Math Place Grade 3 ON Spatial Sense Teachers' Guide - Sample

Contents 2 Math Place Components for the Spatial Sense Kit 4 Spatial Sense Overview 8 Getting Started 9 Geometry 10 Embedding Geometry Throughout the School Day 12 1: Three-Dimensional Objects 95 2: Location, Movement, and Coding Concepts 134 Measurement 136 Embedding Measurement Throughout the School Day 137 3: Introduction to Measurement 150 4: Linear Measurement 183 5: Perimeter and Area 253 6: Mass 276 7: Capacity 302 8: Time 326 References

Spatial Sense Overview What Is Spatial Sense? According to John Van de Walle and LouAnn Lovin, spatial sense is “an intuition about shapes and the relationship among shapes” (Van de Walle & Lovin, 2006a, p. 187). This includes a feel for geometric aspects in the environment that allow individuals to make sense of their world. For example, people need to recognize, measure, and compare geometric attributes of objects in order to make decisions in their daily lives, such as whether a table will fit into a certain space. Marian Small explains that a critical part of spatial sense is visualization, “a process of representing abstract concepts as mental images” which “allow the visualizer to remember and ‘manipulate’ the concepts and to make the concepts meaningful” (Small, 2007, p. 131). Math Place places a particular emphasis on visualization throughout the lessons, as well as in the accompanying Math Talks, in order to build and reinforce students’ visualization skills. Nora Newcombe and Andrea Frick add that spatial thinking “involves the locations of objects, their shapes, their relations to each other, and the paths they take as they move” (Newcombe & Frick, 2010, p. 30). They highlight the dynamic nature of spatial sense: it is not just about naming and measuring static shapes and objects, but knowing how they move and describing their relative positions. Recent Research Research indicates that spatial reasoning is critical to mathematical thinking, and better predicts future math success than verbal and math skills (Moss et al, 2016, p. 10; Wai et al., 2009). This has an impact on our view of mathematical instruction. Doug Clements and Julie Sarama state that “the goal of increasing children’s knowledge of geometry and space is second in importance only to numerical goals” (Clements & Sarama, 2009, p. 116). They also stress that numerical and spatial goals are strongly interrelated. For example, composing and decomposing objects is related to composing and decomposing numbers and quantities. Spatial sense also plays an integral role throughout all areas of math, across other subject areas, and in our everyday lives as we navigate daily routines. The Study of Spatial Sense While some people think that you are either born a spatial thinker or not, research indicates that spatial reasoning is malleable and can be improved with proper instruction and practice (Newcombe & Frick, 2010, p. 30). It is therefore critical that spatial sense is strongly emphasized in the primary grades. The connection between measurement and geometry is strong, as students need to understand a geometric attribute before they can measure it. Measurement is also closely linked to other areas of math, such as counting, the operations of addition and multiplication (used to calculate measurements), and statistics (where measurement data are represented and 4 Spatial Sense

interpreted). Furthermore, measurement spans other subject areas, such as science, visual arts, and physical and health education. Embedding Spatial Sense in Students’ Learning Environments Since spatial sense plays an integral role in our lives, it makes sense to highlight its importance in other math topics and subject areas. Whether it is incidentally talking about shapes encountered in the environment or describing the positions of objects in relation to each other, bringing spatial experiences into real-life contexts will deepen understanding of concepts in a meaningful way. Concepts can also be embedded into regular school routines or reinforced using quick activities (5–10 minutes each) throughout the day. Specific activities are described in the introductions to the Geometry units and the Measurements units. A Balanced Approach A conceptual understanding of mathematics allows students to develop a deep understanding of math concepts, which they can apply to a variety of real- world problems. Marian Small cites research by Carpenter and Lehrer (1999) that explains conceptual understanding as “the development of understanding not only as the linking of new ideas to existing ones, but as the development of richer and more integrative knowledge structures” (Small, 2009, p. 3). A balanced math program includes the implementation of various high-impact instructional practices. As noted in the curriculum, “the thoughtful use of these high-impact instructional practices—including knowing when to use them and how they might be combined to best support the achievement of specific math goals—is an essential component of effective math instruction” (Ontario Ministry of Education, 2020, p. 40). High-impact practices in Math Place lessons include: learning goals, success criteria, and descriptive feedback; direct instruction; problem-solving tasks and experiences; the use of a variety of tools and representations; and small-group instruction using flexible groupings. Math conversations are critical, and throughout the lessons are several Math Talks that give students opportunities to reason and prove their own ideas and to hear and debate the ideas of others. Students see math from other perspectives at the same time as they build their own mathematical understanding and feelings of confidence. It is important for students to develop basic skills and proficiency within the different strands; for example, becoming proficient with skip counting as they determine the total number of tally marks in data. Deliberate practice plays a key role in students internalizing skills and being able to apply them independently in new situations. By using concrete materials and discussing their ideas during Math Talks, students develop mental math strategies that help them visualize concepts and gain automaticity of number facts and calculations. Marian Small also suggests using “rich tasks embedded in real-life experiences of children, and with rich discourse about mathematical ideas” (Small, 2009, p. 3), which aligns with Indigenous teaching that emphasizes “experiential learning, modelling, collaborative activity and teaching for meaning” (Beatty & Blair, 2015, p. 5). 5

