Download Space and Shape (Geometry)

Space and Shape
(Geometry)
General Curriculum Outcomes E:
Students will demonstrate spatial sense and apply
geometric concepts, properties, and relationships.
Revised 2011
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
KSCO: By the end of grade 9,
students will have achieved the
outcomes for primary–grade 6 and
will also be expected to
E1 Students should already be familiar with the concepts and properties
of translations, reflections, rotations, and dilatations. In this course
students will explore these transformations on a coordinate plane,
developing generalizations called mapping rules. Note, students should
discover that for mapping rules for dilatations, the centre of dilatation
must be at (0, 0). Also note that rotations should be restricted to 180°,
centre (0,0).
iii) develop and analyze the
properties of transformations
and use them to identify
relationships involving
geometric figures
SCO: By the end of grade 9,
students will be expected to
E1 interpret, represent, and
apply mapping notation for
transformations on the coordinate plane
Students should develop the mapping rules by responding to leading
questions, examining patterns, and conjecturing and/or making
conclusions. For example, to develop the mapping rule for a translation:
• Give students a triangle ABC on a grid and ask them to record the
coordinates for the three vertices.
−− Tell them to translate vertex A to the right five and up three, and
name the image point A' (read A prime) and record its coordinates.
−− Ask them to complete the next two lines by filling in the blanks:
ºº B
ºº __
__' or (coordinates for B)
C' or ( ___ , ___ )
( ___ , ___ )
(coordinates for C')
−− Ask them to write in words how they could determine the
coordinates for B' and C' without looking at the graph. [Through
discussion, lead them to wording something like “I added five to
the x-coordinate, and three to the y-coordinate.”]
−− Ask them to change that statement to math symbols. They should
write (x + 5, y + 3). [Show them that this can be represented using
the mapping rule (x, y) (x + 5, y + 3)]. Ask them to enunciate
and/or write in complete sentences what it is they have developed
and how it might be applied. Perhaps a leading phrase could
be given to them to help them get started. Have them read and
discuss their statements and edit for clarification.
• Give them the coordinates for a particular geometric figure. Ask them
to draw a specific translation using words to describe the translation,
then have them develop the mapping rule. Ask them to use the
mapping rule to get the image coordinates of another given shape.
• Students should be asked to describe the translation that has or will
occur given a mapping rule, and/or given a diagram.
Similar activities should take place that allow students to discover the
mapping rules for other transformations. For example, ask students to
plot the point A(1, 5) then construct its image after a rotation of 180°,
centre C(0, 0), counter clockwise. Students should join the point A to
the origin C(0, 0), then extend the line making a 180° angle. Make
the distance AC the same as CA', then record the coordinates for A'.
Students should notice that the (1, 5) has been mapped to (–1, –5),
and by doing a few more examples like this, discover the mapping rule
(x, y) (–x, –y).
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
• Mathematics 9: Focus on
Understanding Geometry
Supplement, pp. 6–15
E1.1 Construct RST on a coordinate plane with vertices R(–4, 4),
S(–6, 2), and T(–3, 2). Trace and cut out a copy of RST and label it
R'S'T'.
a) Translate
• rulers
• protractors
R'S'T' four spaces to the left.
i) What are the new coordinates of this triangle?
ii) Compare with the vertices of the original triangle. Write the
mapping rule for the translation.
b) Translate R'S'T' 5 spaces down. Explain why
(x, y) (x – 4, y – 5) would describe the relation between the final
position of the triangle and the original position.
c) Without graphing, determine the coordinates of the image of
RST using this mapping rule: (x, y) (x + 7, y – 3).
E1.2 On a coordinate plane, construct ABC with vertices A(2, 3),
B(0, 0), and C(2, 0). Trace and cut out a copy of ABC but label it
A'B'C’.
a) Explore these mappings and identify which transformation they
represent:
i) (x, y)
(x, –y)
ii) (x, y)
(0.5x, 0.5y)
iii) (x, y)
(x – 3, y + 2)
iv) and as an extension: (x, y)
(–y, x)
b) If possible, write the coordinates for the image triangle in each case.
If not possible, explain why not.
c) Find the area of the image and the pre-image. What do you notice?
And as an extension:
d) What happens to
applied to it?
ABC when the mapping (x, y)
(–2x, -2y) is
e) Write the coordinates of the image. What assumption about the
centre did you make?
Paper/Pencil
E1.3 Ask students to graph y = 2x + 1.
Ask students to
a) Draw the image, using the mapping rule (x, y)
(x, –y).
b) Use the graph to find the equation of the image.
c) Explain how the equation of the image relates to the equation of the
pre-image?
d) Start with the equation y = –3x – 1, and using the same mapping
rule, predict the equation of the image. Check by graphing.
