Ipevo provide FREE software that you can use on your iPhone or iPad to turn it into a visualiser to show what is on your phone directly onto a computer screen and if that is connected to your classroom big screen you will see your iPhone output there.
The software for your iPhone or iPad is called iDocCam and is free at the Apple App Store. Download this onto your iPhone/iPad.
The software for your computer is called IPEVO Visualiser and you can download it FREE from https://www.ipevo.com/software/visualizer
This video shows how to get it all working together.