I like to start this class with a silly song I sang a lot at summer camp. Here's an example: https://www.youtube.com/watch?v=_GiJOURSAt4
Every student receives a worksheet with the nonsense sounds and the proper English words. The lyrics are a little different from the video since I remember it differently from my camp. The students practice a few of the difficult words or phrases first, and then we start the nonsense sounds at a very slow pace, about one beat per second. They copy the pat-clap beats and repeat after me with the rhythm. We repeat the nonsense side another two times, gradually ramping up speed, and then we do the English version the same way. Clarify what a 'mosquito' is (ka in Japanese) and how we deal with them with bug spray right beforehand though.
If the students are having a little trouble being silly with the song, I'll pause and remind them that they need to practice moving their mouths. English sounds very different from Japanese because the muscles in your mouth move differently, so they need to really exaggerate those movements (like I'm doing when I explain this) and be aware of the sounds they're making if they want to be better English speakers. The more they practice, the faster their mouths will move and the easier the sounds will come.
Tongue Twister Telephone
Explain that moving their mouths to make sure they're understood is what we're practicing today. Those long, difficult-to-say sentences are called tongue twisters (hayakuchikotoba).
Split the class into groups. In a class of 20, I'll have 3-4 groups and in a class of 40, I'll have 6. Tape one piece of B4 paper per group onto the board and number them. Make sure the groups know which numbered paper is theirs.
One volunteer from each group will stand up and follow me out into the hallway. The JTE remains in the classroom to ensure no one dies. I'll lead the volunteer group down the hallway a bit and show them one of the tongue twister sentences. Always start out with a simple one, like 'She sees cheese'. The volunteers will take some time to practice the sentence and memorize it, then they head back into the classroom.
When they enter the room, everyone in all the groups must keep their ears closed except for the first listener! The volunteer will ~whisper~ the tongue twister into the listener's ear. For higher-level classes, I'll tell them they can only whisper the sentence three times and that's it. Once the listener has the sentence, the volunteer shuts up, closes their ears, and the listener passes the sentence to the next person. The sentence continues through the group to the last person.
When the last person has received the sentence and thinks they got it down, they stand up and go to their group's paper on the board. They lift the paper up, write the sentence on the board as well as they can, let the paper fall to hide it, then return to their group. They have to write the sentence in English, but spelling doesn't matter! They just have to try their best to match the sounds. They cannot receive any help from their group while they're writing.
When everyone's finished, write the correct tongue twister at the top of the board, then lift the papers to check each group's sentence. I get a lot of laughs here, so definitely ham it up, pronounce it like you're some hoity-toity English professor, have fun with their bad spelling. Even the students who got it monumentally wrong will enjoy it.
Award one point to the team who got the closest to the original sentence. Then practice pronouncing the tongue twister all together before starting a new round with new volunteers.
I have done this lesson with first-years from my very low-level technical high school and low-intermediate girls' high school. Both groups of students have enjoyed the warm-up and the game so far! My tech school students were able to get through 2-3 tongue twisters and my girls got through 4-5.
Sorry for the wall of text--
But I hope this works well for you guys!