# 5.3: Shapes and Spatial Relations

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## Course Competency 5. Examine strategies that teach early math skills.

5.3. describe shapes and spatial relationships.

Geometry sounds like a complex topic to introduce to young children. However, the basic foundations of geometry are introduced in the early childhood years.

### Supporting Geometry

Geometry is the study of shapes and spatial relationships. Children enter preschool with a strong intuitive knowledge about shapes, spatial location, and transformations. They learn about geometry as they move in space and interact with objects in their environment. From infancy they begin to form shape concepts as they explore their environment, observe shapes, and play with different objects. Before they can name and define shapes, very young children are able to match and classify objects based on shape. During the preschool years, children develop a growing understanding of shape and spatial relationships. They learn the names of shapes and start to recognize the attributes of two- and three-dimensional shapes. They also develop an understanding of objects in relation to space, learning to describe an object’s location (e.g., on top, under), direction (e.g., from, up, down) and distance (e.g., near, far).

Please see the Wisconsin Model Early Learning Standards for Mathematical Thinking to review the standards, developmental continuum and sample behaviors of adults.

The information below is from the California Preschool Standards and is again provided as a reference to the age you may see these geometry related skills appear. However, remember the Wisconsin Model Early Learning Standards include the skills for Shapes and spatial relationships in order of development but do not assign a specific age to any skills as all children develop at their own rate and time.

Examples in 1.0 and 2.0 would correlate to WMELS B.EL. 3 Explores, recognizes, and describes shapes and spatial relationships

At around 48 months of age

At around 60 months of age

1.0 Children begin to identify and use common shapes in their everyday environment.

1.0 Children identify and use a variety of shapes in their everyday environment.

1.1 Identify simple two-dimensional shapes, such as a circle and square.

1.1 Identify, describe, and construct a variety of different shapes, including variations of a circle, triangle, rectangle, square, and other shapes.

1.2 Use individual shapes to represent different elements of a picture of design.

1.2 Combine different shapes to create a picture or design.

2.0 Children begin to understand positions in space.

2.0 Children expand their understanding of positions in space.

2.1 Identify positions of objects and people in space, such as in/on/under, up/down, and inside/outside.

2.1 Identify positions of objects and people in space, including in/on/under, up/down, inside/outside, beside/between, and in front/behind.

Teachers can support children’s development of the geometry foundations with the following:

• Refer to shapes and encourage the use of shape names in everyday interactions
• Engage preschool children in conversations about shapes, including both
• Two-dimensional shapes (such as circles, squares, and triangles)
• Three-dimensional shapes (such as spheres, cubes, and cones)
• Provide materials that encourage pre­­school children to explore and manipulate shapes in space
• Include books, games, and other learning materials with shape-related themes in the preschool environment
• Provide preschool children with playful opportunities to explore and represent shapes in a variety of ways
• Present preschool children with many different examples of a type of shape
• Provide materials and equipment to promote spatial sense
• Support preschool children’s spatial sense in everyday interactions
• Provide preschool children with planned experiences to promote the understanding of spatial sense, including
• Songs and games
• Books
• Construction opportunities[14]
##### Vignettes

The teacher had noticed that several children in her group had shown a strong interest in castles. They built castles in the block area, in the sandbox, and even looked for castles in fairy tale books when visiting the library. The teacher suggested that the group build a big castle outside. They started by gathering the materials. The children brought from home different sized boxes and figures or characters to be included in the castle. The teacher also offered big cylinders, cones, building blocks, construction boards, and other materials. The children made different suggestions: “Put all the big boxes here and the small ones on top of them.” ”I put it above this for the roof.” “We can use these for the tower.”

The teacher described their ideas using names of shapes and spatial terms. “So you want to put the small square blocks on top of the big rectangle blocks.” “Are you suggesting using the cylinders to build the tower?” The children enjoyed building the structure, using different shapes and materials, and were proud of it.

