Geometry 2 Part 1 − Introduction - PBworkscommondrive.pbworks.com/f/TG+for+Book+2-1+GE.pdf ·...

2 Teacher’s Guide for Workbook 2

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 non-symmetrical, 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.

Geometry 2 Part 1 − Introduction

3

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 CurriculumLesson 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.

ACTIVITIES

These activities can be used and repeated throughout the unit, for practice and reinforcement.

• 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.”

Geometry 2- Introduction

Probability and Data Management.

CONNECTION

Arts

CONNECTION

Science and Literature

CONNECTION

CONNECTION Literature

Music

CONNECTION

Technology

CONNECTION


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. 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.

G2-1 LinesPages 193-194

CuRRICuLuM ExPECTATIONSOntario: 1m46, 2m5, 2m47WNCP: 1SS2; 2SS8, [CN, V]

VOCABuLARYline open shape square straight triangle curved rectangle closed circle

GoalsStudents will identify straight and curved lines and sides, and open and closed lines.

PRIOR KNOWLEDGE REQuIRED

knows the words “line” and “shape”

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)

ACTIVITY 1

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.

Investigating lengths of straight and curved lines.

ONLINE GuIDE

straight lines curved lines

Probability and Data Management.

CONNECTION

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?

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.

ACTIVITIES 2-3

2. 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.

Have students draw examples of closed and open lines in their journals, using both straight and curved lines.

JOuRNAL

closed path open path

Geometry 2-1

Art

CONNECTION


G2-2 Sides and VerticesPages 195-196

CuRRICuLuM ExPECTATIONSOntario: 2m2, 2m3, 2m5, 2m48WNCP: 2SS8, [CN, R, V]

VOCABuLARY side corner vertex/vertices square rectangle circle triangle

GoalsStudents will identify and count sides and vertices of shapes. They will sort shapes by the number of 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.

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


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)

ACTIVITY 1

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.

Visualizing

PROBLEM SOLVING

vertex

3 vertices

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?

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.

7

Making and investigating conjectures Reflecting on other ways to solve a problem

PROBLEM SOLVING

BLM Sides and Vertices

ExTRA PRACTICEACTIVITIES 2-3

2. 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.

1 vertex 2 vertices 3 vertices 4 vertices

2

1 3

1

3 2

3 vertices

1

33 sides

2side

Geometry 2-2


GoalsStudents will identify squares by their attributes and draw squares.

G2-3 SquaresPage 197

CuRRICuLuM ExPECTATIONSOntario: 1m46; 2m1, 2m5, 2m6, 2m47, 2m48 WNCP: 1SS2, 1SS4; 2SS8, 2SS9 [CN, V]

VOCABuLARY shape vertex/vertices square closed side square corner straight

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.

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


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 squaresBLM Square or Not? (p XXX) and BLM 2 cm Dot Paper (p XXX)

Extra support: BLM The SquareBLM Matching Squares and Not Squares

ONLINE GuIDE

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.

An alternate explanation using an overhead projector.

ONLINE GuIDE

9

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)

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.

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.

Generalizing from examples

PROBLEM SOLVING

More on comparing corners and square corners.

ONLINE GuIDE

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).

A way to reflect on what made the problems on BLM Square or Not? easy or hard.

ONLINE GuIDE

Have students trace an attribute block square in their journals and create a picture incorporating that square.

JOuRNAL

Geometry 2-3

Data Management and Probability-Sorting

CONNECTION


G2-4 RectanglesPages 198-199

CuRRICuLuM ExPECTATIONSOntario: 1m46; 2m2, 2m5, 2m7, 2m47, 2m48 WNCP: 1SS2, 1SS4; 2SS8, 2SS9 [C, CN, R, V]

VOCABuLARY square equal rectangle longer side shorter

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.

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

GoalsStudents will identify and model rectangles. They will compare squares and rectangles.


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

ACTIVITIES 1 & 2

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).

