Geometry 2 Part 1 Introduction

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2 Geometry 2 Part 1 Introduction In this unit, students identify, describe, compare, and manipulate 2-D geometric shapes: circles, squares, rectangles, triangles, and other polygons. Through a wide variety of activities and explorations, students develop and apply their reasoning skills to identify attributes of 2-D shapes. They communicate their understanding by naming and describing shapes and their attributes, including symmetry. They use visualization to help them recognize polygons in various positions and as components of composite shapes and to identify symmetrical shapes and designs. They connect 2-D shapes to faces of 3-D objects in the environment and connect measurement and geometric concepts. As students explore problems concerning geometric relationships, as they play games, and as they create shapes and designs, they use and practise problem-solving strategies such as making and investigating conjectures, reflecting on the reasonableness of an answer, guessing and testing, and making an organized list, among others. Manipulatives Many lesson plans require manipulatives such as geoboards and standard 1-inch pattern blocks. We have also provided BLM Pattern Blocks (p XXX) to help you create manipulatives. BLM Attribute Blocks (p XXX XXX) provides many different shapes with various geometric and non-geometric attributes, such as patterns, colours (shades of black), sizes, curved and straight sides, symmetrical and nonsymmetrical, and so on. We recommend that you reproduce BLM Attribute Blocks onto paper in different colours, laminate the sheets, and cut the shapes out. You can also paste the shapes onto thicker materials, such as cardboard or foam, to create thicker shapes, and then use thickness as an attribute. We will refer to these shapes throughout as attribute blocks. You will often need large paper shapes for demonstrations. Shapes to be traced (but not folded) can be cut from cardboard. Vocabulary As students learn the names of different shapes, add them to your word wall. NOTE: Students do not need to learn the names of all the shapes they encounter. Shapes such as parallelograms and trapezoids are referred to by their proper names in the lesson plans for your reference. Student Readiness Some students may not yet have developed the understanding and skills needed to decode visual information such as geometric shapes. These students may have difficulty copying numbers or letters or drawing shapes because they are unable to identify the important information in what they are looking at. The activities on BLM Geoboards (p XXX XXX) and BLM More Geoboards (in the Online Guide) will develop and strengthen these students visual decoding skills. You can also use geoboards and washers or elastics and have students copy figures you make, beginning with single dots, then lines, then very simple figures on 9-dot geoboards and working up to more complex patterns and shapes on larger arrays. 2 Teacher s Guide for Workbook 2

3 ACTIVITIES These activities can be used and repeated throughout the unit, for practice and reinforcement. Probability and Data Management. Arts Literature Music Technology Science and Literature Sorting shapes. Distribute yarn circles (or use BLM Sorting Circles, p XXX) and collections of shapes (attribute blocks) and have students sort the shapes. For example, put all the squares in the circle, put all the triangles in the circle, and so on. Once students have learned about two or more shapes, they can use two yarn circles (or BLM Sorting into Two Groups, p XXX) to sort by two different criteria simultaneously. For example, put all the squares in one circle and the rectangles in the other. Collages and posters. Invite students to make collages or posters for the different shapes they learn about. They can look for examples of each shape in old catalogues, magazines, and newspapers. They can also look for particular shapes in everyday objects and add drawings to their collages or posters. For example, after learning about squares (G2-1), students could create a poster with the title What Looks Like a Square? or I See Squares. When a Line Bends, A Shape Begins by R.G. Greene. Ten shapes in pictures and verse. Use as an introduction to the strand. You can find a wide variety of songs about shapes online. Search using the keywords songs and shapes. Teach students to create and manipulate circles, squares, rectangles, triangles, and other polygons on the computer, using a word processing or drawing program. The Very Busy Spider by Eric Carle, Spiders and Their Kin by Herbert W. Levi and Lorna R. Levi. Read The Very Busy Spider and then ASK: How does a spider create its web? What geometric shapes can you identify in the process? What shapes do you see in the ready web? Students are likely to identify parts of the web as circles. Ask them to draw the web as spiders build it, without lifting their pencils. Students will quickly realize the web is a spiral, not individual circles. As an extra challenge, discuss the shapes and lines of symmetry that students see in the bodies of spiders and other creatures presented in Spiders and Their Kin (or another junior field guide). Meeting Your Curriculum Lesson G2-6 Circles addresses curriculum expectations for the Grade 1 Ontario curriculum and so is optional for Ontario students. Lessons G2-7 to G2-13, address curriculum expectations which are on the Grades 3 and 4 of WNCP. These lessons are optional for those who follow the WNCP curriculum framework. Geometry 2- Introduction 3

