Posted by: degarcia | November 24, 2008

Geometric and Spatial Reasoning

Geometry is more than studying shapes. It is a way to visually represent algebra and other higher levels of math”

(Bahr & de Garcia, 2010, p.387).


*All information is paraphrased from Bahr and de Garcia (2010) unless explicitly stated otherwise.


Definition of Geometry:


Four Geometric Systems:

••         There are four different types of geometric systems that are covered in the elementary grades:

•o   Topological geometry: mazes and networks, place and order, proximity and relative position, things that change or distort yet remain the same.

•o   Transformational geometry: how shares appear as they are moved in space, transformations.

•o   Euclidean geometry: two and three dimensional figures, points, angles, lines, congruence, similarity, and symmetry.

•o   Coordinate geometry: location and relationship of points and figures within the Cartesian plane

••         Topological and transformational geometry together are considered spatial reasoning and the heart o geometry and all other mathematical strands (Bahr and de Garcia, 2010, p.390).


Explanation of Spatial Reasoning:

  • Spatial reasoning involves the ability to think and reason by comparing, manipulating, and transforming mental pictures. Spatial sense comprises two components: spatial visualization and spatial orientation.
  • o Spatial Visualization: the ability to visually compare shapes that have changes position on the plane or in space.
  • § Includes the study of transformations including a flip, slide, and rotation. Many children have a difficult time understanding that the shape stays the same even when it is moved; they have not developed what Jean Piaget has called conservation.
  • Slide: when an object is slightly moved over.
  • Flip and reflection: when a shape is flipped across a reflection line (produces its mirror image).
  • Symmetry: a single shape that can be divided into two identical pieces is symmetrical.
  • Rotations: turning an object on a rotational point inside or outside the shape. When something is rotated 360 degrees it returns to its original position.

•o   Spatial Orientation: the ability to look at a fixed figure from several different points of view.  (Bahr and de Garcia, 2010, p. 390-394)


Tools and Activities to Use in a Classroom for Spatial Reasoning:

  • Tessellations: these help students to develop slides, flips, and rotations.
  • Petonimoes: these help students to visualize shapes and work with mental rotations and flips.
  • Nets: these are flat shapes that when folded create a three dimensional figure.
  • Isometric dot paper: this allows children to draw three-dimensional figures on a two dimensional surface.
  • Coordinate Grids and Graphing: these helps students with spatial orientation skills
  • Block building
  • Build and identify shapes within other shapes
  • Tangrams
  • Pattern Blocks
  • Geoboards
  • Look at shapes in the classroom and determine what their parts look like ( a square is two triangles, etc.)
  • Solving and creating mazes

Ideas were taken from: Bahr and de Garcia, 2010, p. 392-396.


Explanation of Euclidean Geometry:

••         This is the study of two and three-dimensional shapes. We can identify both the shapes and their properties (Bahr and de Garcia, 2010, p. 396).

•o   Two Dimensional Figures:

•§  Things to keep in mind:

•·         We underestimate the ability for young children to identify more complex shapes such as hexagon, trapezoid, etc.

•·         We normally only show students regular polygons, so students consider irregular polygons to be different shapes.

•o   Points and Lines:

•§  Students need to learn the basic foundational vocabulary for later use.

•·         Point: a nondimensional location in space

•·         Line: a set of points that go on forever in both directions

•·         Ray: a set of points that go on forever in one direction with a point of origination at the other end

•·         Line Segment: a set of points with a specific beginning and a specific end

•·         Intersecting lines: two lines that cross

•·         Parallel lines: two lines that are equidistant and never cross

•·         Perpendicular lines: two lines that meet or intersect at a 90 degree angle

•o   Three-Dimensional Solids:

•§  Solid figures are all around us, start with things that students are familiar with and have seen in their lives like an ice cream cone.

•§  Students need to learn the difference between

•·         Face: the sides

•·         Edge: where two faces come together

•·         Vertices: corners

(Bahr and de Garcia, 2010, p.396-401)

Tools and Activities to Use in the Classroom for Euclidean Geometry:

••         Put up a chart of examples and nonexamples of polygons, have students come up with a working definition and then add more shapes to the piles

••         Use a straw with a pipe cleaner or twist tie to talk about angles that are bigger or smaller than a right angle.

••         Have students sort triangles in any way that they want and see what they notice, then introduce vocabulary.

••         Focus on parallel and perpendicular lines. Have students act them out, draw them, talk about them, etc.

••         Give students a bag of solid figures and have them figure out the shape by feeling and describing to their neighbor.

