Course Description
This course enables students to develop an understanding of mathematical concepts related to algebra, analytic geometry, and measurement and geometry through investigation, the effective use of technology, and abstract reasoning. Students will investigate relationships, which they will then generalize as equations of lines, and will determine the connections between different representations of a linear relation. They will also explore relationships that emerge from the measurement of three-dimensional figures and two-dimensional shapes. Students will reason mathematically and communicate their thinking as they solve multi-step problems.
Overall Provincial Curriculum Expectations
See Ontario math curriculum for grades 9 and 10 for details.
See also this news article for
possible changes to the grade 9 math curriculum.
A: Number Sense and Algebra
- demonstrate an understanding of the exponent rules of multiplication and division, and apply them to simplify expressions;
- manipulate numerical and polynomial expressions, and solve first-degree equations.
B: Linear Relations
- apply data-management techniques to investigate relationships between two variables;
- demonstrate an understanding of the characteristics of a linear relation;
- connect various representations of a linear relation.
C: Analytic Geometry
- determine the relationship between the form of an equation and the shape of its graph with respect to linearity and non-linearity;
- determine, through investigation, the properties of the slope and y-intercept of a linear relation;
- solve problems involving linear relations.
D: Measurement and Geometry
- determine, through investigation, the optimal values of various measurements;
- solve problems involving the measurements of two-dimensional shapes and the surface areas and volumes of three-dimensional figures;
- verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.
Specific Provincial Curriculum Expectations
A1: Operating with Exponents
- – substitute into and evaluate algebraic expressions
involving exponents (i.e., evaluate
expressions involving natural-number
exponents with rational-number bases
- – describe the relationship between the
algebraic and geometric representations of
a single-variable term up to degree three
[i.e., length, which is one dimensional, can
be represented by x; area, which is two
dimensional, can be represented by (x)(x)
or $x^2$; volume, which is three dimensional,
can be represented by (x)(x)(x), $(x^2)(x)$,
or $x^3$];
- – derive, through the investigation and examination
of patterns, the exponent rules for
multiplying and dividing monomials, and
apply these rules in expressions involving
one and two variables with positive
exponents;
- – extend the multiplication rule to derive and
understand the power of a power rule, and
apply it to simplify expressions involving
one and two variables with positive
exponents.
A2: Manipulating Expressions and Solving Equations
- – simplify numerical expressions involving
integers and rational numbers, with and
without the use of technology;
- – solve problems requiring the manipulation
of expressions arising from applications of
percent, ratio, rate, and proportion;
- – relate their understanding of inverse
operations to squaring and taking the
square root, and apply inverse operations
to simplify expressions and solve
equations;
- – add and subtract polynomials with up to
two variables [e.g., (2x – 5) + (3x + 1),
$(3x^2y + 2xy^2) + \\
(4x^2y – 6xy^2)$], using a
variety of tools (e.g., algebra tiles, computer
algebra systems, paper and pencil);
- – multiply a polynomial by a monomial
involving the same variable [e.g., 2x(x + 4),
$2x^2(3x^2 – 2x + 1)$], using a variety of tools
(e.g., algebra tiles, diagrams, computer
algebra systems, paper and pencil);
- – expand and simplify polynomial
expressions involving one variable
[e.g., 2x(4x + 1) – 3x(x + 2)], using a
variety of tools (e.g., algebra tiles,
computer algebra systems, paper and
pencil);
- – solve first-degree equations, including
equations with fractional coefficients,
using a variety of tools (e.g., computer
algebra systems, paper and pencil) and
strategies (e.g., the balance analogy,
algebraic strategies);
- – rearrange formulas involving variables in
the first degree, with and without substitution
(e.g., in analytic geometry, in measurement)
- – solve problems that can be modelled with
first-degree equations, and compare algebraic
methods to other solution methods
B1: Using Data Management to Investigate Relationships
- – interpret the meanings of points on scatter
plots or graphs that represent linear relations,
including scatter plots or graphs in
more than one quadrant [e.g., on a scatter
plot of height versus age, interpret the
point (13, 150) as representing a student
who is 13 years old and 150 cm tall; identify
points on the graph that represent students
who are taller and younger than this
student]
- – pose problems, identify variables, and
formulate hypotheses associated with relationships
between two variables
- – design and carry out an investigation or
experiment involving relationships
between two variables, including the
collection and organization of data, using
appropriate methods, equipment, and/or
technology (e.g., surveying; using measuring
tools, scientific probes, the Internet)
and techniques (e.g.,making tables, drawing
graphs)
- – describe trends and relationships observed
in data, make inferences from data, compare
the inferences with hypotheses about
the data, and explain any differences
between the inferences and the hypotheses
(e.g., describe the trend observed in the
data. Does a relationship seem to exist? Of
what sort? Is the outcome consistent with
your hypothesis? Identify and explain any
outlying pieces of data. Suggest a formula
that relates the variables. How might you
vary this experiment to examine other
relationships?)
