Thank you for your interest in NCTM’s Illuminations. Beginning in mid-April, all Illuminations content will be moving to Interactives will remain openly available and NCTM members will have access to all Illuminations lessons with new filtering and search options. We hope you will continue to utilize and enjoy these resources on

Pin it!
Google Plus

Puzzling Relationships

  • Lesson
Carol Midgett
Location: unknown

Puzzles provide excellent opportunities for students to apply spatial relationships in an entertaining and meaningful way. In the first part of this lesson, the students use concrete materials to develop the skills needed to work with interactive tangram puzzles in the second part of the lesson. The students also further develop their understanding of spatial relationships and develop problem-solving skills.

Once students complete puzzles using concrete materials, introducing them to virtual puzzles enables them to experience turns, rotations, slides, and spatial relationships in a powerful way. These transformations are more apparent in a virtual environment than when using concrete tools.

Make or use commercial overhead transparencies of the Tangram Puzzle Activity Sheet. You will also want to make enough copies of this activity sheet for each student in the class (cardstock is recommended).

Note: If tangrams and readily available in the classroom, you may choose to use those instead of the cut outs.

pdficon Tangram Puzzle Activity Sheet 

Use the overhead projector to model how to fill the puzzles with the tangram pieces. For example, the Tangram School can be filled as shown below.

2751 tangram school 

Discuss how it is necessary to turn over, rotate, or slide pieces to complete a puzzle. Provide each student with the tangram puzzle activity sheet. Have students cut out their pieces (or pre-cut them prior to class), and fill out the Trangram School stencil themselves.

For the next part of the lesson, display the Interactive Tangram Puzzles to the students, using an overhead projector or Smart Board.

appicon Interactive Tangram PuzzlesPuzzlingRelationship IMAGES eExample4.4ScreenShot 

Show the students that what they see on the screen matches the tangram pieces they used in the first part of this lesson. Identify each of the tangram pieces and compare the virtual ones with the concrete ones used by the students. Demonstrate how to select a puzzle piece on the site and move it onto the puzzle outline.

Demonstrate how to "grab" a corner of a shape in order to rotate it and how to turn a shape over using the curved arrow icon (once the tangram is clicked on).

You may also wish to show the students how to change the colors of the shapes. (Selecting a a color and click on the tangram.)

Model for students how to turn over (flip), rotate, and slide their tangram pieces with the concrete tools simultaneous with the electronic tool so that students understand what happens when they make these transformations.

Using the tools and strategies you just demonstrated, complete one of the puzzles on the computer while the students observe. A discussion of what you did and why you did it will help the students hear the vocabulary and consider the thinking required to complete a puzzle.

Have the students work in pairs to complete the electronic tangram puzzles. Encourage them to discuss with each other what needs to be done to fit the pieces to match the outline of the puzzle. (This exercise focuses students' attention on the transformations required to make pieces fit. A turnover, rotation, or slide of pieces may be necessary to complete the puzzle.) This discourse facilitates students' problem solving and reasoning about the process. It connects actions with vocabulary and assists the students in relating what happens with concrete and virtual tools. Circulate the room while students have this discourse, and take note of students who are struggling or becoming proficient.

After students are done, the Print Screen button can be used to print students' work. As a wrap up activity, pairs of students can share their work (what was most difficult, what feature they used the most, how long it took them, etc.).

Assessment Options

  1. By completing tangram puzzles, the students should demonstrate the ability to do the following:
    • Interpret relative positions in space and apply ideas about relative position.
    • Describe and interpret direction and distance in navigating space, and apply ideas about direction and distance.
  2. Making notes about what the students do and do not understand is helpful when planning future instruction for individuals, groups of students, and the class as a whole. Collecting this data for future reference helps you understand the impact of your teaching strategies on students' learning. You may use the Class Notes recording sheet for this purpose.


  1. The students who find it easy to complete the puzzles could create new ones for the class to solve. These may be laminated and saved for future practice.
  2. Have students create more complex tangram puzzles with the interactive. This can be done by pressing the irregular polygon, located at the upper-right hand corner of the interactive.
     PuzzlingRelationship IMAGES ExtensionHardMode 

Questions for Students 

[Answers will vary.]

  1. What did you find when you tried to place the puzzle pieces in place?
  2. What happens when you turn (or rotate) a puzzle piece?
  3. What happens when you turn over a puzzle piece to fit it into place?
  4. How many different ways did you find to fill the puzzle? What did you do to fill the puzzles in different ways? (This question focuses students' attention on different orientations of pieces and whether pieces must be turned over, rotated, or slid to fill the puzzle.)
  5. What did you learn from other students as you watched them put their puzzle together?
  6. If you were teaching this lesson, what would you tell your classmates about how to put the puzzle together?
  7. Why is it necessary to turn over, rotate, or slide the tangram pieces to make them fit?
  8. How did you decide which triangle would fit into the vacant space?
  9. Which shape(s) caused you the most trouble? Why?
  10. What did you learn by using the computer that you did not notice when using the other tangram pieces?

Teacher Reflection 

  • Which students demonstrate an understanding of how to solve puzzles by turning over, rotating, and sliding puzzle pieces? Which did not? What caused them difficulty?
  • What other learning opportunities do these students need? What instructional strategies would provide these learning experiences?
  • Were some students able to use more than one strategy to complete the puzzles?
  • Which students were able to interpret and use relative position in space as a problem-solving strategy?
  • Were all students able to describe how they solved puzzles? If not, what would help them do so?
  • Which students demonstrate an understanding of the relative sizes among the tangram pieces by identifying the varying sizes of the triangles as small, medium, or large? For those who do not, what instructional experiences are needed?
  • Which students do not realize that rotating or sliding changes the orientation of shapes? What experiences would help them learn this?
  • Which students realize that turning over pieces can change the orientation of shapes? If not, what experiences would help them learn this?

Learning Objectives

Students will:

  • Interpret relative positions in space and apply ideas about relative position.
  • Describe and interpret direction and distance in navigating space and apply ideas about direction.
  • Use tangram puzzles to experience turns, rotations, and slides.

NCTM Standards and Expectations

  • Recognize and apply slides, flips, and turns.
  • Create mental images of geometric shapes using spatial memory and spatial visualization.
  • Recognize the attributes of length, volume, weight, area, and time.
  • Understand how to measure using nonstandard and standard units.

Common Core State Standards – Mathematics

-Kindergarten, Geometry

  • CCSS.Math.Content.K.G.A.1
    Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

-Kindergarten, Geometry

  • CCSS.Math.Content.K.G.A.2
    Correctly name shapes regardless of their orientations or overall size.

-Kindergarten, Geometry

  • CCSS.Math.Content.K.G.B.4
    Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/''corners'') and other attributes (e.g., having sides of equal length).

Grade 2, Geometry

  • CCSS.Math.Content.2.G.A.1
    Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.