Shape and Space General Curriculum Outcome E: Students will demonstrate spatial sense and apply geometric concepts, properties and relationships.
Elaboration Instructional Strategies/Suggestions KSCO: By the end of grade 6, students will have achieved the outcomes for entry-grade 3 and will also be expected to i) identify, draw and build physical models of geometric figures and iv) solve problems using geometric relationships and spatial reasoning SCO: By the end of grade 4, students will be expected to E1 draw various nets for rectangular prisms and cubes E2 construct models for various cylinders, cones, prisms, and pyramids E3 construct shapes, given isometric drawings E4 explore relationships among 3-D shapes E1 Students previous experiences have been with cutting out and assembling prepared nets. It is now expected that students will draw their own nets for rectangular prisms, square prisms, and cubes. They will also consider the various possibilities for these nets. Have the students trace on paper the various faces of a cube/ rectangular prism to make its net. Have the students cut out the net and fold it up around the shape to see if it works. Ask them to record this net on grid paper. Have them cut one of the faces off and investigate the possible places it could be reattached to make a new net. Have them record each one on grid paper. E2 Students can cut out prepared nets for cylinders and cones. They can make skeletal models for prisms and pyramids, using rolled newspapers and tape, straws and string, or toothpicks and miniature marshmallows. There are commercial materials available that allow students to snap faces together to build 3-D shapes. E3 In addition to building from drawings of common 3-D shapes, possibly also include drawings of structures in which the representations allow for hidden cubes. For example: The study of geometry helps students represent and make sense of the world. Geometric models provide a perspective from which students can analyse and solve problems, and geometric interpretations can help make an abstract (symbolic) representation more easily understood. (Curriculum and Evaluation Standards for School Mathematics, p. 112) E4 Design explorations in which students will discuss such matters as shapes that have the same number of faces, edges, or vertices how a cone is like/different from any pyramid how a cylinder is like/different from any prism how sizes of 2 rectangular prisms or 2 square prisms compare (informally) Students should be able to discuss the similarities and differences between the prisms and pyramids that share the same name, (e.g., square prism and square pyramid, or hexagonal prism and hexagonal pyramid). 4-68
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Performance E1.1 Tell the students that this diagram is part of a net for a square prism. Ask them to complete the net by drawing the additional faces that would be needed. E1.2 Provide students with a square or rectangular prism and a 11-pin x 11-pin geoboard. Ask them to use elastics to construct a net for the prism. Ask them to discuss how they might move one of the faces to make a new net for the same prism. Have them check by recording the new net on square dot paper and cutting it out. E1.3 Provide the students with a pentomino puzzle piece (a 2-D shape made by joining 5 squares along full sides) that would fold to make a box with no top. Ask them to trace this piece and then add a square for the top of the box. Ask: In how many places can this square be added? (Note: This can be cut from grid paper.) Example: Give them E1 Unit 8: Culminating Work (pp. 339-342) E2 Unit 8: Activity 2 (pp. 327-330) Culminating Work (pp. 339-342) E4 Unit 8: Activity 2 (pp. 327-330) E2.1 Ask students to build a skeletal model of the shape that has these faces: E2.2 Ask students to build a prism that uses nine toothpicks for edges. E2/4.1 Ask students to build skeletal models of two different triangular pyramids. Ask them how they are the same/different? E3.1 Ask the student to use cubes to build two possible structures for this drawing: Interview E4.1 Ask: Who are we? We both have six vertices. We both have some triangular faces. 4-69
Elaboration Instructional Strategies/Suggestions KSCO: By the end of grade 6, students will have achieved the outcomes for entry-grade 3 and will also be expected to i) identify, draw and build physical models of geometric figures iv) solve problems using geometric relationships and spatial reasoning SCO: By the end of grade 4, students will be expected to E5 find all possible composite figures that can be made from a given set of figures E5 Students need many hands-on experiences putting shapes together to make new shapes before they are able to visualize this outcome. They should be encouraged to predict the results before they do the combining and to follow this activity with a pulling apart and visually recombining. Show students two isosceles right triangles (e.g., two small triangles in a tangram set) on an overhead. Have them predict what polygons could be made by combining them with equal sides fully joined. Check their predictions and/or explore other possibilities. Have students investigate all the distinct shapes that can be made from four congruent squares (with sides fully joined). Many students might recognize these as the shapes in Tetris, a computer game. Have groups of students investigate all the distinct shapes that can be made from three pattern blocks (with equal sides fully joined). These could be put on display and/or traced on paper to record. Have students find the midpoints of three sides of a square. Have them join consecutive midpoints so that the square is divided into 2 right triangles and a pentagon. Have them cut the square into these 3 pieces and investigate all the shapes that can be made using them. Have students investigate the distinct shapes that can be made from four cubes with full faces joined. Put these on display. Compare to the four-square activity above. A good way to explore shapes...isto use smaller shapes or tiles to create larger shapes. Different criteria or directions can provide the intended focus to the activity. Among the best materials for this purpose are pattern blocks, but many teacher-made materials can be used. (Elementary School Mathematics, p. 331) Note: These activities involve many aspects of spatial sense, particularly discrimination, position-in-space, perception of spatial relations, and perceptual constancy. Experiences such as these are useful for students to develop visualization ability. 4-70