••         Place a picture of a solid figure on a child’s back and have them ask questions about the physical features until they can name the shape.

••         Make a chart comparing faces, vertices, and edges, so kids can make generalizations.

Ideas taken from Bahr and de Garcia, 2010, p. 396-401.


Coordinate Geometry:

••         This is used to specify locations of points and describe paths between points. The coordinate grid geometrically represents algebra

••         Ordered pairs are composed of the x and y axis coordinates. The x axis is always written first, followed by the y coordinate. For example, (x, y) or (3, 4)(Bahr and de Garcia, 2010, p.402).


Tools and Activities to Use with Coordinate Geometry:

••         Map reading

••         Battleship

••         Create translations or dilations (scale versions): this is a great way to integrate math and art.

Bahr and de Garcia, 2010, p.402

Big Ideas of Geometry:

            Randall Charles has created 21 big ideas that relate to all concepts covered under the umbrella of mathematics. Of the 21 big ideas, three are directed specifically at geometry and spatial reasoning. (Charles, 2005, p.19-20)


To view all of Charles’ mathematical big ideas please click here.


  • 1. Shapes and Solids: Two- and three-dimensional objects with or without curved surfaces can be described, classified, and analyzed by their attributes.

Examples of Mathematical Understandings:

  • Point, line, line segment, and plane are the core attributes of space objects, and real-world situations can be used to think about these attributes.
  • Polygons can be described uniquely by their sides and angles.
  • Polygons can be constructed from or decomposed into other polygons.
  • Triangles and quadrilaterals can be described, categorized, and named based on the relative lengths of their sides and

the sizes of their angles.

  • All polyhedra can be described completely by their faces, edges, and vertices.
  • Some shapes or combinations of shapes can be put together without overlapping to completely cover the plane.
  • There is more than one way to classify most shapes and solids. (Charles, 2005, p.19-20)
  • 2. Orientations and Locations: Objects in space can be oriented in an infinite number of ways, and an object’s location in space can be described quantitatively.

Examples of Mathematical Understandings:

Lines and Line Segments

  • Two distinct lines in the plane are either parallel or intersecting; two distinct lines in space are parallel, intersecting or skew.
  • The angles formed by two intersecting lines in the plane are related in special ways (e.g., vertical angles).
  • A number of degrees can be used to describe the size of an angle’s opening. Some angles have special relationships based on their position or measures (e.g., complementary angles).
  • In the plane, when a line intersects two parallel lines the angles formed are related in special ways.


  • The orientation of an object does not change the other attributes of the object.
  • The Cartesian Coordinate System is a scheme that uses two perpendicular number lines intersecting at 0 on each to name the location of points in the plane; the system can be extended to name points in space.
  • Every point in the plane can be described uniquely by an ordered pair of numbers; the first number tells the distance to the left or right of zero on the horizontal number line; the second tells the distance above or below zero on the vertical number line. (Charles, 2005, p.19-20)
  • 3. Transformations: Objects in space can be transformed in an infinite number of ways, and those transformations can be described and analyzed mathematically.

Examples of Mathematical Understandings:

  • Congruent figures remain congruent through translations, rotations, and reflections.
  • Shapes can be transformed to similar shapes (but larger or smaller) with proportional corresponding sides and congruent corresponding angles
  • Algebraic expressions can be used to generalize transformations for objects in the plane.
  • Some shapes can be divided in half where one half folds exactly on top of the other (line symmetry).
  • Some shapes can be rotated around a point in less than one complete turn and land exactly on top of themselves (rotational symmetry). (Charles, 2005, p.19-20)


Additional Websites and Resources:

  • o Isometric Drawing Tool

  • o Shape Cutter

  • o This website is useful in reviewing the definitions of the geometric shapes. It has links for the different shapes and critical thinking questions that relate each individual shape to the idea of geometry as a whole.
  • o This website is the most interactive of the all. It has virtual geoboards, tangrams, and even tessellations. Each of the programs is easy to use and open in their own window so that students could go back to the main menu without pressing the back button.
  • o This website focuses more on spatial reasoning than geometry. It does have links for tangrams, but it also has games like Tetris that help students figure out the relationship of one shape to another.
  • o In this website students get actual practice in calculating the area and perimeter of different geometric shapes. Rather than sitting down with a worksheet, students can practice with the numbers the computer gives and have multiple chances to find the correct answer.


  • Additional Information/ Comprehensive Websites:
  • o IMAGES: Improving Measurement and Geometry in Elementary School ( This website has lesson plans, research, activities, etc.
  • o Illuminations by the NCTM (National Council for the Teachers of Mathematics)

( This website has more lesson plans and activites.