B2: Understanding Characteristics of Linear Relations
- – construct tables of values, graphs, and
equations, using a variety of tools (e.g.,
graphing calculators, spreadsheets, graphing
software, paper and pencil), to represent
linear relations derived from descriptions
of realistic situations
- – construct tables of values, scatter plots, and
lines or curves of best fit as appropriate,
using a variety of tools (e.g., spreadsheets,
graphing software, graphing calculators,
paper and pencil), for linearly related and
non-linearly related data collected from a
variety of sources (e.g., experiments, electronic
secondary sources, patterning with
concrete materials)
- – identify, through investigation, some properties
of linear relations (i.e., numerically,
the first difference is a constant, which represents
a constant rate of change; graphically,
a straight line represents the relation),
and apply these properties to determine
whether a relation is linear or non-linear;
- – compare the properties of direct variation
and partial variation in applications, and
identify the initial value (e.g., for a relation
described in words, or represented as a
graph or an equation)
- – determine the equation of a line of best fit
for a scatter plot, using an informal process
(e.g., using a movable line in dynamic
statistical software; using a process of trial
and error on a graphing calculator; deterand error on a graphing calculator; determining
the equation of the line joining
two carefully chosen points on the scatter
plot)
B3: Connecting Various Representations of Linear Relations
- – determine values of a linear relation by
using a table of values, by using the equation
of the relation, and by interpolating
or extrapolating from the graph of the
relation
- – describe a situation that would explain the
events illustrated by a given graph of a
relationship between two variables
- – determine other representations of a linear
relation, given one representation (e.g.,
given a numeric model, determine a graphical
model and an algebraic model; given a
graph, determine some points on the graph
and determine an algebraic model);
- – describe the effects on a linear graph and
make the corresponding changes to the
linear equation when the conditions of
the situation they represent are varied
(e.g., given a partial variation graph and
an equation representing the cost of producing
a yearbook, describe how the
graph changes if the cost per book is
altered, describe how the graph changes if
the fixed costs are altered, and make the
corresponding changes to the equation)
C1: Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph
- – determine, through investigation, the characteristics
that distinguish the equation of a
straight line from the equations of nonlinear
relations (e.g., use a graphing calculator
or graphing software to graph a variety
of linear and non-linear relations from
their equations; classify the relations
according to the shapes of their graphs;
connect an equation of degree one to a
linear relation);
- – identify, through investigation, the equation
of a line in any of the forms
y = mx + b, Ax + By + C = 0,
x = a, y = b;
- – express the equation of a line in the form
y = mx + b, given the form
Ax + By + C = 0.
