When did you memorize the patriotic poem, "Paul Reveres ride"?

In the above sentence is the predicate
did memorize? If so what part of the sentence would the word poem be??

Yes, "did memorize" is the main verb or predicate. The basics of this sentence are as follows:

you ... did memorize ... poem

What do you think the word "poem" is doing in this sentence if "you" is the subject and "did memorize" is the verb?

http://www.uottawa.ca/academic/arts/writcent/hypergrammar/rvsentpt.html

In the sentence provided, the predicate is indeed "did memorize." The word "poem" functions as the direct object of the verb phrase "did memorize." Let's break down the sentence and identify the parts:

"When" – adverb, introducing a subordinate clause
"did" – auxiliary verb (inverted with subject for the interrogative form)
"you" – subject
"memorize" – main verb
"the patriotic poem" – direct object, specifying what was memorized

So, the word "poem" is part of the direct object in the sentence.