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Geometry
General Curriculum Outcomes E:
Students will demonstrate spatial sense and apply
geometric concepts, properties, and relationships.
Revised 2010
Specific Curriculum Outcomes, Grade 8
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 investigate and establish the conditions necessary
for uniqueness of a triangle before establishing the conditions necessary
for congruency of triangles. Each pair of students might construct (from
Geo-Strips) or draw, using compasses, rulers, and protractors, or using
Bull’s Eyes, a set of given conditions that would result in a triangle. For
example, one pair of students might each be given an identical set of
three Geo-Strips and be asked to make a triangle, then compare it with
their partner’s, and make a conclusion about their triangles—are they
the same or not? Students might place one on top of the other to see
if they are the same size and shape. (This is how congruence has been
established in earlier grades.) Another pair of students might be asked
to construct a triangle given the measures of three sides, then compare
their triangles. When they conclude that the given conditions will
produce one and only one triangle, that triangle is said to be unique.
They should be asked to reflect on all the sets of conditions that resulted
in a unique triangle, and record this information.
v) draw inferences, deduce
properties, and make
logical deductions in
synthetic (Euclidean) and
transformational geometric
situations
SCO: By the end of grade 8,
students will be expected to
E1 make and apply informal
deductions about the
minimum and sufficient
conditions to guarantee the
uniqueness of a triangle
and the congruency of two
triangles
By conducting several investigations, beginning with a different set
of conditions each time, students should become aware that a unique
triangle results when the given conditions are a set of:
1) three side lengths
2) two side lengths and the included angle measure
3) two angle measures and a specified side length
Students should also explore and understand why a triangle is not
unique given a set of three angle measures. Also, students should explore
the set of conditions that includes two side lengths and one angle
measure that is not the included angle. With this set of conditions, they
will discover that sometimes they can get a unique triangle, but often
they get two triangles or none at all. They should conclude that these
two sets of conditions do not guarantee unique triangles.
Most of the reasoning that students have applied has been inductive.
Making conclusions based on measurement is an example of inductive
reasoning. In grade 7, students inductively reasoned that vertically
opposite angles must be congruent, and alternate interior angles must
be congruent if lines are parallel, and so on. These ideas, that have
been accepted to be true based on inductive reasoning in grade 7, can
be proven deductively. If a student needs to say that two corresponding
angles in a pair of triangles are congruent and they happen to be
vertically opposite angles, that student can say the angles are congruent
and use the theorem “vertically opposite angles are congruent” as their
reason. (This is deductive reasoning, using “if ” and “then” logic.)
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
In the following problems, if lines look like straight angles (180°), then
they are.
• Geo-Strips
Pencil and Paper
• Bulls Eye Compass
E1.1 Examine the following triangles and determine which are unique.
Explain why you think they are unique.
• Mathematics 8: Focus on
Understanding, pp 108–120
a)
b)
c)
d)
• Pattern Blocks
E1.2 Study the following diagrams. For each diagram,
i) write the pairs of corresponding congruent angles and pairs of
corresponding congruent sides, giving reasons why you know they
are congruent
ii) determine the congruent triangles, if any, and write a convincing
reason why you know they are congruent. (If they are not congruent
explain why not.)
a)
b)
c)
d)
e)
f)
E1.3 Draw the diagonals for rectangle ABCD. The diagonals meet at
point E. Ask students to
a) name four pairs of triangles that are congruent
b) explain how they know that each of the pairs of triangles are
congruent
Atlantic Canada Mathematics Curriculum
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Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
Students should be given activities where they are asked to convince
their partner of some conclusion that can be shown to be true using
if-then statements. For example, if given ABC with A = 63° and
B = 27°, then one could say that the third angle must be a right angle
and convince one’s partner by saying that the three angles in a triangle
must total 180°, and if A = 63° and B = 27°, they total 90°, then the
third angle must be 90° since 180°– 90° = 90°. This type of reasoning,
where we ask students to “convince others,” is necessary to help students
develop their understanding of minimum sufficient conditions.
Students should reflect on the conditions that made a triangle unique
and realize that what makes a triangle unique also determines that
triangles are congruent, and now should record the conditions that
would guarantee that two triangles are congruent.
Triangles are congruent if
1) three pairs of corresponding sides are congruent
2) two pairs of corresponding sides and the included angle are
congruent
3) two pairs of corresponding angles and a specified pair of
corresponding sides are congruent
Students should examine pairs of triangles with given measurements and
determine if the given measures are sufficient to guarantee the triangles
are congruent. They should convince their partners using if-then
statements.
If students are given that two triangles are congruent, they will need
some practise recording the corresponding parts of the triangles that are
congruent.
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
E1.4 If ABC
congruent.
