Geometry: K-6 (CCSSM)

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Common Core Content Domain- Geometry: K-6 (CCSSM)

Contributors: Katie Quinlan, Lindsey Gatfield, Suheyla Sarac, and Hannah Suh


Background: Levels of Geometric Thinking

Lead Contributor: Suheyla Sarac
  • Distinguish between Examples vs. Non-Examples

The first step in the development of geometric shapes is students' ability to be able to distinguish between the examples and non-examples of shapes (K.G.4).

  • Visual/Syncretic: Recognize Shapes

Students recognize shapes as wholes (i.e. a figure is a rectangle because it looks like a door) rather than identifying the properties of the shapes.

  • Descriptive: Perceive Attributes of Shapes

Students classify shapes by their attributes, e.g. a square has four sides. They differentiate between geometrically defining attributes (hexagons have six sides) and non-defining attributes (color, size, or orientation)

  • Analytic: Characterize Shapes by their Properties

Students characterize shapes by their properties, e.g. the sum of the angles in a triangle is 180 degrees.

  • Abstract: See Relationship between Classes of Figures

They can classify shapes hierarchically by their properties, i.e. accept classes within classes. e.g. a rectangle is a parallelogram because it has all the properties of parallelograms. e.g. a square is a rectangle because it has all the properties of rectangles

Common Core Strands

Geometric Shapes & Components

2-D Shapes

Lead Contributor: Kathryn Quinlan

Common Core Progression for 2-D figures

Summary of the 2-D Shape Progression:

Kindergarten and first grade are devoted to the general concept of shapes, and students learn about shape vocabulary, recognition, comparison, and attributes. Students learn to distinguish between 2-D and 3-D, then focus on 2-D attributes and a wider range of shapes (ex: hexagons) until 3rd grade. The category of quadrilaterals are introduced in 3rd grade as are a greater range of shapes (ex: rhombuses). Students move to into the concept of angles and delve even deeper into attributes of 2-D shapes in 4th grade. RIght angles and right triangles are formally introduced in the 4th grade. Graphing begins in 5th grade with an expectation that students learn to graph points on the coordinate plane to solve real-world and mathematical problems. Students continue to classify 2-D figures into categories based on their properties. In 6th grade, 3-D shapes are re-introduced for the first time since 1st grade, and continue to focus mainly on graphing and representing and measuring 2-D shapes on the coordinate plane.


  • Students begin to identify and describe shapes in their environment, use informal names for the shapes they recognize and describe the relative position of shapes (K.G.1 and K.G.2).
  • Students name shapes despite differences in the orientation or size of the shape (K.G.2), and learn to distinguish between 2-D and 3-D (K.G.3).
  • Students analyze and compare shapes, in varying sizes and orientations, and use informal language to describe their attributes (i.e., a triangle has three corners) (K.G.4).

1st Grade:

  • Students begin to distinguish between defining attributes (triangles are closed and have three sides) versus non-defining attributes (color or orientation, for example) of 2-D shapes (1.G.1).
  • Students will build or 2-D draw shapes to display these attributes (1.G.2).

2nd Grade:

  • Students will learn to recognize and draw shapes having specific attributes, like a certain number of angles (triangles have three) or a given number of equal faces (2.G.2).
  • Students identify triangles, quadrilaterals, pentagons, hexagons, and cubes (2.G.2).

3rd Grade:

  • Students begin to understand that shapes in different categories can share attributes (for example, both squares and rhombuses have four sides but are different shapes) and that the shared attributes can define a larger category (e.g., quadrilaterals) (3.G.1).
  • Students recognize rhombuses, rectangles, and squares as examples of quadrilaterals (3.G.1).

4th Grade:

  • Students are asked to draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines for the first time in 4th grade. Students also learn to identify these in two-dimensional figures (4.G.1).
  • Additionally, students classify two-dimensional figures based on the presence or absence of these new ideas: parallel or perpendicular lines, the presence or absence of angles of a specified size (4.G.2).
  • Students are introduced to right angles and learn to both identify right angles and recognize right triangles as a category (4.G.2).

5th Grade:

  • Students learn about a pair of perpendicular number lines called axes that define a coordinate system. The intersection of the lines (origin) is at "0" on each line and students learn to locate a given point in the plane by using an ordered pair of numbers, called coordinates. Students also learn to represent real world problems by graphing and learn more about the graphing system in 5th grade (5.G.1 and 5.G.2).
  • Students continue to classify and categorize 2-D shapes with increasing complexity (5.G.3).