C2: Investigating the Properties of Slope
- – determine, through investigation, various
formulas for the slope of a line segment or a line, and use the formulas to determine the slope of a line segment or a line;
- – identify, through investigation with technology,
the geometric significance of m
and b in the equation y = mx + b;
- – determine, through investigation, connections
among the representations of a constant
rate of change of a linear relation
- – identify, through investigation, properties
of the slopes of lines and line segments
(e.g., direction, positive or negative rate of
change, steepness, parallelism, perpendicularity),
using graphing technology to facilitate
investigations, where appropriate
C3: Using the Properties of Linear Relations to Solve Problems
- – graph lines by hand, using a variety
of techniques (e.g., graph $y =\frac{2}{3} x – 4$
using the y-intercept and slope; graph
2x + 3y = 6 using the x- and
y-intercepts);
- – determine the equation of a line from
information about the line (e.g., the slope
and y-intercept; the slope and a point; two points)
- – describe the meaning of the slope and
y-intercept for a linear relation arising
from a realistic situation
- – identify and explain any restrictions on
the variables in a linear relation arising
from a realistic situation (e.g., in the relation
C = 50 + 25n,C is the cost of holding
a party in a hall and n is the number
of guests; n is restricted to whole numbers
of 100 or less, because of the size of the
hall, and C is consequently restricted to
$50 to $2550);
- – determine graphically the point of intersection
of two linear relations, and interpret
the intersection point in the context
of an application
D1: Investigating the Optimal Values of Measurements
- – determine the maximum area of a rectangle
with a given perimeter by constructing
a variety of rectangles, using a variety of
tools (e.g., geoboards, graph paper, toothpicks,
a pre-made dynamic geometry
sketch), and by examining various values
of the area as the side lengths change and
the perimeter remains constant;
- – determine the minimum perimeter of a
rectangle with a given area by constructing
a variety of rectangles, using a variety of
tools (e.g., geoboards, graph paper, a premade
dynamic geometry sketch), and by
examining various values of the side lengths
and the perimeter as the area stays constant;
- – identify, through investigation with a variety
of tools (e.g. concrete materials, computer
software), the effect of varying the
dimensions on the surface area [or volume]
of square-based prisms and cylinders, given
a fixed volume [or surface area];
- – explain the significance of optimal area,
surface area, or volume in various applications
(e.g., the minimum amount of packaging
material; the relationship between
surface area and heat loss);
- – pose and solve problems involving maximization
and minimization of measurements
of geometric shapes and figures (e.g., determine the dimensions of the
rectangular field with the maximum area
that can be enclosed by a fixed amount of
fencing, if the fencing is required on only
three sides)
D2: Solving Problems Involving Perimeter, Area, Surface Area, and Volume
- – relate the geometric representation of the
Pythagorean theorem and the algebraic
representation $a^2 + b^2 = c^2$;
- – solve problems using the Pythagorean
theorem, as required in applications (e.g.,
calculate the height of a cone, given the
radius and the slant height, in order to
determine the volume of the cone);
- – solve problems involving the areas and
perimeters of composite two-dimensional
shapes (i.e., combinations of rectangles,
triangles, parallelograms, trapezoids, and
circles)
- – develop, through investigation (e.g., using
concrete materials), the formulas for the
volume of a pyramid, a cone, and a sphere
- – determine, through investigation, the
relationship for calculating the surface area
of a pyramid (e.g., use the net of a squarebased
pyramid to determine that the
surface area is the area of the square base
plus the areas of the four congruent
triangles);
- – solve problems involving the surface areas
and volumes of prisms, pyramids, cylinders,
cones, and spheres, including composite
figures
D3: Investigating and Applying Geometric Relationships
- – determine, through investigation using a
variety of tools (e.g., dynamic geometry
software, concrete materials), and describe
the properties and relationships of the
interior and exterior angles of triangles,
quadrilaterals, and other polygons, and apply the results to problems involving the
angles of polygons
- – determine, through investigation using a
variety of tools (e.g., dynamic geometry
software, paper folding), and describe
some properties of polygons (e.g., the figure
that results from joining the midpoints
of the sides of a quadrilateral is a parallelogram;
the diagonals of a rectangle bisect
each other; the line segment joining the
midpoints of two sides of a triangle is half
the length of the third side), and apply the
results in problem solving (e.g., given the
width of the base of an A-frame tree
house, determine the length of a horizontal
support beam that is attached half way
up the sloping sides);
- – pose questions about geometric relationships,
investigate them, and present their
findings, using a variety of mathematical
forms (e.g., written explanations, diagrams,
dynamic sketches, formulas, tables)
- – illustrate a statement about a geometric
property by demonstrating the statement
with multiple examples, or deny the statement
on the basis of a counter-example,
with or without the use of dynamic
geometry software