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Performance E5.1 Give students a picture of two congruent isosceles (not right-angled) triangles side by side. Ask them to predict the shapes that could be made by combining them with equal sides fully joined. Ask them to cut out the triangles, check their predictions, and investigate other possibilities, if necessary. Compare these shapes with those made by 2 congruent isosceles right triangles (see Elaboration section). E5.2 Provide students with four squares (two each of two colours). Have them cut each square along one diagonal. Ask them to arrange the eight triangles inside a large square to make a quilt block design. E5.3 Ask students to investigate all the distinct shapes that can be made by combining the square and two small triangles from a tangram puzzle (with equal sides fully joined). Have them record those shapes by tracing on paper. E5.4 Ask students to investigate the different 3-D shapes that can be formed by combining two common 3-D shapes (with congruent faces fully joined). (For example, if they used two congruent rectangular prisms, they could make three different new rectangular prisms by matching different faces.) E5 Unit 5: Activity 1 (pp. 177-180) Paper and Pencil E5.5 Provide students with square dot paper and ask them to draw all the shapes that can be made from Interview E5.6 Provide the student with a variety of polygons made by outlining three combined pattern blocks. Ask him/her to predict which three blocks are in each new shape and check the predictions by using the blocks. 4-71
Elaboration Instructional Strategies/Suggestions KSCO: By the end of grade 6, students will have achieved the outcomes for entry-grade 3 and will also be expected to ii) describe, model and compare 2- and 3-D figures and shapes, explore their properties and classify them in alternative ways SCO: By the end of grade 4, students will be expected to E6 recognize, name, describe and construct acute and obtuse angles E6 Students have had experiences recognizing and naming right angles as square corners in shapes and recognizing angles greater or less than right angles. Have students investigate angles in various shapes, using the corner of a piece of paper as a reference for right angle. (Does it fit the angle of the shape or is the angle greater/less than the corner of the paper?) Students should have experiences that introduce the names acute (less than right) and obtuse (greater than right but less than straight) to describe angles in shapes and angles as entities. They will learn to recognize these angles by their overall appearance, not by their measurements. Students often focus on the length of the arms of the angle rather than their spread; therefore, activities should involve angles with short, long, and a combination of short and long arms. They will need to understand that the arms of the angle could be continued to any length. These activities should be connected to measuring activities with angles. Angles should be presented in a variety of contexts (e.g., angles formed by the two hands of a clock, by the intersection of two roads, and by the blades of scissors or hedge clippers). Have students make various acute angles with pipe cleaners (e.g., almost a right angle, about half a right angle, almost no angle). Show students acute angles (with arms of different lengths) in various positions and of different sizes. Ask them to estimate each as almost right, almost no angle or about half a right angle. Have students find acute angles in various 2-D polygons and on faces of 3-D shapes. Note: Similar activities should be performed with obtuse angles. Students could use any right-angled corner of a piece of paper to check their estimates. Folding this corner in half could also help visualize half a right angle. Have students stand with their arms out straight and then make different types of angles according to your instructions. 4-72
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Performance E6.1 Tell students: Jeri went on a trip with her parents. To amuse herself, she sketched the positions of roads as they met. The following are some of her drawings. Ask: How many of the angles formed were right angles? acute? obtuse? E6 Unit 3: Activities 3 and 4 (pp. 97-104) E6.2 Ask students to classify the angles found on each of the faces of a 3-D shape (e.g., a hexagonal pyramid). E6.3 Have students investigate and name the angles formed when they print various capital letters, (e.g., A, E, L, M, W). E6.4 Ask students to explore the angles in the six different pattern blocks. Which blocks have only acute angles? only obtuse angles? both acute and obtuse angles? only right angles? E6.5 On overhead clocks, prepare 6-8 different times. Displaying them one at a time on an overhead, ask the student to name and describe the angle made by the hands of each clock. Paper and Pencil E6.6 Provide the student with toothpicks of two sizes. Ask him/her to make a display of three angles (one using two short toothpicks, one using two long toothpicks, one using a short and a long toothpick) for each of the following: a) an acute angle that is almost right, b) an obtuse angle that is almost a straight line, c) an acute angle that is almost half a right angle and d) an obtuse angle that is almost a right angle. E6.7 Ask students to combine two or more pattern blocks to make examples of acute, right, and obtuse angles. Have them record by tracing each one on paper. 4-73
