Engaging students: Solving for unknown parts of rectangles and triangles

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Nada Al Ghussain. Her topic, from Pre-Algebra: solving for unknown parts of rectangles and triangles.

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How has this topic appeared in high culture (art, classical music, theatre, etc.)?

In the mathematical classroom it is always easier to engage the left-brainers who excel in numbers, reasoning and logic. My right brainers on the other hand can also shine when engaging them through the underlying foundation of the arts. The Golden ratio \phi = \frac{a+b}{a} = \frac{a}{b} is seen in paintings and architecture. It shows how rectangles and triangles can organize the placement of other shapes and figures in an eye pleasing way. Artists and architectures constantly mapped out their masterpieces on blueprints, which required basic calculations that set up the Golden ratio. Artists using the Golden Rectangle would need to find the missing sides to be able to get the correct proportions for the Golden ratio. This is seen in Leonardo Da Vinci’s “The Last Supper” and in the Parthenon building. Rembrandt solved the third side of an acute triangle before he continued work on his self-portrait. He then drew the line from the apex of the triangle to the base, which cuts into the golden section. Finding the part of a triangle and rectangle contributes to creating masterpieces! Students, left and right brained will see beyond paint, color, and stones. As Luca Pacioli, a contemporary of Da Vinci had said, “Without mathematics there is no art.”

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How does this topic extend what your students should have learned in previous courses?

Beginning Geometry students, Can with little and quick computational work solve for the unknown parts of any given rectangle and triangle. A great starter for a Pythagorean lesson is to get them to find missing parts using their shoes! Middle school students can take off their shoes as they work in groups and form the two legs of a right triangle. Once they compute the hypotenuse students can check it by adding the right amount of shoes. This lets students interact with each other and with the right triangle. They can see which triangle theorems can be formed, and discuss the type of angles found with the right triangle. Going beyond that, students can shoe in the missing sides of the squares. This sets up The Pythagorean theorem. This engagement can be quick or take a whole lesson. Students find different calculations, theorems, and set them up for figuring out the Pythagorean theorem.

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.


Technology is information at our fingertips. Calculator Soup has a Triangle Theorems calculator that can calculate AAA, AAS, ASA, ASS, SAS, and SSS. This would be a great and quick way for students to explore triangles. As a teacher I would ask the students to make an acute SSS triangle using the digits 1through 10 for the sides. I then can ask them if a given side was 20 and the other two were between 1through 10, would I still have an acute triangle? Many quick questions can be used from this calculator. It has the students think about the relationship of the sides and angles as they form triangles. There are also Square, Rectangle, Parallelogram, and a Polygon calculator too. For the parallelogram, different angle measurements can be given to change the side length. Good ways to have students differentiate between rhombus and parallelograms. Calculator Soup is quick visual for students to help them understand the relations between different squares and triangles.










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