How can I improve my tenor playing in a middle school jazz band?

​The obvious, but essential answer is to call a music school in your area and get a jazz saxophone instructor if your family can afford one. Even a few lessons from a good jazz saxophone teacher who is versed in the various styles you will encounter, will be able to help you improve your reading ability as well as prepare you for playing jazz.
With or without a sax teacher you should buy the Lennie Niehaus Book, “Basic Jazz Conception” It’s a fantastic book to help young saxophone players. I have used it for 30 years with my students and there’s never been a better book to teach jazz conception. You will learn how to play swing music and basic jazz articulations as well as the basic rhythms and style of playing jazz saxophone. And best of all, it now comes with a professional saxophone player playing all of the jazz etudes so you can hear how it’s supposed to sound.
Basic Jazz Conception for Saxophone, Vol 2: 12 Jazz Exercises; 10 Jazz Tunes (w/CD): Lennie Niehaus: 9781934638019: Amazon.com: Books
You should do your best to take you sheet music and practice every day at home. Sometimes you can find recorded examples of some of the music you will be playing and listening to a professional player will help you understand how it should sound, whether rhythmically or tonally or stylistically.
Listening to a lot of tenor saxophone players is a great way to improve your playing. These days, there are more books and theory about how to play and improvise jazz, but the way many jazz players learned in the past is through just listening to jazz and imitating what they hear. That is a process called transcribing in the music world. You can do it if you start with something simple like an easy melody you like. Transcribe and write it out the best you can or just do it all from memory. That’s a sure way to improve your playing and prepare you for improvising, which as a tenor saxophone player in a jazz band you will be forced to do at one point or another. What ever work you can do in that direction will definitely help you get prepared for improvising. Jazz improvisation, is all about using your ear. Theory helps too but you can do a lot on your own just transcribing and imitating great players!
Good Luck!

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