in Three-Space
Specific Expectations—Section: Determining Revised: Applying Vector Methods in ThreeIntersections of Lines and Planes in ThreeSpace
Space
(Expectation moved here; formerly GE1.06)
represent vectors in three-space as ordered triples;
represent Cartesian vectors in two-space and in
three-space as ordered pairs or ordered triples;
determine the vector and parametric equations of represent and apply lines in three-space using the
lines in two-space and the vector, parametric, and vector, parametric, and symmetric equations;
symmetric equations of lines in three-space;
determine the intersections of lines in three-space; determine the intersections of lines in three-space
and interpret the results;
determine the cross product of two vectors
represented in component form and as directed line
segments, using a variety of tools (e.g., paper and
pencil, graph paper, technology) and interpret the
results.
determine the vector, parametric, and scalar
develop and apply the vector, parametric, and scalar
equations of planes;
equations of planes;
determine the intersection of a line and a plane in determine the intersection of a plane with a line and
three-space;
interpret the result
(Expectation moved here; formerly GE2.07)
determine the intersection of two or three planes by
determine the intersection of two or three planes setting up and solving a system of linear equations in
by setting up and solving a system of linear
three unknowns, algebraically;
equations in three unknowns;
solve systems of linear equations involving up to Unchanged: solve systems of linear equations
three unknowns, using row reduction of matrices, involving up to three unknowns, using row reduction
with and without the aid of technology;
of matrices, with and without the aid of technology;
(Expectation moved here; formerly GE1.07)
perform the operations of addition, subtraction,
perform the operations of addition, subtraction, scalar multiplication, dot product, and cross product
scalar multiplication, dot product, and cross
on Cartesian vectors and interpret the results.
product on Cartesian vectors.
interpret row reduction of matrices as the creation represent and interpret a linear system using
of a new linear system equivalent to the original; matrices and the process of row reduction; (e.g.,
showing the system 2x + 4y + 2z = 8, z = 3, x + 2y +
2z = 7 may be represented by [1 2 1 4 , 0 0 1 3 , 0 0
0 0];
determine the intersection of two or three planes Moved to Grade 12 Geometry and Algebra - MGA
by setting up and solving a system of linear
4U, Strand: The Geometry and Algebra of Vectors,
equations in three unknowns;
Section: Applying Vector Methods in Three-Space
interpret a system of two linear equations in two interpret a system of two linear equations in two
unknowns and a system of three linear equations unknowns and a system of three linear equations in
in three unknowns geometrically, and relate the three unknowns geometrically, and identify the
Page 5 of 10
Draft of Proposed Senior Mathematics Curriculum
geometrical properties to the type of solution set system as consistent, inconsistent, or dependent.
the system of equations possesses;
solve problems involving the intersections of lines solve problems involving the intersections of lines
and planes, and present the solutions with clarity and planes, and present the solutions with clarity and
and justification.
justification. (e.g. determine the distance between
origin and intersection of a specific line and plane.)
Specific Expectations—Section: New section
added.
New: Investigation of Loci and Conics
Strand: Proof and Problem Solving
Revised: Reasoning and Proving
Moved to this location.
Section: Overall Expectations
(Expectation moved here; formerly PSV.01)
prove properties of plane figures by deductive,
algebraic, and vector methods;
Specific Expectations—Section: Proving
Properties of Plane Figures by Deductive,
Algebraic, and Vector Methods
(Expectation moved here; formerly PS1.06)
demonstrate an understanding of the relationship
between formal proof and the illustration of
properties that is carried out by using dynamic
geometry software.
Unchanged: Overall Expectations
demonstrate a flexibility in applying a variety of
reasoning strategies and recognize the relative merits
of each
prove properties of plane figures by deductive,
algebraic, and vector methods, and prove
mathematical statements by induction;
Revised: Reasoning Mathematically
demonstrate an understanding of inductive reasoning
by making conjectures from a specific set of
examples or observations and justify the reasoning;
Sample Problem: Given a sketch generated using
dynamic geometry software, what generalizations
can be made?
generate counterexamples that can be used to argue
that a conjecture is false;
(Expectation moved here; formerly PS2.03)
make and test conjectures, with and without
use technology effectively in making and testing technology;
conjectures;
solve problems that involve satisfying a given set of
conditions and verify that the solution meets the
conditions; Sample problem: Construct all
quadrilaterals with one pair of opposite angles equal
and one line of symmetry;
compare and describe inductive reasoning and
deductive reasoning
(Expectation moved here; formerly PS2.02)
generate multiple solutions to the same problem and
generate multiple solutions to the same problem; compare the reasoning used (e.g., comparing an
algebraic solution with a geometric solution);
Sample problem: Using vector methods, analytic
geometry and similar triangles, investigate the figure
formed by joining the midpoints of four sides of a
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Draft of Proposed Senior Mathematics Curriculum
quadrilateral.
