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Big Idea Geometry: Spatial sense
and geometric relationships are a means to solve problems and make
sense of a variety of phenomena.
Big Idea Measurement: Measurement is a tool to
quantify a variety of phenomena. |
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4.2 A.
Geometric Properties |
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Descriptive Statement: This includes identifying,
describing and classifying standard geometric object, describing and
comparing properties of geometric objects, making conjectures
concerning them, and using reasoning and proof to verify or refute
conjectures and theorems. Also included here are such concepts as
symmetry, congruence, and similarity. |
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Essential Questions |
Enduring Understandings |
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- How can spatial relationships be described by
careful use of geometric language?
- How do geometric relationships help in solving problems and/or
make sense of phenomena? |
- Geometric properties can be used to construct
geometric figures. (4.5D1; 4.5D2; 4.5E3)
- Geometric relationships provide a means to make sense of a variety
of phenomena. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Use geometric models to represent real-world situations and
objects and to solve problems using those models (e.g., use Pythagorean Theorem
to decide whether an object can fit through a doorway). |
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2.
Draw perspective
views of 3D objects on isometric dot paper, given 2D representations (e.g., nets
or projective views). |
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3.
Apply the
properties of geometric shapes.
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Parallel lines –
transversal, alternate interior angles, corresponding angles
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Triangles
a.
Conditions for congruence
b.
Segment joining midpoints of two
sides is parallel to and half the length of the third side
c.
Triangle Inequality
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Minimal conditions
for a shape to be a special quadrilateral
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Circles –
arcs, central and inscribed angles, chords, tangents
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Self-similarity
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4.
Use reasoning and some form of proof to verify or refute
conjectures and theorems.
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Verification or refutation of proposed proofs
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Simple proofs involving congruent triangles
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Counterexamples to incorrect conjectures
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4.2
B. Transforming Shapes |
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Descriptive Statement: This includes identifying,
describing and classifying standard geometric object, describing and
comparing properties of geometric objects, making conjectures
concerning them, and using reasoning and proof to verify or refute
conjectures and theorems. Also included here are such concepts as
symmetry, congruence, and similarity. |
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Essential Questions |
Enduring Understandings |
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- What situations can be analyzed using
transformations and symmetries? (4.5E1; 4.5E2; 4.5E3) |
- Shape and area can be conserved during
mathematical transformations.. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Determine,
describe, and draw the effect of a transformation, or a sequence of
transformations, on a geometric or algebraic object, and, conversely, determine
whether and how one object can be transformed to another by a transformation or
a sequence of transformations. |
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2.
Recognize three-dimensional figures obtained through
transformations of two-dimensional figures (e.g., cone as rotating an isosceles
triangle about an altitude), using software as an aid to visualization. |
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3.
Determine
whether two or more given shapes can be used to generate a tessellation. |
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4.
Generate and analyze iterative geometric patterns.
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Fractals (e.g., Sierpinski’s Triangle)
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Patterns in areas and perimeters of self-similar figures
·
Outcome of extending iterative process indefinitely
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4.2 C. Coordinate Geometry |
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Descriptive Statement: Coordinate geometry provides an
important connection between geometry and algebra. It facilitates
the visualization of algebraic relationships, as well as an
analytical understanding of geometry. |
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Essential Questions |
Enduring Understandings |
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- How can geometric/algebraic relationships best
be represented and verified? (4.5C2; 4.5D2; 4.5E1; 4.5E2; 4.5F5) |
- Reasoning and/or proof can be used to verify or
refute conjectures or theorems in geometry (4.5D1; 4.5D3; 4.5D4;
4.5D5; 4.5F5)
- Coordinate geometry can be used to represent and verify
geometric/algebraic relationships. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Use coordinate
geometry to represent and verify properties of lines.
·
Distance between
two points
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Midpoint and slope
of a line segment
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Finding the
intersection of two lines
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Lines with the
same slope are parallel
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Lines that are
perpendicular have slopes whose product is –1 |
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2.
Show position and represent motion in the coordinate plane using
vectors.
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Addition and subtraction of vectors
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4.2 D. Units Of Measurement |
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Descriptive Statement: Measurement helps describe our
world using numbers. An understanding of how we attach numbers to
real-world phenomena, familiarity with common measurement units
(e.g., inches, liters, and miles per hour), and a practical
knowledge of measurement tools and techniques are critical for
students' understanding of the world around them. |
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Essential Questions |
Enduring Understandings |
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- How can measurements be used to solve problems?
(4.5A6) |
- Everyday objects have a variety of attributes,
each of which can be measured in many ways.
-What we measure affects how we measure it.
(4.5A4; 4.5A6)
- Measurements can be used to describe, compare, and make sense of
phenomena. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Understand and use the concept of significant digits.
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2.
Choose appropriate
tools and techniques to achieve the specified degree of precision and error
needed in a situation.
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Degree of accuracy
of a given measurement tool
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Finding the
interval in which a computed measure (e.g., area or volume) lies, given the
degree of precision of linear measurements
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4.2 E. Measuring Geometric Objects |
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Descriptive Statement: This area focuses on applying the
knowledge and understandings of units of measurement in order to
actually perform measurement. While students will eventually apply
formulas, it is important they develop and apply strategies that
derive from their understanding of the attributes. In addition to
measuring objects directly, students apply indirect measurement
skills, using, for example, similar triangles and trigonometry. |
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Essential Questions |
Enduring Understandings |
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- How can measurements be used to solve problems?
(4.5A6) |
- Everyday objects have a variety of attributes, each of which can
be measured in many ways.
- What we measure affects how we measure it. (4.5A4; 4.5A6)**
- Measurements can be used to describe, compare, and make sense of
phenomena. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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By the end of Grade 12: |
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1.
Use techniques of indirect measurement to represent and solve
problems.
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Similar triangles
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Pythagorean theorem
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Right triangle trigonometry (sine, cosine, tangent)
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2.
Use a variety of strategies to
determine perimeter and area of plane figures and surface area and volume of 3D
figures.
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Approximation of
area using grids of different sizes
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Finding which
shape has minimal (or maximal) area, perimeter, volume, or surface area under
given conditions using graphing calculators, dynamic geometric software, and/or
spreadsheets
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Estimation of
area, perimeter, volume, and surface area |
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