  • Example Lesson Plans:


Title – Folding Shapes: Are the Sides the Same?

By – Janna Moore

Primary Subject – Math

Grade Level – Grade 1


QCC Reference: M.1.6: Determines figures that are symmetrical by folding.


Instructional Objectives:


      The student will be able to:

      1. Recall the shapes-circles, squares, triangles, ovals, diamonds, and rectangles in various orientations/positions.

      2. Define symmetry.

      3. Fold paper cutout shapes to find the line of symmetry for each shape.

      4. Identify the letters of the alphabet as symmetrical or nonsymmetrical and put the letters in their appropriate boxes that are marked symmetrical and nonsymmetrical.


Materials Needed:


      1. Cutout shapes — circles, squares, triangles, ovals, diamonds, and rectangles — for each student

      2. Flash cards of shapes for students to identify

      3. Two boxes (small) one labeled symmetrical, one labeled nonsymmetrical

      4. Cutout capital letters of the alphabet




Planned Beginning:


      Do you remember all of the shapes we found in the classroom yesterday? What are some of those shapes? Some shapes can be folded in half to make both sides of the fold look the same. These shapes are called symmetrical shapes. The line that is left in the middle when you fold the shape is called the line of symmetry. Not all shapes have a line of symmetry. Some shapes have more than one line of symmetry. We are going to fold some shapes to find out if they are symmetrical.


Activities and Discussion:


      Find the square in your stack of shapes. If we fold the square in half, we can see that it is symmetrical — it is the same on both sides. Can you identify another line of symmetry on the square? Fold your paper to show another line of symmetry — make sure that both sides are equal. How many more lines of symmetry can you find on the square?


      Now find any lines of symmetry on the circle. Does the circle have more than one line of symmetry? Predict how many lines of symmetry the circle would have. Why does the circle have so many lines of symmetry?


      Do the same with the rest of the shapes. When folding the triangle, ask: Why is there only one line of symmetry on the triangle? And when doing the diamond, ask: Compare the diamond’s lines of symmetry to the square’s lines of symmetry. And compare these to the rectangle’s lines of symmetry.


      Now divide the cutout letters of the alphabet between the students. Explain to the students that they will tell if their letter is symmetrical or nonsymmetrical. Show them the boxes and explain that they will put their letter in the correct box after they have decided if the letter is symmetrical or nonsymmetrical.


      One at a time — going through each letter of the alphabet (in order) — have the students tell what letter they have (make sure they can recognize their turn in the order of the alphabet) and tell if it is symmetrical. Ask the students to explain why their letter is or is not symmetrical. Have the students place their letter into the appropriate symmetrical or nonsymmetrical box. (In what box does your letter belong?)


      When they are done placing the letters, ask: Can you think of any other shapes/objects that are symmetrical? How are these shapes/objects symmetrical? If there is time, the students can stand up and tell if the human body is symmetrical. Each student can stand up, one at a time, and the other students can tell if that person is symmetrical and why or why not (If a student has a watch on one wrist but not on the other, that person is not symmetrical).


Planned Ending:


      What does the word “symmetrical” mean? What shapes did we decide were symmetrical? Are all shapes/objects symmetrical? Why? Do some shapes/objects have more than one line of symmetry? Why? What about this ball, is it symmetrical? Next time we are going to talk about some of our shapes that are 3-D, not flat like the ones we used today, and what these shapes are called.




      I would have a test that would have pictures of shapes/letters on it that we discussed in class. The students would have to write yes or no as to whether or not the shape is symmetrical. Then they would draw the lines of symmetry on the shape if the shape has any lines of symmetry.




Multiple Intelligences:


    * Linguistic: The students identify and recognize the letters of the alphabet in the correct order.

    * Mathematical-Logical: The students identify shapes and identify which shapes are symmetrical and nonsymmetrical.

    * Bodily-Kinesthetic: The students fold shapes and they place letters into the correct box. If time permits, students tell if the human body is symmetrical.

    * Spatial: The students pick the correct box to place their letter in.





Grade Levels: 3 – 6


The connections between art and math are strong and frequent, yet few students are aware of them. This geometry lesson is integrated with history and art to engage even the most math resistant of your students and to enlighten everyone about M. C. Escher’s work in tessellations.



40 minutes



Students will:

# follow precise, multistep directions.

# have the opportunity to go beyond the immediate lesson and apply artistic creativity, or learn more about M. C. Escher, his art, or the contributions he made to mathematics.