4 G2-1 Lines Pages Curriculum Expectations Ontario: 1m46, 2m5, 2m47 WNCP: 1SS2; 2SS8, [CN, V] Goals Students will identify straight and curved lines and sides, and open and closed lines. PRIOR KNOWLEDGE REQUIRED knows the words line and shape Vocabulary line shape straight curved closed open square triangle rectangle circle Materials string or rope masking tape large paper square, rectangle, triangle, and circle attribute blocks made from BLM Attribute Blocks (p XXX XXX) BLM Find Closed Lines (p XXX) Curved and straight lines. Draw an assortment of straight line segments (see samples in the margin). Tell students that lines that go straight from one point to another are called straight lines. Draw a few curved line segments. Tell students that a line that isn t straight and has no pointed corners is called a curved line. Have volunteers draw two or three more examples of straight and curved lines. online guide Investigating lengths of straight and curved lines. Probability and Data Management. straight lines Sides of a shape. Show students a large paper square. ASK: What shape is this? If students do not recall the name, SAY: This shape is a square. Run your finger along each of the sides in turn and SAY: These are the sides of the square. ASK: Are the sides of a square straight or curved? (straight) Repeat with a large paper triangle and rectangle. Then show students a large paper circle and ASK: Which type of line makes a circle? Is a circle made with a straight line or a curved line? Show students some shapes with curved and straight sides (from BLM Attribute Blocks, for example). Point to each side of the shape in turn and ASK: Is this side straight or curved? Draw several shapes on the board and have students say which shapes have only straight sides, only curved sides, or both. Then have volunteers draw shapes for classmates to describe. ACTIVITY 1 curved lines Sorting shapes (see page XXX). Give students a large variety of attribute blocks. Have them sort the shapes according to whether they have all straight sides or at least one curved side. 4 Teacher s Guide for Workbook 2

5 Closed and open lines. Draw two curved lines, one closed and one open (see examples in the margin). SAY: These are two paths. Draw a person at some point on the closed path and ASK: What will happen as this person keeps walking along the path where will she end up? (back where she started) Draw a person at one end of the open path and ASK: Where will this person end up as he walks along the path? Will he end up where he started? (no) Why not? (because the path has two ends) Tell students that a path that has no ends is called a closed path. A path that has two ends is called an open path. SAY: We also use the words closed and open to describe lines. Draw several more curved lines (without vertices), both closed and open, and ASK: Is this line closed or open? closed path open path Shapes are closed lines. Draw three sides of a square on the board. ASK: Is this an open line or a closed line? (open) Then add the fourth side to the square. ASK: Is a square an open or closed line? (closed) Will people walking along the sides of a square end up where they started? (yes) If necessary, explain that the corners of the square are not ends. Point out the difference between closed and open lines: If you draw a closed line on paper, it creates a shape that you can cut out. An open line does not create a shape that you can cut out. Draw a shape similar to a square but with one curved side and ASK: Is this a closed line? (yes) Why? (because if you walk along it you will end up where you started) How is this shape different from a square? (It has a curved side.) Repeat with a triangle and a circle. Drawing closed and open lines. On one side of the board, draw a variety of closed lines, including squares, rectangles, triangles, circles, and other shapes with straight and curved sides. ASK: Are these open or closed lines? (closed) Label the shapes Closed lines. Draw a few open lines on the other side of the board. Include a square with a missing side, a triangle with a break in one side, a spiral, and a W shape. Point to each open line and ASK: How is this line different from the closed lines? (It has a break in it; the ends don t join up.) Have volunteers draw some lines of both types. Journal Have students draw examples of closed and open lines in their journals, using both straight and curved lines. Art ACTIVITIES Tic-tac-toe. Students can complete BLM Find Closed Lines. Students can then play an adapted version of Tic-tac-toe (on the board or on scrap paper) where one player draws closed lines and the other draws open lines rather than Xs and Os. 3. Collages and posters (see page XXX). Invite students to make collages or posters showing the different types of lines they have learned about in this lesson. Geometry 2-1 5

6 G2-2 Sides and Vertices Pages Curriculum Expectations Ontario: 2m2, 2m3, 2m5, 2m48 WNCP: 2SS8, [CN, R, V] Goals Students will identify and count sides and vertices of shapes. They will sort shapes by the number of vertices. PRIOR KNOWLEDGE REQUIRED Vocabulary side corner vertex/vertices square rectangle circle triangle knows the word shape can identify the sides of a shape can identify triangles, squares, rectangles, and circles Materials attribute blocks made from BLM Attribute Blocks (p XXX XXX) large paper triangle, square, and rectangle masking tape BLM Sides and Vertices (p XXX) BLM Sorting into Two Groups (p XXX) vertex 3 vertices Introduce vertices. Give students attribute block triangles. Show a large paper triangle. ASK: What shape is this? (triangle) Run your finger along a single side of the triangle and SAY: This straight part is a side. Run your finger all the way around the edge of your triangle and have students do the same with their blocks. ASK: Are there places on the edge of the triangle that feel different from the sides? How do these places feel different? (They are sharper, or pointed.) Point to one of the vertices on your paper triangle and SAY: A corner of a shape is called a vertex. Point to the three vertices of the triangle one by one and SAY: When there is more than one vertex, we say vertices. This triangle has three vertices. Trace your triangle on the board, draw a small arrow pointing to each vertex, and write the terms vertex and vertices as shown. ACTIVITY 1 problem solving Visualizing Counting vertices game. Give pairs a set of attribute blocks. Player 1s close their eyes. Player 2s place a block in their partners hands. Player 1s count the number of vertices on the shape by feel, then open their eyes to check the count. Players swap roles. Students can play this game until they are good at it. Discuss strategies: do not rotate the shape, because you might count some vertices twice; use one hand to hold the shape and the other to count; keep a finger on one corner at all times to know where you started, etc. Also note how the number of vertices of a shape does not change no matter what position the shape is in. Vertices of open lines. Draw any type of angle on the board. Explain to students that a vertex is created when two lines meet at a point. Tell students that the line you just drew has one vertex. ASK: Is this line open 6 Teacher s Guide for Workbook 2

