When we use AirPlay to share our laptop screens to the big classroom displays, we ordinarily use it to "mirror" the laptop's screen, so that the two screens show the same thing.
However, you can also tell AirPlay to use the big classroom display as a second, separate monitor. This gives you more screen real estate to work with, and also allows you to show content to your students on the classroom display while you continue to do other work on your laptop.
Zoom also recognizes this second monitor, so that your in-person and remote students will see the same thing.
Note: If you will be playing a video and want the audio to play through your classroom display and through Zoom, please see the Share Computer Audio Through Zoom article as well; there are some additional steps required.
Here is an 11 minute video demonstrating how to use AirPlay's "Separate Display" feature. This video is very wide; please watch it in fullscreen. If you are already familiar with using your classroom display as a separate monitor, you can skip to the Zoom demonstration at 6:50 in the video.
Configuring AirPlay to "Use As Separate Display
Begin by connecting to your classroom display in the usual way, by going to the AirPlay menu, choosing your classroom, then entering the 4-digit code.
Once your laptop is connected to your classroom display, click the AirPlay menu again, and this time choose Use As Separate Display.
Both screens will flash, then your classroom display will show an empty background image; this means it is ready for you to windows onto the display, by dragging the windows from one side of your laptop's screen onto the corresponding side of the classroom display. Please watch the demonstration video above for more details about how to do this.
Arranging the Displays
You can change which side of your laptop's display is connected to the classroom display. Click the Apple menu in the top-left corner of your screen, then choose System Preferences. Click the Displays item, and then click the Arrangement tab at the top of the window.
Your laptop's window is the smaller of the two rectangles (highlighted above); drag this rectangle around to change how you drag windows from one display onto the other. Please watch the demonstration video above for more details about how to do this.
Share the Separate Display over Zoom
In your Zoom meeting, click the Share Screen button, just as you normally would. This time, however, you should see two options for "Desktop", Desktop 1 and Desktop 2.
Zoom will also show numbers in the corners of both displays, so you know which number is which. Click the desktop which matches your classroom display, then click Share. Now your students at home and your in-class students will all see the same material from your classroom display, and you can continue to do other work on your laptop's screen. Please watch the demonstration video above for more details about this.