Measure and cut four 2"x3" pieces of wood to 7' in height. Cut four 21" pieces out of the fifth 2"x3" wood piece and put them inside the 7' pieces to make two rectangular frames. Secure the wood pieces with 16 drywall screws. You now have two 25"x7' frames.
Cut the felt into two separate 3'x7' strips. Lay one strip onto a clean, flat surface. Center one of the 7' frames on the felt, and fold the excess fabric over the top of the wood. Use a staple gun to fasten the felt to the wood, with each staple approximately 3 inches apart. Repeat this process with the second 7' frame.
Measure and cut four pieces of fiberglass panels to fit securely inside the 7' frames. If you cut 21" sides, you will need to trim 3" from the fiberglass insulation.
Measure and cut the backer board to the exact dimensions of the 7' frame. Nail the backer board to the frame, leaving about 3" between the nails. This will cover the staples from the felt and the fiberglass insulation. Repeat for the second frame. Both panels should now look exactly alike, with felt around the edges, one side covered in felt, and one side covered in backer board.
Align the felt panels of the two 7' frames on top of each other. Screw in three door hinges along the edge, connecting the pieces together. Stand up the panels and open them up to create a triangular recording booth.