Specific Expectations—Section: Using a
Variety of Strategies to Solve Problems
Deleted.
Specific Expectations—Section: Using
Mathematical Induction to Prove Results
(Expectation moved here; formerly PS1.01)
demonstrate an understanding of the principles of
deductive proof (e.g., the role of axioms; the use
of "if . . . then" statements; the use of "if and only
if" statements and the necessity to prove them in
both directions; the fact that the converse of a
proposition differs from the proposition) and of
the relationship of deductive proof to inductive
reasoning;
(Expectation moved here; formerly PS1.02)
prove some properties of plane figures (e.g.,
circles, parallel lines, congruent triangles, right
triangles), using deduction;
Revised: Developing Strategies for Proof
Moved to this location.
describe the principles of deductive proof (e.g., the
selection and role of axioms; the use of "if . . . then"
statements; the use of "if and only if" statements and
the necessity to prove them in both directions; the
fact that the converse of a proposition differs from
the proposition) Sample problem: Given a statement
its supporting argument, state the assumptions used
and determine by investigation or by proof, whether
the converse is true.
prove, using deduction, some properties of plane
figures (i.e., circles, lines, and polygons), ;
prove deductively, angle, chord, tangent and secant
properties associated with the circle;
(Expectation moved here; formerly PS1.03)
justify, using analytic geometry, vector methods,
prove some properties of plane figures (e.g., the transformations and other know properties, some
midpoints of the sides of a quadrilateral are the properties of plane figures (e.g., the midpoints of the
vertices of a parallelogram; the line segment
sides of a quadrilateral are the vertices of a
joining the midpoints of two sides of a triangle is parallelogram; the line segment joining the
parallel to the third side) algebraically, using
midpoints of two sides of a triangle is parallel to the
analytic geometry;
third side) Sample problem: Investigate which lines
are parallel when you join the midpoints of the edges
of a tetrahedron in space.
(Expectation moved here; formerly PS1.05)
prove, using indirect methods, some properties of
prove some properties of plane figures, using
plane figures,
indirect methods;
(Expectation moved here; formerly PS1.04)
prove, using vector methods, some properties of
prove some properties of plane figures, using
plane figures, ;
vector methods;
(Expectation moved here; formerly DM2.01)
demonstrate an ability to prove mathematical
demonstrate an understanding of the principle of statements by applying the principle of mathematical
mathematical induction;
induction and using appropriate conventions and
symbols (e.g., sigma notation); Sample Problem:
Use mathematical induction to prove that 1 + 3 + 5 +
7 + ....... + (2n - 1) = n²;
(Expectation moved here; formerly PS3.03)
research and present the context and reasoning
demonstrate significant learning and the effective associated with famous mathematical problems;
use of skills in tasks such as solving challenging Sample Problem: Compare and contrast the
problems, researching problems, applying
reasoning used in two different proofs of the
Page 7 of 10
Draft of Proposed Senior Mathematics Curriculum
mathematics, creating proofs, using technology
effectively, and presenting course topics or
extensions of course topics.
Pythagorean Theorem
read and analyze given proofs; Sample problem:
Read and describe the reasoning used to prove root 2
is not a rational number.
Specific Expectations—Section: Completing
Significant Problem-Solving Tasks
Independently
Deleted.
Strand: Proof and Problem Solving
Revised: Reasoning and Proving
Moved from this location.
Section: Overall Expectations
prove properties of plane figures by deductive,
algebraic, and vector methods;
solve problems, using a variety of strategies;
complete significant problem-solving tasks
independently.
