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The first M1 increase method leaves a hole. If you doing a project that will be felted, this is okay. The hole will close. Sometimes you want a hole, as in knitting lace. Also, this M1 method can be used to correct a missed yarn over in the previous row. Just mark the place where you missed the yarn over and M1 when you come to it. I call this the M1, left needle. Insert the tip of your left needle beneath the ladder from front to back. Knit this lifted ladder stitch through the front leg. Notice the large hole formed beneath the M1 stitch.
The other method I use does not leave a hole in the knitting. It's a great way to make an M1 increase more invisibly. I call this the M1, right needle. Insert the tip of the right needle beneath the ladder from back to front. Insert the tip of the left needle into the front leg of this stitch and knit in this position. Notice that there is no hole beneath this M1. This M1 is much more difficult, especially inserting the tip of the left needle into the front leg of the stitch, so be patient with yourself! The result is well worth it.