Suggested Resources
DEF state all the corresponding parts that are
E1.5 Have students work in groups, and provide each group with two
straws, one 5 cm long, and the other 7 cm long. Also give them each
a 50° angle made from angle strips. The third side can be any length
necessary to complete the triangle. Experiment by having students place
the 50° angle in different locations so that it is the included angle, and
then the non-included angle.
a) Ask students what they notice and to record their findings.
b) Ask them to compare their findings with those from other groups
for each case explored, and to state conclusions.
c) Ask students where the 50° angle must be placed relative to the two
given sides in order for all groups to produce a unique triangle.
Journal
E1.6
a) Given that MN || ST, and MN = ST,
ask students to write a paragraph to
convince someone that NE = ES.
b) Ask students to replace the given above
with this new given for the same diagram:
MT and NS intersect at E. Ask students
if they can conclude that MN || ST? Explain.
c) On the same diagram, here is a new given:
Given. E is the midpoint of NS, and
MN || ST. Ask students if it can be concluded that
i)
MEN
TES? Explain.
ii)
MEN is not congruent to
ii)
M
TES. Explain.
T? Explain.
iii) MN bisects NS?
Atlantic Canada Mathematics Curriculum
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Specific Curriculum Outcomes, Grade 8
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 have studied translations, reflections and rotations,
and composition of transformations in previous grades. In this
course, students should begin by developing the properties of those
tranformations, then focus on the properties of any rotation, and the
properties of dilatations.
iii) develop and analyze the
properties of transformations
and use them to identify
relationships involving
geometric figures
v) draw inferences, deduce
properties, and make
logical deductions in
synthetic (Euclidean) and
transformational geometric
situations
By copying and cutting out a given triangle, then translating it and
drawing the image, students can be asked to determine the properties of
translations, which include:
• preservation of orientation (if naming the pre-image ABC involves
going around the figure clockwise, then naming the image should
result in A'B'C' if going around the figure in a clockwise direction.)
• congruency and parallelism of corresponding sides
• congruency of corresponding angles
• congruency and parallelism of direction arrows
SCO: By the end of grade 8,
students will be expected to
By drawing the reflected image of a pre-image, students will be using
properties of reflections. These include:
E2 make and apply
generalizations about the
properties of rotations
and dilatations, and use
dilatations in perspective
drawings of various 2-D
shapes
• change in orientation
• congruency of corresponding sides and angles
• the mirror line is the perpendicular bisector of the segments joining
points to their images
• the parellelism of segments joining points to their images
Students should engage in activities that will help them identify
properties of rotations. (Pattypaper or onion paper is recommended).
Students should develop an appreciation of these properties through
guided exploration activities. Properties of rotations include:
• preservation of orientation—Give students figures that have vertices
labelled. Provide direction to rotate the figures through various angle
measures both clockwise and counterclockwise. Students should trace
the pre-image and images and label their vertices. Ask them whether
orientation is preserved for all the rotations and for both directions.
Ask them to make a conclusion about orientation. (Orientation is
preserved.)
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Activity
• Pattern Blocks
E2.1 A'B'C'D' is the image of ABCD after a rotation.
• Power Polygons
a) Locate the centre of rotation
(do not erase your construction lines).
• Mathematics 8: Focus on
Understanding, pp 388–394,
410–417
b) Determine the angle and direction
of rotation.
E2.2 Copy the quadrilateral ABCD and point P onto plain paper, then
draw the dilatation image for ABCD, centre P, scale factor . On the
same paper, draw the dilatation image of RST, centre P, scale factor 3.
E2.3
a) Give each group of students a paper with a drawing of a different
shape and a different centre of rotation. Ask students to draw the
image of the shape after a rotation of 180° counterclockwise. Have
them compare the corresponding sides and angles of the image and
pre-image. (Students should conclude that the corresponding sides
and angles are congruent, and that all of the corresponding sides are
collinear or parallel.)
b) Ask students if they think their conclusions would be true for other
angles of rotation and explain their prediction?
c) Have them try different angles of rotation and make their
conclusions.
d) With these new diagrams, have them draw a ray from the centre
of rotation to a point on the pre-image, and a second ray from the
centre of rotation to its corresponding point on the image. Have
them measure the angle formed. Try it again with a different pair of
corresponding points, and ask students to make a conclusion about
the angles. (They should be equal, and also be equal to the angle of
rotation.)
e) For a 90° rotation have students measure the angle formed by a
segment and its image or the lines that include a segment and its
image. (They should conclude that they are perpendicular.)
Atlantic Canada Mathematics Curriculum
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Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
• congruency of corresponding sides and angles—Using the same
figures, ask students to measure corresponding sides and angles of the
image and pre-image and discuss whether corresponding angles and
sides are congruent after rotation.
• centre of rotation is the intersection of the perpendicular bisectors of
all segments joining corresponding points of the image and pre-image
• equality of all angles formed by rays drawn from the centre through
corresponding points (i.e., If the centre of rotation is called O, then
COC'
BOB'
AOA'.)