Atlantic Canada Mathematics Curriculum
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Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
Students should develop and interpret mapping rules for
• translations
[(x, y) (x + right, y + up)]. The word “right” could be “left” if
the translation moved left. The word “up” could be “down” if the
translation moves down. If the translation was left two and down
three, then the mapping rule would be (x, y) (x–2, y–3)
• reflections in the y- and x- axes
[(x, y) (–x, y) and (x, y) (x, –y) respectively]. Be careful with these
as students often confuse the two.
• dilatations, centre (0, 0) using integer and fractional scale factors
[(x, y) (ax, ay) where ‘a’ is the scale factor]
• 180° rotation, centre (0, 0)
[(x, y) (–x, –y)]
and might extend their experiences to include developing mapping rules
for
• reflections in the lines y = x and y = –x
[(x, y) (y, x) and (x, y) (–y, –x)]
• rotations of 90°, clockwise and counterclockwise
[(x, y) (y, –x) and (x, y) (-y, x)]
Students should be given information about mapping of points,
segments, or shapes and asked to interpret the mapping. That is, they
should be able to tell what they know and about a diagram based on
a given mapping, and determine the mapping rule given the diagram.
These would include any translation, reflections in the x- and y- axes,
a 180° rotation about (0, 0), and a dilatation with centre (0, 0) using
integer and fractional scale factors.
For example:
• A transformation on a quadrilateral takes place based on the mapping
(x, y) (x, –y). Describe the shape, orientation and position of the
image of the quadrilateral, and what transformation has occurred.
Suppose that one point on the quadrilateral is (4, –5). What are the
coordinates of the image points? [This describes a reflection in the
x-axis. The orientation is reversed, the image is still a quadrilateral but
its position is reflected across the x-axis from its pre-image. The image
of (4, –5) would be the point with coordinates (4, 5).]
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
E1.4 Ask students to examine the quadrilateral ABCD in the graph
below.
Ask students to
a) State the coordinates of the 4 vertices
b) Perform the following transformations, one after the other, to each
successive image.
i) (x, y)
(–x, y)
ii) (x, y)
(0.5x, 0.5y)
iii) (x, y)
(–x, –y)
c) State the coordinates of the final image of each point A, B, C,
and D.
d) Show how you could find those image coordinates using only the
mapping rules.
e) Discuss whether the image is congruent to, or similar to, the
pre-image, or neither.
Extension
E1.5
a) Marla said that the mapping rule for the diagram on the left below
would be (x, y) (–x + 2, y). Explain how you know that the image
is correct.
b) Determine the mapping rule for the figure on the right.
Atlantic Canada Mathematics Curriculum
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Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
KSCO: By the end of grade 9,
students will have achieved the
outcomes for primary–grade 6 and
will also be expected to
E2 Students will be expected to use properties of transformations
to convince someone that a particular transformation is that
transformation. Students need to understand that when examining a
diagram (below, left) that looks like it might be a 180° rotation about
the centre point M, doesn’t mean that it necessarily is.
v) draw inferences, deduce
properties, and make
logical deductions in
synthetic (Euclidean) and
transformational geometric
situations
SCO: By the end of grade 9,
students will be expected to
E2 make and apply informal
deductions about the
minimum sufficient
conditions to guarantee a
translation, reflection, and a
180° rotation
Also, diagrams can be drawn so that it may seem like there is no obvious
transformation (above, right), but the properties of a rotation and the
given information about the diagram (MF = MC, BF = CE, AB = DE,
E) may lead to the conclusion that indeed one triangle is the
and B
image of the other after a 180° rotation. If students can use properties of
a rotation to explain why one triangle is the image of the other triangle,
then the transformation (in this case, a 180° rotation about centre
M) is as stated. If not enough properties can be stated to guarantee a
particular transformation then this transformation does not exist. For
example, in the diagram on the left above, there is no information given
about side lengths or parallelism. Thus, we don’t even know that A maps
to the point C, so there is not enough information to guarantee any
transformation.
Students must explore to determine the minimum sufficient conditions
to guarantee that a reflection, or rotation of 180°, or a translation will
occur.
The following are minimum sufficient conditions for ...
Translation:
• The line segments that join corresponding points are in the same
direction (parallel) and are congruent.
Rotation 180°:
• The line segments that join corresponding points intersect each other
at a common midpoint.
Reflection:
• The line segments that join corresponding points have a common
perpendicular bisector.
These are not the only minimum sufficient conditions that students
might need. For example, consider a pair of triangles with a common
vertex: if a pair of corresponding sides are congruent and parallel (or
collinear), and the midpoint of a line segment joining corresponding
points is known, then this guarantees that the midpoint is the centre of
a 180° rotation mapping one triangle onto the other.
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
• Mathematics 9: Focus on
Understanding Geometry
Supplement, pp 16–23
E2.1 Ask student to work in pairs.
a) On a blank piece of paper have each pair of students draw any
triangle and label it ABC. They have only a compass and straight
edge.
b) Ask them to use centre C and 180° rotation properties to produce a
triangle that they are convinced is congruent to the original triangle,
ABC. Have them record their constructions using step 1, step 2,
and so on until they are sure they have a congruent image. Tell them
that the student pair with the fewest steps is the winner.
