That's how Sanchez got into baseball, a sport he'd not played before as a youngster. His father, Sigfredo, and his brothers were baseball players, but Jonathan liked basketball. Oh, there was some pickup softball around the neighborhood, but that was all.
But the day that Sanchez was throwing rocks at the mangoes, his left arm must have looked pretty good to a man passing by.
"There was a guy who was a coach. He said, 'Do you want to play baseball?' But I told him I never played baseball before. He said, 'Do you want to try?'" Sanchez recalled.
However, the team, called the Dodgers, played about 25 minutes away from Sanchez's house and he didn't have a ride to get to practice.
"So the coach got the team to come to the little field next to my house, because he wanted me to try to play," Sanchez said. "I played the outfield, got a couple hits and made a good play and they signed me up. The next day I started playing baseball."
And kept playing, through amateur ball in Puerto Rico and through four seasons at Ohio Dominican University in Columbus, and in 2004 he was drafted in the 27th round by the San Francisco Giants. He made it to the Majors in 2006, pitched a no-hitter in 2009, reached the World Series in 2010 and now is starting a new phase of his career with Kansas City. Obtained in a trade for outfielder Melky Cabrera, he figures to be the third starter in the Royals' rotation.
But back to Puerto Rico. By age 15, Sanchez felt he wanted to do more than play the outfield and began pitching the second game of weekend doubleheaders. That felt more like his niche.
"I made the decision," he said. "I knew I wasn't going to hit a lot of homers. I was skinny. I had power, but not like a home-run guy. So I said, 'I'm a lefty, I throw hard, so I can do it as a pitcher.' So I changed."
While Sanchez could plunk a mango with a rock, throwing a baseball over the plate did not always come that easily. When he throws, the ball takes on a life of its own, and breaks and snakes every which way but loose.
That might help explain why, when he helped take the Giants to the World Series championship in 2010, he also led the National League with 96 walks. But there's a positive factor to all that movement, too. It makes a lot of hitters go chasing and make outs.
"I think it's natural," Sanchez said. "God gave those pitches, and they happen to be good pitches. I know I'm a strong, tall guy, but he gave me good pitches to throw."
Most pitchers don't prosper when they stray out of the strike zone but Royals manager Ned Yost calls Sanchez a different story. Why?
"His deception, his delivery, the way the ball jumps out of his hand," Yost said. "You watch Sanchez on tape and he gets more missed hits, weak fly balls, weak ground balls, swings and misses on pitches outside of the strike zone than anybody else we have."
Even so, Yost believes Sanchez, whose record in six years with the Giants was eight games under .500 (38-46), can be more productive with better command of his pitches. With that in mind, the Royals are making slight adjustments in his delivery, although they don't want that dynamic stuff to disappear.
"He makes good hitters take bad swings," Yost said.
In the same season that Sanchez issued 96 walks, he also struck out 205 batters in 193 1/3 innings. In fact, his average of 9.37 strikeouts per nine innings is the third-highest in the Majors since 2008.
"All my pitches are strikeout pitches," he said after one of his Arizona outings. "I'm not the kind of guy that goes out there and out of a hundred pitches throws 90 strikes. I can make you chase out of the zone, and I'm going to throw strikes so you've got to be ready."
Sanchez's next turn to pitch comes on Thursday, but the Royals won't use him in their Cactus League game against the Angels. After all, that's the team that he'll face in the third game of the season at Anaheim. No sense in letting the opposition get a preview.
So instead, he'll pitch against some Royals Minor Leaguers at the Surprise complex just to build his pitch count. That will put him in line to face the Padres in an exhibition game next Tuesday in San Diego.
Sanchez is one of six children -- he has three brothers and two sisters -- and his mother, Xiomara, helps him care for his son Christian.
"She used to work making clothes, but since I got to the big leagues, I told her she wasn't going to work anymore. She did everything for us when I was little," Sanchez said.
His father works in a business managerial capacity in Puerto Rico.
"He's still playing today, in old-timers' baseball in Puerto Rico," Sanchez said. "He pitches, plays third, plays everything. He likes baseball. When we talk, he always tells me how they did in baseball."
Sigfredo Sanchez, according to his son, was a hard-throwing right-hander in his youth.
"He had a couple of scouts come to see him throw, but his dad said, 'No, he had to work,'" Sanchez said. "I think that's one of the reasons I'm here, because God didn't give him the chance to play, but God gave him a son that's in the big leagues."
Maybe that wouldn't have happened, though, if one day Jonathan hadn't taken aim at a mango tree.