7 or closed? (open) Draw the open lines shown in the margin and ASK: How many vertices do these open lines have? Then draw some closed lines with two or three curved sides (e.g., circle, semi-circle, heart, half moon). ASK: Are these lines open or closed? (closed) How many sides and vertices do these shapes have? 1 vertex 2 vertices 3 vertices 4 vertices problem solving Making and investigating conjectures Reflecting on other ways to solve a problem vertices side Extra practice BLM Sides and Vertices 1 3 sides Counting vertices. Use masking tape to create a polygon (a closed line with straight sides) with seven or eight vertices on the floor, and let students guess how many vertices it has. Then ask volunteers to stand at the vertices and ask another volunteer to count how many students are standing. ASK: Was your guess right? Repeat for more polygons (but do not use this term yet). Then solve the problem a different way: instead of counting volunteers, label the vertices with numbered cards (1, 2, 3, and so on) and have students verify that the answers are the same. Draw a large triangle on the board and number each vertex (as shown at left). ASK: How many vertices does the triangle have? (3) How do you know? Repeat, this time beginning at a different vertex. ASK: Does it matter which vertex you start counting at? (no) Repeat with a square. Then draw a variety of other polygons and have volunteers count the vertices by numbering them. Bonus Draw a circle. ASK: How many vertices does a circle have? (none) Counting sides. You can now explain that a side of a shape is a line joining two vertices. Use diagrams to show students how to count the sides of shapes by numbering them, as they did vertices. ACTIVITIES Sorting shapes (see page XXX). Give each student 8 or 9 attribute blocks (see sample group below; the shapes appear on pages 1 and 5 of BLM Attribute Blocks) and a 2-circle sorting chart (such as on BLM Sorting into Two Groups). Ask students to sort the shapes into the groups 4 vertices and 3 vertices. ASK: How many sides do shapes with 4 vertices have? Repeat for shapes with 3 vertices. Then ASK: Are there shapes that stay outside the sorting circles? How many sides and vertices do these shapes have? 3. Counting sides and vertices. Ask students to draw various shapes on grid paper. Then have them swap their drawings with a partner and count the sides and vertices in each other s shapes. Geometry 2-2 7

8 G2-3 Squares Page 197 Curriculum Expectations Ontario: 1m46; 2m1, 2m5, 2m6, 2m47, 2m48 WNCP: 1SS2, 1SS4; 2SS8, 2SS9 [CN, V] Vocabulary shape vertex/vertices square closed side square corner straight Goals Students will identify squares by their attributes and draw squares. PRIOR KNOWLEDGE REQUIRED knows the word shape can identify squares visually can identify straight and closed lines can identify and count sides and vertices of shapes Materials large paper shapes (squares and others) BLM Finding Squares (p XXX) and pattern block squares BLM Square or Not? (p XXX) and BLM 2 cm Dot Paper (p XXX) online guide Extra support: BLM The Square BLM Matching Squares and Not Squares online guide An alternate explanation using an overhead projector. Being a square does not depend on size, colour, or pattern. Show students a large paper square. Ask them to name the shape. Divide the board in two. Trace the paper square on one side of the board so that the bottom side is parallel to the ground. ASK: Is this a square? How do you know? (It is the shape of a square.) Draw a smaller square in the same position (i.e., bottom side parallel to the ground) and ask the same questions. Draw more squares of varying sizes and colours, but again not rotated. Add a pattern (e.g., a dotted pattern) to some squares. ASK: Do the dots change the shape? Is it still a square? Label the shapes you ve drawn Squares. Being a square does not depend on position. Affix your large paper square to the board in a slightly rotated position (so that the bottom side is not parallel to the ground). Trace and then remove the paper square. ASK: Is this shape a square? How do you know? Point out, if necessary, that the shape is the same square, only turned slightly. Repeat several times, increasing the angle of rotation, until the square is standing on a vertex. Emphasize that the shape did not change; you only turned it. ACTIVITY 1 Give each student a copy of BLM Finding Squares and a pattern block square. Students use the square block to determine which shapes on the BLM are squares and cross out those that are not. The letters on the remaining shapes form the word square, providing an opportunity for self-checking. Squares have 4 sides and 4 vertices. Have a volunteer number and count the sides of one of the squares drawn on the board. ASK: How many sides does the square have? Repeat for vertices. Repeat for a square of a different size, and a square that is rotated. ASK: Do all squares have 4 sides 8 Teacher s Guide for Workbook 2