• Bull’s Eye
• protractor
• ruler
c) Ask each pair to summarize their constructions and share with
another pair to find the fewest steps to guarantee congruence. Ask
different groups to share with the whole class (look for groups that
have done this differently).
d) Ask students, working independently, to draw another ABC on a
new piece of paper, but this time include a point M near C on BC.
Then have them follow the following directions given by the teacher
(teacher lead).
i) use M as the centre of rotation.
ii) locate the image of C when rotated 180°, call the point F.
iii) At F make a ray parallel to CA so that it looks like a ray that is
the image of CA in a 180° rotation about the centre M.
iv) Locate the point D on the ray beginning at F so that D is the
image of A.
v) Ask students to share with another, then with the whole group,
how they found the point D. [Some may have measured (with
ruler or compass) from C to A and made DF = CA, other may
have drawn a line from A through M to intersect the ray drawn
from F — it is important to share all these methods].
vi) Working in pairs, ask students to think about what they would
construct next to ensure that they end up with a triangle DEF
that is the image of ABC after a rotation of 180° about centre
M. Ask them to do this, and record their steps.
vii) Ask pairs of students to compare their results from (vi) with
another pair, then have groups report to the whole class about
how they completed the exercise. [Some may find E making
BM = ME, others may make a ray from D parallel to AB to
intersect AC at E — it is important to consider all possibilities].
Atlantic Canada Mathematics Curriculum
9-81
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
To help students clarify that what looks like a transformation may not
necessarily be one, have them examine diagrams that look like successful
transformations with enough information given that guarantees
that they are transformations, and by examining those that look like
transformations but do not have sufficient information to complete
the transformation. For example, a diagram is given in which two
line segments joining corresponding points of a triangular shape and
its transformed image are equal in length and in the same direction
(parallel), then students should be able to say that the transformation
is either a translation or a rotation of 180°. If a third such line is also
congruent and in the same direction (parallel) as the other two segments
then that guarantees a translation.
In the diagram on the left below, if we know that CF = AD and
CF AD then we know that AC maps onto DF in a translation from
C to F, but we still need to know that CF = BE before we can say that
the point B maps onto E. In the diagram on the right, if we know that
AB = CD and AB CD, then the diagram suggests that this might be a
rotation rather than a translation, but we need to know that A, M, and
C are co-linear and AM = MC before we can say that there is a rotation
about centre M that maps AB onto CD
Students can also determine the minimum sufficient conditions to
guarantee a particular transformation by playing games, like “10
questions.” In a group of four, one student holds the drawing of the
image and pre-image of a particular transformation. The other three,
in order, ask questions about which properties are true in the drawing.
For example the first question might be “does the orientation hold?” If
the answer is “yes” then students can eliminate reflection. If the answer
is “no,” that eliminates translations and rotations, but is not sufficient
information to guarantee reflection (other reflection properties, such as
congruency, must be true as well). From the answers to the questions
students should record information and try to determine (using as few
properties as possible) which transformation the diagram shows.
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
e) Ask students to work independently, examine the diagrams below,
and make the most precise statement they can about the image of A
in each case, after a rotation of 180° about the point M:
Paper/Pencil
E2.2 Ask students to examine the following diagrams and make the
most precise statement they can about the image for the point A for a
reflection RM:
a)
b)
c)
d)
E2.3 Ask students to examine the diagrams given and make the most
precise statement they can about the image of the points A and B in
each case and to state whether any transformation is guaranteed.
a)
b)
Given: A-B-D-E
c)
Given: A-B-D-E
CF=AD=BE
d)
Given: A-B-D-E
CA FD
Journal
Given: A-B-D-E
AC DF
DC EF
E2.4 The teacher told Frank that if he was to do a reflection in the line
RC, A would map onto B. Frank didn’t think this was correct because
BM is not given equal in length to MA. Ask students what they would
say to Frank to help him understand.
Atlantic Canada Mathematics Curriculum
9-83
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
KSCO: By the end of grade 9,
students will have achieved the
outcomes for primary–grade 6 and
will also be expected to
E3 Just as students have learned some of the minimum sufficient
conditions for making two triangles congruent, they need to determine
the minimum sufficient conditions to guarantee that triangles are
similar.
ii) compare and classify geometric
figures, understand and apply
geometric properties and
relationships, and represent
geometric figures via coordinates
Students should begin by reviewing what is meant by two triangles
being similar. That is, they should discover that two triangles are similar
when they have the same shape. This occurs when the corresponding
angles are congruent, and the corresponding sides are proportional.
Students have learned that if corresponding sides are proportional, then
the ratios of corresponding sides will be equivalent.
iv) represent and solve abstract and
real-world problems in terms of
2- and 3-D geometric models
v) draw inferences, deduce
properties, and make
logical deductions in
synthetic (Euclidean) and
transformational geometric
situations
SCO: By the end of grade 9,
students will be expected to
E3 make and apply informal
deductions about the
minimum sufficient
conditions to guarantee the
similarity of two triangles
They may remember from previous study of dilatations that the image
was similar to the pre-image because their ratios of their corresponding
sides were proportional, and their corresponding angles were congruent.