6th Grade:

  • Students draw polygons in the coordinate plane and use coordinates to find the length of sides and other attributes of shapes. Students continue to apply these techniques in the context of solving real-world and mathematical problems (6.G.3).

Understanding the Concept of Shape
Sorting Activity
A useful activity for distinguishing between examples and non-examples of shapes.

Image From: Progressions for the Common Core

Recording Classroom Shapes Activity
An example of a recording sheet for identifying shapes in the environment, appropriate for Kindergarten students.

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Classroom activities for learning about the concept of shape
These are activities that help students to develop a flexible understanding of shapes and to recognize shapes in their environments. Developmentally appropriate for students ages 5-6.

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Image From: Hannibal, 1999.
Lesson plan and assessment for using poetry to develop the concept of shape.
Students will explore geometric figures and positional words by reading the poem "Shapes" from A Light in the Attic, by Shel Silverstein. Students then create their own illustration of the poem.

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Image From: "A Light in the Attic" by Shel Silverstein
Classification Activity - Properties Heirarchy
This is a great activity for fifth-graders learning to classify 2-D shapes based on properties. It is important for students to understand what defines a shape category and teachers should model for students how to identify the integral attributes which define a shape's membership in a particular category (Hannibal, 1999).

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Image From: Progressions for the Common Core.

"Guess My Rule" Game
A game for classification of 2-D figures based on known properties and helpful for visualization skills, particularly appropriate for 4th-5th grades.

Image From: Progressions for the Common Core
Three-sided 2-D Shapes
Introducing 3-sided shapes
Suggestions for how to introduce three-sided shapes to students to give a flexible understanding, using examples in different sizes and orientations.

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Image From: Hannibal, 1999.
Four-sided 2-D Shapes
Categorizing Quadrilaterals by Properties
A classification visual for 3rd graders (3.G.1) for placing different kinds of quadrilaterals into categories based on shared attributes. Students should recognize the attributes that each category shares and understand that some of the quadrilaterals do not fit in any category. This visual aid also shows shapes in various orientations.

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Image From: Progressions for the Common Core

  • Lesson plan: through the use of dynamic software, students examine properties of rectangles and parallelograms (identifying what distinguishes a rectangle from a more general parallelogram).:
Angles and Other Attributes
A unit on simple angles that includes assessments:
Graphing and the Coordinate Plane
A "Describe the Graph" lesson plan that incorporates a short story writing element, appropriate for grades 5-6:

Technology Links for 2-D Shapes
Concepts of Shape:
NCTM Illuminations Shape Sorter
Another useful activity for distinguishing between examples and non-examples of shapes, which provides shapes in different sizes and different orientations.

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NCTM Geoboard Applet
This applet promotes understanding the concept of triangle and the properties of polygons. Students must explain how shapes on their geoboards are alike and different.
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Image From: NCTM

Turtle Logo for kids is a type of computer programming language that can teach students about angles. Logo has a pointer, an icon of a turtle, that draws representations of points, line segments, and shapes. Commands can be used to make simple shapes or lines (ex: “Rotate the Turtle <n> degrees to the right.”):

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Image From: Logo's website
Assessment Resources for 2-D Shapes
A geometry assessment that includes 2-D shapes assessment items:
NCTM's Teaching Mathematics Problem generator is a resource for geometry assessment items. Teachers can select the "shapes" topic and generate anywhere from 5-30 assessment items for shapes.
A shapes item example:
  • The supermarket is full of objects having different shapes. The next time you are in the grocery store, take time to notice the various food items, packaging, and storage displays. What are some of the different shapes you can see? How many different shapes of crackers can you fi nd? Choose some crackers that have interesting shapes and bring these to school to compare them with crackers your classmates found. Group the crackers by their attributes, such as shape and size.

3D Shapes

Lead Contributor: Suheyla Sarac

Common Core Progression for 3-D figures

3D shapes is introduced in the primary grades as a way to talk about the world we live in being three-dimensional. Then the concentration shifts to 2D figure in grades 2 through 5. After they get a grasp of 2D shapes, the focus is back to 3D shapes starting in 6th grade.


  • Students begin their exploration of 3D figures (solid) by firstly distinguishing them apart from 2D figures (K.G.3), using informal language, e.g. balls, boxes (K.G.4).
  • Students then begin to name and describe 3D shapes with mathematical vocabulary, e.g. sphere, cylinder (K.G.1).

1st Grade:

  • Compose 3D shape to create a composite shape and compose new shapes from the composite shape (1.G.2).