Specific Expectations—Section: Proving
Properties of Plane Figures by Deductive,
Algebraic, and Vector Methods
demonstrate an understanding of the principles of
deductive proof (e.g., the role of axioms; the use
of "if . . . then" statements; the use of "if and only
if" statements and the necessity to prove them in
both directions; the fact that the converse of a
proposition differs from the proposition) and of
the relationship of deductive proof to inductive
reasoning;
prove some properties of plane figures (e.g.,
circles, parallel lines, congruent triangles, right
triangles), using deduction;
prove some properties of plane figures (e.g., the
midpoints of the sides of a quadrilateral are the
vertices of a parallelogram; the line segment
joining the midpoints of two sides of a triangle is
parallel to the third side) algebraically, using
analytic geometry;
prove some properties of plane figures, using
vector methods;
prove some properties of plane figures, using
indirect methods;
Unchanged: Overall Expectations
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: Reasoning and Proving, Section: Overall
Expectations, Moved w Strand
Deleted.
Deleted.
Revised: Reasoning Mathematically
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: Reasoning and Proving, Section:
Developing Strategies for Proof, Moved w Strand
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: Reasoning and Proving, Section:
Developing Strategies for Proof, Moved w Strand
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: Reasoning and Proving, Section:
Developing Strategies for Proof, Moved w Strand
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: Reasoning and Proving, Section:
Developing Strategies for Proof
Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: Reasoning and Proving, Section:
Developing Strategies for Proof, Moved w Strand
Page 8 of 10
Draft of Proposed Senior Mathematics Curriculum
demonstrate an understanding of the relationship Moved to Grade 12 Geometry and Algebra - MGA
between formal proof and the illustration of
4U, Strand: Reasoning and Proving, Section:
properties that is carried out by using dynamic
Reasoning Mathematically, Moved w Strand
geometry software.
Specific Expectations—Section: Using a
Deleted.
Variety of Strategies to Solve Problems
solve problems by effectively combining a variety Deleted.
of problem-solving strategies (e.g., brainstorming,
considering cases, choosing
algebraic/geometric/vector or direct/indirect
approaches, working backwards, visualizing by
using concrete materials or diagrams or software,
iterating, varying parameters, creating a model,
introducing a coordinate system);
generate multiple solutions to the same problem; Moved to Grade 12 Geometry and Algebra - MGA
4U, Strand: Reasoning and Proving, Section:
Reasoning Mathematically, Moved w Strand
use technology effectively in making and testing Moved to Grade 12 Geometry and Algebra - MGA
conjectures;
4U, Strand: Reasoning and Proving, Section:
Reasoning Mathematically, Moved w Strand
solve complex problems and present the solutions Deleted.
with clarity and justification.
Specific Expectations—Section: Completing
Deleted.
Significant Problem-Solving Tasks
Independently
solve problems of significance, working
Deleted.
independently, as individuals and in small groups;
solve problems requiring effort over extended
Deleted.
periods of time;
demonstrate significant learning and the effective Moved to Grade 12 Geometry and Algebra - MGA
use of skills in tasks such as solving challenging 4U, Strand: Reasoning and Proving, Section:
problems, researching problems, applying
Developing Strategies for Proof, Moved w Strand
mathematics, creating proofs, using technology
effectively, and presenting course topics or
extensions of course topics.
Strand: Discrete Mathematics
Deleted.
Section: Overall Expectations
solve problems, using counting techniques;
prove results, using mathematical induction.
Specific Expectations—Section: Using
Counting Techniques
solve problems, using the additive and
multiplicative counting principles;
Page 9 of 10
Draft of Proposed Senior Mathematics Curriculum
express the answers to permutation and
combination problems, using standard
combinatorial symbols [e.g. (n choose r), P(n, r)];
evaluate expressions involving factorial notation,
using appropriate methods (e.g., evaluate
mentally, by hand, by using a calculator);
solve problems involving permutations and
combinations, including problems that require the
consideration of cases;
explain solutions to counting problems with
clarity and precision;
describe the connections between Pascal’s
triangle, values of (n choose r), and values for the
binomial coefficients;
solve problems, using the binomial theorem to
determine terms in the expansion of a binomial.
Specific Expectations—Section: Using
Mathematical Induction to Prove Results
Moved from this location.
demonstrate an understanding of the principle of Moved to Grade 12 Geometry and Algebra - MGA
mathematical induction;
4U, Strand: Reasoning and Proving, Section:
Developing Strategies for Proof
use sigma notation to represent a series or the sum Deleted.
of a series;
prove the formulas for the sums of series, using Deleted.
mathematical induction;
prove the binomial theorem, using mathematical Deleted.
induction;
prove relationships between the coefficients in
Deleted.
Pascal’s triangle, by mathematical induction and
directly.
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Draft of Proposed Senior Mathematics Curriculum