Elaboration Instructional Strategies/Suggestions KSCO: By the end of grade 6, students will have achieved the outcomes for entry-grade 3 and will also be expected to ii) describe, model and compare 2- and 3-D figures and shapes, explore their properties and classify them in alternative ways SCO: By the end of grade 4, students will be expected to E7 recognize, name, describe and construct equilateral, isosceles and scalene triangles A teacher s questioning techniques and language in directing students thinking are critical to the students development of an understanding of geometric relationships. Students should be challenged to analyze their thought processes and explanations. (Curriculum and Evaluation Standards for School Mathematics, p. 113) 4-74 E7 Students should have guided explorations to discover patterns (properties) of different types of triangles. Prepare pictures on cards or cutouts of several examples of these three kinds of triangles. Ask the students to sort them into three groups. Ask them to explain their sort. Often, they will sort them by how their sides look, without knowing the actual names. If so, this will lead to a focus on measuring and comparing the sides, and noting common properties to which the names equilateral, isosceles, and scalene can be attached. (If not, the teacher may sort them, ask the students to determine the sorting rule, and do other explorations.) Mix up a set of triangles and have students sort them by numbers of lines of reflective symmetry. Folding, or using Miras, should be part of this exploration. Ask students what they notice about the resulting sets of triangles. Sorting and exploring for lines of symmetry should lead to these patterns (properties): a) equilateral triangles have three lines of symmetry, b) isosceles triangles have one line of symmetry and c) scalene triangles have no symmetry. Exploring triangles by folding to compare and measuring the angles could lead to discovering these patterns: a) all angles in equilateral triangles are equal, b) two angles in isosceles triangles are equal and c) all angles in scalene triangles are different. Provide various lengths of strips of paper for students to make triangles that can be classified and their properties discussed. Provide students with straws, scissors, and string to thread through the straws. Ask them to cut and assemble one example of each kind of triangle. Note: A property of a set of shapes is a characteristic that all members of the set have in common; for example, if a shape is classified as an isosceles triangle, it can only be assumed that it has one line of reflective symmetry or two equal sides. An equilateral triangle could be placed under the isosceles heading because it has these two properties; its additional properties are of no account, except that it would be better classified as equilateral where more pro-perties can be assumed. However, it is recommended that students see these as two distinct shapes at this point in their development.
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Performance E7.1 Ask students to draw, on squared paper, an example of each member of the quadrilateral family: square, rectangle, rhombus, parallelogram, trapezoid, and kite. Have them make a copy of each of these figures. Ask them to draw in one diagonal in each figure in one set of drawings. Ask them to describe and name the triangles formed in each figure. Ask: Are they congruent? Did you get the same results as your neighbour? Ask them to draw in the other diagonal in each figure in the other set of drawings. Ask them to describe and name the triangles formed in each figure. Ask: Are they the same as you got for the first diagonal drawn in each figure? E7.2 Provide students with two congruent equilateral triangles, two congruent isosceles triangles, and two congruent scalene triangles. Ask them to predict what polygons could be made by combining two congruent equilateral triangles. Have them check their prediction. (Repeat with the other pairs.) Which pair of congruent triangles produced the greatest variety of polygons? E7.3 Ask students to explore a variety of 3-D shapes to find triangular faces. Have them trace around them on paper and classify the triangles drawn. E7.4 Ask the student to draw a right triangle. Ask him/her to draw the image of the right triangle, using a Mira and one of the sides forming the right angle as a mirror line. Ask: What is the name of the triangle formed by the two triangles? Presentation E7.5 Provide groups of three students with 2m of yarn. Ask them to stand in formation to make each kind of triangle. Ask them to explain how they are sure that each one they are showing is correct. Portfolio E7.6 Provide students with ten toothpicks, all the same length. Ask them to investigate how many different triangles can be made using from as few as three up to as many as ten toothpicks for each triangle. Toothpicks may only meet at their endpoints. Have students draw to record and put the name of each type of triangle under its drawing. (See example at right.) Isoceles E7 Unit 3: Activities 1-4 (pp. 89-104) 4-75