Have students trace an attribute block square and rectangle in their journals and create a picture incorporating that square and rectangle.

JOuRNAL

JOuRNAL

Measurement

CONNECTION

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.

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.

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?

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.

ACTIVITY 3

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.

Making and investigating conjectures

PROBLEM SOLVING

Connecting

PROBLEM SOLVING

Connecting

PROBLEM SOLVING

Geometry 2-4

Probability and Data Management−Sorting

CONNECTION


G2-5 TrianglesPage 200

CuRRICuLuM ExPECTATIONSOntario: 1m46; 2m5, 2m6, 2m7, 2m47, 2m48WNCP: 1SS2, 1SS4; 2SS8, 2SS9, [C, V]

VOCABuLARY triangle side vertex/vertices

GoalsStudents will identify and model triangles.


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.

ACTIVITIES 1-2

1. 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?

A matching activity for students who need more practice identifying triangles.

ONLINE GuIDE

13

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.”)

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.

Ask students to trace some attribute block triangles and create a picture incorporating the shapes.

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.

Extra support: BLM The Triangle presents only equilateral triangles of different sizes.

ONLINE GuIDE

ACTIVITY 3

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.

Geometry 2-5

JOuRNAL

Making a model

PROBLEM SOLVING


G2-6 CirclesPage 201

CuRRICuLuM ExPECTATIONSOntario: 1m46; 2m1, 2m5, optional WNCP: 1SS2, 1SS4; 2SS8, 2SS9 [CN, V]

VOCABuLARY circle side oval vertex/vertices

GoalsStudents will identify and model circles.


knows the word “shape” can identify circles visually can identify curved and closed lines

MATERIALS

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

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.

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

An introduction to ovals, with activities and Measurement connection. Activity comparing circles and ovals.

ONLINE GuIDE

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.”

Extra practice: BLM Identifying Shapes

ONLINE GuIDE

Ask students to trace several coins or circular objects such as cups and create a picture incorporating the circles.

JOuRNAL

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.)

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 2-4

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.)


PROBLEM SOLVING

Selecting tools and strategies

PROBLEM SOLVING

Extra practice: BLM Tracing Shapes Tracing a square, rectangle, triangle, and circle.

ONLINE GuIDE

ACTIVITIES 5-6

5. 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.

Literature— So Many Circles, So Many Squares by Tana Hoban (Photos of circles and squares in the environment.)

CONNECTION

Literature

CONNECTION

Geometry 2-6


G2-7 PolygonsPages 202-204

CuRRICuLuM ExPECTATIONSOntario: 2m6, 2m7, 2m48WNCP: optional, [C, CN, V], 3SS7

VOCABuLARY polygon triangle quadrilateral pentagon hexagon heptagon octagon

GoalsStudents will identify and model polygons.


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)

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

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.

Have students draw two different polygons and two different shapes that are not polygons in their journals.

JOuRNAL

Probability and Data Management−sorting

CONNECTION

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

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.

BLM Identifying PolygonsBLM Space Polygons

ExTRA PRACTICE

Geometry 2-7

Modelling

PROBLEM SOLVING


G2-8 About PolygonsPages 202-204

CuRRICuLuM ExPECTATIONSOntario: 2m5, 2m6, 2m48WNCP: optional [CN, V], 3SS7

VOCABuLARY polygon triangle quadrilateral pentagon hexagon heptagon octagon

GoalsStudents will identify and model polygons.


can identify and count sides and vertices can identify polygons and know the vocabulary words

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.

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.

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

A strategic game in which partners begin and complete drawings of polygons. Includes a discussion of possible strategies.

ONLINE GuIDE

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).

12-page BLM More Geoboards provides exercises of increasing difficulty for copying dots and lines between sheets of dot paper or geoboards.

ONLINE GuIDE

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)

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.

Extensions1. 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.

ACTIVITIES 2-4

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.

