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Into the classrooms!
How many triangles can you make with these four segments?
Try it: play with the segments, combine them, rotate them, observe and discover how many triangles you can build.
Is there any combination of three colors that you haven’t been able to make?
Space reserved for thinking…
You haven’t been able to make triangles of all possible colors, right? Why do you think this happens?
It’s because, to build a triangle, the sum of the lengths of the two shorter sides must be greater than the length of the longest side. If not, no triangle is possible: the three segments don’t form a triangle.
Had you ever thought about this before? Congratulations! You just discovered the triangle inequality theorem 😊
Look at how children do it:
In Early Childhood Education, three-year-old children already explore, manipulate and realize that the length of segments affects the shape of the triangle. They may not yet know how to name it, but they do perceive through manipulation that not all sets of three segments can form a triangle. It doesn’t close for them!
After the first steps, a journey follows that extends throughout all schooling: building the meaning of triangle.
From here on, everything becomes interconnected: the triangle inequality in elementary school, angles, areas and even the Pythagorean theorem. Each new discovery is anchored in previous ones and opens the door to new learning.
Let’s see it in action: let’s enter the classrooms!
Join us to discover, through a small window, how the triangle learning trajectory is built in math classes.
Where it all begins: in an early childhood classroom
Very small chairs, a colorful carpet and children sitting on the floor. We are in a Kindergarten class (5 years old) in Mariemont, San Juan Unified School District, in California. The goal of today’s session is to discover triangles. How do you work with such an abstract concept with such young children?
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Let’s notice: the session begins with a story that gives meaning to the new learning. This narrative creates a meaningful context and invites children to get involved, proposing a challenge that motivates them to build knowledge actively.
After the story, comes the moment of free exploration. Through the same segments that students have seen in the story, in small groups they build shapes. There are no rigid instructions, but rather it is a moment of free and guided exploration. They try combining the segments, creating open figures, closed figures, quadrilaterals and, of course, triangles.
This manipulation is essential: it allows them to internalize the properties of shapes in a physical way, long before being able to verbalize them.
While all this is happening, the teacher circulates around the classroom and manages the activity. She asks questions to evoke knowledge: “What have you built?”, “How many sides does it have?”, “Can these three segments form a triangle? Why not?”.
It is at this moment when, organically, the seed of the triangle inequality appears. A group of children tries to join one long segment with two short ones. After trying several times, they conclude that the segments “don’t reach”. The teacher, instead of answering directly, asks: “What if you try changing the long segment for a shorter one? What happens now?”.
Through this conversation, the children themselves realize that not all combinations of three segments form triangles. They don’t name the theorem, but they have already encountered the triangle inequality.
From seed to theorem: the triangle inequality in elementary school
Let’s jump forward. If seeds are planted in early childhood education, in elementary school they are watered with mathematical richness. We enter a 5th grade (10 years old) class. The goal of this session is to classify triangles according to their properties.
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The session begins with a group conversation to formally remember what a polygon is. Students contribute properties: “It’s a shape that isn’t round”, “it has straight sides”, “it’s a flat shape”, “it has vertices”. Together they build a solid definition, while the teacher helps them clarify what they say so they use correct mathematical vocabulary.
Once they define what a polygon is, it’s time to see what types of polygons they can build. And here, notice how they use the same material as in early childhood education: the segments. The exploration is similar, but now the goal is not just to see “what happens”, but to understand why it happens.
And here great discoveries emerge:
- Rigidity: students realize that polygons with more than three sides are deformable, while the triangle is rigid. This observation is no longer a simple anecdote; they try to verbalize it by saying that, in a quadrilateral, “the opposite sides don’t touch at any vertex”, which allows movement, while in the triangle “all touch with all”.
Now they focus on forming triangles from four different segments. In this exploration, as teachers, it’s important to encourage students to be systematic and use strategies to find all possible solutions.
- The triangle inequality: The question “can you always make a triangle?” appears again, but this time the answer is more analytical. Even one student uses the actual measurement of the segments to be more precise with her answer. And no, not always can the joining of three segments create a triangle. Students verbalize the general condition: “The sum of the two shorter sides must be greater than the long side”. And this is how they end up formulating the triangle inequality theorem in their own words, and the teacher only has to name it.
Finally, with all these properties on the table, it’s time to classify. The teacher asks them to sort the triangles they have built. What can they focus on? The answer comes from them: the length of the sides and the angles. Thus, they begin to classify them as equilateral, isosceles and scalene, and as acute-angled, right-angled and obtuse-angled.
The leap to abstraction: the triangle inequality in middle school
If seeds are planted in early childhood education and watered with mathematical richness in elementary school, in middle school comes the time to harvest that knowledge. Now, students test what they know, formulate conjectures, verify them and reflect on them.
In this 8th grade session, the goal is to explore all possibilities of triangles with integer sides and fixed perimeter. And they are asked to build triangles with toothpicks.
First, students observe two shapes and share what they see as the same and what’s different. What about you? What do you see?
Some of the ideas that students usually share are: “They have the same perimeter.” “They are isosceles.” “The one on the left looks like a right triangle…” And we take the opportunity to check it.
Do you think there would be more triangles with the same number of toothpicks? And how many could you make with perimeter 10?
This question opens the door to investigation. Students are asked to build triangles with simple materials, like toothpicks, to measure them and test their conjectures. At this moment, they must remember that it is necessary to be systematic to obtain all possible results.
Usually, they establish a connection with number sense and transform the geometric problem into a numerical problem, where what they are looking for is equivalent to finding the decompositions of the number 10 into three addends.
Here, only the most systematic ones will be able to be sure they haven’t missed any:
{1, 1, 8}, {1, 2, 7}, {1, 3, 6}, {1, 4, 5}, {2, 2, 6}, {2, 3, 5}, {2, 4, 4}, {3, 3, 4}
And that’s exactly what we’re looking for, so they fall into the trap. Because then they think there are 8 different triangles with perimeter 10 toothpicks, but here is where they have to recall the idea of the triangle inequality. They know that some of those combinations of segments won’t form a triangle, so they must return to the geometric context of the problem to finish finding the solution.
