Mix the sourdough discard, salt, baby oil, and cream of tartar in a medium size saucepan.
Over medium heat, cook the playdough mixture, stirring continuously until it begins to thicken. Keep a close eye on it, so it doesn't cook to the bottom. Once it starts to form into a ball while mixing, remove it from the heat and let it cool a bit.
Once it is cool enough to the touch, turn it out onto a clean, dry surface and knead it until it's smooth. If the dough is too sticky, you may need to add flour a little at a time, or if the mixture is too crumbly and dry, add warm water a tablespoon at a time until you reach the consistency you like.
Add any food coloring at this stage. Divide the dough into 4-6 portions and add in your desired dyes. The brand and type of food coloring you use will determine how much you need to add to the dough. I did 10-15 drops or 1/8 tsp worth of the gel food coloring and 20-30 drops of the regular food coloring. If you want to use natural dyes, a few I have tried are cinnamon and turmeric. There are lots of other natural dyes; those are the ones I had on hand. I highly recommend using gloves for adding any dyes.
Store in an air-tight bag or container. Homemade playdough can last anywhere from 2 weeks up to a few months, depending on how much it has been used and how well it has been stored.