Four Triangles

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Students will be able to determine the number of figures that satisfy the Four Triangle problem and sort their figures given a particular criterion.

Big Idea

Students will engage in a hands-on activity through which they will be introduced to the notion of proof by exhaustion as well as one of the big ideas of geometry: classifying and differentiating.

Warm-Up: Figure Out the Digits

10 minutes

Since this is the first official warm-up of the year, I want to make my classroom expectations clear:

  • By the time the bell rings, all materials should be out
  • If a Warm-Up is posted, students should be working on that
  • If there is no Warm-Up, students should have homework and begin making corrections with answer key

I tell students I expect them to work on this puzzle on their own.  Giving students independent think time allows students to have something thoughtful to discuss with their group members while limiting any one student dominating the conversation.

As I circulate the room, I look for a student or group of students to articulate how they solved the problem; often times, this involves them sharing out the equations that “unlocked” the puzzle for them as well as the order in which they figured out the rest of the variables

When I debrief the warm-up, I ask student volunteers to present parts of the puzzle.  I give positive reinforcement after students present and encourage students in the audience to make connections between student presentations.  I give specific praise to particular statements that demonstrate justification or logical reasoning.  Some of these phrases include:

  • “Because if ______________ is _____________, then…" 
  • “____ = _____ because…” 
  • “So then…” as a way to extend some logical reasoning and connect different pieces of information.

Expectations for Group Work

12 minutes

Today we will complete The Four Triangles Problem, a rich, hands-on task, that engages students in constructing viable arguments and critiquing the reasoning of others (MP3) as students build their four-triangle figures by considering different cases for how the triangles can be arranged.  In groupwork, however, there is a tendency for some to dominate the discussion while others disengage.  For this reason, I introduce group roles and stress why the roles are important to the group's work to increase all students' participation in the task.

Expectations for Group Product (posted on whiteboard):

  1. Complete set of all figures that can be made using the four triangles
  2. Convincing argument that you have found all possible figures. The reason cannot be "we can't find anymore" as the goal is for students to prove, by exhaustion, that they have found all the figures



The Four Triangle Problem

60 minutes

I introduce the Four Triangle Problem with a whole class demonstration, taking one square piece of paper, folding it along one if its diagonals, and cutting the square into two congruent isosceles triangles. My Four Triangles Launch-Final video demonstrates how I do this. Since this is an introductory lesson to geometry, I ask questions like "are these triangles the same?" or "will these triangles completely cover each other exactly?" and "how do you know?" to foster student's understanding of congruence.

Then, I introduce the problem using only two triangles.  I ask the class to figure out how many distinct figures they can make using only two of these triangles, with the rule that one side of each triangle must completely touch another complete side of the other triangle.  (There are three solutions: a square, a bigger isosceles triangle, and a parallelogram.)  I ask the class questions like, "how do you know there are no other possible figures?" and facilitate a short whole-class discussion.

After this discussion, pass out several square pieces of paper to each group.  Students will fold along one diagonal of the square and cut along this diagonal to create two congruent isosceles right triangles.  The question to answer is "how many figures can be made using only four triangles?"  (This is where students engage with MP7 as they look for and make use of structure.)


  1. One complete side of each triangle must touch another complete side of another triangle

  2. Congruent figures in different positions are the same

  3. What you will produce: each group will produce one set, taping the triangles together

  4. When you think you have found them all, see if you can come up with a convincing argument that you have found them all--the argument cannot be "we cannot find anymore"

  5. Check in with the teacher when every member of the group is prepared to explain the group's strategy for solving the Four Triangle Problem

In the next lesson, each group will be given a characteristic by which they will sort their figures (I have provided some example for number of sides, lines of symmetry, rotational symmetry, convex or concave) and produce a group poster that conveys their solution to the Four Triangle Problem.

Resource Note: A version of this problem can be found at  This problem can be done in an elementary classroom however it is an excellent way to introduce group norms and get students to communicate using geometry vocabulary at any grade level.

Debrief and Exit Ticket

10 minutes

I try to build mathematical community in the classroom by debriefing group work behaviors and key strategies and insights that emerged from various groups.  Some of the group work behaviors I try to look for and highlight include:

  • Making eye contact around the group
  • Keeping papers and materials in the center so everyone can see
  • Asking if others understand and/or need help
  • Asking if others have an idea
  • Offering an idea or suggestion
  • Evaluating each other's work and making suggestions

The debrief of this lesson offers a great opportunity to highlight how groups visualized and built a "base" upon which the rest of the four-triangle figures can be built.  

Today's Exit Ticket: Each student should choose one of the figures their group created to sketch.  Then, the student should write everything he/she knows about the shape, using appropriate geometry vocabulary (e.g., number of sides, name of the polygon, convex/concave, lines of symmetry, rotational symmetry, etc.)