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Counting Embedded Figures

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This grades 7-12 activity allows students to look for patterns within the given data. After looking at the pattern, the student should be able to form generalizations for the problem. Furthermore, this activity sharpens the algebraic skills of the students. The problem sharpens visualization skills.

The problem of counting the number of squares on a checkerboard is a classic.  It is frequently used as an example of the problem-solving strategies "think of a simpler problem" and  "look for patterns" (Krulik and Rudnick 1989; Sonnabend 1993).  In addition, the problem sharpens visualization skills because students are challenged to see the smaller squares of various sizes embedded in the larger square.

This activity begins with the embedded-square problem and offers several extensions.  As students work through these problems, they should appreciate the importance of systematically organizing their data. For instance, in working the embedded rectangle problem on sheet 2, many students correctly count rectangles of various sizes but overlook a particular type of rectangle and, therefore, fail to get the correct total.

By encouraging students to look for patterns, these problems help develop algebraic thinking.

The language in which students express their conclusions may vary from verbal to symbolic. When you ask the class to describe the pattern for the number of embedded figures on the first activity sheet, the responses will differ from student to student:

  • You add up the square numbers until you get to the size of your big square.
  • The number of squares in an n-by-n square is 12 + 22 +...+n2 ;
  • More advanced students may even recognize that this expression is equivalent to n(n+1)(2n+1)/6. 

Because these problems require visualization skills, some students may encounter difficulty. You can help these students compensate for this difficulty in two ways.

  1. One benefit of cooperative learning is that individuals with different learning styles can complement each other.  When students work in groups, a field-independent person can guide a field-dependent person to see the embedded figure, for instance, by tracing the figure on the paper with a finger or pencil. 
  2. Teachers can give students templates for cutting out copies of some figures. 
    These copies can them be moved around and super-imposed on the original figure.  This process enables the student not only to identify the figure as embedded but also to determine how many positions the figure occupies within the larger figure. The template contains figures related to the first three activity sheets. If possible, duplicate the template on paper of a different color from the activity sheets to give visual contrast.  Heavier paper or card stock will produce template pieces that are stiffer and thus more effective.  An alternative is to have students copy the template onto tracing paper.

Getting Students Started

Working through the first activity sheet, students should feel comfortable finding the different squares with their group or partner.

pdficon Counting Squares Activity Sheet 

A few sets to be considered on the first sheet are as follows:
There are 6 ways to place a 1x2 rectangle on a 3x3 square:

1097 1x2 on 3x3 

There are 4 ways to place a 2x2 rectangle on a 3x3 square:

1097 2x2 on 3x3 

On the second activity sheet, rectangles are used instead of squares.

pdficon Counting Rectangles Activity Sheet 

Finding generalizations for the problems on sheets 3 and 4 will challenge even the most sophisticated students.  In fact, the author knows of no simple, closed-form formula for either problem.

On the third activity sheet, the shapes are triangles, and the students have to be sure to really find all the embedded figures at each size. For these figures, inverting the embedded triangles yields even more embedded shapes.

pdficon Counting Triangles Activity Sheet 

A few sets to be considered on the third sheet are as follows:
There are 6 ways to place a "position up" triangle of side length 2 in a triangle of length 4:

1097 2 on 4 tri 

On the figures on the fourth activity sheet, yield many embedded triangles of each size.

pdficon Counting Triangles in Squares Activity Sheet 

A few sets to be considered on the fourth sheet are as follows:
There are 9 ways to place a triangle of side length 2 in a square of length 4:

1097 ex2on4trib 

Have students work in groups or pairs. Each sheet guides the students through several problems of increasing complexity and then asks them to look for patterns and make predictions. Details concerning each sheet are found in the answer key.

Each problem is independent of the others. The sequence progresses roughly from easy on sheet 1 to difficult on sheet 4. The four problems need not be presented at the same time; one possible plan follows.  Sheets 1 and 2 may be a one-day lesson, with questions 3 and 4 from sheet 2 assigned for homework.  Sheet 3 may then be used later in the year as an enrichment lesson and sheet 4 assigned as a special project or for extra credit.