• for 180° rotations, corresponding sides are parallel or collinear
Students have drawn dilatation images in grade 6 and have had some
discussion about the conclusions they could make about the image
and pre-images. In this course students should engage in activities to
help identify properties of dilatations. They might use pattern blocks
to model triangles of different sizes that are dilatation images of oneanother. Students should develop an appreciation of the following
properties through guided exploration activities.
• all points except the dilatation centre change position
• the dialation centre, a point, and its image are collinear
• the ratio of the distances from the centre to corresponding points
equals the ratio of the corresponding sides of a shape and their
respective images (This ratio is known as the scale factor, which is
a side length from the image
calculated by the ratio )
pre–image’s corresponding side length
• segments in the image are parallel to the corresponding segments in
the pre-image, and “r” times as long or short.
• a shape and its image are similar (on page 8-78 are two dilatations;
on the left, from an external centre, O, and on the right, from an
internal centre, O)
Activities should be chosen so that students can draw and recognize
dilatations of shapes using a variety of centres of dilatation (inside,
outside and on the figure). As an extension, students can explore what
happens when a negative scale factor is used.
Students should make three-dimensional drawings of two-dimensional
shapes. By joining each vertex of a 2-D shape to a point, called the
vanishing point, then drawing a corresponding 2-D shape with vertices
on the lines drawn to the vanishing point, and with sides parallel to the
given shape, students will have drawn a one-point perspective drawing.
Students should apply the connection between the centre of a dilatation
and the vanishing point in perspective drawings of various 2-D shapes.
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Paper/Pencil
Suggested Resources
E2.4 A photograph measures 5 cm by 7 cm. It is enlarged so that it is
2.5 times as large.
a) What are the new dimensions if this enlargement factor is referring
to its dimensions?
b) What are the new dimensions if this enlargement factor is referring
to its area?
E2.5 Determine the scale factor for each of the following dialations:
a)
b)
i) In part (a) if A'C' = 4 cm, how long is AC? Explain how you
know.
ii) In part (b) if the area of ABC is 1 cm2, then what is the
area of A'B'C'? Explain how you know.
E2.6 For a dilatation, scale factor 2, centre P,
a) M will map onto what point? Explain how you know.
b) N will map onto what point?
c) What conclusion can be made about the length MN in relation to
QR? Explain.
d) What conclusion can be made about MN and QR? Explain how
you know.
e) If MN is called the mid-line of the PQR, what statement can be
made about the mid-line of a triangle.
E2.7 Create a rectangular prism by constructing a perspective drawing
of a rectangle.
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Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
This part of the outcome could occur while addressing E7. Students
could be asked to examine a map plan, construct the shape, and draw
isometric and perspective diagrams. In the diagram below, the shape
ABCD is dilatated, centre O, to obtain the image PQRS, or is the solid
formed by the object and image a perspective drawing with vanishing
point O?
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
E2.8 Create a picture of a room using the given rectangle and vanishing
point. Your room should have a door with a window, and a window,
with trim around each.
Journal
E2.9 Write about whether congruence, area, length, and orientation are
preserved as they pertain to enlargements and reductions. Use diagrams
to help support your writing.
Atlantic Canada Mathematics Curriculum
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Specific Curriculum Outcomes, Grade 8
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 In grade 6 students observed figures that are similar, discussed what
makes them similar (side lengths with equal ratios, and corresponding
angles congruent), and drew similar shapes larger or smaller than given
shapes.
ii) compare and classify geometric
figures, understand and apply
geometric properties and
relationships, and represent
geometric figures via coordinates
Students should recognize through investigation the properties of similar
triangles, that is,
iv) represent and solve abstract and
real-world problems in terms of
2- and 3-D geometry models
For example, give students a number of shapes, some of which are
similar, that they can measure, and ask them to make conclusions about
which shapes are similar and why. Other examples of investigations: ask
students to examine pairs of shapes that look similar, but whose given
measures do not allow students to conclude that they are similar; give
students diagrams that have some information that might lead them to
think the shapes are similar,
but other information that
contradicts that. For example,
in the diagram, the ratio of two
pairs of corresponding sides
are equal, but students will
measure to note that B is not
congruent to E. They should also note that the third pair of sides are
not in proportion with the given pairs of sides.
v) draw inferences, deduce
properties, and make
logical deductions in
synthetic (Euclidean) and
transformational geometric
situations
SCO: By the end of grade 8,
students will be expected to
E3 make and apply
generalizations about the
properties of similar 2-D
shapes
• corresponding angles are congruent
• corresponding sides are proportional
Another activity might be to ask students what else might be needed to
make a pair of shapes similar. For example, give students a pair of shapes
like those above, and ask students to say what they would need to do
to make the two shapes similar. [A possible response might be that they
E, or they might say that they need to make
would make sure the B
AC longer so that the ratio AC is 1:2, like the other corresponding side
DF
ratios.
