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Though I am not a real sports guy by any stretch of the word I believe that sports and physical activity have a tremendous amount of benefits for kids of all ages, especially toddlers whose limited hand eye coordination can benefit greatly from practicing certain sports and activities.
For little kids you will want to stick with games that don’t really involve catching a ball, they have slower reflexes and poor ability to estimate speed and spacial distance of objects so tossing a ball at the face of a toddler is likely to just bonk him on the nose and he will have little ability to improve as his reflexes are just too slow at this age to adapt.
Soccer is one of the most common and perhaps one of the best games for little kids as they use all core muscles in their body and kicking and blocking a ball with their bodies and feet is much easier for them than catching with their hands. Just remember to set small field sizes and small goals for them since their little bodies can’t cover a wide area. My three year old loves to play “kick ball” and will often run around kicking the ball at goals attempting to out maneuver his six year old brother.
Fencing & Sword Play
While little kids can’t catch balls very easily they are just learning how to react to objects coming toward them. Playing sword fights with foam swords and shields is an ideal way for them to improve their agility and reflexes because you can tailor the speed of your blows to how fast they can respond. You can increase your speeds slightly to keep them a little challenged and help them learn to swing and deliberately let them score more hits than misses to keep them engaged and having fun. My kids love engaging me in sword play and Nerf makes some fine Foam Swords that you can safely use without worrying about cutting or bruising each other. (Though an eager and aggressive toddler can hit pack a whallup when they hit you with those foam swords, particularly when you are hit in the particulars.)
Running & Track and Field
Kids love to run and having running competitions can boost their endurance and help them become better sprinters. Just remember that small kids trip very often and should always were fitting sneakers, never have your kid run in flip flops or sandals which are more likely to make them trip. Keep in mind that you will want to have them wear long pants (preferably jeans which help absorb impact and reduce scrapes). If you live near grassy area’s or beaches, running along sand and grass will help avoid injuries when falls occur. If you must run on concrete then keep even steady paces rather than racing as fast as possible.
There are so many other simple physical activities and sports you can do and I recommend bouncing a beach ball or basketball for little ones, bowling plastic balls at plastic pins (heavy bowling balls are not a good mix for a three year old who can crush their little fingers and toes). Just make sure you reduce the weight and size of objects appropriately when involving your little kids and opt for foam sports equipment wherever possible to reduce the risk of injuries.