[L/O include picture of a school bus]

After doing Activity 2, students can draw pictures of what they saw and highlight and name the polygons in their pictures.

JOuRNAL

AT HOME

Geometry 2-8

Social Studies, Literature

CONNECTION

Literature

CONNECTION


G2-9 Matching ShapesPages 205-207

CuRRICuLuM ExPECTATIONSOntario: 1m50; 2m3, 2m7, 2m51, 2m53 WNCP: optional, [C,V]; 4SS5

VOCABuLARY match exactly same kind of shape square rectangle triangle hexagon

GoalsStudents will decompose 2-D shapes and identify reflection symmetry in diagrams of shapes and real-life objects.


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

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.

BLM Matching Shapes

ExTRA SuPPORT:

21

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.

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.

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.

Reflecting on other ways to solve the problem

PROBLEM SOLVING

ACTIVITY 2

Have students use Miras to check whether some objects, such as letters on the JUMP Math workbook cover, have matching parts.

E

Nothing “sticks out”Fold This part “sticks out”

Geometry 2-9


G2-10 Lines of SymmetryPages 210-212

CuRRICuLuM ExPECTATIONSOntario: 1m50; 2m5, 2m6, 2m51WNCP: optional, [CN, R], 4SS5

VOCABuLARY match exactly line of symmetry symmetrical shape mirror line square rectangle triangle quadrilateral

GoalsStudents will identify shapes with a line of symmetry, draw lines of symmetry in given shapes, and identify reflection symmetry in real-life objects.


can fold paper can identify matching shapes (size and shape)

MATERIALS

paper squares, right triangles, and right trapezoids for students BLM Triangle and Pentagon (p XXX)Miras attribute blocks made from BLM Attribute Blocks (p XXX–XXX)rulers or straight edges clipboards, paper, and pencils

Review exact matching. Give students paper squares and have them fold the squares so that the parts match exactly. Students can fold the squares in different ways. Discuss how nothing “sticks out” when the parts match exactly, and how the parts are of the same size and shape.

Introduce lines of symmetry. Have students unfold their squares and explain that the line along the fold is called a line of symmetry. This line indicates that the parts of the shape match exactly when folded over.

Explore lines of symmetry. Show students a rectangular sheet of paper. Invite a volunteer to fold it in half so that the vertices match. Point out that when the sheet is folded in this way, the sides match also. The parts of the rectangle match exactly—one part is exactly the same as the other. The line along the fold is a line of symmetry. Now fold the rectangle along the diagonal. Do the parts match exactly? (no) Is this line a line of symmetry? (no) Are both parts the same size and shape? (yes) Explain that even though the shapes would match if cut and turned, in symmetry cutting is not allowed. Only folding is allowed.

Introduce symmetrical shapes. Explain that a shape that has a line of symmetry is called symmetrical. ASK: Are all shapes symmetrical? Students can fold paper scalene triangles and right trapezoids to see that some triangles and quadrilaterals, for example, have no line of symmetry. (To create a scalene triangle, cut a rectangular sheet of paper along the diagonal. To create a right trapezoid, cut a diagonal slice off one end of a rectangular sheet of paper.)

23

Check symmetry with a Mira. Show students a Mira. Ask a volunteer to explain what it does and to describe how students used Miras during the previous lesson. Draw a symmetrical heart or a stick person on the board and ask students where and how they would place a Mira to check that this picture has two matching parts. Invite a volunteer to show how to place a Mira and to draw its location on the board. ASK: What do we call the line you drew? (a line of symmetry) Explain that a line of symmetry is also called a mirror line.