Students should be exposed to a variety of situations involving similar
triangles including those that result from dilatations, those that
are varied in their positions when drawn, as well as those that are
overlapping and non-overlapping. Students should be able to use the
properties of similar triangles to find the measures of missing sides and
angles. For example,
• Ask students to find AC, if PR = 2.9 cm
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
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 8: Focus on
Understanding, pp 395–402
E3.1 All the quadrilaterals in this figure are rectangles, and
ABCD ~ EFGD. Ask students to use a ruler to measure the sides, to
help answer the following questions.
a) AD = CD . Why?
ED
GD
b) With regard to similarity, what can be c
oncluded about ADGH and DCIE ? Explain.
c) If AD = CD then HG = ?
ED
GD
IC
d) With regard to similarity, what can be concluded about AEFH and
IFGC? Explain.
E3.2 Arrange students in groups of three. Provide each student in the
group with a set of strips (these can be cut from plastic stir strips, strips
used for stirring coffee, or use Geo-Strips) 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 its angles. 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, and to test their
predictions.
Pencil and Paper
E3.3
a) Are the two triangles in the diagram below similar? Justify.
b) If enough information is available, find the width of the river. If
there is not enough information, what other information is needed?
c) Are the two triangles in the figure below right, similar? Justify.
d) If enough information is available, find the height of the tree.
Atlantic Canada Mathematics Curriculum
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Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
This topic lends itself well to real-life situations such as finding the
height of buildings and trees and finding distances that are normally
difficult to measure directly, such as the distance across a river, pond, or
wetland.
Using dilatations of various 2-D figures is a very good way to explore
the properties of similarity for shapes with 4, 5, and more sides. For
example, give different groups of students different rectangles drawn
in different positions on a page. Assign each group a different centre of
dilatation, and a different scale factor, including fractions and negative
numbers.
• Ask all students to draw the dilatated image and to measure the
lengths of the sides and angles of both the given rectangle and its
image after the dilatation.
• Ask students to examine the ratios of the perimeter and area of the
image to the perimeter and area of the pre-image and compare their
answer to the scale factor. Have them discuss in their groups why this
happens.
• Referring to the fact that the image in a dilatation is similar to the
pre-image ask students to write about the properties of two similar
rectangles. Ask them if these properties would be the same for a
parallelogram and its image; a regular hexagon and its image; a
regular pentagon and its image. (The properties that make rectangles
similar: same shape [equal angles], and ratios of lengths to widths are
the same.)
A large group discussion after the activity should consolidate all the
properties of similar figures. These include
• Corresponding sides are parallel.
• The ratio of lengths of corresponding sides is equal to the dilatation
scale factor.
• Corresponding angles have equal measures (are congruent).
• Corresponding figures are similar.
• Corresponding figures have the same orientation.
• All points, except the centre of dilatation, change position.
• Lines through corresponding points pass through the centre of
dilatation.
• The ratio of the lengths of line segments joining corresponding points
to the centre of dilatation is equal to the dilatation scale factor.
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Journal
E3.4 Ask students to write a paragraph or two to their best friend
explaining to them why when you compare two similar polygons, one
with sides three times as long as the corresponding sides of the other,
that the larger one will have an area that is nine times as great. Tell them
to include a diagram in their explanation.
E3.5 Ask students if each set of figures is similar. Ask them to justify
their decisions.
a) b)
Atlantic Canada Mathematics Curriculum
8-83
Specific Curriculum Outcomes, Grade 8
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 Introduction to angle and segment bisection, perpendicularity,
construction of isosceles and equilateral triangles, and all the
quadrilaterals all occurred in grade 7. Various technologies were used:
i) construct and analyze 2- and
3-D models, using a variety of
materials and tools
• transparent mirror (Mira or Math-Vu Mirror)
iii) develop and analyze the
properties of transformations
and use them to identify
relationships involving
geometric figures
v) draw inferences, deduce
properties, and make
logical deductions in
synthetic (Euclidean) and
transformational geometric
situations
SCO: By the end of grade 8,
students will be expected to
E4 perform various 2-D
constructions and apply the
properties of transformations
to these constructions
• paper for folding and/or tracing (onion paper or patty paper is best)
• geoboards and dot/grid paper
• compass and straightedge, and Bull’s Eyes
• appropriate computer software (optional).
While investigating the conditions for unique and congruent triangles,
students could use some constructions. For example, in E1 when
investigating uniqueness of a triangle, students may be copying
angles and making segments the same length, and in E2 they may be
constructing perpendicular bisectors. Teachers should observe students
while they are constructing, not just looking at their final results.
Students in grade 8 should be using constructions to investigate and/
or apply the properties of transformations. For example, they will use
segment bisection and perpendiculars to create reflection images, they
will use rulers, protractors, (or Bull’s Eyes) to create angles of rotation,
and they will use the properties of reflection to find the mirror line
and perpendicular bisectors to find the centres of rotation, either with
mirrors or with rulers and compasses, or Bull’s Eyes.