daltMehurseearsitn5bohg–neT1g,tahoi5nlekrnfswrdienamhegyea.monyreibnveeeunrtduetshsoeefdraeat Math Talks There are numerous Math Talks linked to the lessons in Spatial Sense which support the understanding of math concepts through purposeful discussion, help to reinforce and extend the learning, and offer opportunities for further investigation. (For more on Math Talks, see the Overview Guide.) In order to maximize student participation and active listening, you can strategically integrate the following ‘math talk moves’ into all discussions (adapted from Chapin, O’Connor & Canavan Anderson, 2009). Math Talk Moves Example Talk Move Description Wait Time Teacher waits after posing a question before – Wait at least 10 seconds after posing a calling on a student so all students can think. question. – If a student has trouble expressing, say “Take your time.” Repeating Teacher asks students to repeat or restate what “Who can say what said in their own another student has said so more people hear words?” the idea. This encourages active listening. Revoicing Teacher restates a student’s idea to clarify and “So you are saying…. Is that what you were emphasize and then asks if the restatement is saying?” correct. This can be especially helpful for ELLs. Adding On Teacher encourages students to expand upon a “Can someone add on to what proposed idea. It encourages students to listen just said?” to peers. Reasoning Teacher asks students to respond to other “Who agrees? Who disagrees?” students’ comments by contributing and “You agree/disagree because justifying their own ideas. (sentence starter) .” aabeStwtcnLnBeenhtayehduaucronaeodcoiswlkrnuhedtubnaroeganireuniyravhnbtgkgseugowglgerestSunilhSetdetmtoehrhkdscoetteioh.imouldaarlmesgenltit-hsaoyleEpietksnmratsboeitaksme.omoertiensispo,kstns,sa,l Social-Emotional Learning Skills and Positive Attitudes in Mathematics 6 Spatial Sense Math Place offers opportunities to build and reinforce social-emotional learning skills beginning with two introductory lessons, which set the tone for nurturing and developing these important skills and attitudes in students. These lessons can be used at the beginning of the year to establish what the skills are and to develop the criteria for building them. The pertinent messages can be regularly reinforced throughout the year using the prompts and suggestions embedded in many Spatial Sense lessons. For interview prompts and questions to build social- emotional learning skills and positive attitudes, see the Overview Guide or the Teacher’s Website. The two introductory lessons include • “Your Fantastic Elastic Brain: Read Aloud,” which is available online. The book is included in the Number & Financial Literacy grade three kit. If you don’t have the kit, you can check your school library for a copy of the book, Your Fantastic Elastic Brain, or you can order it from Scholastic Canada.

• The second lesson, “Thinking Like a Mathematician,” can be found in the Overview Guide. The Mathematical Modelling Process The Mathematical Modelling Process falls within the Algebra strand of the curriculum but is embedded throughout Math Place because problem solving is foundational in all mathematical learning. Students move through the process of mathematical modelling as they create and adapt models to solve real-life problems, make decisions, or deepen their understanding of math concepts. They use critical and creative thinking, and apply social-emotional learning skills as they develop, test, and refine their models. The four components of the Mathematical Modelling Process are as follows: • Understand the Problem • Analyse the Situation • Create a Model • Analyse and Assess the Model Mathematical modelling is fluid and iterative—students move between and among the components as they refine their model. For example, as students apply the model they created, they may find it necessary to revisit the problem or re-analyse the situation to gain more information. Students in the primary grades can be guided through the process, to become aware of their own thinking and ways of organizing and to channel their approaches to solving problems. There are lessons that suggest ways of highlighting the process using an anchor chart of the four components; these suggestions can be adapted to reflect the process your students go through as they solve problems. 7

Getting Started The resource covers two topics, Geometry and Measurement, over the course of eight units. The order of the units/topics is flexible and can be altered to suit your needs. It is beneficial to spread them out throughout the year, and intersperse them with units from other areas of math. In this way, the learning is spiralled so the concepts and skills are reviewed, further developed, and reinforced. This allows students to more fully internalize the big ideas presented throughout the resource. Unit Description Lessons 1 Geometry 1–15 2 Three-Dimensional Objects 1–11 Location, Movement, and Coding Concepts 3 Measurement 1–3 4 Introduction to Measurement 1–9 5 1–16 6 Linear Measurement 1–6 7 1–7 8 Perimeter and Area 1–5 Mass Capacity Time The Geometry units can be taught at any time of year since they do not rely heavily on previous knowledge from other areas of math. Measurement is more reliant on other mathematical concepts, such as counting and understanding quantities up to 1000. It is therefore beneficial to teach the measurement units after a unit on quantities to 500 or 1000. The Measurement units are self-contained and can be taught in any order. After introducing the topic to students (Unit 3), it is recommended that you complete Unit 4, Linear Measurement, since its content will be familiar to students from their studies in grade two. They can use prior knowledge and make connections. 8 Spatial Sense

Geometry About the Math In grade three, students expand their knowledge of three-dimensional objects through a variety of hands-on activities. Experiences such as composing and decomposing three-dimensional structures allow them to kinesthetically learn more about geometric attributes and to more accurately compare three- dimensional objects. This in turn helps students to sort objects using one or two attributes. Through sorting activities, students learn about the properties that define which categories an object belongs to and which it does not. Grade three students also identify congruency in three-dimensional objects. Visualization is critical as students manipulate and rotate 3D objects in their mind. Finally, grade three students learn about location and movement by giving and following multistep instructions which include distances and turns. Students investigate coding concepts as they write and execute their own code (a sequence of instructions), and as they read and alter existing code to explore how sequence affects the outcome. Students can also create codes that result in repetitive movements, or a loop. Levels of Geometric Thinking Based on their research, Pierre van Hiele and Dina van Hiele-Geldof created a five-level Theory of Geometric Thinking (Levels 0–4) that describes the general developmental path that people follow as they acquire ideas about geometry (Van de Walle & Lovin, 2006a, p. 188). Van de Walle and Lovin make it clear that the levels are not age-dependent, noting that some adults never progress beyond Level 0. Instead, “Geometric experience is the greatest single factor influencing advancement through the levels. Activities that permit students to explore, talk about, and interact with content at the next level, while increasing their experiences at their current level, have the best chance of advancing the level of thought for those students” (Van de Walle & Lovin, 2006a, p. 190). Children in the primary grades typically work within Level 0 and Level 1. • At Level 0, Visualization, students identify a shape based on how it ‘looks’ overall. For example, when naming a shape, students claim “it is a square because it looks like a square” rather than define it by its geometric properties. Students’ ideas are based on their early real-life experience with geometry. • At Level 1, Analysis, students begin to see shapes and figures as belonging to particular groups based on similar properties. For example, in order to be a square, a shape must have four straight equal sides, and four right angles. Students at this level also distinguish between defining attributes, such as those described above, and other attributes, such as colour and orientation, that do not affect the categorization of the shape. Many grade three students work within Level 0. As they investigate, sort, and classify more and more three-dimensional objects, they start to transition from Level 0 to Level 1. Geometry 9

