ותעל הצפרדע ותכס את ארץ מצרים אמר ר' אלעזר צפרדע אחת היתה השריצה ומלאה כל ארץ מצרים כתנאי רבי עקיבא אומר צפרדע אחת היתה ומלאה כל ארץ מצרים אמר לו רבי אלעזר בן עזריה עקיבא מה לך אצל הגדה כלה מדברותיך ולך אצל נגעים ואהלות צפרדע אחת היתה שרקה להם והם באו:
השריצה - ממעיה ויצאו ולדות:
שרקה להם - ושמעו קולה כל הצפרדעים שבעולם והם באו:
Shemos Rabba 10,4 quotes this medrash in Sanhedrin word for word, with no mention of anyone hitting the frog.
The frogs did not come out of the mouth rather as regular frog spawn from its innards (see Rashi Quoted) which all miraculously grew into frogs which covered the whole land according to Rabbi Akiva.
Rabbi Elazar ben Azaria's interpretation that that the frog croaked and Rashi explains all the frogs heard its voice, and that caused the other frogs to jump out of the Nile so all the frogs already existed according to Rabbi Elazar Ben Azaria.According to this Medrash There was no hitting a frog.
So this interpretation could be a mistaken hybrid between the 2 opinions and Rashis opinion that the frog was hit see bellow, and when hit, the frog's mouth caused the other frogs to come out, which pre school thought that meant they spawned from the mouth when in actuality according to R' Elazar ben Azaria they came from the river as it was a "natural occurrence", and according to Rabbi Akiva this was one big frog that had miraculous offspring through regular spawning which covered the entire land of Egypt.
Rashi on the torah Shemos 8,2 seams to quote a different Medrash's opinion:
צפרדע אחת היתה, והיו מכין אותה והיא מתזת נחילים נחילים -
There was 1 frog and the Mitzrim hit the frog and it split open and out spurted river-loads of frogs
This opinion does mention hitting a frog, but thatthey spurted out from the place of the wounds which means the frogs came out of the many wounds as it says מכין plural, if they happened to smite the frog on the mouth then frogs could of gushed out from the mouth as well but they did not uniquely come out of the mouth and there is no evidence that one of the wounds was on the mouth.
So here the mistake of the preschool teacher is to falsly attribute the gushing out of frogs uniquely from the mouth.