OD’ = 0.3OD
OA’ = 2.5OA
OE’ = 0.3OE
OB’ = 2.5OB
OF’ = 0.3OF
OC’ = 2.5OC
They should understand from previous study that congruent
means exactly the same size, whereas similar means different size
or proportional size, including the ratio 1:1, or congruence. Thus
congruent triangles are similar with a ratio of corresponding side lengths
equal to one.
Next students, should investigate whether in two triangles they need
to have all pairs of corresponding angles congruent or all pairs of
corresponding sides proportional, to make them similar, or are there
minimum sufficient conditions under which similarity is guaranteed.
To do this students can work in small groups with various instruments
for measurement (rulers, protractors, Bull’s Eye compasses, and
manipulatives such as the sets of Geo-Strips. For example see Activity
E3.2 on the Worthwhile Tasks page.
Students will discover that two triangles are be similar when
• two pairs of corresponding angles are congruent (AA~) [having two
pairs means the third pair must be congruent since the angle sum in a
triangle is 180°], OR
• one pair of corresponding angles congruent, and the ratios of the
two pairs of corresponding sides that include those angles must be
proportional (SAS~), OR
• three pairs of corresponding sides proportional (SSS~)
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Paper/Pencil
• Mathematics 9: Focus on
Understanding Geometry
Supplement, pp. 24–31
E3.1 Ask students if the information given below is enough to
determine that the two triangles are similar and to justify their decisions:
a) Given: AD = CB, and DC = BA
• Bull’s Eye
b) In diagram below, AD = 3 cm, BC = 5 cm
CD = 4 cm, AB = 6 cm
• ruler
c) Given: B
• Geo-Strips
C
Performance
E3.2
a) Ask students to examine the diagram given below and to write a
correct proportion statement concerning the given side lengths.
b) Ask students if this is sufficient information to conclude that the
triangles are similar.
c) After the students agree that the answer to (b) is no, ask students
how they can tell by looking at this accurately drawn diagram that
the triangles cannot be similar.
d) Ask students to redraw the two triangles with the given side lengths,
E. Have them measure and record the lengths AC
but make B
and DF, examine the ratio DF , and now make a conjecture based
AC
on all of this information about the similarity of the two triangles.
e) Ask them to check their conjecture by measuring the other two
pairs of corresponding angles.
f ) From this information and using their conjectures, ask students to
make a conclusion about two triangles that have only three pairs of
corresponding sides that are proportional; and another about two
triangles that have two pairs of corresponding sides proportional
and the included angles congruent.
Atlantic Canada Mathematics Curriculum
9-85
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
Note: students should understand the relationships that exist between
corresponding sides of similar triangles. That is, for ABC and PQR,
ABC ~ PQR,
then AB = BC = AC
PQ QR PR
if
Also, since the two triangles shown are similar, the ratios of side lengths
within one triangle are equal to the ratios of the corresponding side
lengths within the other triangle. That is, students should be able to
conclude that AB PQ
=
or AB = PQ or AC = PR
PR
BC QR
BC QR
AC
When similarity between two triangles can be confirmed, ratios such as
these may be useful in solving problems.
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
E3.3 In groups of three, provide each student in the group with a set
of plastic (or at least stiff ) strips (these can be cut from stir sticks or
purchased commercially) as follows:
• Student A: 3 cm, 4 cm, 5 cm
• Student B: 6 cm, 8 cm, 10 cm
• Student C: 9 cm, 12 cm, 15 cm
a) Ask each student to form a triangle and measure each angle. Ask
them to compare angle measures.
b) Ask student to compare the lengths of each of the sides of the
triangles. Ask them to predict the lengths of the sides of another
triangle that will have the same angle measures, justify, and test their
prediction.
E3.4 The greater the supply of a product, the lower the price it will
bring, as shown in the graph.
In the graph, CP OB and PD AO .
a) Ask students to name any pairs of similar triangles on the graph.
Ask them to explain how they know they are similar.
b) Ask students to complete each of the following proportions by
adding an equivalent ratio.
i)
AC = ?
CO
ii) AC = ?
PD
iii) AO = ?
OB
v) Ask students to explain why AC = OD .
CO DB
iv) OD = ?
OB
Journal
E3.5 Ask students to decide if either of the following statements is true
and to explain why:
i) If two triangles are congruent then they are also similar.
ii) If two triangles are similar then they are also congruent.
E3.6 Marla discovered that the three pairs of corresponding sides of
two triangles had the ratio 1:1. She asked whether the triangles were
congruent or similar. Ask students to write an explanation to help
Marla.
