Breadboarding
Breadboards are white plastic panels covered with holes. The holes along each side are connected vertically; holes in the middle are connected horizontally. It's easy to connect electronic components into a circuit using a breadboard. You connect components by plugging them into the board. You can use short wires to connect together rows or columns as needed. Breadboards make it easy to test new circuits. It's also easy to insert switches or LEDs in the circuits so you can test them during prototyping. You can take out the switches and LEDs once the circuit is tested.
Half Adders
Adders are the classic project for combining logic circuits. A half adder gives the sum and carry for two binary digits. You can make these with an XOR gate and an AND gate. The inputs to the half adder go to the inputs of both the XOR and the AND gate. The output of the XOR gate is the sum output of the half added, and the output of the AND gate is the carry output of the half adder.
Full Adders
Full adders add three binary bits. A full adder is made from two half adders and an OR gate. Two of the inputs go to the first half adder; the sum output of the first half adder along with the third input to the full adder go to the inputs of the second half adder. The carryes from both half adders go to the input to the OR gate. The output of the second half adder is the sum output for the full adder, and the output of the OR gate is the carry output for the full adder.
Accumulators
Accumulators -- aka ripple carry adders -- do all of the arithmetic in computers. Accumulators consist of a series of full adders where the carry from one full adder becomes the third input to the next full adder. Accumulators add together two binary strings. You subtract by inverting one string and then adding. Multiplication is a series of additions, and division is a series of subtraction. The steps of these patterns are controlled by logic gates. By the time you build a working accumulator, you will be an accomplished logic circuit designer.