9 problem solving Generalizing from examples and 4 vertices? (yes) Does the number of sides and vertices change when a square is turned? (no) Squares have straight sides. Draw a shape like a square but with one side curving inwards. ASK: Is this a square? (no) Why not? (one side is curved) Have a volunteer fix the drawing so that it is a square. ASK: Do all squares have all straight sides? (yes) A square is a closed line. Draw a square with a small break in one side. ASK: is this a closed line or an open line? (open) Is this a square? (no) Why not? (One side has a break in it.) Have a volunteer fix the drawing so that it is a square. ASK: Is a square a closed or an open line? (closed) online guide More on comparing corners and square corners. Data Management and Probability-Sorting online guide A way to reflect on what made the problems on BLM Square or Not? easy or hard. Squares have square corners. Draw an acute, right, and obtuse angle on the board (but simply refer to them as corners for now). Ask for a volunteer to circle the corner that matches the corners on all the squares. Tell students that squares have a special type of corner that we call a square corner. Sorting shapes. One by one, show students several large paper shapes, both squares and not squares (e.g., different rectangles, a parallelogram, a circle, a triangle, a trapezoid, a pentagon but students will describe these shapes simply as not squares ). Include one or two shapes with curved sides. Decide as a class whether each shape is a square or not and affix it to the appropriate half of the board. Encourage students to look at and count the sides and vertices of the shapes to help them decide. (The fact that the sides of a square are all of equal length is taught in the next lesson; if a student suggests this idea now, tell the class that you will check this later.) Continue with shapes drawn on the board, include one or two open lines; draw the first three-four examples yourself and invite volunteers to draw others. Prompt students to use different sizes and patterns. ACTIVITIES 2 & 3 2. BLM Square or Not? asks students to add the missing side to various shapes and decide whether the completed shapes are squares. Students should first predict which shapes will be squares. Point out that you can rotate the paper to check visually whether the shape is a square. 3. Have students draw squares of various sizes and orientations on BLM 2 cm Dot Paper. Also, have students draw some shapes that have 4 sides and 4 vertices but are not squares (these may be many types of quadrilaterals or have at least one curved side). Journal Have students trace an attribute block square in their journals and create a picture incorporating that square. Squares in the environment. After students have completed the worksheets, point to various objects around the room and ask if these shapes look like squares. Students can compare the faces of 3-D objects with the squares drawn on the board. Then have volunteers identify other objects that include squares. Geometry 2-3 9

10 G2-4 Rectangles Pages Curriculum Expectations Ontario: 1m46; 2m2, 2m5, 2m7, 2m47, 2m48 WNCP: 1SS2, 1SS4; 2SS8, 2SS9 [C, CN, R, V] Goals Students will identify and model rectangles. They will compare squares and rectangles. PRIOR KNOWLEDGE REQUIRED Vocabulary square rectangle side equal longer shorter knows the word shape can identify rectangles and squares visually can identify straight and closed lines can identify and count sides and vertices of shapes can perform direct and indirect comparisons of length can use a ruler to measure Materials large paper rectangle and square pattern block squares BLM 2 cm Dot Paper (p XXX) BLM Rectangles or Squares?(p XXX) and scissors; rulers and strips of paper (optional) scissors and sheets of paper Journal Have students trace an attribute block square and rectangle in their journals and create a picture incorporating that square and rectangle. Identifying rectangles. Introduce rectangles as you did squares: hold up a large paper rectangle and identify it by name (rectangle); trace the rectangle on one side of the board; draw more rectangles on the board (different sizes, colours, patterns) and label them Rectangles ; use prompts to elicit that rectangles are closed shapes with 4 straight sides, 4 vertices, and all square corners; turn (rotate) a rectangle several times. Emphasize that being a rectangle does not depend on pattern, colour, size, or position. ACTIVITIES 1 & 2 Journal 1. Modelling rectangles. Have students create rectangles using up to 6 pattern block squares. Have them trace their rectangles in their journals and tell how the rectangles are different. (SAMPLE ANSWER: This rectangle is 3 squares wide and 2 squares tall, the other one is wider: it is 6 squares wide, but only 1 square tall.) 2. Drawing rectangles. Students can draw a variety of rectangles on BLM 2 cm Dot Paper. Also, have students draw some shapes that have 4 sides and 4 vertices but are not rectangles (these may be many types of quadrilateral or have at least one curved side). Measurement Length of sides in a square and rectangle. Hold up the large paper rectangle. Run your finger along one of the shorter sides and ask students how they could check which sides of the rectangle are longer than this side. Point to one of the longer sides. ASK: Is it longer? How can we check? Fold the shorter side down toward the longer side to compare. Explain 10 Teacher s Guide for Workbook 2

11 the meaning of the word equal in this context: two sides are equal if they have the same length. Repeat the folding activity with a square. Point to one of the shorter sides of the rectangle and ASK: How many sides of the rectangle are the same length as this side? Point to one side of the square and ASK: How many sides are the same length as this side? Fold shapes as required to check. problem solving Making and investigating conjectures Distinguishing between squares and rectangles by direct comparison of sides. Ask your students if they think their observations (squares have 4 equal sides, rectangles have 2 equal short sides and 2 equal longer sides) will be true for any rectangle and any square. Have students use Activity 3 to check. ACTIVITY 3 Probability and Data Management Sorting Sorting. Give students BLM Rectangles or Squares? and have them cut out the shapes. Ask students to sort the shapes (excluding the two mystery shapes) into rectangles and squares and then check, by folding, how many equal sides the squares and rectangles have. Then ask students to check whether the mystery shapes are rectangles or squares. Alternatively, students can use rulers to measure the sides or strips of paper to compare side lengths indirectly (as in ME2-4). Ask students to explain their findings. Encourage the use of correct terminology, such as sides, equal, longer, shorter, and so on. Distinguishing between squares and rectangles by measuring sides. Draw several rectangles and squares on the board. (Some should be slightly rotated, to make it harder to see whether they are rectangles or squares.) Ask students how they could check whether these are squares or rectangles. Record their ideas and invite volunteers to measure the sides. Assess student s understanding. ASK: What is the difference between a square and a rectangle? ( A rectangle has two long sides and two short sides and a square has four sides that are the same would be a very good answer. If a student says only that rectangles have longer sides, accept that answer, as it shows partial understanding, then prompt the student to elaborate.) Draw a rectangle as well as a square with sides longer than the longer side of the rectangle, and ASK: Does this rectangle have longer sides? Longer than what? Longer than the sides of the square, or longer than the shorter sides of the rectangle itself? problem solving Connecting problem solving Connecting Geometry 2-4 Applying knowledge of squares. Give students scissors and sheets of paper. ASK: What shape is the sheet of paper? (rectangle) How do you know? Teach your students how to create a square from a sheet of paper by folding. ASK: How do you know that the shape you made is a square? Squares and rectangles in the environment. After students have completed the worksheets, point to various objects around the room (doors, windows, books, and so on) and ask if these shapes look like squares or rectangles. Students can compare the faces of 3-D objects in the classroom with the squares and rectangles drawn on the board. Then have volunteers identify other objects that include squares and rectangles. 11