6th Grade:

  • Students make 3D shapes from nets. They describe the shapes of the faces and number of faces, edges, and vertices solid figures have.
  • Students develop visualization skills for components of 3D shapes that are not visible from a given viewpoint.
  • Distinguish between units used to measure volume and area/length.
Primary Grades K-1
Sorting Activity

Geometric thinking begins with play (Van Hiele, 1999). It is important to make the lessons, especially at primary grades, play like.

A useful activity for distinguishing between examples and non-examples of shapes Sorting activities are helpful for students to distinguish between 2D and 3D shapes. Tangible, real-life objects are recommended before a computer software program is used.

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Photo Sort Activity

Sorting can also be used to help students distinguish between the different 3D shapes. They can sort photos of real life objects into different categories of 3D shapes.

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Block Center

Building in the Block Center is helpful at this stage to help students have experience with the shapes. Draw their attention to the solid figures and their faces. They can begin to model objects in their environment, e.g. building a simple representation of the classroom using unit blocks and/or other solids.

Foil Fun

Identify and describe the plane shapes found in objects. Explore and describe faces, edges, and corners of 2D and 3D objects make, name and describe polygons and other plane shapes
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Making Shapes

Making their own shapes help students understand the properties of shapes better. Different materials can be used.

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Feely Bags

To help students recognize 3D figures, describe attributes of solids and to make comparisons between different solids using mathematical language feely bags activity is a great tool. Place solid figures into a bag, have kids feel it, describe it, and name it.

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Technology Links:

Video that explains solid figures, the difference between 2D and 3D. There are also additional resources like lesson plans and activities.
2D to 3D morphing - helps students see the connection between the two.
Upper Grades – 6th
Guess What Shape I can Make

A great activity to relate plane shapes to solid figures is to have students first guess all the plane shapes a solid figure has. Then have the students press 3D figures into clay to make impressions of each face of the 3D figure.

Shapes with sticks

Classify plane shapes and prisms by their spatial features. Construct models of polyhedra using everyday materials and use the terms faces, edges and vertices to describe models of polyhedra
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Technology Links:

A great site that covers the difference between 2D and 3D shapes as well as different attributes of 3D shapes. Includes questions to assess student understanding.

Match Nets and Solids

Students are required to make 3D figures from nets and be able to distinguish what figure a net will result in and vice-versa. After giving students concrete opportunities in making 3D figures from different nets, you can assess their knowledge more abstractly.

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Image From: Math Expressions – Houghton Mifflin

Technology Links:

A great game that covers faces, edges, and vertices of 3D shapes as well as nets of 3D shapes.
End-of Unit Assessment Tool - Jeopardy Games

3D shapes have faces (sides), edges and vertices (corners). A net is what a 3D shape would look like if it were opened out flat.

Draw Pictures of Solids

Students need to develop visualization skills for components of 3D shapes that are not visible from a given viewpoint. Again, it is best to start with more concrete examples. Blocks or connecting cubes can be used to give students the opportunity to grasp the idea and move on to abstract representations on paper.

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Image From: Math Expressions – Houghton Mifflin

Technology Links:

Website that explains 3D shapes in detail with real world examples.
A great game that asks students to figure out the viewpoints of different characters in the game.

Composing & Decomposing

Main Contributor: Lindsey Gatfield


Composing and Decomposing geometric shapes is one of the major components of elementary geometry. In geometry, composing is the idea of putting two or more shapes together to create a new, larger shape. Decomposing then is taking away a part of a shape by covering the total shape with a part or taking shapes apart to see what smaller shapes they can be decomposed into. Composing and decomposing geometric shapes builds a knowledge of properties in students, such as being able to recognize equal length and equal angles in various shapes.

Primary Grades (K-2) Progression Overview

  • In the first three years of composing geometric shapes, students progress from being unable to compose shapes at all to developing the skills required to build and recognize more difficult composite shapes.
  • In Kindergarten, students build shapes from components, compose shapes to build pictures and designs, combine shapes, solve problem such as deciding which piece will fit into a space in a puzzle, and use geometric motions such as sliding, flipping, and turning.
Image From: Commoncoreprogression
  • In 1st grade, students learn to perceive a combination of shapes as a single new shape and to solve shape puzzles and construct designs with shapes.
  • By the end of 2nd grade,students can build the same shape from different parts, can solve puzzles with numerous pieces, and intentionally substitute arrangements of smaller shapes for larger shapes.
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Image From: Commoncoreprogression

Technology Resources

1. Students can match shapes with their outlines in online puzzles using This allows them to begin to recognize the ways to match angles and edges of shapes.

