Below are a few ideas for activities that you can do with your one-year-old.
Make a Doll Bed
Take a shoebox and scarf out of your closet and get a pillowcase or piece of fabric. Ask your child to pick out a small doll or stuffed animal. Now, make a big deal out of picking out a special place for your child to put her baby to sleep. Let her wash her baby if she wants and make the bed the way she wants to. Encourage her to feed and sing to the baby. Your toddler may want to do this over and over, changing songs and what she does until she is certain she has created a perfect routine for her baby.
Be an Animal
Get down on all fours and ask your child to do the same. Now, have her imitate you as you pretend to be different animals. As you do this throughout this year, she will be able to name the animals you pretend to be (dog, cat, bear, lion, tiger, pig, sheep, and even a two-legged duck). You can move throughout the house, having her follow you and behaving like different animals. Eventually you'll be able to turn this into a guessing game, as she's able to name the animals. Then, she'll want you to guess which one she is.
Balance on a Ball
Toddlers love exercise balls. They can roll them around. You can also hold your child on its top (she should be sitting) and gently roll the ball, keeping her steady. You can also put her on her belly and do the same thing. You can play games with the ball, too. Roll it back and forth between you, or have her chase it around the room.
Cookie Sheet Magnets
Take a cookie sheet and some magnets and let your child play with the magnets on the cookie sheet. Make sure the magnets aren't too strong (refrigerator magnets are good) because she may have trouble moving them on her own when she's this young.
Go for a walk with your toddler and bring a small shopping bag along with you. Let her collect things, such as pebbles and sticks, to bring home. If you want, you can glue what she brings home to paper, or just leave them in her bag to sort through another day.