Students should use constructions to determine if a drawing is of a
particular transformation. For example, they would draw the translation
arrows and measure them, then construct a transversal and, check
corresponding angles to determine parallel lines. They might construct
segments from points to their images, drawing a line through their
midpoints to check to see if the drawing is a reflection.
Students should also be constructing transformations of shapes to create
polygons, then use the properties to convince their partners that they
have created that polygon. For example constructing a 180° rotation
using the midpoint of one side of a triangle to create a parallelogram.
Also, they might use a mirror to construct an isosceles triangle by
placing the mirror so that a point maps onto another point, then
choosing a third point along the beveled edge and connect all three
points to create the isosceles triangle. They should convince their
partners, both verbally and in writing, referring to properties, that these
constructions are accurate.
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Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Pencil and Paper
• Bull’s Eye Compass
E4.1 Ask students to locate and draw the mirror line in this diagram.
Have them record their steps, and then to explain how they know they
have done it correctly.
• Geometry Transparent Mirrors
• Mathematics 8: Focus on
Understanding, pp 121–130
E4.2 By construction, find the centre of rotation, and determine the
measure of the angle of rotation, if A'B'C' is the image of the
ABC.
E4.3 Marla told Jeff that in an isosceles triangle the bisector of the
vertex angle is also the perpendicular bisector of the base. Ask students
to find a way to confirm this using a construction, and explain how they
did it.
Performance
E4.4 Ask one student in each group to explain how constructions could
be used to turn a triangle into a parallelogram, and the parallelogram
into a rectangle. Have the other students listen to the instructions, then
do it.
E4.5 Ask one student in each pair to begin by marking two points A,
and B, on a page. Place a mirror so that the image of A lands on B.
Draw a line with a sharp pencil along the beveled edge of the mirror.
Find two more points on the beveled edge, name them C and D. Join
the four points to create a quadrilateral. Now the second partner, who
has been watching, has to name the shape and explain how they know
for sure what shape it is.
E4.6 Activity 1:
Sheriff ’s badges have traditionally been in the shape of a 5-point star.
In the nineteenth century 6-point stars were very common, probably
because they were easier to make. (Join every second point on a regular
hexagon.) The 5-point star is based on the regular pentagon.
a) In the middle of a blank piece of paper have students draw a circle
(some use radius 5 cm, some use radius 6 cm, and others use radius
7 cm), with a Bull’s Eye. Draw a diameter that cuts the circle at
Atlantic Canada Mathematics Curriculum
8-85
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
8-86
Elaboration—Instructional Strategies/Suggestions
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
points A and C, and label the centre of the circle O. Draw another
diameter BD through O at right angles to AC.
Suggested Resources
• Bull’s Eye Compass
b) Bisect the radius OA. Label the midpoint M.
c) Using M as the centre, and MB as the radius, swing an arc through
the radius OC. Label the intersection point P.
d) Using B as the centre, and BP as the radius, swing an arc through
the circle, near C. Label the intersection point Q.
e) Draw the segment BQ. This is one side of a regular pentagon
inscribed in the circle. Using the Bull’s Eye and the same radius,
mark off the remaining four sides around the circle. You have
constructed a regular pentagon.
E4.7 Activity 2:
Ask students to begin with a square ABCD with side length about 3 or
4 cm, and to follow these directions:
a) Construct the midpoint of side AB
and name it M.
b) Reflect M in the side AD so that B-A-P
where P is the image of M.
c) At P construct a 60° angle with a compass.
To do this
i) with centre P make arc RS
ii) Using radius P and centre R, cut
arc RS at T with a new arc TQ
iii) draw PT to intersect AD at N
iv) explain why RPN is 60°
d) At M, construct a 60° angle so that
BMZ is 60°, and remove BMZ
from the diagram.
e) Translate M to P, and C to D. Draw
the image of PMCD.
f ) Construct the image of the whole
diagram by reflecting it in the line DC.
h) Marla calls the resulting diagram her Block.
Comment on the Block. Is it the beginning of
a tessellation? Describe the transformations
that created this design.
i) Describe, using transformations, how you would continue with this
Block to fill the whole page so that the finished product would be a
tessellation.
Atlantic Canada Mathematics Curriculum
8-87
Specific Curriculum Outcomes, Grade 8
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 A regular polygon is a figure in which all sides are congruent, and
all angles are congruent. Regular polygons will be the bases of prisms
that students will study while working towards outcome E6. Students
can use tables like the one shown to organize information about various
properties of regular polygons up to and including the dodecagon.
Using their tables they can observe patterns and generalize using “n-gon”
notation, for properties such as the ones indicated by the last four
headings in the table.