Atlantic Canada Mathematics Curriculum
9-87
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
KSCO: By the end of grade 9,
students will have achieved the
outcomes for primary–grade 6 and
will also be expected to
E4 The properties that students will discover through guided
exploration using activities like those found on the Worthwhile Tasks
pages are:
i) construct and analyze 2- and
3-D models, using a variety of
materials and tools
iv) represent and solve abstract and
real-world problems in terms of
2- and 3-D geometric models
SCO: By the end of grade 9,
students will be expected to
E4 make and apply
generalizations about the
properties of Platonic Solids
• regular polyhedra are constructed using only one regular polygon for
its faces
• why there are only five regular polyhedra
• every regular polyhedron has a dual
• the order of rotational symmetry of a polyhedron is the total of the
order of rotational symmetry about all its axes
• the number of planes of symmetry that each regular polyhedron has
• all dihedral angles, within a regular polyhedron, have the same
measure
• each vertex of a Platonic Solid has vertex regularity
In grade 7 students learned how to construct, name, and describe the
Platonic Solids. In grade 7 a regular polyhedron was defined as having
all congruent regular polygonal faces, and vertex regularity. Students
should be familiar with and understand the term vertex regularity.
Student should begin by constructing the five regular polyhedra (the
Platonic Solids) using only regular polygons and building solids with
vertex regularity. Students begin with equilateral triangles, then squares,
then pentagons. They should understand (from their studies in grade
7, outcome E3, E4, E5, and E6) and be able to articulate why it is
impossible to make a regular polyhedra using regular hexagons as faces,
or polyhedra whose faces are regular polygons that have six or more than
six edges. In grade 9 students learn why there are only 5 platonic solids.
There are many characteristics that students should study, one of which
is symmetry, in particular rotational symmetry. A 3-D figure is turned
about an axis of rotation to determine its rotational symmetry. For
example, a straw or pipe cleaner stuck through a pair of opposite vertices
of the cube becomes the axis of rotation. As the viewer looks down the
axis of symmetry and rotates the cube on the axis, it takes on positions
exactly the same as the original position three times in one complete
rotation, thus being called a solid with rotational symmetry through the
vertices of order 3. Students should determine how many different axes
of rotational symmetry through the vertices the cube has. [Hint: how
many pairs of opposite vertices are there?]
There are other axes of rotational symmetry such as when the axis
connects the midpoints of two opposite edges, or the centre points of
two opposite faces. Students should explore only the tetrahedron and
cube and determine how many rotational axes each has and the total
order of rotational symmetry.
9-88
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
• Mathematics 9: Focus on
Understanding Geometry
Supplement, pp. 58–65
E4.1 Give each student a sheet of paper with pictures of the five
Platonic solids (and have models available). Ask students to
a) test the properties that make these solids regular
b) examine the pictures or the five models, and list as many reasons as
possible why these are called regular polyhedra
c) explain why it is impossible for a regular polyhedron to have faces
that are regular hexagons
• Polydron pieces
• Polydron Framework pieces
• The Visual Geometry Project
presents “The Platonic Solids”
(DVD)
d) explain why it is impossible for a regular polyhedron to have faces
that have more than six sides
E4.2 Divide students into working groups. Give out a sheet of paper
that contains pictures of tessellations (regular, semi-regular, and some
non-regular) to each group. Also, give out a second piece of paper with
at least two large regular tesselations. Ask students to
a) mark the vertex configuration at each vertex on sheet number 1
b) determine whether each tessellation on sheet one is regular or semiregular, or neither, and to state why
c) identify, on sheet 2, all the reasons why these tessellations are regular
(Name each tessellation using words and numbers.)
d) find the centre of each polygon in the tessellations
e) use a pencil and straight edge to join all the centres of adjacent
polygons (The resulting tessellation is called a dual of the original
tessellation. Name the duals using words and numbers.)
f ) count the number of faces, edges, and vertices for the tessellating
polygon for each of the tessellations, and organize the information
in a table with headings: Name of polygon, Number of Faces,
Number of Edges, and Number of Vertices (Have students identify
the pattern that determines dual pairs.)
g) determine if the dual of a regular tessellation must be a regular
tessellation
h) determine if there are any self-duals
Atlantic Canada Mathematics Curriculum
9-89
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
A 3-D figure has reflective symmetry about a plane if the plane cuts
the figure into two mirror images. Students can determine how many
planes of symmetry the tetrahedron and cube each have. They should
also investigate to see if there is some connection between the axes of
symmetry and the planes of symmetry (see activities E4.5, E4.6, and
E4.7).