# be able to understand and define the following terms: tessellation, polygon, angle, plane, vertex, and adjacent.

# create a concrete model of a tessellation.



# Scissors, tape, 11″ x 14″ paper, crayons, black fine-tip pen

# Worksheets: Creating Tesselations and Shapes



1. Introduce key vocabulary words: tessellation, polygon, angle, plane, vertex and adjacent. Ask students to tell you what they know about the word tessellation. Discuss the three basic attributes of tessellations:


  # First, they are repeated patterns. Ask students to find examples of repeated patterns in the room. Generate a list of the words one could use to describe these patterns. Tell students that while those are repeated patterns, only some are tessellations because tessellations are a very specific kind of pattern.

  # Second, tessellations do not have gaps or overlaps. If students have pointed to a pattern in the room that has a gap or an overlap in it, point out that it does not fit the definition of a tessellation.

  # Third, tessellations can continue on a plane forever. Define plane (use a concrete example in the room) and show students how the pattern could continue on that plane if it were to go on beyond the confines of the building (e.g., it could continue as a pattern on the ceiling without any gaps or overlaps even if the ceiling were to continue forever, far beyond the walls of your school).


2. Provide students with the Shapes worksheet, which has a copy of a square, a rectangle,a rhombus, and a hexagon on it. (These were chosen because each tessellates.) Using the Student Directions worksheet, demonstrate how to transform a shape into something that will also tessellate.



# Note how the students follow multistep directions as well as how they cut and trace (manual dexterity).

# To assess an understanding of the vocabulary, create a quiz at or ask them to perform another project that requires an understanding of the terms. (For example, ask them to tell you who is adjacent to them or ask them to label the top right vertex of a shape you provide.)

# Have your students teach another class how to tessellate.



# Younger students can discover for themselves what shapes tessellate using pattern blocks and lots of space. They will notice that only some, not all, can make a pattern that would fit all three of the criteria.

# Encourage students to experiment to see if they can discover other ways to make shapes tessellate.

# Teach students about the history of tessellations and show examples. If you can have students point out the three features of tessellations, it will help to make their understanding more concrete and it will also review the definition. Tessellations have been used all around the world for many years. The earliest tessellations we can find come from Islamic art circa 3000 BC. There are examples from medieval European art as well (e.g., stained glass patterns).

# Use Web resources to extend the lesson:


  # Enter your class in one of several online tessellation contests.

  # Look at American folk art that uses tessellations (such as quilts).

  # Tessellations were popularized by M. C. Escher.

  # Research M. C. Escher, Penrose, and other “Recreational mathematicians.”


Discovering the Magical Pi


Grade Levels: 6 – 8


In this lesson plan, students will use data on the circumference and diameter of various objects to calculate pi. The exciting aspect of this lesson is that no matter the size or nature of the circular objects measured, the ratio of the circumference to the diameter will come out the same: pi!



80 minutes



Students will:

# measure the circumference and diameter of a variety of circular objects.

# organize the data in a table or chart.

# calculate pi – the ratio of circumference to diameter (C/D).



# Rulers or yardsticks, preferably metric

# White string cut in two lengths – 30 cm (approximately 1 ft) and 60 cm (approximately 2 ft) Note: Length of string pieces may vary depending on the circular objects chosen

# A variety of circular objects that would allow for easy measurement of circumference and diameter, such as canned goods (soup, beans, fruit, etc.), oatmeal container top, Frisbee, cake pan, plate, coffee mug bottom, any lid to a glass jar (baby food, pasta sauce, etc.), a half-dollar or quarter; it is also very powerful to have some “natural” circular objects, such as a sand dollar, half a well-wrapped melon or orange or other piece of “sturdy” fruit

# Calculators

# Worksheet: Circular Measurements

# Worksheet: Mystery Ratio



This lesson is broken into two parts (each lasting approximately 40 minutes).


Part 1

1. Introduce key vocabulary:

# circumference = the distance around a circle

# diameter = the line segment that goes through the center of the circle and has endpoints on the edge of the circle

# ratio = a way of comparing two numbers – the ratio of a to b is a/b

# pi = the ratio of the circumference to the diameter of a circle; pi is approximately 3.1416 (Note: Do not introduce pi to students until after they have done the exploration.)


2. In the room, make a few stations of three or four circular objects each. Put students in groups of two or three (try to have same number of groups as stations). Give each group a 30-cm string, a 60-cm string, and a ruler or yardstick.


3. Give each student a Circular Measurements worksheet. Read the column headings with them and make sure they understand the vocabulary. Demonstrate circumference and diameter with an object.