Elaboration Instructional Strategies/Suggestions KSCO: By the end of grade 6, students will have achieved the outcomes for entry-grade 3 and will also be expected to ii) describe, model and compare 2- and 3-D figures and shapes, explore their properties and classify them in alternative ways SCO: By the end of grade 4, students will be expected to E8 make generalizations about the angle, side length, and parallel side properties of the various quadrilaterals E9 sort quadrilaterals under property headings E8/E9 Initially, students identified shapes by their overall appearance. While many of their properties have been implied, it is now the goal to make the properties of some shapes explicit. Squares, rectangles, parallelograms, rhombi, trapezoids, and kites should each be analysed separately so that students, through hands-on investigations, can see and describe the patterns that are properties of these quadrilaterals. Teachers should prepare a series of questions to guide the students investigations. Provide students with pictures or cutouts of a variety of rectangles. Ask them to compare opposite sides by sight and by folding over to compare directly (reflective symmetry: see E12). Have them measure the four sides in centimetres, and compare their measurements with those of other students. Have them describe the four angles of their rectangles. Ask: Are any of the sides parallel in your rectangle? Which ones? Is this the same for other students rectangles? Show the students a cutout of a large rectangle and ask them what one could say about the sides and angles. Have them write a summary, including some examples, of what they know about rectangles. Students should then be given application problems that require the use of these properties. (See the note on p. 76 for a discussion of property. ) Some Properties of the Quadrilateral Family At this level students begin to appreciate that the reason a collection of shapes goes together has something to do with properties. It makes sense to define and sort shapes by these properties rather than by appearances. (Elementary School Mathematics, p. 325) 4-76
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Performance E8.1 Provide students with four toothpicks or straws, all the same length. Ask: What different quadrilaterals can be made using all four? Using your knowledge of properties, what can you say about the sides and angles of each of these shapes? E8.2 Provide students with four toothpicks or straws, two each of two different lengths. Ask: What different quadrilaterals can be made using all four? What can you say about the sides and angles of each of these shapes? E9.1 Provide students with cutouts of several quadrilaterals. Ask them to select a property card (use the property headings on the chart in the Elaboration section) and place any/all of their cutouts that have this property with the card. Select another property card. Ask: Do any of them also have this property? Explain. E8.3 Have a variety of lengths of geostrips available. Assign the student a particular quadrilateral to make. Observe how he/she selects materials and builds it. Ask him/her to describe the properties the shape has. Paper and Pencil E8.4 A farmer wants to know the distance around his rectangular field. Ask students to explain the fewest number of measurements he would need to make. Why? If the field is square, then how many measurements would he need to make? Interview E8.5 Ask: What is the same and what is different about the sides of a kite and of a parallelogram? E9.2 Ask: Who am I? I am a quadrilateral. All of my sides are equal. None of my angles is a right angle. Presentation E8.6 Have lengths of rope or string available, four in one length, two each in two other lengths. Have groups of four students choose a quadrilateral to make, have them select the ropes they will need, and have them stand, holding these ropes, to make the desired quadrilateral. Have them describe the properties of the shape. E8 Unit 3: Activity 4 (pp. 101-104) Unit 8: Activity 3 (pp. 331-334) E9 Unit 3: Activity 4 (pp. 101-104) 4-77
Elaboration Instructional Strategies/Suggestions KSCO: By the end of grade 6, students will have achieved the outcomes for entry-grade 3 and will also be expected to ii) describe, model and compare 2- and 3-D figures and shapes, explore their properties and classify them in alternative ways SCO: By the end of grade 4, students will be expected to E10 make generalizations about the numbers of vertices, edges, and faces of various prisms and pyramids, cones, and cylinders E10 Through guided explorations, students should begin to identify some properties of 3-D shapes. Prior experiences stacking 2-D shapes to make prisms should have established the uniform nature of these shapes. In a similar way, stacking circles would result in a cylinder which has this same uniformity. Have students make, using toothpicks and marshmallows, skeletal models of prisms (triangular, rectangular, pentagonal, hexagonal) and record their findings in table form. By looking for patterns in this table and thinking about how the skeletal models were made, the students should find these patterns for prisms: a) the number of vertices for any prism is two times the number associated with its name (e.g., for an octagonal prism, 2x8 or 16 is its number of vertices because the vertices come from the 2 bases); b) the number of edges is three times the number associated with its name because the edges come from the 2 bases plus the edges that join the 2 bases and c) the number of faces is two more than the number associated with its name, two for the bases plus one each for the faces that join corresponding sides of the bases. Similar explorations with pyramids and their skeletons should result in finding these patterns: a) the number of faces is equal to the number of vertices, and both are one more than the number associated with the name because a triangular face starts from each side of the base; also, the base itself is a face, and the vertices are the vertices of the base, plus the single vertex (e.g., a pentagonal pyramid has 5 + 1or6vertices and faces) and b) the number of edges is two times the number associated with the name because each side of the base is an edge; also, there is an edge from each vertex of the base to the single vertex. Cylinders have two faces and one surface with no vertices; cones have one face, one surface, and one vertex. They have no edges. 4-78
