What activities can I do with my grandmother to stimulate cognitive and motor skills?

Ask her to "help" with chores - here are a few things we did with my mom and grandma. What ever you do be sure to get it at her level. Both my mom and grandma were short and as they aged became bent from arthritis. They could not reach as high or as far as me so I used all kinds of small tables, used chairs as tabletops - anything to get the work closer to them.
peel potatoes - put a lap table or cutting board on her lap with a bowl of potatoes and a peeler (use the thick handled kind), ask her to peel them all - then boil them up for a meal. You can also do this with apples, cucumbers - blanched tomatoes.
folding laundry - save up all the clean towels and park her in front of the dryer with an extra chair for the folded goods.
dry dishes - put towels and an extra strainer on the table or get them a higher stool to reach the counter
stir gravy - get a step stool with a seat and back rest on top, park her in front of the gravy and ask her to stir. I suggest doing this ONLY with an electric range, never with gas range and open flame and ONLY if she's able to understand that it's hot and risky...
- animal plant and national geography channel - IF they're interested - these programs present new info and often it's quite interesting, leave these on in the background so they pick up on it whether they realize it or not.
- the old standby's playing cards, cross words, search word puzzles; if her cognition is going - coloring books with pencils, crayons or water soluable markers. one friend set up a large tablet on the wall for her mom to draw with washable markers. She had to remove it 6 months later because mom began to draw on the walls and it took too much time to wash it off. Be sure to use "fat' pens so her grip is better.
Did she crochet or knit? I find crochet easier and knitting needles intimidating. You can find crochet hooks of all sizes if her grip is starting to go.
- potted plants on patio or deck - ask her to deadhead the old blooms, this also gets them outside for fresh air and sunshine.
I'm sure others will have great ideas. These worked for my mom and grandma for a long time. BEst of luck
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Just thought of some things that the nursing home has done with my mom - have her assemble a "percolator" or stove top coffee pot with coffee, play chinese checkers or the toddlers games where they put objects in like shaped holes (she gets frustrated with this because she realizes it's for kids).. for grandma, we got her some dolls and doll clothes but that was later on when she was pretty far gone. It was weird to us but made her happy so what's the harm.

At one point we also sat down with grandma (early on) and went through photo albums and had her tell us the names and histories of those in the pics. Often this was great because she gave us a lot of interesting stories about ages old farming equipment, techniques, canning techniques and recipe ideas. I also asked her how she baked beans, pies, made pickles, canned various foods, preserved food, baked bread - all kinds of stuff that we take for granted now. During the middle of her decline these things seemed fresh in her mind, it's like she'd just done it. She also did a lot of embroidery and cross stitch but as she got worse we kept getting bigger and bigger needles because we feared she'd hurt herself. If your elder is not nice to others anything like that is a bad idea.

My mother always swore by crossword puzzles. Also playing card games.

My Mom loves jigsaw puzzles (the 300 piece variety)...she goes through several a week. She has some fine motor skill problems, but is still able to work the puzzles. She finds them creative and relaxing. I don't know what I would do without them - there is so little that she can do now.