12 G2-5 Triangles Page 200 Curriculum Expectations Ontario: 1m46; 2m5, 2m6, 2m7, 2m47, 2m48 WNCP: 1SS2, 1SS4; 2SS8, 2SS9, [C, V] Vocabulary triangle side vertex/vertices Goals Students will identify and model triangles. PRIOR KNOWLEDGE REQUIRED knows the word shape can identify triangles visually can identify straight and closed lines can identify and count sides and vertices of shapes Materials large paper equilateral triangle BLM Triangles and Not Triangles (p XXX) attribute blocks made from BLM Attribute Blocks (p XXX) geoboards BLM Find the Triangles (p XXX) paper or plastic drinking straws and scissors photographs or magazines including triangular shapes Identifying triangles. Hold up a large paper equilateral triangle and identify it by name (triangle); trace the triangle on one side of the board; draw more equilateral triangles (different sizes, colours, patterns) and label them Triangles ; use prompts to elicit that triangles are closed shapes with 3 straight sides and 3 vertices; turn (rotate) an equilateral triangle several times. Emphasize that being a triangle does not depend on pattern, colour, size, or position. Non-equilateral triangles. Draw a large variety of triangles (or have BLM Triangles and Not Triangles photocopied on a transparency and display the triangles only, covering the bottom half of the sheet). Explain that all these shapes are triangles even though they look quite different. ASK: What is the same about all the triangles? (They have 3 straight sides and 3 corners.) How do the triangles change in each row? (They get taller or fatter, they lean to the right, and so on.) Students can also show with their hands or whole bodies how the triangles change. online guide A matching activity for students who need more practice identifying triangles. ACTIVITIES Sorting shapes (see page XXX). Give your students various attribute blocks (squares, rectangles, triangles, and other shapes with straight sides only from BLM Attribute Blocks) and ask them to sort the blocks into Triangles and Not Triangles. ASK: How many vertices do all the triangles have? Are there triangles that do not have 3 vertices? How many sides do all the triangles have? 12 Teacher s Guide for Workbook 2

13 2. Modelling triangles. Have students create triangles on geoboards. Ask students to move one vertex of the triangle. ASK: Is the new shape still a triangle? Encourage students to make more changes to their triangles, so that the resulting shapes have sides of different lengths. online guide Extra support: BLM The Triangle presents only equilateral triangles of different sizes. Draw a large variety of non-triangles (or show the bottom half of BLM Triangles and Not Triangles). ASK: Why aren t these shapes triangles? (curved sides, more than 3 sides or vertices) Hold up some attribute blocks, both triangles and not triangles, and ask students where they should be placed on the BLM. Then invite volunteers to draw shapes on the board and decide as a class whether the shapes are triangles or not. Ask students if they can draw a shape that has 3 sides and 3 vertices but is not a triangle (these will need to have at least one curved side). For additional practice, draw groups of two or three shapes where only one is a triangle and ask which shape is a triangle. (Or, have students complete BLM Find the Triangles. The letters in the triangles produce the word beaver. ) ACTIVITY 3 PROBLEM SOLVING Making a model Modelling triangles. Divide students into groups of three. Give each student a set of 3 straws with relative lengths as shown below (each student in a group should have a different set) and ask students to use their straws to make a triangle. If possible, give each student straws of their own colour, so that they do not mix their straws. Ask students to share their triangles within their groups. How are their triangles different? How are they the same? Then, students can trade straws to make other triangles. You can also give students more straws and scissors and have them cut the straws to lengths of their choosing to make other triangles. Students could glue their triangles to construction paper and create a class display. Triangles in the environment. Discuss with students where they see triangles in the environment (e.g., half a slice of bread cut diagonally, tile patterns, part of a roof). Ask students to identify triangular shapes around the room and in photographs or magazines. Journal Ask students to trace some attribute block triangles and create a picture incorporating the shapes. Geometry