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2. Students can use multiple smaller shapes to compose larger shapes using online Tangrams. This is another application on

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3. Students can use online pattern blocks to compose and decompose shapes. This allows them to see how many smaller shapes can make up a larger shape. This online tool is provided by Learning Today.

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Curriculum Resources

1. Students can use pattern block to compose larger shapes. They can also decompose the larger shapes to see which smaller shapes fit in the picture. This is a resource from McRuffy Press.

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Image From: McRuffyPress

2. Teachers can purchase a Geometry Activity Box from LakeShore Learning Store. With this activity box, students will be able to use pattern blocks to complete puzzles, as well as compose smaller shapes to build a house. This box will also allow students to master other important concepts in Geometry

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Image From: LakeShoreLearning

3. LakeShore Learning Store also has Geometry folder games for the elementary level. There are games that include different shapes, and help students to master the concepts of composing and decomposing. This helps students by actually being able to manipulate the shapes.

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Image From: LakeShoreLearning

Assessment Resources

1. An assessment like this one allows teachers to see how students can decompose a larger shape to identify smaller shapes that make up that larger shape.

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Image From: 1stGradeGeometryAssessment

2. This assessment is similar to the previous one, but gives students options to choose from. It also provides an example to use for 3d objects. This assessment could also be used in higher grades.

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Image From: 1stGradeGeometryAssessment

Upper Grades (3-6) Progression Overview

  • In terms of composing and decomposing geometric shapes, the emphasis in the geometry curriculum for 3rd-6th grade is highly developing these concepts in order to understand multiplication, area, volume, and coordinate plane.
  • In 3rd grade, students begin with finding all different compositions of a set of shapes involved and illustrate properties of multiplication by viewing rows and columns. Skills developed from 3rd-5th grades provide the foundation for students to be able to understand formulas, decompose to find area, and compose and decompose polyhedral solids by the end of 6th grade.

Curriculum Resources

1. This puzzle demonstrates the 3rd grade skill in which students find all possible compositions of a set of shapes. This puzzle has at least three ways that students can compose the pieces to make a rocket ship image. This puzzle is from Big Cartel, but puzzles like these can be found at any toy or learning store.

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Image From: BigCartel

2. This geometry kit from LakeShore Learning is an example of many geo-boards that students can use to compose shapes and figure out area.

Image From: LakeShoreLearning

3. Illustrations like this one demonstrate the 6th grade skill of students being able to decompose a parallelogram into two triangles. This helps students to understand the concept of decomposing shapes to find the area of the composed shape.

Image From: OneMathematicalCat

Techology Resources

1. This online puzzle provided by Kids Math Games Online allows students to compose a larger shape using many smaller shapes. Students have to take into consideration angles and sides to see which pieces go where. This is a similar puzzle to ones a teacher may use in the primary grades, but its complexity is at a level for older students.

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Image From: KidsMathGamesOnline

2. This online tool from Shodor allows student to decompose shapes into square units to find the area. Different shapes are given to the students and then they can check their answer to see if they are correct. This could also be used as an assessment tool.

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3. Khan Academy is an extraordinary math courseware that teaches students new math skills, reviews with them, and assesses their learning. It also keeps track of their progress and understanding, so that the student or teacher/parent can follow along. Khan Academy has a lot of amazing resources, videos, and assessments for geometry subjects, especially for area and volume.

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Assessment Resources

1. An assessment tool where teachers can give students questions such as this one. This is for higher grade levels, where students are decomposing shapes to find area. This resource is called Aleks Assessments.

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2. This online investigation program is great for fifth or sixth graders. It goes through six geometry concepts and then assesses the student's knowledge afterwards. This tool is provided by the University of Connecticut.

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Spatial Relations & Structuring

Lead Contributor: Hannah Suh


  • Early composition and decomposition of shape is a foundation for spatial structuring
  • Spatial structuring is the mental operation of constructing an organization or form for an object or set of objects in space, a form of abstraction, the process of selecting, coordinating, unifying, and registering in memory a set of mental objects and actions.

