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Apply wood glue to 1 of the short boards. Press the glue side of the short board to the top edge of the backing. Clamp the boards together with a C-clamp for 1 hour. Drive 1 inch (2.5 cm) nails into the 2 ends of the board. Attach the rest of the boards to the backing to form the frame.
Take 1 of the shorter length of 1 by 4 in (2.5 by 10.2 cm) boards that will form either the top or the bottom of the shadow box frame and squeeze a line of wood glue across the bottom edge of it. Use enough glue to form an even and consistent line along the bottom. Wood glue is available at home improvement stores, at hardware stores, and online. Press the short board at either short end of the plywood backing so the edges of the board and the backing are flush against each other. Hold the board in place for about 10 seconds so it sticks. Gently release pressure from the board so the glue holds it in place. Tighten a C-clamp on top of the board to apply pressure while the glue bonds the 2 pieces of wood together. Leave the clamped pieces of wood alone to allow the glue to dry and bond them together. After an hour, release the clamp and gently wiggle the wood to make sure it’s held securely by the glue.  You could also use a jaw clamp or a screw clamp as well. You can find C-clamps at your local hardware store and online. Check the packaging of the wood glue for specific drying times. Use a hammer to drive the nails through the plywood backing and into the thicker board. Drive nails at both ends of the board to secure it to the backing. Use light hammer strokes to drive the nails into the wood so you don’t crack the frame. Apply wood glue to the bottom edge of a board, press it to the backing, and then clamp it into place with a C-clamp. When the glue dries, drive nails through the backing and into the top and bottom of the board. Continue working until all of the sides are connected to the backing.