# Curriculum

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```					Sustaining
Quality
Curriculum
Learning Tools
Learning Tools

The revised curriculum supports
the use of technology and
manipulatives as tools for teaching
and learning mathematics.
Learning Tools
“Students need to develop the ability
to select the appropriate
electronic tools, manipulatives
and computational strategies to
perform particular mathematical
ideas and to solve problems”
Draft Introduction Curriculum Document Grades 9 – 10
Mathematics, 2005
Learning Tools
“Information and communication
technology (ICT) provides a range of
tools that can significantly extend and
enrich teachers’ instructional
strategies and support students’
learning in mathematics.”
Draft Introduction Curriculum Document Grades 9 – 10
Mathematics, 2005
Learning Tools: Ministry support documents

“Technology, when used properly,
enhances and extends
mathematical thinking by allowing
students to focus on decision
making, reflection, reasoning, and
problem solving.”
TIPS Research Technology: Summaries
National Council of Teachers of Mathematics. (2000). Principles and standards
for school mathematics. Reston, VA: Author.
Learning Tools: Ministry support documents

In an effective mathematics program,
students learn in the presence of
technology. Technology should influence
the mathematics content taught and how
it is taught. Powerful assistive and
enabling computer and handheld
technologies should be used seamlessly
in teaching, learning, and assessment.”
(p. 47)
Learning Tools: Ministry support

Dynamic Geometry Software
(note: also licensed for students !)

Dynamic Statistical Software:

and others
Learning Tools
come in many forms,
we will look briefly at:

•TinkerPlots
• Computer Algebra Systems
• Geosolids – manipulatives
Learning Tools
Algebra tiles (grade 7 – 10)
The sum of the exterior angle of a
c

a

drag me

b

Geometer’s
Learning Tools

Area of a parallelogram (grade 6)
Learning Tools – Dynamic Geometry Software
1997 CURRICULUM             SPRING 2005 DRAFT
• understand the           • develop, through
relationship between       investigation, the relationship
the area of a              for the area of triangle and
parallelogram and the      the area of a parallelogram,
area of a triangle,        generalizing to develop the
between the area of a      formulae, using a variety of
triangle and the area of   tools (e.g., concrete
a rectangle and the        materials, dynamic
area of triangle and the   geometry software, grid
area of a                  paper, paper folding);
parallelogram;
Learning Tools – Dynamic Statistics Software

• TinkerPlots allows teachers and students to
focus on reasoning about and with data as
opposed to merely computing statistical
measures or constructing models
• As such, Tinkerplots allows students to
become data analysts, able to dynamically
interact and reason with rich, relevant data
sets

• pose a relevant question, design an experiment or
survey and collect data to compare two attributes
or characteristics using a variety of data
management tools and strategies
• determine, through investigation, the appropriate
measure of central tendency (i.e., mean, median or
mode) needed to compare sets of data (e.g., in
hockey, compare heights or weights of players on
defence with forwards)
Are NHL defensemen the heaviest
players?
This data set is
from the NHL
website from the
spring of 2004. It
includes the top 100
players by points.
Each dot represents
one player. The blue
triangles mark the
mean and the red
marks mark the
median for each
group.
Computer Algebra Systems
something new….
Referred to as one of a
“variety of tools”
in the revised Grades 9 and 10
Ontario Mathematics Curriculum
Mini Demo on
Computer Algebra Systems

Curriculum Connections
MFM1P Specific Expectation
(revised)
• solve first-degree equations with non-fractional
coefficients, by applying algebraic reasoning using a
variety of tools (e.g., the balance analogy, algebra tiles,
computer algebra systems, paper and pencil) (Sample
problem: Solve 2x + 7 = 6x - 1 using the balance analogy)
MFM2P Specific Expectation
(revised)
• expand and simplify second-degree polynomial
expressions in one variable that consist of the product of
two binomials [e.g., (2x + 3)(x - 4)] or the square a
binomial [e.g., (x + 3)2], aided by a variety of tools (e.g.,
algebra tiles, computer algebra systems, paper and pencil)
and strategies (e.g. patterning)
MPM2D Specific Expectation
(revised)
• factor polynomial expressions involving common factors,
trinomials, and differences of squares [e.g., 2x2 + 4x, 2x -
2y + ax - ay, x2 - x - 6, 2a2 + 11a + 5, 4x2 - 25], using a
variety of tools (e.g., concrete materials, computer algebra
systems, paper and pencil) and strategies (e.g., patterning)
Learning Tools
“Students should be encouraged to
select and use concrete learning tools
to make models of mathematical
ideas…making their own models is a
powerful means of building
understanding and explaining their
thinking to others.”
Draft Introduction Curriculum Document Grades 9 – 10
Mathematics, 2005
“Manipulatives should be used
effectively and selectively as a means
of facilitating the ongoing transition
from the concrete (physical and
visual) stage in student learning, to
the more abstract knowledge and
deep understanding.”