Number of
diagonals
from one
vertex
Total
number of
diagonals
Interior
angle sum
Exterior
angle sum
v) draw inferences, deduce
properties, and make
logical deductions in
synthetic (Euclidean) and
transformational geometric
situations
Name
iii) develop and analyze the
properties of transformations
and use them to identify
relationships involving
geometric figures
Regular Polygons
Number of
Sides
ii) compare and classify geometric
figures, understand and apply
geometric properties and
relationships, and represent
geometric figures via coordinates
3
triangle
0
0
180°
360°
4
quadrilateral
1
2
2 ×180°
360°
5
pentagon
2
5
3 ×180°
6
hexagon
3
9
4 ×180°
heptagon
octagon
nonagon
SCO: By the end of grade 8,
students will be expected to
E5 make and apply
generalizations about
the properties of regular
polygons
decagon
hendecagon
dodecagon
n
n-gon
n–3
n × (n – 3)
(n - 2)180°
2
Students might want to add columns to the above table to keep track of
which regular polygons have opposite sides parallel, whether the regular
polygons are convex or concave, what is the measure of their central
angles, and their exterior angles.
Symmetry of the various regular polygons can be explored to determine
if the number of lines of symmetry is related to the number of sides, and
if there is a connection between line symmetry and rotational symmetry.
In previous grades students have explored rotational symmetry
(sometimes called point symmetry) and discussed what rotational
symmetry of order 3, or 4, or 5 means. For example, the regular
pentagon has 5 lines of symmetry.
Initial
Position
8-88
After one clockwise rotation
of 260°/5 = 72°
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
• Geo-Strips
E5.1 Ask students to sketch any triangle and, through a series of
reflections, determine if a regular polygon can be produced. Ask
students to consider what characteristics would be necessary in a triangle
so that, through a series of reflections, a regular polygon is produced.
(Possible answer: When the triangle is isosceles, and the reflections are
across the equal sides, a regular polygon is produced, as long as the
product of the number of sides and the measure of each vertex angle is
360°.)
• Bull’s Eye Compass
• Geometry Transparent Mirrors
• Mathematics 8: Focus on
Understanding, pp 121–130
E5.2 Ask students to sketch a series of regular polygons with 3 to 12
sides.
a) Have students work in pairs to find the number of lines of
symmetry that can be drawn for each polygon. Ask them if any
patterns emerge and to explain their answers.
b) Have students determine that the centre of the polygon is the
intersection of the lines of symmetry. Will they get the same point
as centre if they find the intersection of the perpendicular bisectors
of each side? When two adjacent vertices of the polygon are joined
to the centre, the angle formed is called the central angle. Ask
students to record the measures of all the central angles.
c) Ask students to explain what happens to the size of the central angle
as the number of sides of a regular polygon increases.
d) Ask students to explain what happens to the size of the angles of a
polygon as the number of sides of a regular polygon increases.
e) As a result of (c) and (d) ask students what pattern seems to emerge.
(Possible answer: As the number of sides of a regular polygon
increases, the central angle measure decreases and the polygon’s
interior angle measures increase.)
f ) Ask students how they would find the area of the regular polygon
using the triangles formed when the vertices are joined to the
centre.
g) Ask students to find the area of a loonie.
E5.3 Ask students to speculate about the
sum of the measures of the exterior angles
of polygons, and then ask them to explore
and determine relationships. Ask them to use
the given chart to keep track of data, and to
determine as many relationships as possible.
(An exterior angle is defined as an angle formed when one side of a
polygon is extended at any vertex.)
Atlantic Canada Mathematics Curriculum
8-89
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
Ask students to use the intersection point of the lines of symmetry
as the centre of rotational symmetry. Since the figure will rotate onto
itself 5 times in one revolution, the pentagon is said to have rotational
symmetry of order 5, whereas the equilateral triangle has rotational
symmetry of order 3. Also students might use the intersection of the
lines of symmetry to draw the circumscribed circle, and inscribed circle
of the polygon.
8-90
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Name of
regular
polygon
# of
sides
# of
exterior
angles
measure
of each
central
angle
measure
of each
interior
angle
measure
of each
exterior
angle
sum
of all
exterior
angles
hexagon
6
6
60°
120°
60°
360°
Pencil and Paper
E5.4 Using the relationship established between the number of sides
and the angle measures in a polygon, find
a) the number of sides if the sum of the interior angles is 1620°
b) the number of sides if the measure of each interior angle of a regular
polygon is 144°
c) the sum of the interior angles if there are 15 sides
d) the measure of each interior angle of a regular 12-gon
E5.5 Using the relationship established between the number of sides of
a regular polygon and the number of diagonals that can be drawn from
one vertex, find
a) the total number of diagonals that can be drawn from one vertex in
a regular 20-gon
b) the number of sides a regular polygon has if 14 diagonals can be
drawn from one vertex
c) the number of sides a regular polygon has if a total of 27 diagonals
can be drawn
E5.6 Draw and name each figure. Include the line(s) or point of
symmetry if possible.
a) A triangle with no lines of symmetry.
b) A quadrilateral with one line of symmetry.
c) A quadrilateral with four lines of symmetry.
d) A triangle with rotational symmetry.
e) An octagon with rotational symmetry.
f ) A quadrilateral with rotational symmetry of order two.
g) A polygon with rotational symmetry of order five.
