space and shape (geometry) - cape breton€¦ · space and shape (geometry) general curriculum...

Space and Shape (Geometry)

General Curriculum Outcomes E:

Students will demonstrate spatial sense and apply geometric concepts, properties, and relationships.

Revised 2011

AtlAntic cAnAdA MAtheMAtics curriculuM

specific curriculuM OutcOMes, GrAde 9

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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

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 co-ordinate plane

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).

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:

º B __' or (coordinates for B) ( ___ , ___ )

º __ 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).


AtlAntic cAnAdA MAtheMAtics curriculuM 9-77


Suggested ResourcesWorthwhile Tasks for Instruction and/or Assessment

Performance

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 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 ABC when the mapping (x, y) (–2x, -2y) is applied to it?

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.

• Mathematics 9: Focus on Understanding Geometry Supplement, pp. 6–15

• rulers

• protractors



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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).]





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.



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v) draw inferences, deduce properties, and make logical deductions in synthetic (Euclidean) and transformational geometric situations


E2 make and apply informal deductions about the minimum sufficient conditions to guarantee a translation, reflection, and a 180° rotation

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.

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, and B E) may lead to the conclusion that indeed one triangle is the 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.





Performance

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.

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].

• Mathematics 9: Focus on Understanding Geometry Supplement, pp 16–23

• Bull’s Eye

• protractor

• ruler



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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.





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)

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)

c) d)

c) d)

Given: A-B-D-E Given: A-B-D-E CF=AD=BE

Given: A-B-D-E AC DF DC EF

Given: A-B-D-E CA FD

Journal

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.



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ii) compare and classify geometric figures, understand and apply geometric properties and relationships, and represent geometric figures via coordinates

iv) represent and solve abstract and real-world problems in terms of 2- and 3-D geometric models



E3 make and apply informal deductions about the minimum sufficient conditions to guarantee the similarity of two triangles

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.

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.

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.

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~)

OA’ = 2.5OAOB’ = 2.5OBOC’ = 2.5OC

OD’ = 0.3ODOE’ = 0.3OEOF’ = 0.3OF





Paper/Pencil

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

b) In diagram below, AD = 3 cm, BC = 5 cmCD = 4 cm, AB = 6 cm

c) Given: B 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, but make B E. Have them measure and record the lengths AC and DF, examine the ratio , and now make a conjecture based

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.

DFAC


• Bull’s Eye

• Geo-Strips

• ruler



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Note: students should understand the relationships that exist between corresponding sides of similar triangles. That is, for ABC and PQR,

if ABC ~ PQR,

then

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

When similarity between two triangles can be confirmed, ratios such as these may be useful in solving problems.

=AB BC ACPQ QR PR

=

AB PQ AB PQ AC PRAC PR BC QR BC QR

= or or= =





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) = ? ii) = ? iii) = ? iv) = ?

v) Ask students to explain why = .

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.

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 .

ACCO

ACPD

AOOBACCO

ODDB

ODOB



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i) construct and analyze 2- and 3-D models, using a variety of materials and tools



E4 make and apply generalizations about the properties of Platonic Solids

E4 The properties that students will discover through guided exploration using activities like those found on the Worthwhile Tasks pages are:

• 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.





Performance

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

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 semi-regular, 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


• Polydron pieces

• Polydron Framework pieces

• The Visual Geometry Project presents “The Platonic Solids” (DVD)



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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 # of faces # of vertices

tetrahedron 4 4

cube 6 8

octahedron 8 6

dodecahedron 12 20

icosahedron 20 12





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 # of vertices # of faces # of edges

tetrahedron 4 4 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.)



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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.)



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i) construct and analyze 2- and 3-D models, using a variety of materials and tools



E5 solve problems using 3-D shapes using visualization, reasoning, and geometric modeling

E5 Students should model, using Polydrons, and study various non-regular 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.

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.





Performance

E5.1 a) Ask students to build a square based pyramid.

b) This polyhedron is not regular. Ask students which properties it lacks.

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.


• Polydron pieces



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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.





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.



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E6 recognize, name, describe, and represent arcs, chords, tangents, central angles, inscribed angles and circumscribed angles, and make generalizations about their relationships in circles

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.

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.





Pencil/Paper

E6.1 Determine the measure of each indicated angle or arc. Explain your answers.

a) b)

c) d)

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.


• Bull’s Eye

• ruler



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• 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.





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 end-point 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.

space and shape (geometry) - cape breton€¦ · space and shape (geometry) general curriculum...

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