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Performance E10.1 Ask students to build skeletal models of 3-D shapes that satisfy each of these conditions: a) a prism with 12 edges, b) a pyramid with seven faces, c) a shape with eight faces and eight vertices and d) a shape with eight faces and 16 vertices. Paper and Pencil E10.2 Place an octagonal prism and a cylinder side by side. Ask students to write a comparison of them, mentioning things that are the same and that are different. E10.3 Ask students to explain why the number of faces and vertices of a hexagonal pyramid is one more than the number associated with its name. Interview E10.4 Ask the student to use the terms vertices, edges, and faces to describe an octagonal prism. E10.5 Ask: What shape am I? I have five faces and five vertices. One of my faces is different from the other four. E10.6 Ask: a) What prism am I? I have 12 vertices. b) What prism am I? I have 15 edges. c) What prism am I? I have 12 faces. d) What pyramid am I? I have 11 faces. e) What pyramid am I? I have 12 edges. E10 Unit 8: Activity 2 (pp. 327-330) 4-79
Elaboration Instructional Strategies/Suggestions KSCO: By the end of grade 6, students will have achieved the outcomes for entry-grade 3 and will also be expected to iii) investigate and predict the results of transformations and begin to use them to compare shapes and explain geometric concepts (e.g., symmetry and similarity) SCO: By the end of grade 4, students will be expected to E11 predict and confirm the results of various 2-D figures under slides, reflections, and quarter/half turns E12 make generalizations about the reflective symmetry property of the various quadrilaterals E11 Encouraging students to think in terms of sliding, flipping, and turning shapes is a very useful strategy to help with their visualization in geometry. Provide students with a right triangle and have them try to visualize the images of this triangle reflected in each of its three sides. What shape do this triangle and its image make together in each case? (Answers: 2 different isosceles triangles and a kite). Have students, working in pairs, place two geoboards side by side. Have them make a triangle on the left-hand geoboard. Explain that this geoboard will be rotated a quarter turn clockwise. Ask them to try to make a triangle on the right-hand geoboard that will look like the other triangle after it is rotated. Repeat this activity several times, using other shapes and including half turns for some of the rotations. Have students draw a square on squared dot paper. What shape would result from combining this square and its image after a slide along any of its sides? Have students use shapes from the pattern blocks, tangrams, logic blocks, and other sources, to predict and confirm the results of the various transformations. Give the students pictures of shapes and their images under various transformations. Have them predict what the relationships are and then confirm, using tracing paper. E12 Have students draw on squared dot paper examples of the different quadrilaterals (square, rectangle, rhombus, parallelogram, trapezoid, and kite) and cut them out. Have them fold the shapes to find any lines of symmetry. Ideally, there would be a number of different examples of each shape in the class. Ask: Do all quadrilaterals of the same type appear to have the same number of lines of reflective symmetry? (Another investigation could involve using the Mira with pictures of the shapes.) 4-80
Worthwhile Tasks for Instruction and/or Assessment Suggested Resources Performance E11.1 Have students explore the different polygons that can be made with each of the six different pattern blocks under reflections in their sides and under half-turns about the midpoint of each side. Have them predict by visualizing; then, using two of each block, do the transformations, and trace around each result. Ask: Which blocks produced only one shape under all these transformations? Which block produced the most shapes? Have them examine all the shapes made under these transformations. Ask: Could any of them also be described as a pattern block under a slide? E11.2 Have students work in pairs, with a geoboard between them. One student of each pair should make a shape on the geoboard and ask the other student to make its slide, reflected, or turned image on his/her geoboard. Have them check by using a Mira (for reflection) or turning the original (for a turn) after the visualized image has been placed on the geoboard. Have the pairs of students change roles. E11.3 Place the large triangle from the tangram set on the overhead. Ask the students to visualize and sketch each shape formed by reflecting this triangle in each of its sides. How many different shapes were made? Interview E12.1 Tell the student that someone says he/she has a quadrilateral with three lines of symmetry. Ask him/her to explain how this could happen. E12.2 Ask the student to name each shape, to state how many lines of symmetry each has, and to show where they are. Portfolio E12.3 Ask the student to fold a piece of paper in half and in half again, and to cut out from the corner of the double fold a polygon that will unfold to make a rhombus. Have him/her repeat to make a rectangle. 4-81
Elaboration Instructional Strategies/Suggestions 4-82