Embedding Geometry Throughout the School Day There are many ways to embed geometry concepts in daily routines. Below are brief activities that can engage students while the class waits in line to go somewhere or when there are five minutes at the end of a period. Some of the longer activities can be carried out when students need a break or a change of pace. • Word Wall: Geometry involves a great deal of new vocabulary. Continually add to your Math Word Wall as vocabulary arises. Ensure that all students can see the Math Word Wall and accompany the vocabulary with visuals to support the learning. Use your Math Word Wall for 5 to 10 minutes every day. Give a definition and have students locate the matching word. Ask for an example in the classroom. Alternatively, give the word and have students explain its meaning and provide a real-life example. The more you do this, the better students will become at recognizing the words, understanding their meanings, and knowing where to find them on the Math Word Wall when they want to use them or be reminded of what they mean. • Do a daily sort with your students so they will pay attention to attributes and the characteristics within each category. For example, sort the students by a secret rule, such as the types of shoes they are wearing (e.g., laces versus Velcro). You can also have each student pick an object from the room, sort the resulting collection according to a particular attribute, and have students guess the sorting rule. (e.g., Sort objects by whether their faces are curved or flat. You can create a third pile for objects that have both curved and flat faces.) • Drama: Challenge students to represent certain 3D objects with their bodies. Have them transition from one object to a new object (e.g., move from being a sphere into being a cone). Challenge them to imagine and mimic holding different objects in their arms. Discuss why certain objects are held differently. • Visual Arts: Students can use recycled materials to build three-dimensional structures. This requires using spatial reasoning to ensure that certain objects will support and balance other objects. • Play ‘I Spy.’ Select various objects and describe their attributes (e.g., I spy something that has 5 flat faces). • Do a daily visualization task. For example, have students visualize a simple structure made of connecting cubes as you verbally describe it. You may also pick an object in the room and have students visualize what it would look like if they imagined it upside down. 10 Spatial Sense

• Show students a photo of interesting objects that are in their environment. (You can quickly find such pictures on the Internet.) Show the photo for a brief time and have students study it. After it is hidden, ask students what they remember. Show the picture again and have students look for something specific. Reinforce vocabulary that describes the attributes of objects in the picture, as well as positional language that describes where objects are in relation to each other. • Show students a tray with some objects on it for a short period of time. Cover the tray. Ask students what objects they saw and where they were in relation to each other. Ask what shapes the objects were. • Quick Scavenger Hunt: Give students quick challenges, such as, “Bring something with a curved surface to the meeting place.” • Pose a daily riddle. For example: “I am long and thin. I have a vertex at one end. I have a curved surface. You use me in class. What am I?” • Block Building Centre: Constructing with various types of blocks is a great way to develop spatial reasoning skills and the ability to physically and mentally rotate shapes and objects. Have a building centre set up throughout the year, and vary the types of blocks and materials that students can use at the centre. Geometry 11

Three-Dimensional Objects PMraotcheesmseast:ical About the rccrPeeoorpamnorsnbemoelsenecumitnnnintigicsgnaoa,gtlnrv,inedinfglpg,er,cotviningg,, ssterlaetcetginiegstools and An understanding of 3D geometry is critical, both for living in a three- dimensional world and working in several occupations. For example, Maddcvtcicirautetmtmoeyruortrinlcvrbtheipennieteunc,nad,,dVetsbsnrpee,oisoiogoorr,f,ct,tillnpnsateaaasoamrat,tnpbom,ellcg,h,up,v,i3l,e2repleevcaDlrrDiyr,elsetrtre,e,efuyy,a,ttdf,x:as,mwat,gtqlrhcioieizuader-,eaen,,c,ergue-lbe, e, , architects and engineers need to visualize and mentally manipulate 3D right objects to understand how they will look and operate from different perspectives. Construction workers need to know the form and function For the modifiable of objects and materials to build stable structures. Therefore, it is worth Home Connections letter investing time in the study of 3D geometry so our students can function and Observational in their lives and be prepared for the future. Assessment Tracking Sheets for this unit, see The study of three-dimensional geometry extends beyond simply the Teacher’s Website: naming objects. Students need to be able to identify and describe 3D scholastic.ca/ objects according to their attributes, and they must also learn to education/mathplace/ distinguish between defining attributes, or geometric properties Password: SP9f5H (e.g., number and shape of faces), and non-defining attributes (e.g., colour, size, orientation). This is an important distinction since geometric properties define the different classes of objects, while other attributes do not. For example, a cube must have six congruent faces, but its colour does not affect how it is categorized. Sorting experiences help students make these distinctions. Students also need to be able to compose and decompose 3D objects, using visualization and mental rotation beforehand to develop intentional thinking about their placement. For example, students can build models and skeletons of prisms and pyramids, using concrete materials. They can build cylinders out of nets. They can build structures out of 3D objects, identify the 2D shapes within them, and determine congruency. Joan Moss and her colleagues emphasize the importance of being able to think in both two and three dimensions since they “are complementary to one another, and it is important for students to come to see and understand how the two kinds of geometry are related” (Moss et al., 2016, p. 144). It is important to explicitly discuss the differences between two and three dimensions, and to provide concrete examples and non-examples that allow students to make comparisons. 12 Spatial Sense

