To build a half fraction design we're going first to build the full 2 in the power of k-1 design and then we're going to write the last column by using the defining relation of the design. Let's take our example of the 2 in the power of 3-1 design with identity ABC. A full 2 in the power of 3-1 design has the size of a 2 in the power of 2 design: 4 runs. In the standard order, we have column A: minus plus minus plus, and column B: minus minus plus plus. Now we have to use the defining relation for the third column. We have that C is C times the identity. The identity is ABC, this is AB times C squared, that is AB. Then column C is given by column A times column B: minus times minus is a plus, plus times minus, is a minus, minus and plus, a minus, and plus and plus, a plus. In the first row, we have treatment C, in the second treatment A, then treatment B, and treatment ABC. Let's now take the example of 2 in the power of 3-1 design with identity minus ABC. As a suggestion, pause the video and try to do it by yourself before resuming the lesson. First, we build a full 2 in the power of 2 design and then we use the defining relation for the next column. C is equal to C times minus ABC, that is minus AB times C to the square, that is minus AB. The third column C will be given by minus column A times column B. For the first row, minus with minus is a plus, with minus a minus. Then we have minus plus plus and minus, and treatments (1), ac, bc, and ab.