During circle time, the teacher invited children to describe the castle and how it was built. “Look at the castle you built. Can you tell me what it looks like?” Children were encouraged to use spatial words and the names of shapes in their talk. The activity evolved into a long-term project. The children kept adding more pieces to the s

Geometry is a branch of mathematics that studies the sizes, shapes, positions, and dimensions of things. “Geometric and spatial thinking are important in and of themselves, because they connect mathematics and the physical world… and because they support the development of number and arithmetic concepts and skills” (Progressions for the Common Core State Standards in Mathematics, 2013). The Progressions document goes on to say,

“learning geometry cannot progress in the same way as learning number, where the size of the numbers is gradually increased and new kinds of numbers are considered later. In learning about shapes, it is important to vary the examples in many ways so that students do not learn limited concepts that they must later unlearn. From Kindergarten on, students experience all of the properties of shapes that they will study in Grades K–7, recognizing and working with these properties in increasingly sophisticated ways. The Standards describe particular aspects on which students at that grade level work systematically, deeply, and extensively, building on related experiences in previous years” (Progressions for the Common Core State Standards in Mathematics, 2013).

Former NCTM President J. Michael Shaughnessy said, “If algebra is the language of mathematics, geometry is the glue that connects it” (Shaughnessy, 2011). Additionally, geometry “covers the skills and concepts of visualization, spatial reasoning and representation, and analyzing characteristics and properties of two- and three-dimensional shapes and their relationships” (Lappan, 1999). In the Kansas Mathematics Standards, geometry spans every grade level from kindergarten to grade eight; it first begins with spatial sense, an intuition about shapes and the relationships between them including an ability to recognize, visualize, represent, and transform geometric shapes.

## Van Hiele Levels for Teaching Geometry

The development of geometric thinking comes from Pierre van Hiele and Dina van Hiele-Geldof. The van Hieles identified five levels of geometric thinking through which students pass. Most elementary students are at levels 0 or 1 and some middle school students are at level 2. The levels are developmental – children of any age begin at level 0 and progress to the next level through experiences with geometric ideas (Van de Walle, Karp, Bay-Williams, 2019).

### Level 0: Visualization

Students begin by recognizing shapes by their whole appearance, but not exact properties. For example, students see a door as a rectangle or a clown’s hat as a triangle, but may not be able to recognize the shape if it is rotated. The emphasis at Level 0 is on shapes that students can observe, feel, build/compose, or take apart/decompose. The goal of Level 0 is to explore how shapes are alike and how they are different and use these ideas to create classes of shapes (Van de Walle, Karp, Bay-Williams, 2019).

### Level 1: Analysis

At this level, students start to learn and identify parts of figures and can describe a shape’s properties. Additionally, students at this level understand that shapes in one group have the same properties. For example, students know that parallelograms have opposite sides that are parallel and can talk about the properties of all parallelograms, not just this one. Students at Level 1 use physical models and drawing of shapes and use the properties of shapes such as symmetry, classification, and congruent sides and angles.

### Level 2: Informal Deduction/Abstraction

Students at Level 2 start to recognize the relationship between properties of shapes and develop relationships between these properties. Students will consider if-then reasoning, such as “If all four angles are right angles, the shape must be a rectangle. If it is a square, all angles are right angles. If it is a square, then it must be a rectangle.” Level 2 includes informal logical reasoning and should be encouraged to ask “Why?” or “What if?” Additionally, Level 2 tasks emphasize logical reasoning.

### Level 3: Formal Deduction

At this level, students analyze informal arguments and are capable of more complex geometric concepts. A student at this level is usually in high school.

### Level 4: Rigor

The last level of geometric reasoning is the ability to compare geometric results in different axiomatic systems; they see geometry in the abstract. A student at this level is usually a college mathematics major.

Read this short article from NRICH for some activities required at each of the levels of geometric thinking.

In order to support students as they move from a Level 0 to a Level 1, teachers should focus on the following:

• Focus on the properties of shapes rather than the identification of those shapes,
• Challenge students to test their ideas about shapes using a variety of examples, and
• Provide multiple opportunities for students to draw, build, make, compose, and decompose shapes.