14 G2-6 Circles Page 201 Curriculum Expectations Ontario: 1m46; 2m1, 2m5, optional WNCP: 1SS2, 1SS4; 2SS8, 2SS9 [CN, V] Goals Students will identify and model circles. PRIOR KNOWLEDGE REQUIRED Materials Vocabulary circle side oval vertex/vertices knows the word shape can identify circles visually can identify curved and closed lines large paper circle and square attribute blocks made from BLM Attribute Blocks (p XXX) various objects (see Activity 2) various circular objects to trace pencils, large paper clips, thumb tacks or push pins, elastic bands Journal Ask students to trace several coins or circular objects such as cups and create a picture incorporating the circles. online guide An introduction to ovals, with activities and Measurement. Activity comparing circles and ovals. online guide Extra practice: BLM Identifying Shapes Identifying circles. Hold up a large paper circle and identify it by name; trace the circle on one side of the board; draw more circles (different sizes, colours, patterns) and label them Circles ; use prompts to elicit that circles are closed shapes that have a curved edge and no corners/ vertices. Emphasize that being a circle does not depend on pattern, colour, or size. Title the other half of the board Not Circles. Draw a large variety of polygons and closed shapes with at least one curved side (but no ovals yet). ASK: Why aren t these shapes circles? (they have straight sides and/or vertices) Ovals are not circles. On the Not Circles side of the board, draw an oval. ASK: Is this a closed shape? (yes) Does it have a curved edge? (yes) Does it have any corners/vertices? (no) Is it a circle? (no) Why not? (It is more squashed-looking. It doesn t look the same all the way around. It isn t round.) Tell students that shapes that look like squashed circles are called ovals. Draw a few more ovals, and then have volunteers add some. ACTIVITY 1 Sorting shapes (see page XXX). Give your students various attribute blocks (circles and other shapes from BLM Attribute Blocks) and ask them to sort the blocks into Circles and Not Circles. Rotating a circle. Trace a large paper square on the board (among the non-circles). Rotate the square a bit and place it back on the board. ASK: Does the square fit in the outline now? (no) Trace a large paper circle on the board. Mark a large dot on the edge of the paper circle and place the circle into its outline with the dot upwards. Show that it fits. SAY: I am going to turn the circle a little bit as I did with the square. Do you think the circle will still fit in the outline? Let students predict, then check that the circle fits back into the outline. Ask students to explain how the circle and the square are different and why they think the circle fits back in the outline when 14 Teacher s Guide for Workbook 2

15 the square does not. (The circle does not have any vertices, its sides are curved, and it looks the same no matter how you turn it.) ACTIVITIES 2-4 problem solving Making and investigating conjectures online guide Extra practice: BLM Tracing Shapes Tracing a square, rectangle, triangle, and circle. problem solving Selecting tools and strategies 2. Modelling circles. Give small groups of students various objects, such as string, strips of paper, connecting cubes, popsicle sticks, blocks, pipe cleaners, toothpicks, and so on. Ask students to predict which materials are good for making circles and then test their predictions. Have students record their predictions and results in a chart. Provide the headings for the chart ( Object, Good for circle, Bad for circle ) and show students how to use pictures and checkmarks to fill it in. 3. Tracing circles. Have students trace circular objects of various sizes (e.g., coins, plastic cups, paper plates, CDs, attribute blocks) to draw circles. They can make a class display of their drawings. 4. Drawing circles. Show students how to draw a circle using two pencils and a paper clip, as shown below. Give each student a large paperclip and have students try drawing circles using this method. Alternatively, students can use a thumbtack or push pin instead of the second pencil. Discuss with students how you could draw larger circles (for example, replace the paper clip with a loop of string). You can also have students try replacing the paper clip with an elastic band. ASK: Why doesn t it work when you use an elastic band instead of a paper clip? (The elastic stretches, so the circle isn t the same all the way around.) Literature So Many Circles, So Many Squares by Tana Hoban (Photos of circles and squares in the environment.) Literature Circles in the environment. Discuss with students where they see circles in the environment (wheels, clocks, coins, and so on). Ask students to identify circular shapes around the classroom. ACTIVITIES On a walk in the surroundings (indoors or outdoors), have students identify objects shaped like squares, rectangles, triangles, and circle. Students can have clipboards to draw simplified images of the objects they see. 6. Read the message in The Secret Birthday Message by Eric Carle to students. Discuss with them what object each symbol in the message could be. ASK: What do you think the present is? Where is it hidden? Read the book together to check the guesses. Geometry

16 G2-7 Polygons Pages Curriculum Expectations Ontario: 2m6, 2m7, 2m48 WNCP: optional, [C, CN, V], 3SS7 Vocabulary polygon triangle quadrilateral pentagon hexagon heptagon octagon Goals Students will identify and model polygons. PRIOR KNOWLEDGE REQUIRED knows the word shape can identify straight and closed lines can identify and count sides and vertices of shapes Materials attribute blocks made from BLM Attribute Blocks (p XXX XXX) cards made from BLM Matching Polygons (pp XXX XXX) index cards with polygon names and pictures geoboards BLM I Have, Who Has? Blank Cards (p XXX) BLM Identifying Polygons (pp XXX) BLM Space Polygons (p XXX) Journal Have students draw two different polygons and two different shapes that are not polygons in their journals. Introduce polygons. Divide the board into two. On one side, draw a variety of polygons, including regular polygons (shapes with equal sides and equal angles) and irregular polygons. Label the shapes Polygons, reading the label aloud. Label the other side of the board Not Polygons. Draw a pentagon with a little part of one side missing on the Not Polygons side and ASK: How is this line is different from the polygons? (It is not closed.) Repeat with a shape with one curved side, a shape with a tail, and a line (closed or open) that intersects itself. Add a variety of other non-polygons and ASK: What is the same about all the polygons? Are they all closed shapes? Do they all have straight sides? Conclude by telling students that all shapes that have straight sides and are closed lines are called polygons. not polygons ACTIVITY Probability and Data Management sorting Sorting shapes (see page XXX). Give students many different attribute blocks and have them sort the shapes into polygons and not polygons. (You might include the shape cards from BLM Matching Polygons to provide a larger variety of shapes.) Then ask students to put aside all the shapes that are not polygons and sort the polygons by the number of sides polygons with 3 sides, polygons with 4 sides, and so on. Students can compare their results with partners. 16 Teacher s Guide for Workbook 2