Symmetry in the environment. Have students brainstorm and record a list of shapes and objects, such as letters of the alphabet, that students predict have one or more lines of symmetry. Then list shapes that students think do not have a line of symmetry. Draw some of the shapes on the board and ask volunteers to look for and draw a possible line of symmetry. Draw letters in different ways to show that some letters can have or not have a line of symmetry, e.g., B or B, A or A

ACTIVITY 1

How many lines of symmetry? Ask students to compare the ways they folded their squares earlier. How many ways can they fold a square so that the parts match exactly? Give students more paper squares and let them experiment to find out. How many lines of symmetry does a square have? (Students can trace a line of symmetry along the fold once they discover it to avoid counting it twice.) Repeat with other shapes, such as rectangles, equilateral triangles, and regular pentagons. (A large triangle and pentagon are provided on BLM Triangle and Pentagon.)

Bonus Try a circle!

ACTIVITY 3

Go for a walk in the neighbourhood and have students find examples of symmetry in the environment. If possible, students should each have a clipboard, a piece of paper, and a pencil to sketch what they see.

ACTIVITY 2

Sorting (see page XXX). Give students an assortment of shapes from BLM Attribute Blocks and ask them to determine which shapes from their collection have lines of symmetry. Let them check the shapes with Miras, trace the shapes in their journals, and use a ruler or straight edge to draw the lines of symmetry.

JOuRNAL

Geometry 2-10

Making and Investigating conjectures

PROBLEM SOLVING

24

G2-11 Creating Symmetrical ShapesPage 213

CuRRICuLuM ExPECTATIONSOntario: 1m57; 2m1, 2m3, 2m7, 2m51, 2m52, 2m58WNCP: optional, [C, R, V], 4SS5

VOCABuLARY line of symmetry symmetrical shape square rectangle circle triangle quadrilateral hexagon

GoalsStudents will create and describe symmetrical shapes and symmetrical designs composed of 2-D shapes.


can fold paper can identify polygons understands the concept of symmetry

MATERIALS

large paper rectangle paper and scissors sheets of paper with a straight line down the middle (see Activity 1) and pattern blocks (or blocks made from BLM Pattern Blocks p XXX)Miras BLM 2 cm Grid Paper (p XXX) and pencil crayons

Creating symmetrical shapes and patterns. Hold up a large paper rectangle. Fold it in four along the lines of symmetry. Ask your students to identify the folded shape. Unfold once and ask students to identify the new shape. Unfold completely and ask students to identify the shape again. Refold the paper and cut the folded vertex as shown (each end of the cut-line should be the same distance from the vertex).

ASK: What shape will the hole be when I unfold the paper? Unfold once and let students adjust their guesses. Then unfold completely to check their predictions. Ask students to predict how they could produce holes of different shapes (e.g., diamond, circle, rectangle). Let them experiment with paper and scissors.

Bonus How would you cut to create a star-shaped hole?Bonus Making snowflakes. Fold a sheet of paper 3 times into

8 parts. Cut out some small pieces from the sides. For each piece you cut off, predict the shape of the hole it produces. Unfold in stages and check your predictions. How does the way you fold the sheet of paper affect the number of lines of symmetry of your snowflake?

ACTIVITY 1

Give pairs a sheet of paper with a straight line drawn down the middle (either vertically or horizontally) and a set of pattern blocks. Player 1

Guessing, checking, and revising

PROBLEM SOLVING

cut

Teacher’s Guide for Workbook 2

25

Have pairs display their designs from Activity 1 and describe them to classmates, naming the shapes used and explaining how they know the design is symmetrical. Students do not need to know the terms “rhombus” and “trapezoid,” they can refer to the rhombuses and the trapezoid as the blue, tan, and red quadrilaterals. Ask students to consider whether any of the designs have more than one line of symmetry.

Draw a 4 × 4 square as shown at left. ASK: Is this design symmetrical? (yes) Colour the design asymmetrically as in 2. ASK: Is the design still symmetrical? How do you know? Tell students that the design is symmetrical in its shape but not in its colouring. ASK: How can we change the design so that it is symmetrical in colouring too? Point out that a diagonal line of symmetry could be used. Have a volunteer colour the design so that it is symmetrical with a diagonal line of symmetry. Note that colours and patterns are important in art and design, but in geometry, when we compare shapes or decide if they are symmetrical, the colour of the shapes doesn’t matter.

