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Concordia Online Education

Math Manipulatives Lesson Plan




2, 3, 4  

Lesson Summary

Students use two-dimensional illustrations to create three-dimensional models. Using cubes, the students figure the number of cubes used to create the object in the two-dimensional illustration.  Students learn that not all cubes are visible in the two-dimensional illustrations that are used to build the three-dimensional model.  Through multiple experiences creating the visual representations with cubes, students should eventually be able to count the number of cubes in a stack correctly with out building a model.  Divide the lesson activities over a period of two to three days.

Estimated Duration:    Two hours

To develop visualization skills, students need opportunities to work with three-dimensional models and two-dimensional visual representations. Building models from pictures allows students to visualize and interpret the two-dimensional representation. Students understand the limitations of a two-dimensional representation when they identify the missing attributes of the actual model, such as unseen cubes in a tower.

Students benefit from experiences in which they can mentally “see” attributes of three-dimensional shapes and verify those predictions physically (NCTM, 2000).


  • Distribute How Many Cubes?, Attachment A, to each student. Have students find the number of cubes it would take to build the model in each picture. Ask questions to reveal their thinking such as:
  1. How did you find the number of cubes?
  2. Are all of the cubes used visible?
  3. How did you know how many were not visible?

Scoring Guidelines:

Assess understanding of spatial sense, connecting two-dimensional representations and three-dimensional models. Students who find the correct number of cubes for eight or nine of the drawings have met grade-level expectations. Focus instruction on drawing the views and building models given the various points of view (see Extension).

Answer Key:

1. 11 cubes                  2. 15 cubes                  3. 6 cubes                    4. 9 cubes

5. 5 cubes                    6. 8 cubes                    7. 9 cubes                   8. 9 cubes

9. 8 cubes


This is a performance assessment. Have additional work available for the students to complete while they wait to be assessed.

  • Assign three of the examples on page two of Cube Towers, Attachment B to each student. Distribute cubes to students and tell them to build a model of the three towers in the pictures.
  • Have students raise their hand after they complete a tower. Assess the tower for accuracy.

Scoring Guidelines:

Use anecdotal notes to assess understanding of building models. Students meet expectations when they use the appropriate number of blocks and have accurate dimensions as shown in the picture. Differentiate and provide intervention using the complexity of the towers. Assign simple and challenging towers according to the students’ needs. If results are inconsistent, assign additional towers to assess.

Instructional Procedures:

1.   Distribute sets of cubes and page one of Cube Towers, Attachment B to each pair of students.

2.   Have students look at the first example (Tower A). Have students describe the dimensions of the tower. Ask questions such as:

  • How tall is the tower?
  • How many cubes wide is the tower?
  • What is the shape of the tower?
  • What would this tower look like if you were viewing it from the top?

3.   Have students predict the number of cubes used to build the tower, then build the tower with the predicted number of cubes. Observe and record notes about student performance, identifying students who count and use the number of cubes actually seen.

4.   Have students share their tower. Students who count only the cubes they see will find that their tower does not look like the tower in the picture. Ask questions such as:

  • How many cubes are in this tower?
  • Could you see all of the cubes?
  • How did you figure the number of cubes that were missing?

5.   Have the students complete additional towers. Ask them to predict, build and share with a partner or in small groups. Observe students as they work.

6.   Summarize the lesson and skill. Make sure students understand that not all cubes are shown in the two-dimensional representation to make the three-dimensional model.


M.Ed. in Curriculum & Instruction: Math



Part Two

7.   Select a tower from Attachment B and have students build the tower. Have students observe the tower from different points of view. Ask questions to describe the shape of the tower from the different points of view.

  • What is the shape of this tower when looking from the side?
  • What is the shape of the tower looking down from above?
  • What is the shape of the back view of the tower?
  • Are any of the views alike? (front and back, top and bottom, both sides)

8.   Provide students grid paper. Ask students how they would draw the side view of the tower. Model drawing the side view; explain that each square on the grid paper represents one block of the model. Have students draw the top view and share their drawing with a partner, then discuss as a class.

9.   Have students build additional towers. Ask questions to describe the tower and identify the number of cubes not shown.

  • How tall is the tower?
  • Are there any cubes that are not visible? How many?
  • What is the shape of the base or bottom of the tower? How many cubes make the base?

10. Allow students to build a model of the tower and draw the top, front and side view. Observe students who used the incorrect number of cubes when building Tower A. Record notes regarding change in understanding or continued misconceptions.

11. Have students view each other’s towers and compare for accuracy. Observe students as they work and provide assistance as needed.

12. Lead a discussion about building models from pictures. Ask questions and allow students to reflect with partners or in small groups. Have students write what they learned in a journal.

  • Which is a model, the cube tower or the grid paper?
  • What is the same in the model of the cubes and the illustration on the paper?
  • What is different in the model of the cubes and the illustration on the paper?
  • Were some models harder to build than others?  Why?
  • When looking at an illustration of the figure, what things do you need to remember?  (count the hidden cubes)

Differentiated Instructional Support:


  • Use simplified towers, begin with towers showing all cubes in the drawing.
  • Use blocks that snap together so they hold shape if bumped or fine motor skills are not completely developed.
  • Draw towers from different points of view using grid paper.
  • Post questions from the lesson about towers. Have students ask each other questions about the towers built.


  • Identify objects from the classroom and have students build a model using the cubes. For example, have students build a model of a crayon box. Ask students if they can see the entire box.
  1. What do they need to do to see the entire box?
  2. How is the box like the model of the cubes?
  • Build and display a figure with cubes such as a rectangular prism. Have students identify objects in the room that is the same as the figure.
  • Have students work in pairs. Each student builds a tower without letting the partner see the tower. One student describes the dimensions and views of the tower, while the other builds a model of the tower. Students can compare towers. If towers are not alike, have students discuss the descriptions and possible ways to describe more accurately.
    • On a grid, shown on a transparency, draw the shape of the sides of a tower from different views. Have students build towers that have sides that are the same as the views.


Homework Connections:

      • Have students practice identifying the number of blocks used to build the models from Attachments A or B.
      • Have students draw different views on grid paper of selected towers on page one of Attachment B.

For the student: cubes, grid paper

Research Connections:

National Council of Teachers of Mathematics. Principles and Standards for School Mathematics.Reston,Va.: NCTM, Inc., 2000.


Attachment A, How Many Cubes?

Attachment B, Cube Towers

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