Lesson Topic Page 1 Shapes at Play 14 19 2 Understanding 2D and 3D: Paper Chain Magic 26 30 3 Read Aloud: Mummy Math: First Reading 35 39 4 Mummy Math: Second Reading 42 44 5 2D Shapes in 3D Objects 46 53 6 and 7 Exploring 3D Objects 62 67 6 Attributes and Properties of 3D Objects 69 75 7 Describing 3D Objects 79 87 8 Sorting 3D Objects 89 92 9 Building 3D Objects with Modelling Clay 10 Will It Stand? 11 and 12 Building with 3D Materials 11 Cube Configurations 12 Building from a Plan 13 Guided Math Lesson: Investigating Nets with 3D Packaging 14 and 15 Optical Illusions 14 Analysing Optical Illusions 15 Creating Optical Illusions Three-Dimensional Objects 13

1Lesson Shapes at Play Math Spatial Sense Curriculum Expectations • E1.1 sort, construct, and identify cubes, prisms, pyramids, cylinders, and cones by comparing their faces, edges, vertices, and angles • E1.2 compose and decompose various structures, and identify the two- dimensional shapes and three-dimensional objects that these structures contain • E1.3 identify congruent lengths, angles, and faces of three-dimensional objects by mentally and physically matching them, and determine if the objects are congruent Teacher Possible Learning Goals Look-Fors • Identifies and compares two-dimensional shapes and three-dimensional PMraotcheesmseast:ical rccrPeeoorpamnorsnbemoelsenecumitnnnintigicsgnaoa,gtlnrvinedinfglpg,er,cotviningg,, objects in real life and describes their attributes and properties using geometric vocabulary • Distinguishes between attributes and properties of shapes and objects • Distinguishes between two-dimensional shapes and three-dimensional objects • Describes some shapes by some of their geometric properties (e.g., number of sides, edges, and vertices) • Distinguishes between attributes that do not change or affect a shape or object and geometric properties, which do • Identifies some two-dimensional shapes on three-dimensional objects and either names them or describes them by their properties • Compares various shapes and objects by identifying some similarities and some differences Math Vocabulary: About the Lesson ttsqwhpvfhuryaeoaearrc-apetddeeme-irsmdxis,l,iiadmee(vtt,dnereecigrsranotaieniosnclsg,niee,oela,seantl,c,cn,ayr.g2l)el,,ilDnec3p,tdsDraie,snrm,gl,e, This introductory lesson is intended to spark discussion about shapes and objects that are in students’ environments. It can also be used to assess what students know, reveal any misconceptions they may have, and identify areas to focus on in the lessons ahead. 14 Spatial Sense

Materials: Minds On (10 minutes) “Shapes at Play” • Show students the spread “Shapes at Play” in the big book. Read the title. (pages 2–3 in the Spatial Sense big book Ask students what they think the title means and what it has to do with and little books) the pictures. Time: 45–65 minutes • Ask what the theme of the pictures is and why the pictures all belong together. (e.g., they all depict playground equipment) • Ask students what playground equipment they like best and why. Ask if there is a piece of equipment that they had difficulty using at first but have now mastered. Ask what they had to do in order to get better at it. • You may want to keep a running list of geometric vocabulary that arises during the discussion that follows. These terms can be added to the Math Word Wall. Working On It (Whole Group) (20 minutes per session) NOTE: There are more pictures and prompts than can be covered in one session. Select the pictures that best meet the needs of your students. You may decide to discuss only two or three pictures, and to focus on the remaining pictures in another session or in Math Talks at a later date. The order in which you discuss the pictures is optional, but it is suggested that you start with the picture at the top right of page 3 (prompts have been provided for that picture first) because it gives clear examples for distinguishing between 2D shapes and 3D objects (e.g., the game board is painted on a surface and can’t be lifted up). Prompts for the remaining pictures are given in sequence, clockwise around the page. Game Board on Pavement and Circular Climbing Apparatus (page 3, top right) • Ask students what they see. Ask what is two-dimensional and what is three- dimensional in the picture. Discuss the difference between two-dimensional things and three-dimensional things (e.g., two-dimensional things are on the surface and have only length and width, while three-dimensional things have length, width, and depth/height). • Ask students whether the game board painted on the ground is two- dimensional or three-dimensional and why they think so. Ask what activity the game board might be used for and why they think so. Ask if they have seen other game boards painted on the ground. Ask what the overall shape of the game board is and what shapes they see within it. Have students name and describe the shapes they identify. Ask what shape the four squares in a row make. (e.g., quadrilateral, rectangle) Have students describe the angles in these shapes. Ask what other shape is on the game board and how it is different from the squares. (e.g., a semicircle; it has a curved side) Have students visualize another semicircle at the opposite end of the game board and ask what overall shape the game board would have now. • Ask about the different ways in which the three-dimensional tunnel frame could be used, and whether students have ever used something like this. Ask what three-dimensional object the tunnel looks like. (cylinder) Ask what two-dimensional shapes students can see in each ‘ring’ in the tunnel, or in Three-Dimensional Objects 15

