If I understand the challenge correctly, its something that can be referenced by link.
If that is the case, then perhaps give instructions under the caption "Challenge a friend!" and under it "Send this link to a friend to challenge them in this SAT Battle Royal or whatever!" With a textbox with a short link for them to copy with a button nearby that will copy the link to the clipboard (nice for phone users).
Then students will have the option to send it via email or text message or facebook or whatever means they prefer.
I think that would be received better, since logging into email takes effort, text message is a pretty effortless way to communicate, and people tend to be reluctant to enter email addresses into form fields, especially emails that aren't their own.