Teaching author's purpose to any age student can be a challenging task. I taught high school English for many years and helping students identify the audience certainly made it easier to determine the author's purpose. I like your acronym PIEE and imagine it would aid some students in remembering your list of writing purposes.
I recommend some lessons in Thinkfinity that may guide you in your search for materials to teach the author's purpose to third graders.
You may find it helpful to go to the Thinkfinity Resources page and use the keywords search on the right side of the page to locate materials such as lessons, interactives, worksheets, and much more. I recommend using words like writing, information, persuasion, and opinion; then select grade 3, and you will have numerous results. You may want to expand your search to grades 2, 4, and 5 to take advantage of additional resources.
Let us know which resources you like and how effective they are as you use them with your students.
I always taught it... as Author's Purpose is as easy as PIE. Persuade, Informational, Entertain. We ate pumpkin pie. We also created huge pies and drew three lines for the three sections. Then my students found examples from texts, magazines, and newspapers to put in each piece of the pie.