17 Introduce the names of polygons. Explain that mathematicians give special names to polygons according to the number of sides and vertices they have. Present the names (see vocabulary) on word cards, explain each one (e.g., hexagons have 6 straight sides and 6 vertices), and invite volunteers to assign each card as a label for the appropriate group created in Activity 1. Students may notice that the Quadrilaterals group includes shapes that they have described as squares and rectangles. Explain that, just as there is more than one word that describes a student (e.g., boy/ girl, child, person), many shapes have more than one name. Tell students that squares and rectangles are special types of quadrilaterals. Draw a pentagon on the board. ASK: How many sides does this polygon have? What is the name for this polygon? How do you know that this is a pentagon? If students do not say that it has 5 sides and 5 vertices and all sides are straight, draw another pentagon (different size, colour, or pattern), and ask what the two pentagons have in common. Repeat with a triangle and a quadrilateral that is neither a rectangle nor a square (e.g., a parallelogram, rhombus, or trapezoid). Finally, draw a square and a rectangle. For each, ASK: How many sides does this polygon have? What is a polygon with 4 sides called? (quadrilateral) What is another name for this shape? (square/rectangle) ACTIVITIES 2-5 Extra practice BLM Identifying Polygons BLM Space Polygons problem solving Modelling 2. Modelling polygons. Use a geoboard to create a shape (do each shape introduced in the lesson). Ask students to re-create the exact same shape on their geoboards. 3. I Have, Who Has? (see page XXX). Have students play in groups of six. Use BLM I Have, Who Has? Blank Cards to create cards with polygon names (triangle, quadrilateral, pentagon, hexagon, heptagon, and octagon) on one half and pictures of various polygons on the other. (You can use the shapes from BLM Matching Polygons.) Vary the polygons between the groups so that students can play the same game with different shapes over the course of several lessons, to reinforce the names of the shapes. 4. Students will need the name cards from BLM Matching Polygons and geoboards. Player 1 draws a card at random and creates the polygon on a geoboard. Player 2 has to identify the polygon. 5. Memory (see page XXX). Use the cards from BLM Matching Polygons. A matching pair would be a polygon and its name. Geometry

18 G2-8 About Polygons Pages Curriculum Expectations Ontario: 2m5, 2m6, 2m48 WNCP: optional [CN, V], 3SS7 Goals Students will identify and model polygons. PRIOR KNOWLEDGE REQUIRED can identify and count sides and vertices can identify polygons and know the vocabulary words Vocabulary polygon triangle quadrilateral pentagon hexagon heptagon octagon Materials BLM Naming Polygons (p XXX) rulers or BLM Polygons and Geoboards (p XXX) geoboards and dot paper (or BLM 2 cm Dot Paper p XXX) BLM Geoboards (pp XXX XXX) clipboards, paper, and pencils magazines BLM Shapes in Flags (p XXX) Naming polygons. Draw a chart on the board that summarizes what students learned in the last lesson: the names of the polygons studied (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon) and the number of sides and vertices in each. Draw various polygons with 3 to 8 sides, including a square and a rectangle, and have students name each shape (and/or have students complete BLM Naming Polygons). Remind students that, in geometry, shapes often have more than one name; for example, all squares and rectangles have four sides, so all squares and rectangles are quadrilaterals. online guide A strategic game in which partners begin and complete drawings of polygons. Includes a discussion of possible strategies. online guide 12-page BLM More Geoboards provides exercises of increasing difficulty for copying dots and lines between sheets of dot paper or geoboards. Drawing polygons. Provide students with rulers. Have students draw an example of each type of polygon. (These drawings could be used to create a class display.) Alternatively, have students complete BLM Polygons and Geoboards. ACTIVITY 1 Students can create polygons on geoboards and then copy the polygons onto dot paper. If students have trouble copying shapes, they can practise copying dots and lines from one sheet of dot paper to another. Sample exercises, and blank geoboards on which to create more, are provided on BLM Geoboards. When students are comfortable with the exercises on BLM Geoboards, they can copy polygons from geoboard to geoboard, first using geoboards of the same size and later geoboards of different sizes (that is, geoboards with a different distance between pegs). Polygons in the environment. Point at the faces of various objects in the room and have students identify which type of polygon is similar to 18 Teacher s Guide for Workbook 2

