The exchanges went on for a third day despite Egyptian calls for an immediate cease-fire, The Jerusalem Post reported.
The Israeli air force struck a motorbike in southern Gaza, the newspaper said, seriously injuring Mohammed Rashdan. The Israeli military said he is a member of a Gaza-based Global Jihad and one of the planners of an attack along the Egyptian border Monday.
The Post said Aleb Armilat, who assisted Rashdan in planning the attack, was killed in the strike.
Wednesday evening, the Israeli air force struck what it said were two Hamas training camps. Palestinians said two teenagers were injured.
In an Israeli airstrike on a rocket launching cell in the northern Gaza Strip, a 14-year-old was killed. The newspaper said the teenager was identified as Momen al-Adam. Two other people were injured.
The Wednesday airstrikes followed eight bombings late Tuesday night throughout the Strip, the newspaper said.
An Israeli military source told the newspaper the involvement of Global Jihad in the attacks was not surprising. The source said the groups did not obey Hamas, and might not abide by a new cease-fire.
The Israeli Home Front Command banned classes in schools that are not reinforced and are located near the Gaza border -- an indication more attacks were expected Thursday.
Palestinian militants fired 55 rockets into southern Israel throughout Wednesday, damaging homes and sending eight civilians into shock. One rocket was intercepted by Israel's Iron Dome counter-rocket defense system, one three operational Iron Dome batteries deployed in the south.
Yasser Othman, Egypt's ambassador to the Palestinian Authority, said Cairo is working with Israel and the Palestinian factions in Gaza to halt the violence, proposing a "calm-breeds-calm" solution, Israel Radio said. Othman said it could take a few days for the lull to take effect, the report said.