Another characteristic of Platonic Solids is duality. A second polyhedron
whose vertices touch the midpoints of the faces of a given polyhedron
is called its dual. Students could begin by exploring, naming, and
describing 2-D duals that result when joining the centres of tessellated
polygons, then apply what they learn about the 2-D duals to the
Platonic solids. When exploring the Platonic Solids for duality, they
will determine that the cube and the octahedron are duals. They could
watch the video on the Platonic Solids (see 4th column), which gives
an animated presentation of the concept of duality. They could then
determine other possible dual pairs either by construction, or by looking
for patterns after completing a table like the following:
Name
9-90
# of faces
# of vertices
tetrahedron
4
4
cube
6
8
octahedron
8
6
dodecahedron
12
20
icosahedron
20
12
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
E4.3 Give each group an acetate net for a transparent cube (7.4 cm
square), and a paper net for the matching octahedron (each side of
the equilateral triangle is 6.4 cm with another concentric dashed-line
equilateral triangle (side length 4.8 cm) drawn within for folding). Ask
students to
a) construct each solid
b) complete the following table:
polyhedron
tetrahedron
# of vertices # of faces
4
4
# of edges
6
cube
octahedron
dodecahedron
icosahedron
c) determine a pattern about some pairs of regular polyhedra in the
above table
d) open the transparent cube and place the octahedron inside so that
each vertex of the octahedron touches the centre of each face of
the cube (Describe the position of the edges of the octahedron in
relation to the cube.)
e) tilt the cube, looking directly down at one corner (Describe the
position of the vertex at the corner in relation to the closest face of
the octahedron inside. These two solids are called duals. Write a
definition that determines a pair of dual regular polyhedra. [Note
to Teachers: After students have written their definition, have them
share what they have written, then give them time to edit and
complete the definition, ensuring that all have a correct version.])
f ) describe how to build the dual for the cube by attaching faces to
vertices of the cube (Determine what solid is the resulting dual.)
g) describe how to draw the dual for the octahedron by joining the
midpoints of adjacent sides
E4.4 Each group of students should have a transparent net for the cube
or construct the cube from Polydrons or Polydron Frameworks and a
few elastics. Ask students to
a) make the cube
b) place an elastic around the cube so that it touches four edges, and
is perpendicular to the edges. (Determine the number of faces that
the elastic touches. Imagine a plane (outlined by the elastic) slicing
the cube into two pieces. The cross section is the new face produced
from the slice. Determine the shape of the cross section.)
c) consider some other planes that form square cross sections of the
cube (Draw some. Determine if all square cross sections of the cube
are congruent.)
Atlantic Canada Mathematics Curriculum
9-91
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
9-92
Elaboration—Instructional Strategies/Suggestions
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
E4.5 (this is a continuation of E4.4) Ask students to
a) change the location of the elastic so that it forms a cross-section
that is rectangular, but not square (Draw some. Determine which
rectangular cross section has the largest area, and explain how they
know.)
b) determine if any of the positions of the elastic in the above
exploration resulted in a plane of symmetry (Explain.)
c) describe how to place the elastic so that all planes of symmetry are
determined (Draw their pictures. Add the axes of symmetry to the
picture. What statement can be made about a possible connection
between the axes of symmetry and the planes of symmetry for this
cube?)
Paper and Pencil
E4.6 Marla said that the tetrahedron is a dual of itself.
a) Ask students to justify whether Marla is correct or not.
b) Ask students if there is another Platonic Solid that is a dual of itself.
c) Ask students to find any Platonic Solid that is the dual of any other,
and to explain in words how they know.
d) Ask students to explain how duality affects Euler’s Law.
E4.7 The picture shows one corner of a regular dodecahedron. Ask
students to
a) sketch a diagram the same as this one including the one face of the
dual solid, and determine what polygon is represented by this face
b) determine the number of vertices on a dodecahedron
c) determine how many faces the dual will have
d) determine what polyhedron is the dual of a regular dodecahedron
E4.8 Ask students to draw and describe the dual for the tetrahedron.
Have them name the resulting polyhedron that is the dual of the
tetrahedron. (Note to Teachers: Have students determine the midpoint
of each face, then join the midpoints of adjacent faces.)
Atlantic Canada Mathematics Curriculum
9-93
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
KSCO: By the end of grade 9,
students will have achieved the
outcomes for primary–grade 6 and
will also be expected to
E5 Students should model, using Polydrons, and study various nonregular polyhedra that can be constructed with regular polygonal
faces—uniform prisms and antiprisms, deltrahedra, dipyramids, and
Archimedian Solids—to develop their abilities to visualize, reason, and
to solve problems.
i) construct and analyze 2- and
3-D models, using a variety of
materials and tools
iv) represent and solve abstract and
real-world problems in terms of
2- and 3-D geometric models
SCO: By the end of grade 9,
students will be expected to
E5 solve problems using 3-D
shapes using visualization,
reasoning, and geometric
modeling
Students should be encouraged to build polyhedra from nets or partial
nets, as the patterns that are viewed in the nets will strengthen their
geometric visual memory. And as an alternative, students could be
asked to take different polyhedra and collapse them to form a net. Draw
the net and look at all the different possible nets drawn by classmates
for that shape. Students should discuss which of the nets are easiest to
visualize and remember. Visualizing and describing what they see are
very important activities.
