One recent morning proved quite informative and interactive for Class 7 Math students as they advanced through their in-depth study of geometry.
“In the last class, they learned the Pythagorean theorem conceptually,” explained Head of Middle School Mathematics Sarah Davies. “It has a lot to do with area. So, rather than just give them the algebra, they experimented with squares to determine which could make right triangles.”
Students arranged squares with different lengths and areas so that three together made an exact right triangle – a triangle with one angle at 90 degrees – by touching their vertices. After a period of experimenting, they recorded their findings.
“One of the things I noticed about triple sets of squares that were positioned to create a right triangle successfully was that the three areas were related,” one student shared. “I learned that for right triangles, if you have the lengths of the shorter sides, you could find the length of the hypotenuse. I also learned how to algebraically represent this. I finally got a better understanding of this equation.” By observing patterns, making connections and using critical thinking, these perceptive mathematicians discovered the Pythagorean theorem all on their own.
During the classes that followed, they put their newfound knowledge to the test with various challenges. “Try these in any order to practice what’s comfortable for you,” Ms. Davies commented as she distributed a worksheet. Students were instructed to draw a smiley face next to each that was easy and a star next to those that they felt were more confusing.
Using colorful “geoboards” (a mathematical manipulative used to explore basic concepts in geometry) and several rubber bands, the students worked diligently, either as a group or individually, to construct the correct shapes and use the right formulas. “Can you make an equilateral triangle?” one student said quietly to herself as she read a challenge aloud. “Yes, I can!” she confirmed, before placing the rubber bands on pegs equidistant from each other.
Ms. Davies circled the room, quick to aid anyone with questions or confirm if a challenge was completed properly. At one table, students collaborated to find the length of the longest side, or the hypotenuse, of a right triangle. “We have to use the Pythagorean theorem!” said one student, to which Ms. Davies nodded approvingly.
Other students examined a challenge that read: “Make a square that has an area of 36 units squared. What is the length of the diagonal of this square?” Due to the puzzled looks, Ms. Davies prompted, “What are the lengths of each side? How do you know?” After a moment, one student said, “The square root, that would be 6 by 6!” and began scribbling an equation on her worksheet.
Some of the other complex challenges included: “Create three different scalene triangles,” “Create an isosceles triangle with a base length of 2 units squared,” and “Create a triangle that has an area of 5 units squared.”
“I’d like to report a genius in the midst!” joked one enthusiastic student, who had correctly executed a tougher challenge.
As the lesson came to a close, the students made it clear they were eager —and excited—to take on the challenges their next class would offer.