Geometric radicals

Note: Shamelessly borrowed from Henri Picciotto's Geometry Labs - activities for grades 8-11. His website has a lot of great stuff:


Posing the problem

What does "different" mean? There are two possible options:
    • different side lengths
    • different areas

Since the objective of the lesson is to introduce students to square roots and to manipulate radical expressions, it appears that the latter definition of "different" is more useful, as students do not yet know how to find the sides of all squares that can be drawn on the given grid.

Questions to ask

Most students first draw "horizontal" squares. Some will draw slanted, but in the event that none do, how do we lead them to those possibilities?
  1. Looking at the column of areas in the table the student has filled out: "Hmmm, looks like you're missing some numbers here. How could you get them?"
  2. "Where is a 4 square?" - there will be a discussion of what a 4 square might be. There are two possibilities, either a square with area 4 or a square with side length 4. Let's choose the former. "Where is an 8 square?"
  3. "Do you remember seeing squares in real life (nature?)? What did they look like?"
  4. Inadvertently turn their paper so that the sides of the square are no longer horizontal.
  5. You can provide a square "tile" (post its, or cut out) and ask the students if they can fit it on the grid somewhere.
  6. "Where could the sides of your squares be? What lines can we draw on this grid?"


  • You can start with smaller grids and enlarge it slowly.
  • If starting with a smaller grid you can say explicitly that there are more squares than just _ (whatever the number of "horizontal" squares is).