In class, I drew the following figure on the board:
I asked the students if they could determine the length of the diagonal d given the information we’ve been recently covering. I’ve been talking to them about special right triangles (i.e. 45-45-90 and 30-60-90 triangles) at length to help them gain a deeper understanding of their properties as I prepare them for trigonometry next year. I even showed them how to use the distance formula to find the length of the hypotenuse. Yet, with all this knowledge and understanding, they weren’t even sure how to begin working on the problem.
I thought about this over the weekend and wondered if there were a different way, a more engaging way, that I could use to help them through this problem. So, I went to the store and bought several spools of string. I divided the class into groups of four and gave them each a spool of string. Then, I marked two opposite corners of the classroom and presented the challenge.
Cut the string provided using only one cut so it may touch both corners when pulled tight. The groups could use any of the measuring devices provided (i.e. 12-inch ruler, yard stick, or measuring tape).
This required them to figure out how to use the measurements of the floor and the walls in determining the length of the string. By giving them the condition of only being able to cut the string once, they had to attend to precision (see CCSS.MATH.PRACTICE.MP6). Also, with a variety of measuring devices provided, they needed to determine which device would give them the most accurate measurement (see CCSS.MATH.PRACTICE.MP5).
Needless to say, this offered the students a different way of thinking about finding the diagonal of a cube. Some were able to derive the distance formula on their own as they worked through the calculations. Others admitted that they understand more now about square roots and working with triangles.