Atlantic Canada Mathematics Curriculum
8-91
Specific Curriculum Outcomes, Grade 8
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 Students in grade 4 have drawn nets for various rectangular prisms
and cubes and constructed skeletal models of various cylinders, cones,
prisms and pyramids, and have explored relationships amongst these
solids. In grade 5, students identified various cross-sections of cubes and
rectangular prisms, and continued drawing nets of pyramids and prisms.
In grade 6, students described various cross-sections of cones, cylinders,
pyramids and prisms, and explored planes of symmetry in 3-D shapes.
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 geometry models
In this course, students should revisit the sets of polyhedra known
as cones, cylinders, prisms and pyramids. They should extend their
understandings of cone and cylinder to include some
not-so-conventional looking cones and cylinders (see picture).
SCO: By the end of grade 8,
students will be expected to
Cones
E6 recognize, name, describe,
and make and apply
generalizations about
the properties of prisms,
pyramids, cylinders, and
cones
Cylinders
Cones have a flat base and a point not on the base called an apex. Points
on the base are connected to the apex by straight line segments that
must lie on the lateral surface. Cylinders have congruent parallel flat
bases. Corresponding points on the bases are connected by straight line
segments that must lie on the lateral surface. Cones, cylinders, prisms,
and pyramids can be described as right or oblique.
Students should examine, name, and describe the many different cones
(realizing that their name comes from the polygonal shape of their base),
and demonstrate understanding of the connections between pyramids
and cones, and between prisms and cylinders. Prisms and cylinders are
called right prisms and right cylinders when their faces are perpendicular
to their bases. Right pyramids and cones occur when the line joining
the vertex of a pyramid, and the apex of a cone, to the centre of the
base is perpendicular to the base. Oblique shapes result when the
perpendicularity is lost.
Students should understand that prisms and pyramids can be
categorized as polyhedra. That is, a pyramid, all 4 of whose faces are
triangles can be called a tetrahedron, and a square-based pyramid would
be a pentahedron.
8-92
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
• Polydrons
E6.1 The figures below are pyramids that have polygons as bases. Ask
students to
• Geoblocks
• Geometric Solids
• Mathematics 8: Focus on
Understanding, pp 403–409
a) draw the polygon that is the base for each
b) react to this statement: A cone is a pyramid
c) draw the prism that has each base
E6.2 Ask students to
a) examine a right hexagonal prism whose base is a regular hexagon
b) draw a line of symmetry on the base
c) imagine the line being extended to a plane that cuts through the
prism (What does it do to the prism? What would this plane be
called? [Ans.: a plane of symmetry]. How many like this one are
there for this prism? [Ans.: there are 6 altogether]. Are there any
other for this prism? [Yes, there is another that cuts the prism in
half, but the plane is parallel to the bases.])
d) determine if there is a line that could be called the axis of
symmetry (Where would this line be located? Are there other axes
of symmetry? Describe the order of rotational symmetry for this
hexagon.)
e) explain if all the answers above are true for an oblique hexagonal
prism
E6.3 Ask students to explore both right and oblique cylinders,
pyramids, and cones for planes and axes of symmetry. Keep track of this
information and the information learned in the activity in E6.2 in a
table.
Pencil/paper
E6.4 Which of the following is not a cone? Explain your thinking.
a)
b)
c)
d)
e)
f)
E6.5 Which of the following is not a prism? Explain your thinking.
a)
b)
c)
d)
e)
Atlantic Canada Mathematics Curriculum
f)
8-93
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
To develop a connection between pyramids and cones, ask students to
examine a set of regular pyramids and determine what geometric solid a
pyramid becomes more and more like as the number of sides in its base
increases. (cone).
Similarly the set of prisms below have the same polygon bases as the
pyramids above.
Ask students to determine what geometric solid a prism becomes more
and more like as the number of sides of its base increase. (cylinder)
Students should explore prisms, pyramids, cylinders and cones to
determine what kind of symmetries exist (planes of symmetry, and axes
of symmetry). How does one determine the planes of symmetry if they
exist for these solids? (Planes of symmetry of a prism are determined
by the lines of symmetry on its bases). Students should know how
to determine which solids have rotational symmetry, or an axis of
symmetry, and what order of rotational symmetry might exist.
8-94
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
E6.6 What am I?
a) I have one lateral face.
b) I have one base that is shaped like an oval or the outline of an egg.
c) My point is above one end of my base.