Answers are provided here.


This lesson plan has been adapted from a lesson by Timothy V. Craine, which appeared in the October 1994 edition of the Mathematics Teacher.


Assessment Options 

  1. As students work in groups, listen to their discussions to determine the strategies they use to visualize the embedded figures.  Students who tend to rely on template figures may be field dependent.  These students may need extra assistance when processing visual information, such as from a computer screen or graphing calculator.
  2. Watch students' responses to question 4 on sheet 2.  Some students may be reluctant to classify a square as a rectangle, which indicates that they are thinking at a lower van Hiele level than is assumed for high school-level geometry. (Crowley 1987).
  3. Ask students to write about their strategies for solving these problems.
  4. Ask students to create their own embedded-figure problems.  Their designs need not be limited to squares and triangles, nor must they be symmetric.  Have students trade papers and solve each other's problems or create a challenge for other students on a hallway bulletin board.  Possible extensions include finding trapezoids or parallelograms on sheet 4, drawing altitudes on sheet 3, or creating a grid of equilateral triangles and hexagons.  In addition, students may want to investigate fractal-like images, such as the Sierpinski carpet.


  1. Probe for deeper geometric understanding by asking such questions as "Why are there more smaller squares than larger ones?" "Why is the number of squares of a particular size on sheet 1 always a square number?"
  2. Use these activities to review fractions and proportion.  In terms of area, what fraction of the whole is a square, a triangle, or a rectangle?  What fractions of type 1/k can be represented by coloring in a certain number of embedded shapes?  Which figure can you use to illustrate a certain fraction?
  3. Use these activities to review fractions and proportion.  In terms of area, what fraction of the whole is a square, a triangle, or a rectangle?  What fractions of type 1/k can be represented by coloring in a certain number of embedded shapes?  Which figure can you use to illustrate a certain fraction?
  4. In the Counting Rectangles Activity, some students will notice that 15 and 21 are triangular numbers. In general, the nth triangular number is the sum of the first n positive integers. 
    T= 1 + 2 + ... + n = n(n+1)/2

    A generalization for this problem is that the total number of rectangles in an m row x n column rectangle is Tm × T = mn(m+1)(n+1)/4.


Questions for Students 

  1. Ask students to describe how they discovered their patterns. Encourage and validate a variety of appropriate responses.
  2. How can we represent the patterns that we found with algebra? With sentences?

Counting Squares  

1. Ask students to look for a pattern in questions 1‑3. Make a prediction for the total number of squares in a 5×5 square. Students should test their predictions to find 55 squares total (25+16+9+4+1).

[Answers will vary.] 

Counting Rectangles  

1. Tell students to draw a 3 row × 4 column rectangle. Find the number of each type of rectangle. Organize the data in a table. Look for patterns. Find the total number of rectangles.

[Students should identify 60 rectangles total.]

2. Ask students to predict the number of rectangles in a 6 row ×5 column rectangle. They should use the pattern found for the 3 row ×:4 column rectangle. Once again, they can use a table to organize their data.

[Students should identify 315 rectangles.]

Teacher Reflection 

  • Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were those adjustments effective?
  • What, if any, issues arose with classroom management? How did you correct them? If you use this lesson in the future, what could you do to prevent these problems?
  • What were some of the ways that the students illustrated that they were actively engaged in the learning process?

Learning Objectives

Students will be able to:

  • Analyze situations, check for patterns within the given information.
  • Practice finding generalizations for the problems.
  • Work in small groups encouraging classmates and communicating thoughts.

NCTM Standards and Expectations

  • Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
  • Relate and compare different forms of representation for a relationship.
  • Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
  • Use geometric models to represent and explain numerical and algebraic relationships.
  • Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.
  • Analyze properties and determine attributes of two- and three-dimensional objects.
  • Investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.
  • Visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP2
    Reason abstractly and quantitatively.
  • 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.
  • CCSS.Math.Practice.MP8
    Look for and express regularity in repeated reasoning.