ExtensionSymmetry in three dimensions. Hold up a figure built from connecting cubes. ASK: How many cubes are in this figure? Hold it up to the board as shown and ASK: What would happen if the board was a mirror? What would you see in the mirror? How many cubes do you need to produce the figure in the mirror? (You could point out that the figure in the mirror is called a reflection.) Ask a volunteer to produce a figure that is like the figure and its reflection combined. ASK: How many cubes will the volunteer need? Check by holding the initial shape to a Mira.

Reflecting on the reasonableness of an answer

PROBLEM SOLVING

ACTIVITY 2

Give students copies of BLM 2 cm Grid Paper, pencil crayons, and Miras. Have students create designs that are symmetrical in shape but not in colour, and designs that are symmetrical in both shape and colour. Students can use a Mira to check the symmetry of their.

12

2 13

3

More extensions.

ONLINE GuIDE

1 2 3

Geometry 2-11

places a pattern block so that it touches the line. Player 2 places a block on the other side of the line to make the design symmetrical. Then Player 2 places a second block on her side of the page such that the two blocks have a common side. Now Player 1 must add a block to his side of the page to maintain the symmetry and then add another block. Players continue placing pairs of pattern blocks in turn. At the end, ask students how they could check whether the design is symmetrical (SAMPLE ANSWER: using a Mira.) ExAMPLE: Three turns were taken.


G2-12 Breaking and Creating ShapesPage 214

CuRRICuLuM ExPECTATIONSOntario: 1m52; 2m1, 2m2, 2m6, 2m7, 2m48, 2m53WNCP: optional, [C, R, V]

VOCABuLARY square rectangle side vertex/vertices polygon triangle quadrilateral pentagon hexagon heptagon octagon

GoalsStudents will compose and decompose polygons.


can fold paper can identify and count sides and vertices can identify different polygons by name can perform direct comparisons of length

MATERIALS

paper squares for students overhead projector pattern blocks (or blocks made from BLM Pattern Blocks p XXX) large paper acute isosceles triangle pairs of triangles from BLM Pairs of Triangles (p XXX)BLM Quadrilaterals (p XXX)

Create different shapes from a given shape. Give each student a square piece of paper. Ask them to fold the paper so that when they unfold it, they see two triangles. ASK: Can you make other shapes just by folding the paper a different way? (yes) How many sides and vertices do your shapes have? ExAMPLES:

2 rectangles a triangle and 2 quadrilaterals a triangle a quadrilateral and a pentagon

Create given shapes from simple 2-D shapes. Distribute pattern blocks to students. Place two pattern block triangles side by side on the overhead so that students see the shadow of a rhombus on the screen. (Make sure students do not see the triangles—cover the overhead with a sheet of paper while you arrange the shapes.) Ask students to identify the shape (it is a quadrilateral) and to find a pattern block that has the same shape as the shadow. When students identify the right shape, place a blue rhombus on the overhead to show that it does in fact produce the same shape. Explain that you used two other shapes to produce the quadrilateral shape of the shadow. Show your students the triangles. Ask them to place two triangles so that they produce the same shape as the shadow. (They can use the rhombus pattern block to help them.)

Place two different pattern blocks side by side on the overhead (again, don’t let students see the shapes). Ask students to choose two pattern blocks and to place them so that the figure they create has the same shape as the shadow. When students are comfortable working with two blocks, make

Visualizing

PROBLEM SOLVING

27

shapes with three blocks, then four blocks—but do not tell students how many blocks you used. Each time, ask students to re-create the shape and identify the resulting polygon. ExAMPLES: Encourage multiple solutions whenever possible. Students with different pattern block combinations can check whether their shapes match by superimposing their arrangements.