Kamii, Lewis, & Kirkland (2001)
Learning Tools
“Even at the secondary level
manipulatives are necessary tools for
supporting the effective learning of
mathematics.”

Draft Introduction Curriculum Document Grades 9 – 10
Mathematics, 2005
Learning Tools
Manipulatives:
To assist with understanding,
connecting, remembering….
To develop concepts, to relate concepts
and procedures….
And they are fun…
Learning Tools

To assist with understanding,
connecting, remembering….
To develop concepts, to relate concepts
and procedures….
And they are memorable, engaging…
Learning Tools
Measurement and Geometry, revised expectation

• develop, through investigation
(e.g., using concrete materials), the
formula for the volume of a
pyramid, a cone, and a sphere…
Learning Tools
Measurement and Geometry DEMO

Develop the formula for the volume of a sphere
Learning Tools
Develop the formula for the volume of a sphere

1. Use relational solids and pouring material to note that
the volume of a cone is the same as the volume of a
hemisphere (with corresponding dimensions)
Using “math language” Volume (cone) = ½ Volume (sphere)
Therefore            2(Volume (cone)) = Volume (sphere)
2. We already know the formula for the volume of a
cone (due to previous investigation with relational
solids)    Volume (cone) = Volume (cylinder) /3
Develop the formula for the volume of a sphere,

We already know the formula for the volume of a cylinder
(due to previous investigations in grade 8)
Volume (cone) = Volume (cylinder) /3
Volume (cylinder) = (Area Base) (height)
Volume (cone) = (Area Base) (height)/3
2(Volume (cone)) = Volume (sphere)
2(Area Base) (height) /3= Volume (sphere)
2(r2)(h)/3= Volume (sphere)
BUT 2r = h

2(r2)(2r)/3= Volume (sphere)
4(r3)/3= Volume (sphere)
Learning Tools

Some learning tools are referred to
in the expectations, as “e.g.’s”
Learning Tools in the Expectations
Revised MPM 1D               Revised MPM 2D
polynomials with up to two   investigation (e.g., using
variables (e.g.,…) using a   dynamic geometry
variety of tools (e.g.,      software, by paper
algebra tiles, computer      folding) some
algebra systems, paper       characteristics, and
and pencil)                  properties of geometric
figures (e.g., medians in a
triangle, similar figures
constructed on the sides
of a right triangle)
Learning Tools in the Expectations
Revised MFM 1P/2P                      Revised MFM 2P
-multiply a polynomial by a          -determine, through
monomial in the same variable        investigation, the relationship
… (e.g.,…) using a variety of        for finding the surface area of a
tools (e.g., algebra tiles,          pyramid (e.g., use the net of a
drawings, computer algebra           square based pyramid to
systems, paper and pencil)           determine that the surface
2P:                                  area is the area of the square
-factor trinomials of the form x^2   base, plus the areas of the four
+ bx + c (e.g.,…) using a variety    congruent triangles)
of tools (e.g., algebra tiles,       -determine graphically the
computer algebra systems,            point of intersection of two
paper and pencil) and strategies     linear relations (i.e., using
(patterning)                         graph paper, using
technology)…
Learning Tools Activity
Each table will be assigned a
amongst yourselves.
Look through the course/strand of the
document and find reference to
learning tools
hint: look for “using a variety of tools”
Learning Tools Activity
Make a list of the tools referred to.
Summarize on chart paper
(course/strand/tools referred to)
for this strand that are not included.
Learning Tools Activity
Learning Tools Connection
How are learning tools used in this
problem to assist with the teaching
and/or learning of mathematics?
Are these other tools which could be
incorporated? How?
PARKING LOT
- E + NING
Home Extension
Be prepared to discuss the following
questions tomorrow:
Home Extension
1. What aspect of the revised
curriculum do you find most
positive?
2. What aspect of the revised
curriculum are of concern to
you, and how might these
concerns be overcome?
Assessment of the day

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