Math Unmuddled: Triangles

For children entering Grades 6, 7, 8 in 2021-22

A good grasp of basic Mathematics is important in today’s world, irrespective of the career one chooses to pursue. While Math is typically associated with Science and Engineering, it is hard to think of any line of work that does not involve an understanding of school-level Math (at least till grade 10). Unfortunately, many students struggle with the subject and even develop a fear of Math. The most common reason for this is that they attempt to remember many disconnected rules and procedures, and this makes Math seem more difficult than it is.

The GenWise team has seen many instances of this in their experience. For example, we have asked grade 2/3 students to perform a division like 391/17. After they answer this correctly and say ‘23”, we ask them to multiply 23 by 17. We find that there are always some students who start performing this multiplication instead of immediately seeing that the answer has to be 391 (because multiplication is the inverse of division). When underlying concepts such as these are understood, there is ‘less to remember’ and Math becomes simple and unmuddled. The ‘Math Unmuddled’ series takes up topics that pose common difficulties for children and clarifies these to build strong conceptual foundations.

This course in the series addresses one aspect of basic Geometry - Triangles. Under this topic, here are some concepts that students are expected to learn and then apply, whether or not they have actually spent time working with triangles!

Properties of a triangle

• A triangle has three sides, three angles, and three vertices.
• The sum of all internal angles of a triangle is always equal to 1800. This is called the angle sum property of a triangle.
• The sum of the length of any two sides of a triangle is greater than the length of the third side

But without having spent enough time handling and visualising different kinds of  triangles, this exercise is sure to become either boring or a tough climb or both.

According to the Van Hiele model of how children learn Geometry, students (irrespective of their age) need to spend sufficient time at different levels to be able to make sense of and actually learn geometry.

The first one, stage 0, is the Visualisation stage, in which a student has prototypes of some shapes in his mind and learns to classify an individual shape based on the whole appearance of the shape. For example, a student in this stage would be able to identify a square in its most familiar orientation, but might not if it were rotated by 45 degrees.

Only after this does a student enter stage 1, Analysis, in which the properties of the shapes are articulated and families of shapes are seen as having a certain set of properties. For example, a student in this stage could say: All sides of a square are equal.

It is only after these two are crossed, can they arrive at the stage of Abstraction, in which a student can think in terms of properties being related - either within a shape or across shapes. For example, all squares are rectangles.

When students are expected to understand and articulate geometry concepts at stages 1 and 2 without having spent enough time in stages 0 and 1 respectively, we set them up to fail. A child should comfortably be able to answer questions such as these if he is ready for stage 2.

Without a picture, how would you describe a parallelogram to somebody who has never seen one before? This course will give the student an opportunity to engage with Visualisation and guided Analysis to clarify the concepts around a triangle. It will address the following topics:

• Classification of triangles
• Attributes of triangles
• Congruence
• Angle sum property

References:

• Crowley, Mary L. "The Van Hiele Model of the Development of Geomemc Thought." Learning and Teaching Geometry, K-12, 1987 Yearbook of the NCTM by Mary Montgomery Lindquist, pp.1-16.
• https://en.wikipedia.org/wiki/Van_Hiele_model
• https://www.mathgiraffe.com/blog/understanding-van-hiele-levels-for-geometry

Note:

The course is not a substitute for learning the topic in school. It is assumed that students already have some familiarity with the topic. Also, in the limited time available in this course, it is not possible to focus on practice. Students must internalize what they learn in the course through subsequent practice, under the guidance of a teacher or a parent.