19 the shape of the face. Some objects could be examined from different sides; for example, a toilet paper roll looks like a rectangle from the side and a circle from the top. Invite students to spot distinct polygonal shapes in objects. For example, what polygons can we find on a soccer ball? (pentagons and hexagons) ACTIVITIES 2-4 [L/O include picture of a school bus] Journal After doing Activity 2, students can draw pictures of what they saw and highlight and name the polygons in their pictures. At home Social Studies, Literature Literature 2. On a walk through the neighbourhood, have students point at various objects and describe their shape as polygons. For example, the stop sign is an octagon, the city bus looks like a rectangle from the side, the school bus is similar to a hexagon, and so on. Students can have clipboards to draw simplified polygonal images of the objects they see. 3. Collages and posters (see page XXX). Read The Shape of Things by Dayle Ann Dodds, which shows how circles, squares, and triangles are part of everyday objects. Then have students create their own 2-D collage of everyday objects that are made up of various shapes. They can use cut-out shapes, trace pattern blocks, or cut illustrations and photos out of magazines. 4. Have students identify the shapes in flags on BLM Shapes in Flags. Students choose a country connected with their family history and find its flag. They identify and count the geometric shapes they see on the flag and find the country on the map. A Cloak for a Dreamer by A. Friedman. A tailor s son makes a cloak using circles, which doesn t work. The family then changes the circles to hexagons and comes to terms with the fact that the son is not meant to be a tailor and should pursue his own dreams. Students can create their own cloaks, first with pattern blocks and then on paper with pencil crayons or shapes cut from construction paper. The Wing on a Flea by E. Emberley. Introduction to basic shapes in real life situations, using rhyming text. A Star in My Orange by D.M. Rau. Forms in nature. Extensions 1. Word search puzzles on BLM Word Search Polygons (p XXX). The answer is I am done the word search puzzle! 2. BLM Crossword Polygons (p XXX). The answer is polygon. 3. Can students think of any object that is a regular heptagon? If available, show students coins or pictures of coins in that shape. Some students might bring such coins from home. EXAMPLES: UK 50 or 20 pence, Ghana, 200 cedis, Latvia, 1 lat, Mauritius, 10 rupees. Geometry

20 G2-9 Matching Shapes Pages Curriculum Expectations Ontario: 1m50; 2m3, 2m7, 2m51, 2m53 WNCP: optional, [C,V]; 4SS5 Vocabulary match exactly same kind of shape square rectangle triangle hexagon Goals Students will decompose 2-D shapes and identify reflection symmetry in diagrams of shapes and real-life objects. PRIOR KNOWLEDGE REQUIRED can fold paper can identify squares, rectangles, triangles, and hexagons knows the concepts larger and smaller Materials overhead projector pattern block triangles, squares, and hexagons (or blocks made from BLM Pattern Blocks p XXX) large paper right triangle paper squares for students shapes from BLM Shapes to Fold (p XXX XXX) BLM Matching Shapes (p XXX) Miras Extra Support: BLM Matching Shapes Identifying parts of shapes. Place a pattern block square and a pattern block triangle side by side on the overhead (so that the shapes have a common side). ASK: Which pattern block shapes did I use to create the shadow on the board? Invite volunteers to trace the shadow and draw a line showing where the blocks meet. Ask students to identify the shapes of the blocks. Point out that the shape on the board is made from two different kinds of blocks a triangle and a square. Repeat with more shapes (EXAMPLES: two triangles, a triangle and a hexagon, a hexagon and a square). Same kind of part, different size. Draw a rectangle on the board and divide it into two unequal rectangles, as shown below. Invite volunteers to shade or trace each part of the rectangle in a different colour. Ask students to identify the shape of each part. Explain that since the parts are both rectangles, we say that they are the same kind of shape. ASK: Are the parts the same size? Repeat with the shape below. Folding shapes so that parts match exactly. Show your students a right triangle created from a rectangular sheet of paper by cutting the rectangle along the diagonal. Fold the triangle once as shown below. Show students how the two parts do not match one part sticks out. From one side, the wider part of the triangle covers the narrower part, but when you flip the folded shape over, the narrower part doesn t cover the wider part. 20 Teacher s Guide for Workbook 2

21 Fold Nothing sticks out This part sticks out Give each student a paper square. Ask them to fold the square so that one of the vertices matches up with any other vertex. (All students don t have to and likely won t fold the square the same way.) ASK: Does the top part of your folded square cover the bottom part? Flip the folded square over does the top part cover the bottom part now, too? Unfold your square what parts do you see? Are the shapes of the same kind? Are the shapes of the same size? How do you know? (nothing sticks out ) What can you say about the parts of the square? (They are the same, or equal. Remind students of the meaning of the word equal.) Explain that in this case we say that the parts of the square match exactly. Ask students to check whether they folded the square so that the parts match exactly. If not, ask them to fold the square again so that the parts match exactly. Bonus Find another way to fold the square so that the parts match exactly. ACTIVITY 1 Give students several shapes from BLM Shapes to Fold and ask them to fold the shapes along the dotted lines. Do the parts match exactly? For the shapes where the parts do not match, discuss how students can trace and cut to make the shapes match, then have them test their predictions. problem solving Reflecting on other ways to solve the problem Miras. Ask students what object they know that can show an exact match of anything. (a mirror) Give students Miras and ASK: What does this kind of mirror do? Let them experiment with personal objects. When your students understand that this mirror is transparent, show them the way to check whether parts of objects match exactly. For example, you could write a large E (as shown) on the board and ask whether the top part is exactly the same as the bottom part. ASK: if you could fold the board, would the top of this E fall precisely on the bottom? How could we check? Take guesses, then show your students how to check this using the Mira. Students should clearly see that the parts do not match. E ACTIVITY 2 Have students use Miras to check whether some objects, such as letters on the JUMP Math workbook cover, have matching parts. Geometry