the opening of each ring. (circles) Ask how cylinders are different from circles. Ask what smaller cylinders they can see within this climbing structure. (e.g., between two rings) Ask whether circles make up any parts of a cylinder. Show a real cylinder and have students identify the circles on its surface. (at the ends) Spider Web (page 3, middle) • Ask how this piece of equipment works. Ask what shape students see in the centre. Ask what individual shapes the ropes make. (e.g., quadrilaterals, trapezoids) Ask how many sides these shapes have. Ask what happens to the trapezoids as you move out from the centre. (e.g., they get bigger) Ask whether this changes the shapes. (e.g., no, they are still trapezoids) • Ask how many trapezoids there are in one complete turn round the centre. (8) Ask students to look at the bigger shape made by all of the trapezoids together. How many sides does this bigger shape have? (8) What is the name for this shape? (octagon) Elevated Playhouse (page 3, bottom) • Ask students if they have ever seen or played in a house like this. Ask what makes it fun and what they can do inside. • Ask students how they know that the playhouse is three-dimensional. (e.g., it has length, width, and depth) Ask what three-dimensional object the house most closely resembles, without including the roof (e.g., a prism, a cube) and why they think so. Ask what shape the front and back sides of the house are. (pentagons) Explain that the whole house is called a ‘pentagonal prism.’ Explain that the house with the roof and the house without the roof are both ‘prisms’ because they have matching ends. Tell students that they will learn more about ‘prisms’ in the days ahead. • Ask students what shapes they see on the surface of the sides of the house. Ask if the shapes on the surface are two-dimensional or three-dimensional. Ask what is different about the windows and whether they can be considered to be rectangles. (e.g., no, because the top side is curved) • Ask students to find different-shaped triangles. Ask why they are all triangles. Ask what other shapes or three-dimensional objects they see in the picture. Pentagonal Climber (page 2, bottom right) • Ask what is different about this climber compared to the others pictured. (e.g., it has solid sides, or faces, and the others have sides with open spaces, or sides made of ropes) Ask which type of climber they might find more challenging to use and why. • Ask what shape the faces of the boxes are (pentagon) and how many sides the faces have. Draw attention to the blue design painted on one of the faces. Ask how many sides it has. Ask what a shape with 10 sides might be called. Tell them that it is a ‘decagon.’ • Have students visualize and predict how many faces each box may have and why they think so. Ask whether there are more than five sides, fewer than twelve sides, etc. Tell students that it is a ‘dodecahedron’ and has 12 sides. 16 Spatial Sense

Tower with Slide (page 2, bottom left) • Ask students what they find interesting about the climbing tower and slide. Ask how they would climb up the tower. • Ask what shapes they see within the equipment. Ask how they could describe the slide. • Ask students to describe the tower. Ask what shape the sides would be if they were solid. • Draw attention to the roof of the tower. Ask which three-dimensional object it most resembles (pyramid) and why students think so. Have them compare the shape of this pyramid to the pyramid climber immediately above it. Ask what shape the base of the roof is and how they could give the pyramid a more specific name. (square-based pyramid) Ask how the roof (pyramid) is the same as and different from the triangular climbing set (prism, top right of page 2). (e.g., Same: Both have triangular faces and a four-sided shape [quadrilateral] for their bases. Different: The prism has matching end faces. The prism’s sides don’t all meet at a common vertex the way the pyramid’s sides do.) Pyramid Climber (page 2, top left) • Ask students if they have ever seen equipment like this and how it can be used. • Ask what three-dimensional object the climber most closely resembles and why they think so. (pyramid) Ask what shapes they see on the surface of the climber. • Have students visualize what shape the base of the climber would be if it was solid. Discuss different suggestions. (Some may think it would be a square since there seem to be four posts going into the ground, while others may think it is a pentagon if they envision five posts.) Ask what shapes the sides of the apparatus would be if they were solid. (triangles) Explain that the sides of a three-dimensional object are known as ‘faces.’ Ask what is common to all of the triangular faces here. (they all meet at a common point at the top) Explain that the point is known as a ‘vertex.’ • Ask how the pyramid climber is similar to and different from the cone climber on page 3. (Same: common vertex. Different: square or pentagonal base versus a circle; triangular sides versus a curved side) Triangular and Curved Climbing Sets (page 2, top right) • Ask how the two pieces of equipment in the picture can be used. • Ask students how they know that the triangular piece of equipment is three- dimensional. Ask what three-dimensional object it most resembles (prism) and why they think so. (e.g., there are two matching ends) Ask what shape the matching ends are. Ask what students think the full name of this object is. (triangular prism) Ask what shape the other sides that make up a triangular prism are. (rectangles) Three-Dimensional Objects 17

• Ask what shapes the ropes make. • Draw attention to the piece of equipment on the left side of the picture. Have students visualize what the whole piece looks like, even though only part of it is visible. Ask what three-dimensional object it probably most resembles. (e.g., a half sphere) Ask what shapes they see on the surface. • Ask why triangles and squares are good shapes to use in these climbers. (e.g., they fit together well with no spaces) Cone Climber (page 3, top left) • Ask if students have ever seen a climber like this and what would be challenging about using it. • Ask what three-dimensional object the climber most resembles and why they think so. (cone) Ask what two-dimensional faces they see. (e.g., It has one circular face. The other part is not a face because it is not flat.) Ask what other objects they have seen in their environment that are cone-shaped. • Ask what shapes the ropes make in the bottom five rows. Ask what happens to the shapes moving from the bottom to the top. Ask whether the shapes are still quadrilaterals (four-sided) even though they are different sizes. Ask what shapes the ropes make at the very top of the climbing apparatus. Ask why they couldn’t have four sides, too. (e.g., the ropes comes to a point, or vertex, at the top) Consolidation (15 minutes) • Revisit your running list of geometric vocabulary that arose during the lesson, or make a list now. Put the words into two groups: those that describe two-dimensional shapes and those that describe three-dimensional objects. These words can be added to the Math Word Wall. • Building Social-Emotional Learning Skills: Identification and Management of Emotions; Stress Management and Coping: Ask students how they feel after discussing the shapes and objects in the big book. Explain that if they feel overwhelmed by all the new or unfamiliar words and ideas, they have lots of time to learn more about them and to become comfortable with them. Invite students to suggest words to be added to the Math Word Wall. As a class, spend 5 or 10 minutes a day reviewing those words by playing some word games. Students can better manage their stress when they know that there are learning strategies in place to help them. 18 Spatial Sense