Create quadrilaterals from triangles. Show your students a pair of pattern block triangles. Review with students the shape they create (a quadrilateral). Ask students to try connecting the two pattern block triangles at different sides. ASK: Do all the sides match? (yes) Why? (The sides are all the same length.) If I turn or flip one of the triangles and connect the triangles at a different side, will the shape of the quadrilateral that I make change? Let students predict the results, then check. ASK: Do you think you will get the same result with other triangles? Show a paper isosceles triangle and ASK: How is the shape of this triangle different from the shape of the pattern block triangle? Are the sides of this triangle all the same length? (no) Trace the isosceles triangle on the board twice to form one of the quadrilaterals shown. Explain that to make a quadrilateral from two triangles that are the same, you have to join sides that are the same length. ASK: How many different quadrilaterals can we make? Have students suggest other ways you could join two copies of the paper triangle to make a different quadrilateral, and trace the different arrangements on the board. You could discuss with students how these quadrilaterals are different. (One has four equal sides and two lines of symmetry; the other two have two pairs of equal sides, but one has a line of symmetry and the other doesn’t).

ACTIVITY

Divide students into groups of 5. Give each student a pair of congruent triangles from BLM Pairs of Triangles, so that each student in the group works with a different triangle. Students should create all possible shapes made by joining the triangles so that they share a side, and then trace their shapes on paper. Then have students compare their findings. ASK: Do some arrangements produce the same shapes? Are all the shapes quadrilaterals? (No, some triangles produce larger triangles.) Did you get any rectangles or squares?

Now students will do the opposite: given quadrilaterals, they will identify the triangles that can be used to make them. Display BLM Quadrilaterals on a transparency. ASK: Who has the two triangles that make up the first quadrilateral? Invite volunteers to lay their triangles on the overhead transparency to check. Repeat with the other shapes. To make this activity more challenging, ask each group to mix up their triangles first.


PROBLEM SOLVING

Geometry 2-12

Visualizing

PROBLEM SOLVING


ACTIVITIES 1-2

1. Set a divider (e.g., an upright open binder) between each pair of students. Player 1 creates a design from pattern blocks and describes her design to her partner, who has to re-create the design according to the description. Then players compare their designs.

2. Students can create designs from pattern blocks; trace the outlines (without showing which pattern blocks were used); and exchange their designs with partners, who then fill the designs with pattern blocks. Students can then compare the designs they make on the outlines with the original designs. A harder task would be for students to tell their partners how many of each shape they used and for partners to use the same numbers of shapes. Bonus Create a symmetrical design. Extra Bonus Create a design with two or more lines of symmetry.

G2-13 Making Polygon ShapesPage 215

CuRRICuLuM ExPECTATIONSOntario: 1m54; 2m1, 2m5, 2m7, 2m52, 2m53, 2m54 WNCP: optional, [C, CN, T]

VOCABuLARY polygon square rectangle triangle quadrilateral hexagon line of symmetry

GoalsStudents will create polygons from pattern blocks and tangram pieces (tans).

Create designs from pattern blocks and describe designs. Show your students the hexagonal shape at left created from pattern blocks. Ask them to describe the shape. PROMPTS: How many blocks of each kind are used in this shape? Which shape is on the top? Which shape is on the bottom? (Encourage the use of “above,” “below,” and “between” if students are familiar with these terms from the previous year.) Does the picture have a line of symmetry? Is there only one line of symmetry? Repeat with another simple design, such as a flower from a hexagon and six squares or a house from two squares, a trapezoid, and a triangle.

Identifying shapes within other shapes.

ONLINE GuIDE


can identify different polygons by name understands the concept of symmetry

MATERIALS

pattern blocks (or blocks made from BLM Pattern Blocks (p XXX) dividers (1 per pair of students) BLM Tangram (p XXX) and scissorsBLM Tangram Puzzles (pp XXX–XXX) and BLM Tangram Puzzles— Answers (p XXX)

29

Tangrams. Give students a copy of BLM Tangram. The shapes that make up this square are called tans. Have students cut out the shapes and solve the following sequence of puzzles:

1. Create a square using only one of the shapes, then two. CHALLENGE: Create a square using three shapes (two small triangles and one medium triangle). Can you draw lines of symmetry in the designs?