Primary Grades (K-2) Progression Overview

  • Students discuss shapes, orientations, and the relative positions of objects using terms such as "above, below, next to, behind, in front of, and beside".
  • Students use spatial reasoning competencies, their growing knowledge of 3D shapes, and their ability to compose them to model objects in their environments
    • For example: Building a simple representation of the classroom using unit blocks and/or other solids
  • Students learn to see an object, such as a row, in two ways:
    1. As a composite of multiple squares
    2. As a single entity (a row of units)
  • Such spatial structuring precedes meaningful mathematical use of the structures, including multiplication, area, volume, and the coordinate plane.
  • Spatial structuring can be further developed with several activities with grids, including:
    1. Battleship Game
    2. Copying designs drawn on grid paper by placing manipulative squares and right triangles onto copies of the grid
    3. Creating and drawing designs on grid paper; Exchange with a partner and try to copy each others' designs
    4. Tessellations

Upper Grades (3-6) Progression Overview

  • Students develop more competence in spatially structuring rectangular arrays
  • Students explore line segments, lengths, perpendicularity, and parallelism on different types of grids (rectangular and triangular grids)
  • Students extend their spatial structuring in two ways:
    1. Learn to spatially structure in three dimensions and use this understanding to find volume
    2. Extend their knowledge of the coordinate plane, understanding the continuous nature of 2D space and the role of fractions in specifying locations in that space
  • Students solve mathematical and real-world problems using coordinates (whole number ordered pairs)
  • Students develop visualization skills connected to their mathematical concepts as they recognize the existence of, and visualize, components of 3D shapes that are not visible from a given viewpoint
  • Use drawings and physical models to learn to identify:
    1. lines perpendicular to a plane
    2. lines parallel to a plane
    3. plane passing through three given points
    4. plane perpendicular to a given line at a given point

Curriculum Examples

Students practice drawing tessellations   
Image From: Math Expressions 2nd Gr

Image From: Math Expressions 2nd Gr

Students practice copying and creating designs on grid paper   
Image From: Progressions for the Common Core

Students practice drawing square units inside a rectangular plane   
Area Square Units.png
Image From: Math Expressions 2nd Gr

Students practice counting square units on a grid   
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Image From: Math Expressions 2nd Gr

Curriculum Resources

Pre-K through 2nd Grade

Students practice moving a ladybug on a plane   
Image From: NCTM Illuminations

3rd through 6th Grade

Students use pattern blocks to create a design for their clubhouse   
Image From: NCTM Illuminations

Technology Resources

  • NCTM General Purpose Tool Offers a free downloadable interactive geometry platform that constructs, measures, manipulates, transforms (translations, reflections, rotations, dilations) and animates geometric figures including lines, circles, angles, perpendiculars, bisectors, tangents and vectors

Geometry Games

  • Shape Inlay Create a geometric work of art by adding shapes to the board as quickly as possible
  • Building Blocks Arrange the blocks to fit in the given shapes
  • Shape Mods Reflect, rotate, and translate your way through 18 levels of shape shifting geometry fun
  • Design a Party Use area and perimeter skills on a gird to design a part
  • Cube Perspective Count the number of cubes on the board. Don't forget the hidden ones
  • Locate the Aliens Identify the coordinates of points on a graph
  • Space Boy to the Rescue Locate the coordinates on a plane

Assessment Items

Example assessment problem requiring students to apply skills and knowledge of arranging shapes on a plane   
Image From: Smarter Balanced Assessment Consortium

Image From: Smarter Balanced Assessment Consortium
Image From: Smarter Balanced Assessment Consortium

Useful Links


Barkley, C. A. & Cruz, S. (2001). Geometry through beadwork designs. Teaching Children Mathematics, 7(6), 362-367.

  • This article is about ethnomathematics - studying mathematics within the context of culture. It goes through a 5-stage lesson based on Native American beadwork design. It address an inquiry-investigative approach to teaching.

Battista, M. T. (2002). Learning geometry in a dynamic computer environment. Teaching Children Mathematics, 8(6), 333-339.

  • This article discussed the benefits of students developing personally meaningful geometric concepts and ways of reasoning that enable them to carefully analyze spatial problems and situations. The author elaborated on the van Hiele levels of geometric thinking and also discussed the benefits of using the Shape Makers Interactive Geometry Microworld program, which is a special add-on to Geometer's Sketchpad.

Hannibal, M. (1999). Young children's developing understanding of geometric shapes. Teaching Children Mathematics, 5(6), 353-357.

  • This article outlines the ways that early childhood educators should introduce basic shapes to young children. The author provides research-based activities and suggestions for fostering an initial understanding of shape categories and attributes.

Munier, V., Devichi, C., & Merle, H. (2008). A physical situation as a way to teach angle. Teaching Children Mathematics, 14(7), 402-407.