9Lesson Building 3D Objects with Modelling Clay Math Spatial Sense Curriculum Expectations • E1.1 sort, construct, and identify cubes, prisms, pyramids, cylinders, and cones by comparing their faces, edges, vertices, and angles • E1.2 compose and decompose various structures, and identify the two- dimensional shapes and three-dimensional objects that these structures contain • E1.3 identify congruent lengths, angles, and faces of three-dimensional objects by mentally and physically matching them, and determine if the objects are congruent Teacher Possible Learning Goal Look-Fors • Builds 3D objects, names them, and describes their properties using Previous Experience with Concepts: descriptive geometric vocabulary Students are familiar with the geometric properties • Creates various 3D objects using modelling clay, from memory, using a model, of several 3D objects. or by feeling the properties • Explains or shows how 3D objects are the same and different • Describes their objects by the number and shapes of faces (optional: by the number of vertices, edges, and/or angles) • Compares their models to their peers’ models using geometric vocabulary PMraotcheesmseast:ical About the arPenrpodrbeplseroemvnitnsinoggl,v,inrega, soning ssccteoorlmanetncmeetgicunitenginsictgoa,otrilensfglae,ncdting, Students can reinforce their understanding of properties by building three-dimensional objects. Doing so encourages them to not only verbally identify properties, but to create them. By building, students can also explore how objects within categories can differ. Initially, students may need to refer frequently to a model while building. In later activities, they can be challenged to build objects from memory. This kinesthetic experience helps students form and refine mental images of objects, which can later be retrieved to solve problems. continued on next page Three-Dimensional Objects 53

Angles and Vertices in 3D Objects Working with angles and vertices when discussing 3D objects can be confusing. The curriculum defines an angle as “a shape formed by two rays or two line segments with a common endpoint” and a vertex as the “common endpoint of the two line segments or rays of an angle” (Ontario Ministry of Education, 2020). In a two-dimensional shape, such as a square, there are angles and vertices wherever two sides meet. The angles are a measurement of the relationship of between line segments on a flat surface. Two-dimensional shapes with straight sides (polygons) can be described by the number and length of their sides, and the number and size of their angles. Three-dimensional objects are described by the number of their faces, edges, and vertices (or corners). The edges are where two faces meet and the vertices are where three edges meet. We can find and describe the angles in the two-dimensional faces on three-dimensional objects, where the sides of those faces meet. So a cube has 8 vertices and 24 right angles (4 right angles on each of its 6 square faces). Math Vocabulary: angle side vertex pctdevrwyiidelmsoirgnm-teeddic,nseimep,sr,siyaoe,crnnnaovgasmenlilreeo,itsd,enf,a,cxacul,ebtseh,,ree- edge face About the Lesson Students will build 3D objects using modelling clay. They can build the objects from memory, checking back against a model if/as needed. They can compare their objects using geometric properties. Materials: Minds On (10 minutes) various geometric solids and other 3D • Show students some of the 3D objects from previous lessons (Lessons 6–8). objects, modelling clay, cards or paper for Review some of their geometric properties and how feeling the objects labelling sorting groups helped students to identify what they looked like. Review some of the geometric vocabulary used. Time: 50 minutes Working On It (20 minutes) • Challenge students to build several 3D objects out of modelling clay. Students can work in groups of four to share resources, but can build objects individually. • Once finished, group members can compare their creations. They can also sort them, choosing the properties that will make up their sorting rule. 54 Spatial Sense

Differentiation • Some students may be able to build the objects from memory, while others may need to have concrete examples directly in front of them to serve as a model. • Students who need more of a challenge can select an object from an opaque bag and then build it based only on what they feel. In this way, they cannot rely on their sense of sight. • Some students may benefit from focusing only on selected objects, such as prisms and pyramids, since this would be less overwhelming. Assessment Opportunities Observations: Take note of whether students can build objects from memory or whether they need to refer to a model. If they are building from memory, they have probably internalized strong mental images of the objects and their properties. If students are always using objects as models to build their own, encourage them to start building from memory. Conversations: Prompts to use with students who want to build from a model before they start building: – Is this the object you are building? Let’s just hide it for a little while. – What do you remember about the object you are building? Does it have flat faces or curved faces? Did it have any vertices? – Let’s put the object in the bag. Feel it, and try to answer some more questions: ° How many faces does it have? Are they flat or curved? Are they all the same? ° How many vertices do you feel? – Start building the object again, starting with one face. If you need to, you can reach into the bag and feel the object again. Materials: Consolidation (20 minutes) various of materials (e.g., straws, • Have a gallery walk. Group members can sort their objects and then leave toothpicks, marshmallows, modelling clay), Polydrons them in different piles without any labels. • As a class, move from one group’s piles to another’s. Have students predict what the sorting rule is and why they think so. The group members can explain their sorting rule once their peers have made their predictions. Create sorting labels before leaving each station. Further Practice • Have students make skeletons of prisms and pyramids using various materials. For example, they can use straws for the edges and pieces of modelling clay for the vertices. Alternatively, they can use toothpicks and marshmallows. Have students determine the number and shapes of faces, and the number of edges and vertices. Three-Dimensional Objects 55

• Polydrons are commercial manipulatives that can be used to create nets of three-dimensional objects. They snap together easily and can be folded or unfolded to be either two-dimensional or three-dimensional. Have students create three-dimensional objects using Polydrons. In this way, students can concentrate more on identifying geometric properties rather than on accurately producing 3D objects. You can also give students folded Polydrons and have them decompose the objects to explore the 2D shapes that make up their faces. Materials: Math Talk: Find My Object geometric solids Adapted from the lesson “What’s My Shape?” by Van de Walle and Lovin Teaching Tip (2006a, p. 195). Math Focus: Identifying 3D objects by their geometric properties Integrate the math talk moves (see page 6) Let’s Talk throughout Math Talks to maximize student Have students sit in a circle around five objects from the bag in Lesson 7 and/ participation and or the collection used in Lesson 8 (e.g., cone, cylinder, cube, rectangular prism, active listening. square pyramid). Select the prompts that best meet the needs of your students. 56 Spatial Sense • I am thinking of one of the objects in the middle of the circle. You need to figure out which one. You can ask me questions about the mystery object, but I can only answer ‘yes’ or ‘no.’ One more rule is that you can’t point to an object and ask “Is it this one?” until you are sure that you know the mystery object. • You can ask me questions now. If a question is repeated, tell students that you have answered it already and ask who remembers the answer. Here is a sample sequence of questions and answers: – Does it have curved faces? No – Does it have square faces? Yes – Are all of the faces square? No – Does it have 12 edges? Yes – We think it is the rectangular prism. • What did you find challenging about this game? (e.g., knowing what questions to ask, remembering the questions that were asked and the answers) • Let’s go back to the first question: “Does it have curved sides?” I answered ‘no.’ What did you know? (e.g., We knew that it couldn’t be the cone or the cylinder.) You knew which ones couldn’t be the mystery object. What could you do to help you remember that? (e.g., We could move them to a different pile.) • We are going to play the game again except, this time, I am going to add more objects. You can add prisms and pyramids with different-shaped bases. You may want to arrange the objects differently as I answer your questions. • After students have asked the first question and you have provided an answer, ask them what they now know about which object(s) it isn’t. This should prompt them to start sorting or removing some objects. You may need to repeat this prompt one or two more times. • Play the game again, except have one of the students pick the mystery object and answer the questions.

• Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: When you asked your questions, I sometimes said that what you were thinking was not correct. For example, you may have thought that the shape was the cylinder so you asked if it had curved surfaces and I said no. How can knowing what the mystery object isn’t help you solve the problem? We always learn something when our guesses or predictions are not correct. So we can learn a lot by making mistakes. Arts Math Talk: Shapes in Indigenous Curriculum Expectations Creations Math Focus: Identifying two-dimensional and three-dimensional shapes in Indigenous creations and how they affect the images and impressions of the artwork Visual Arts • D3 Exploring Forms and Cultural Contexts: demonstrate an understanding of a variety of art forms, styles, and techniques from the past and present, and their social and/or community contexts • Elements of Design: shape and form, space, colour, texture Materials: About the “Shape Art” (pages 8–9 in the Spatial Sense The aesthetics of two- and three-dimensional shapes found in nature big book and inspire First Nations, Métis, and Inuit artists to create pieces that convey little books) stories, messages, and feelings. Artists carefully plan how they construct three-dimensional structures and combine shapes to tell their stories, often using repetition or symmetry to create patterns. This spread in the big book displays several pieces of art from various First Nations, Métis, and Inuit communities across Ontario and Canada. It must be remembered that there are many communities that are diverse, as is their artwork. The art plays an integral role in the various cultures, telling stories that reflect family history, details about the families, and their spiritual connections. Typically, the artwork reflects harmony and balance in nature and often features animals found in the environment, such as seals, bears, beavers, and eagles. Some of the three-dimensional pieces are created from materials found in the local environment. They are created and shaped using techniques that have been passed down from generation to generation. Several of the pieces are functional as well as artistic. For example, birchbark baskets can be used to carry and store items. They can also be used as cooking pots. The skill of the artist is critical to ensure that the items can properly perform their function, as well as reflect the traditional way in which they were made. The artists also creatively incorporate designs that convey the stories. continued on next page Three-Dimensional Objects 57

Some of the pieces of art are stylized, rather than depicting how the animals or objects realistically appear. The artists use various shapes to achieve this technique, many of which are inspired from the natural environment. For example, the ovoid, which is a curvy, rounded rectangle, may have been derived from the white circle around the orca whale’s eye or from the shapes that are on the backs of skate fish. Other common shapes include the U shape, which looks like a filled in U that can vary in thickness. There are also split U shapes, which are U shapes divided by a distinctive T shape. S shapes are also used, often forming a repeating pattern. Many of the shapes are outlined in a curvy, black line known as the formline. Colours are also taken from nature, including red, blue-green, white, and yellow, and in the past were made from natural materials. Let’s Talk Select the prompts that best meet the needs and interests of your students. • Display the “Shape Art” spread. What do you see on this page and who do you think created the pieces of art? These are the works of First Nations, Métis, and Inuit artists across Ontario and Canada. They depict life, both past and present, in various cultures. • What do many of the pieces of art have in common? (e.g., bright colours, a variety of shapes, elements of nature) • Are these pieces two-dimensional or three-dimensional? How do you know? • What two-dimensional shapes make up the three-dimensional artwork? • What do you notice about the two-dimensional shapes that decorate the outside of the pieces? Eagle Mask – Bert Smith (Kwakwaka’wakw) • What do you think this picture shows? Why do you think so? Is it three- dimensional or two-dimensional? How would you describe the shape of this three-dimensional mask? • What material is the mask made from? How do you think it was made? It is made from wood found in the local environment. The artist carved the mask using special tools and applying techniques that he learned from his elders. • What do you notice about the shape of the two sides of the mask? (e.g., The left and right sides are symmetrical.) What challenges are involved with making a three-dimensional mask that is symmetrical? • What two-dimensional shapes do you see on the mask? What shapes can you see? Point to an ovoid shape. This shape is an ovoid and it is a rounded, curvy rectangle. Where else do you see ovoid shapes? It is thought that people in past First Nations, Métis, and Inuit communities may have adopted this shape from the animals in their environment, such as the white circle around the eye of the orca whale or from the shapes on the back of the skate fish. 58 Spatial Sense