Students should discover that uniform prisms and uniform antiprisms
(prisms made using regular polygons) have vertex regularity (each vertex
of the solid is the same). For example, a prism has an n-gon as its two
bases and parallelograms for the other faces. A uniform prism has an
n-gon as its two bases and squares for the other faces. Each vertex is
then, denoted by {4,4,n}. Similarly, a uniform antiprism, whose sides are
triangles instead of parallelograms, would have each vertex denoted by
{3,3,3,n} depending on which n-gon was its base.
When examining uniform prisms students might be asked to construct
the net for a prism given the configuration for each vertex and the
number of vertices. When the students use the net to form the solid,
they should be asked to name and describe the solid, and to check to
see if there is vertex regularity. They then might be given a net (of a
uniform antiprism) and be asked to predict the solid that would result
from the net, then form the solid to check their prediction. They should
describe the solid and check for vertex regularity. They should discuss
how this solid differs from a uniform prism, and why it’s called an
antiprism.They should look for relationships amoung the faces, edges,
and vertices.
As students explore the various regular and non-regular polyhedra they
should be introduced to solids that have alternative names such as the
deltahedron (a polyhedron with only equilateral triangle faces) and the
dipyramid (a polyhedra with all triangular faces formed when placing
two pyramids base to base). They should note that there are various
deltahedra with varying number of faces, and that some of these are
dipyramids. They should also explore solids that can be both convex and
concave.
9-94
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
• Mathematics 9: Focus on
Understanding Geometry
E5.1
Supplement, pp 49–57
a) Ask students to build a square based pyramid.
b) This polyhedron is not regular. Ask students which properties it lacks. • Polydron pieces
c) Ask students to build a prism using two triangles and three squares.
Name the solid.
d) Ask students to build a prism using two regular pentagons and five
squares. The prisms in (c) and (d) are not regular polyhedra. Ask
students which properties they lack.
e) Ask students to build a solid using 2 regular hexagons and 12
equilateral triangles. This shape is called an uniform hexagonal
antiprism. Ask students to compare it to a hexagonal prism, describing
how it is different, and how it is the same.
f) Ask students to describe in detail how they can build a uniform
pentagonal antiprism.
E5.2
a) Ask students to construct a box that has pentagons for its top and base
and square faces.
b) Ask students to name the solid made in (a).
c) Ask students to make a second solid with pentagonal bases, but
congruent equilateral triangle faces, and a third solid using congruent
isosceles triangles as its faces.
d) Ask students the name of these solids, and have them compare them
to the uniform pentagonal prism made in (a). How are they the same?
How are they different?
e) Ask students to complete the table and state any relationships that
seem to emerge. V is the number of vertices, F is the number of faces
(including bases), E is the number of edges, and T is the number of
triangular pieces (other than the bases):
V
F
E
T
square antiprism
triangular antiprism
pentagonal antiprism
hexagonal antiprism
octagonal antiprism
n-gonal antiprism
f ) Ask students to make another uniform antiprism using a different
polygon for a base. Ask them to determine if this antiprism will hold
more or less uncooked popcorn than its corresponding uniform
prism with the same height.
g) Ask students to determine if all uniform prisms fill a space (no gaps).
Ask them if all uniform antiprisms fill a space. Fill a space means ...
can the solid be fitted to itself one or more times to form a new solid
with no gaps.
Atlantic Canada Mathematics Curriculum
9-95
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
Students should build various Archimedean Solids (semi-regular
polyhedra) given nets or partial nets. These nets would be created
using two or more regular polygons. Examine these pictures as starting
configurations. The figure below suggests that students should make two
or more congruent 3-D shapes given six regular polygons and a {6,6,5}
configuration. They should pick up these shapes and try to fit them
together to make a semi-regular polyhedra.
Students should understand that the 13 semi-regular polyhedra (called
the Archimedean Solids) can be obtained from the five regular polyhedra
(the Platonic Solids) by the appropriate cutting of corners (truncating).
Truncation actually means “the changing of one shape into another by
altering the corners.” For example the four corners of a square can be cut
back to make an octagon, so the cube (which is made from 6 congruent
squares) when truncated at each corner, will become the semi-regular
polyhedron called the truncated cube. Each vertex has regularity since
each vertex is formed by a triangle and two octagons.
9-96
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
E5.3 Ask students to build a solid by connecting two squares and two
equilateral triangles alternately at each vertex. Continue until they have a
closed shape.
a) Ask students how many squares were used.
b) Ask students how many triangles were used.
c) Ask students to determine if this solid is regular, semi-regular, or
neither. Explain.
d) Ask students to explain why they think that this shape has the name
cuboctahedron.
E5.4 Beginning with the solid constructed in E5.3, above, ask students to
a) break it down to form a net. (Use a template to record the net on
paper.)
b) record at least three different nets
c) compare their nets with each other
d) determine which of the recorded nets allows then to visualize the
cuboctahedron, and have them explain their thinking
E5.5
a) Ask students to examine the diagram below.
b) Ask them to predict, without folding, whether it will make a solid.
c) Ask students to use Polydrons to check their prediction. Ask them
how many vertices there are and if the vertices have the same
configuration.
d) Ask students to visualize and predict what they would have to add to
complete the solid.
e) Ask students to make another shape like they one they have and
connect them together to make a closed solid with vertex regularity.
e) Ask student to identify the solid.