2. Create a square using four triangles. (HINT: One of the large triangles is not required.) ANSWER: See Question 6.

3. Find two ways to create the shape at left using only two shapes.

4. Now distribute copies of BLM Tangram Puzzles. The pictures on the BLM are made up of tangram shapes. Can students match their tangram shapes to the outlines on the BLM to create the pictures?

5. Make the task a little more difficult: instead of matching shapes to the outlines on the BLM, give students small versions of the pictures (see BLM Tangram Puzzles—Answers) and have them create the shapes with reference to the pictures. Ask students to start by identifying the largest triangle in each picture and placing the largest triangle from their set of tans in the same position. Then have them add the other shapes one by one, identifying the exact position of each new shape in relation to the shapes already in place. Students can decide what they think the Mystery tangram puzzle represents.

6. Ask students to create squares using from one to five tangram shapes. ASK: How many different squares can you create? Can you create different rectangles? Other quadrilaterals, pentagons, and hexagons? SAMPLE ANSWERS:

7. Have students use tangram shapes to create a picture of their own.

Extension A shape is missing from the square. Pick out the shape that completes the square.

ACTIVITIES 3-4

3. Students can use online tangram applets to create and solve puzzles. There is also a wealth of tangram puzzles available online. Search using the terms “tangram applet” and “tangram puzzle.”

4. Grandfather Tang’s Story by A. Tompert. Grandfather tells a story of a shape-shifting fox. Give students outlines of shapes presented in the book and have them place tangram pieces over top to create animals.

Geometry 2-13

Bonus In steps 5–7, students can trace the outlines of their shapes and have partners re-create their shapes.

Technology

CONNECTION

Literature

CONNECTION

Visualizing

PROBLEM SOLVING


G2-14 Problems and PuzzlesPage 216

NOTE: The problems and puzzles on the worksheet are suitable for students who have been taught lessons G2-6 to G2-12.

If your students have trouble creating the 12-sided polygon in the problem on the worksheet, given them BLM Polygon with 12 Sides (p XXX), on which the polygon is drawn to scale, and have them cover the polygon with pattern blocks. The second copy of the same polygon is provided so that students can cover the shape in a different way.

ACTIVITY

Always, Sometimes, Never: Use cards like the ones on BLM Always, Sometimes, Never (p XXX–XXX) and attribute blocks made from page XXX of BLM Attribute Blocks. Each player picks a card with a statement at random and decides whether the statement is always true, sometimes true, or never true. If the answer is always or never, the player sorts the cards to illustrate the answer. If the answer is sometimes, the player has to provide two shapes from the set, so that the statement is true for one card and wrong for the other. Students can check each other’s answers.

ExAMPLE: “Triangles are striped.” This is sometimes true, so the player has to produce two triangles, one striped and one not striped. “Triangles have 3 sides” is always true; the player can sort the shapes into “triangles” and “not triangles” (to show that the triangles all have 3 sides) or sort the shapes into “shapes with 3 sides” and “shapes that do not have 3 sides” (to show that all triangles are in the set of shapes with 3 sides).

NOTE: The cards on the first page of BLM Always, Sometimes, Never have easier properties and use only shapes required by both the Ontario and the WNCP curricula. The cards on the second page use harder properties and shapes that are required by the Ontario curriculum only.

Using logical reasoning, Visualizing

PROBLEM SOLVING

Geometry 2 Part 1 − Introduction - PBworkscommondrive.pbworks.com/f/TG+for+Book+2-1+GE.pdf ·...

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Transcript of Geometry 2 Part 1 − Introduction - PBworkscommondrive.pbworks.com/f/TG+for+Book+2-1+GE.pdf ·...