4. Explain to students that they are to work with their group to measure the circumference and diameter of the circular objects at each station. This is a good time to have a student come up to the front of the class with you and demonstrate how to do this with the string. One of the challenges for students is estimating the center of the circle when they measure diameter. Demonstrate accurate and less accurate ways of measuring, and ask students which are best.


5. Have kids go to their stations. Using the Circular Measurements worksheet, they should enter the object name and then proceed with measurements. (Time permitting, they can also take two readings for each measurement to test accuracy.) All students should record the object name, circumference, and diameter before proceeding to the next object. They are not to fill in the Mystery Ratio.


6. When finished, students should proceed to the next station to measure other objects.


Part II

1. Ask students to sit with their groups and look over their data. This could be a time that you pick one or two objects and ask for data from different groups. It helps students understand the concepts when they see the variation in measurement and discuss why this happens.


2. Next, tell students to fill in the Mystery Ratio (circumference to diameter, or C/D) on the Mystery Ratio worksheet. They should enter that in the column heading.


3. Using their calculators, students should now calculate C/D for each object. Have them take the ratio to the hundredths place.


4. After they have finished their calculations, give each student a Mystery Ratio worksheet. The students answer the questions in their small groups. The “Aha!” moment is usually more potent in the small groups because more kids can come to it at different times.


5. Do a quick wrap-up before class ends. Whether or not natural objects were used, it is nice to talk about pi in nature. How does mother nature know how to “grow” a circular object? Pi must be a pretty important number!



# Students should be able to define all terms.

# Evaluate Circular Measurements worksheet for organization of data in the chart and calculations.

# Assess Mystery Ratio worksheet to check answers to questions.

# Students can write a letter to someone explaining the activity and what they discovered.




Lots of great pi links. Also includes lists of pi carried out to lots of digits. Kids can try memorizing digits using mnemonics and other methods.


A brief summary on pi, its mystery, and its occurrence in nature.


Check this site out to do an extension on finding the circumference of the planets given the diameters.


A site that trains students to memorize pi’s digits. <



National Council of Teachers of Mathematics (standards for grades 6-8)


# applies techniques and tools to accurately find measurements (measurement).

# solves problems involving circumference and diameter (geometry).

# generalizes from a pattern of observations made in particular cases and makes a conjecture (problem solving).

# recognizes and applies geometric ideas to everyday life (connections to the world).

 This website shows the lesson plan outline for teaching a lesson on the area of a circle in relation to the area of a parallelogram


Task Development

Question:  Create a drawing of an object that includes at least:

            5 shapes with 4 sides

            4 shapes with 3 sides

            3 shapes with no edges

After you have created your drawing, color the 4 sided shapes blue, 3 sided shapes red, and none-edged shapes yellow.  Make a key at the bottom of your paper for these colors.  On the other side of the paper create a graph that shows how many there are of each shape and label the axis with the shape names.  Be prepared to share your pictures and labels with the class.


This task is for a first grade classroom and meets the 1st grade state core standard in mathematics for standard three objective 1, indicator a.  These state, “Students will understand simple geometry and measurement concepts as well as collect, represent, and draw conclusions from data.  Identify, describe, and create simple geometric figures.  Name, create, and sort geometric plane figures (i.e., circle, triangle, rectangle, square, trapezoid, rhombus, parallelogram, hexagon).”  They will also meet standard 3 objective 3 indicators a and b.  “Collect, organize, and represent simple data.  Collect and represent data using tables, tally marks, pictographs, and bar graphs.  Describe and interpret data.”


The students will be creating a picture out of geometric plane figures and sorting them in a graph on the opposite side of their paper.  They will then be required to explain their work to the class, labeling the shapes with proper terminology.


Kinestetics – Students may cut and trace the shapes that they want to include in their drawing from objects of that shape in the classroom.

Algebraically – Students will be using the colors to represent the shapes in their picture and must create a key using the shape and color.

Numerically – Students will write the number of sides of each shape in their graph

Verbally – Students can talk to one another about the shapes and will be required to share their picture with the class using proper geometric terminology for the shapes they have included.

Graphically – Students will graph the number of shapes they used on the opposite side of their paper, labeling the axis with the names of the shapes and the number of shapes.



Bahr, D.L., & de Garcia, L.A. (2010). Elementary mathematics is anything but elementary.Belmont, CA: Wadsworth, Cengage Learning.


Charles, R.I. (2005).Big ideas and understandings as the foundation for elementary and middle school math. Journal of Mathematics Educational Leadership. 7, 9-24.


Weisstein, Eric W. “Geometry.” From MathWorld–A Wolfram Web Resource.



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