## Shapes and their Properties

Students should first focus on the location and position of shapes in order to develop a variety of skills that will contribute to their spatial thinking. In kindergarten, students are expected to describe the position of shapes in the environment using the terms above, below, beside, in front of, behind, and next to. Additionally, students develop spatial sense by connecting geometric shapes to their everyday lives and shapes in their environment.

Students must have experience with a variety of two- and three-dimensional shapes. Additionally, triangles should be shown in several forms and not always with the vertex at the top or the base horizontal with the bottom of the paper (Van de Walle, Karp, & Bay-Williams, 2019).

Kindergarten students learn to describe shapes by their attributes and should be given multiple opportunities to build physical models. When students build and manipulate shapes, they can learn to explore and describe the number of sides and corners (vertices). Ask students to describe shapes by the properties, such as “squares have four sides of equal length.”

### Two-dimensional shapes

It is in kindergarten that students learn to distinguish between two- and three-dimensional shapes. Two-dimensional shapes are flat and can be measured in only two ways such as length and width. Examples of two-dimensional shapes are squares, circles, triangles, etc.

Classifying shapes begins in kindergarten. And when students sort and classify polygons, they should determine the groupings, not the teacher. In second grade, students focus on triangles, quadrilaterals, pentagons, and hexagons. Third grade students think about the subcategories of quadrilaterals, and by fifth grade, they “understand the attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.” For example, a square is a rectangle and a square is also a rhombus.

### Composing and Decomposing

Students need multiple opportunities to explore how shapes fit together to form larger shapes (compose) and how larger shapes can be made of smaller shapes (decompose). In kindergarten, students move beyond identifying and classifying shapes to creating new shapes using two or more shapes. This concept is foundational to students’ development of translation (move or slide), rotation, and reflection (flip). Chilldren can first be exposed to many of these concepts while playing in the block area.

In first grade, students compose and decompose plane figures and determine which attributes are defining and non-defining. Defining attributes are those properties that help to define a shape, such as the number of angles, number of sides, length of sides, etc. Non-defining attributes are properties that do not define a shape such as color, location, position, etc. An example is a square. All squares must be closed figures and have 4 equal sides and all angles 90o; these are defining attributes. Squares can be different colors and be turned in a different direction; these are non-defining attributes.

Example:

Which of the following is a square? And how do you know?

• A square has 4 sides.
• All four sides are the same length.
• All angles are 90 degrees.
• Therefore, the red figure is a square.

Be sure to give students many opportunities with a variety of manipulatives to explore and build shapes. These manipulatives could be paper shapes, pattern blocks, color tiles, tangrams, or geoboards with colored rubber bands (or you can use virtual geoboards).

When students are composing and decomposing both plane and solid figures, they are also building an understanding of part-whole relationships and compose and decompose numbers. Furthermore, it is in first grade that students partition regions. Students need multiple opportunities to use the words halves, fourths, and quarters, which is critical to understanding fractions. In second grade, students work with halves, thirds, fourths; and teachers will help students make the connections that a whole is two halves, three thirds, or four fourths.

In third grade, students partition shapes into equal portions. For example, partition a square into four equal parts.

Each of the above squares are partitioned into fourths, and each part has the same area.

## Three-dimensional shapes

Kindergarten students learned to distinguish between two- and three-dimensional shapes. Three-dimensional shapes have three dimensions, such as height, width, and depth. Examples of three-dimensional shapes are cubes, prisms, rectangular prisms, pyramids, etc. Additionally, it is important to note that the faces of three-dimensional shapes can be named as specific two-dimensional shapes. Give students multiple opportunities to work with pictorial representations, concrete objects, and technology as they develop their understanding of both two and three-dimensional shapes.

5.3: Shapes and Spatial Relations is shared under a CC BY license and was authored, remixed, and/or curated by Janet Stramel & Vicki Tanck (Northeast Wisconsin Technical College).