• Where do you see U shapes? Many First Nations, Métis, and Inuit artists use U shapes. How are they like the ovoid shapes? There are also split U shapes, like the U has been split in the middle. Where do you see them? • What do you think the colours on the mask represent? The colours represent elements in nature. • What symmetry do you see in the design of the mask? What kind of line of symmetry is represented? (vertical) Basket – Juliana Alexander (Splatsin te Secwepemc) • What do you see in this picture? How would you describe the shape of this basket? How is it different from a cylinder and from a prism? (e.g., the bottom has sharp corners, the sides are curved) • What do you think this basket is made from and why do you think so? This basket is made from birchbark. How do you know that it is made from birchbark? (e.g., The horizontal lines on the basket are like horizontal markings on birchbark trees.) Why would birchbark be a good material for a basket? (e.g., It is light-weight.) • What do you think the baskets may be used for? (e.g., carrying or storing items) They can also be used as cooking pots. What other feature must the basket have in order to be used for cooking? (It needs to be waterproof.) The bark has a substance that is much like wax, which makes it waterproof. • The skill and expertise of making birchbark baskets have been handed down from generation to generation. Building a basket takes time, patience, and respect for the environment. What do you think happens first? (e.g., find and remove the birchbark) It needs to be carefully removed from the tree in late spring so the tree is not harmed. • How do you think the basket is made? The birchbark is carefully heated so it bends easily and then it is folded into a desired shape. It is then sewn together with spruce root, which is strong like thread. The spruce root also makes up the rim of the basket. What is the function of the rim of the basket? (e.g., to give the basket strength and keep the shape) • What other shapes do you think the baskets may be made into? Why? Heron at Sunrise – Joe Greene (Cayuga) • What do you see? Is it two-dimensional or three-dimensional? How do you know? • What animal is represented in the art? What features of the heron has the artist highlighted in his work? (e.g., its long beak and long, flexible neck) What does this tell you about some of the habits of the heron? (e.g., it fishes with its long beak; its long neck can help it quickly catch fish under the water) • What shapes has the artist used to show these features? (e.g., cylinder-like shapes that have been bent) • Where do you see edges in the sculpture? (e.g., in the beak) What do you think the artist is trying to tell us about the heron’s beak by including edges? continued on next page Three-Dimensional Objects 59

• What material do you think this sculpture is made from? It is made from a stone called ‘soapstone.’ It is used for sculpting because it is very soft and easy to carve with tools such as chisels. • What two-dimensional shapes has the artist included on the sculpture? (e.g., triangles on the neck) What patterns are made with the triangles? • Once the artist carves the stone into the shapes he wants, he uses sandpaper to make the surface smooth. He then paints any features, such as the head, which was painted with walnut dye that was made from trees in the area. • How does this sculpture make you feel? How does it make you feel about the heron? • Would you consider this a modern or traditional piece of art? Why? Guardian of the Spirit Dreamer – Thomas Maracle (Kanien’kehá:ka [Mohawk]) • What do you see in this sculpture? How does it make you feel? • Is this piece of art two-dimensional or three-dimensional? How do you know? • This was created by Thomas Maracle, a member of the Mohawk community. He has been involved in Indigenous art since he was a child. He learned many skills and techniques from his parents who were also artists. • Thomas Maracle is known for his ability to use many different types, shapes, and sizes of stone and create images from them. Visualize a piece of stone. What do you have to do to create an image? (e.g., take stone away) Artists must be careful not to take too much stone away because they cannot put it back once it has been chipped away. • This sculpture is actually made from bronze rather than from stone. Bronze is made from a mixture of copper and tin. • This sculpture is called “Guardian of the Spirit Dreamer.” What is a guardian? Who do you think is the guardian and who do you think is the spirit dreamer? How would you describe the relationship between the guardian and the spirit dreamer? Why? • Why do you think the artist has the eyes of the eagle open and the eyes of the spirit dreamer closed? Why are the eyes such important features on this sculpture? • What shapes do you see? How does the artist use different shapes, lines, and textures to show different types of feathers on the eagle? Hunter and Seal – Willie Pamialok (Inuit) • What do you see in this sculpture? What do you find interesting about it? • This sculpture is made of soapstone, just like the sculpture of the heron. How do you think these pieces of soapstone are the same and how are they different? (e.g., they are both soft for carving; they are very different colours) There are many different colours of soapstone and artists choose the type according to what is available in their area and what best suits the image they are trying to create. 60 Spatial Sense

• The artist starts with a block of soapstone and carves it to make the images. How is this different from creating a piece of art with paint? (e.g., with a painting, you add paint to create a picture and with a sculpture you take away stone to create an image) • What shapes do you see in this sculpture? Does the artist use more rounded shapes or shapes with edges? Why do you think so? Where do you see edges? How do the edges help to define the shapes? • Once the artist has carved the shapes he wants, he sands it, first with coarse sandpaper, and then uses finer and finer sandpaper. With each different type of sandpaper, the soapstone gets smoother and shinier. Sculptors often polish the finished product with oil to make it even shinier. • What does the sculpture tell you about the traditional life of the Inuit? Why do you think the seal is so important to the Inuit? The hunters use all parts of the seal to help them in their survival. For example, they eat the meat, they use the skins to make clothing and waterproof footwear, and they use the fat to make oil to burn in lamps. Moccasin boots – Sarah Timewell (Métis) • What do you see in this picture? What are the moccasins used for? Why are they also considered a piece of art? The way that moccasins are made is considered an art and the methods have been passed down from generation to generation. The beadwork patterns are also a form of art that has been used for generations. • What parts of the moccasins are three-dimensional and what parts are two-dimensional? • What basic three-dimensional shapes do you think the maker of the moccasins used in the design? (e.g., cylinders, oval shapes) What would the maker need to know before making the moccasins? (e.g., the size of the wearer’s feet) • What do you think the moccasins are made from? They are made from animal hides or skins. The skins have been cleaned, stretched, and dried. Why do you think that animal skins would be good for making footwear? (e.g., the skins are tough and waterproof) • This artist decorated the moccasins with beads. Traditionally, the designs often reflect the heritage and personal history of the wearer. The beading techniques have been handed down from generation to generation. • Is the beadwork two-dimensional or three-dimensional? The beads are actually three-dimensional because they have depth, but they make a two-dimensional pattern on the top surface of the moccasins. • What patterns and designs do you see in the beadwork? Turn and talk to your partner. • What two-dimensional shapes do you see? (e.g., circles, triangles, squares) • Which shapes have symmetry? What kind of symmetry do they have? • How does the bead design make you feel? Three-Dimensional Objects 61