Journal
E6.7 Given that the base of a pyramid is a regular polygon, ask
students how many different pyramids are possible if the other faces are
equilateral triangles? Explain.
E6.8 Ask students to explain in words the difference between a
pentagonal pyramid and a pentagonal prism.
E6.9 Ask students to explain the connection between the lines of
symmetry for a prism and the axis of symmetry for the same prism.
Atlantic Canada Mathematics Curriculum
8-95
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
SCO: By the end of grade 8,
students will be expected to
In earlier grades (grade 4, E2, and E3) students have built threedimensional structures using connecting cube shapes from threedimensional drawings on isometric dot paper. They have also made
(grade 5, E3) three-dimensional drawings of cubes on isometric dot
paper, and interpreted these drawings to find possible “missing” cubes.
As well, students have made (grade 6, E2) orthographic views (2-D
views of 3-D shapes) showing front, top, and right views of a given
Cube-A-Link structure and represented that as a mat plan. Also,
they have progressed in the opposite direction making Cube-A-Link
structures from orthographic views and from mat plans.
E7 draw isometric and
orthographic views of 3-D
shapes and construct 3-D
models from these views
E7 To address this outcome, students should re-address all of the above
but use “non-cube” shapes as their models. Shapes such as those found
in the GeoBlock set would be perfect for this.
Students might begin by sketching some simple figures (parallel lines,
parallel angles, triangles, and donut squares) on dot grid paper, and on
isometric dot paper, then finally on plain paper.
Also, they might review the various techniques of
sketching, but using the geoblocks as their models. For
example, they could be given a block, like the R2 Block
(see above), a rectangular prism, and be asked to make
the three orthographic views (the top, the front, and the
side) of it. They should be given orthographic views and
asked to find the block that fits the drawing.
Which block do these views represent? Is
there any other block?
They could be asked to draw more than one
set of orthographic views for any block.
8-96
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
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
• Cube-a-links
E7.1 Ask students to examine each set of orthographic views, construct
the object, and draw the isometric drawings to represent it.
• Geoblocks
a)
b)
• Mathematics 8: Focus on
Understanding, pp 410–417
E7.2 Ask student to
a) create a figure using 3 geoblocks
b) draw and label the top view, front view, and side view of the figure
c) have a partner build the figure and compare it to the original
d) use the sketches from the partner to build a figure that has the
same top view, but has different side and/or front views (Draw a
perspective drawing of it on grid paper.)
E7.3 Ask students to complete the sketches started here:
a)
b)
Use a geoblock
other than a T3
c)
Use a geoblock other
than the T9 and T2
E7.4 Draw the front, top, and side views for a P Geoblock.
Interview
E7.5
a) A particular geoblock has all three (top, front, and side) views
identical.
i) Ask students what type of geoblock it is. Explain.
ii) Ask students what type of block has only two views the same.
b) Ask students how many different sets of orthographic views there
can be for
i) R1
ii)
S5
iii)
C4
c) Ask students how many different sets of orthographic views there
can be for a T2. Have students sketch them all using a regular grid.
Atlantic Canada Mathematics Curriculum
8-97
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Outcomes
Elaboration—Instructional Strategies/Suggestions
Students should be able to use multiple blocks to form a figure. They
should sketch the orthographic views and give them to their partners to
have them build the figure from the sketches.
Students should then move on to isometric (perspective drawing)
sketching of the various blocks in the set, beginning with the cube to
remind themselves of the importance of positioning of the object to
assist in the drawing. (That is, hold the block at eye level so that the
block is parallel to the floor. Rotate it 45° so that the vertical edge is
directly in front of one eye. Close the other eye and tilt the cube (C2)
downward, then upward to see the two views shown here.
At first give students the beginning of the sketches for the square prisms,
and have them complete the isometric drawings. Have them move on to
the rectangular prisms, then the triangular prisms.
Finally, students should sketch the 3-D objects on ordinary grid paper
(perspective drawing). If they begin with the familiar cube, they can
have discussions about the fact that even though the edges of a cube
are the same length, when sketched on regular grid paper, some edges
appear shorter than others. Have students complete started sketches of
the various blocks using regular grid paper.
8-98
Atlantic Canada Mathematics Curriculum
Specific Curriculum Outcomes, Grade 8
GCO E: Students will demonstrate spatial sense and apply geometric concepts, properties, and
relationships.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
E7.6
a) Ask students to trace around the edges of the yellow regular
hexagon.
b) Make a perspective drawing of the hexagon.
c) Describe in words to your friend on the phone the procedure you
followed to make the drawing in part (b).
d) Make isometric and orthographic drawings of the solid in part (b).
Name the solid.
Journal
E7.7 Ask students to find a pair of geoblocks that are similar, and
explain why they think they are similar. Sketch them on a regular grid.
Atlantic Canada Mathematics Curriculum
8-99