Paper/Pencil
E5.6
a) If an antiprism has a base with n sides, ask students how the numbers
of its faces, vertices, and edges can be expressed in terms of n.
b) Ask students if Euler’s Formula is true for all uniform antiprisms.
Show why or why not.
Atlantic Canada Mathematics Curriculum
9-97
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
KSCO: By the end of grade 9,
students will have achieved the
outcomes for primary–grade 6 and
will also be expected to
E6 Examining circles, the parts of the circles, the language of circles, and
the relationships that exist amongst the parts of the circle, and making
conjectures is the focus for the achievement of this outcome.
v) draw inferences, deduce
properties, and make
logical deductions in
synthetic (Euclidean) and
transformational geometric
situations
SCO: By the end of grade 9,
students will be expected to
E6 recognize, name, describe,
and represent arcs, chords,
tangents, central angles,
inscribed angles and
circumscribed angles, and
make generalizations about
their relationships in circles
Through guided activities involving paper folding and the use of
measurement devices like rulers, and Bull’s Eye compasses, through
discussion, and by making conjectures, students will achieve
understanding of the following:
• A circle is a set of points equidistant from one point, called the center
of the circle.
• The distance around the circle is called the circumference.
• There are an infinite number of mirror lines on a circle.
• The segment formed by joining the two points where the mirror
line touches the circle is called a diameter, half of which is the radius
(from the centre to the point on the circle).
• The centre of the circle is the intersection of two diameters.
• The words radius and diameter can also refer to the lengths of the
radius and diameter.
• Chords are segments joining any two points on the circle.
• The perpendicular bisectors of any two chords will intersect at the
centre of the circle.
• Chords of equal length will be the same distance from the centre.
• A central angle is the angle formed by two radii of a circle.
• Part of the circle is called an arc.
• The measure of an arc is that of the central angle that intercepts that
arc.
• If an arc measures more than 180° then it is called a major arc; if less
than 180°, then it is called a minor arc.
• When central angles are congruent, then so are the chords and the
corresponding arcs that are intercepted by those same central angles.
• An angle that is formed by joining three points on the circle is called
an inscribed angle.
• The measure of the inscribed angle is one-half the measure of the
central angle that intercepts that same arc, and thus all inscribed
angles intercepting the same arc or congruent arcs must be congruent.
9-98
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Pencil/Paper
• Mathematics 9: Focus on
Understanding Geometry
Supplement, pp 40–48
E6.1 Determine the measure of each indicated angle or arc. Explain
your answers.
a)
b)
c)
d)
• Bull’s Eye
• ruler
E6.2 For each diagram, which inscribed angles are congruent? Explain
how you know.
a)
b)
c)
Journal
E6.3 In the circle below with center at O, AB = 26 units, BC = 24 units.
What is the length of the segment from O, perpendicular to BC?
E6.4 When a radius is drawn perpendicular to a chord in a circle what
other relationships are there between the radius and the chord? Explain
how you know.
E6.5 A mother wants to cut a hole in the centre of her circular patio
table in which to insert an umbrella. Explain to her how to find the
centre of the table.
Atlantic Canada Mathematics Curriculum
9-99
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
• A tangent to a circle is a line drawn to touch a circle at just one point.
• A radius drawn to a point of tangency is perpendicular to the tangent.
• Two tangents drawn from an external point to two different points
on the circle must be the same length, and they form a circumscribed
angle.
• The opposite angles in the quadrilateral formed by a circumscribed
angle and a central angle that intersect at the points of tangency are
supplementary.
9-100
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 9
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Problem Solving
E6.7 Using what you know about congruent triangles, draw a diagram
and convince your partner that if a line passes through the centre of a
circle and is perpendicular to a chord then it must intersect the chord at
its midpoint.
Performance
E6.6
a) Draw a circle with the quadrilateral ABCD inscribed in it like the
picture below.
b) Measure each of the four angles in the quadrilateral.
c) Find the sum of A and C, and the sum of B and D. What do
you notice?
d) Repeat this same activity using a slightly larger circle, then again
using a slightly smaller circle.
e) Make a conjecture about the opposite angles of a cyclic
quadrilateral.
E6.8
a) Draw a circle with radius at least 4 cm.
b) Name the diameter AB.
c) From a point C on the circle draw two chords, one to each endpoint of the diameter.
d) Measure the angle at C.
e) Draw another chord parallel to CA from B to intersect the circle at
D. Join D to A and measure the angle at D.
f ) Make a conjecture creating angles inscribed by the diameter in
semi-circles.
g) Make another conjecture about how one could create a rectangle
with only a circle and a straight edge.
Atlantic Canada Mathematics Curriculum
9-101