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Trace a rectangle onto medium-density fiberboard. Cut the board along your traced lines. Smooth out the edges with sandpaper. Apply one coat of white latex primer to one side of the board. Paint the board with two coats of chalkboard paint. Fit the chalkboard into the back of the frame. Secure the chalkboard in place. Hang the chalkboard by the hook attached to the frame's backing.
The rectangle must be the same size as the opening of the frame. If the frame is new, take out the paper insert that comes with it and trace the outline onto your board. You can also trace the inside glass window. Draw the outline on the board using a pencil.  If you have an old frame with no glass, use a ruler to measure the back opening of the frame. Draw a rectangle with the same dimensions on your board. Do not use the dimensions from the front opening of your frame. Plywood also works if you don't have medium-density fiberboard. Use an electric jigsaw or manual hand saw to cut along the outline you traced. If you are unable to do the work yourself, bring the dimensions to the hardware store and ask a worker in the lumber section to trim the board down for you. After cutting the board, use rough sandpaper to remove any splintering edges around the perimeter. You may also need to use the sandpaper to shave away some of the board if it will not fit into the frame perfectly. Use a large sponge brush or standard flat painter's brush to brush on a coat of primer. Let the primer dry completely before applying the chalkboard paint. The primer will make it easier for the paint to stick to the surface of the board. Use a painter's brush or roller to apply two even coats of black chalkboard paint to the primed board. Even coats will reduce the amount of ridges that develop on the surface after the paint dries.  Allow the paint to dry completely between coats. Chalkboard paint is formulated to develop the surface texture of a chalkboard upon drying. It can then be written on with chalk. Applying two coats will make the effect stronger. Insert it with the chalkboard side facing the front. It should slide right in, similar to if you were replacing the glass in the frame. If the backing will fit inside the frame with the board, slide it behind the board to secure everything in place. If you cannot use the backing, use masking tape or packing tape to fix the back of the board to the back of the frame. Alternatively, you could use heavy staples to attach thick twine or thin rope to the top two corners of the frame and hang the chalkboard on a hook using this string.