Author:
Melody Casey
Subject:
Mathematics
Material Type:
Lesson Plan
Level:
Middle School
Grade:
8
Tags:
  • GEDB
  • Global Education
    License:
    Creative Commons Attribution Non-Commercial
    Language:
    English

    Education Standards

    GEDB Pythagoras and His Theorum (Lesson 5 of 6)

    GEDB Pythagoras and His Theorum (Lesson 5 of 6)

    Overview

    Students will use a variation of the Pythagorean Theorem to find the diagonal measure of a three dimensional figure. This lesson was developed by Michael Tinker as part of their completion of the North Carolina Global Educator Digital Badge program. This lesson plan has been vetted at the local and state level for standards alignment, Global Education focus, and content accuracy.

    Lesson Plan

    Description

    Students will use a variation of the Pythagorean Theorem to find the diagonal measure of a three dimensional figure.


    Content

    Learning Targets and Criteria for Success

    Objectives:

    Students will take the information they have gathered in the prior lessons and will make calculations based on certain scenarios regarding three dimensional figures.


    Learning Tasks and Practice

    Lesson Procedures: 

    By now the students are experts on using the Pythagorean Theorem so we will now throw them a curve.  A Xerox box is sitting on a table at the front of the room and I will tell them that they will be finding the diagonal measure (the distance from a top corner to the diagonally opposite bottom corner) and they will use the theorem to do so.  They will be given a worksheet with another example on it but trying to follow a 3D figure on a 2D surface is very difficult so I put together this very rough hands on example.  Since it is very difficult for them to see inside this box, I bring up small groups one at a time while the others are looking over the sheet with the other example.  I explain that they can find this measure but it must be done in two steps.  I cut out a piece of tag board that is the triangle of the bottom of the box if it was cut diagonally.  I have given them the length, width, and height of the box and now ask that they find the hypotenuse of this triangle #1.  This is easy enough for them and everyone agrees on the answer.  I then take another triangle #2 I have cut which will fit in the box diagonally so that the top of it is up against one of the box’s top corners and the other end at its’ opposite bottom corner.  We talk about how the hypotenuse of this triangle is the diagonal that we are trying to find and this is the final answer they are trying to arrive at.  At first, they think they can’t do it because they only have one of the legs for #2 which equates to the height of the box.  Fairly quickly, though, with both triangles in place, they see that the hypotenuse of the first triangle is the leg of the second triangle.  Now that they have two measures (the two legs) they can find the hypotenuse (the diagonal measure of the box.  There is a lot of grumbling over how much work has to be done to calculate this one answer and, as we do as much as possible in my class, I tell them that there is a shortcut if they look hard enough.  After all the groups have made their way through the process we talk about the shortcut and if anyone has figured it out.  With six groups per class and five classes there is usually a couple who figure it out and then they really get grumpy.  The trick is that, with three dimensional figures such as this, there is a unique variation of the Pythagorean Theorem that can be used.  With normal questions, the triangle being worked on is a two-dimensional figure so we have two legs (a and b) with a hypotenuse of c.  These boxes are three dimensional so we need to add another leg and, therefore, another variable to the equation.  Now we have a, b, and c as legs and d as the hypotenuse so our formula is a^2+b^2+c^2=d^2.  They are mad because this can be figured out in a fraction of the time it takes to do the two-step approach.  They do know that this is done on purpose as they have to know how to do the “long” way and if I showed them the shortcut first they wouldn’t pay attention to the other way.


    Collecting and Documenting Evidence of Learning

    Closure:

    Students will be assigned a number of three-dimensional problems.  They will be instructed to do at least one using the two-step approach making sure to show all steps and all their work.  Of course they are going to want to do them using the shortcut method so at least they’ll have one to use to remind them how to do it the slower way.