The offensive has set off the heaviest fighting between Israel and the Islamic militant group since an eight-day battle in November 2012.
Militants unleashed rocket salvoes deep into Israeli territory and Israel mobilised thousands of forces along the Gaza border for a possible ground invasion into the Palestinian territory.
Israel's defence minister Moshe Ya'alon has warned the offensive will be long-term. Since it began on Wednesday, Israel has attacked more than 400 sites in Gaza, killing at least 41 people.
The strikes came after militants fired more than 160 rockets at Israel, including one that reached the northern Israeli city of Hadera, 60 miles from Gaza, for the first time.
The army said it attacked more than 160 sites in Gaza yesterday including 118 concealed rocket-launching sites, six Hamas compounds, 10 militant command centres, weapons storage facilities and 10 tunnels used for militant activity and to ferry supplies in from Egypt.
Gaza health official Ashraf al Kedra said the new airstrikes killed one militant in south Gaza; an 80-year-old woman; the son, wife and neighbour of a Hamas militant; and three others.
Israel's army said it targeted a militant with the Islamic Jihad group who had launched rockets toward Israel. Islamic Jihad claimed one of its militants was killed with his mother and four siblings but Mr al Kedra said they were all civilians.
Israel and Hamas are bitter enemies but until recently had been observing a truce that ended the previous hostilities in 2012.
Air raid sirens sounded in Tel Aviv and Israel's south early yesterday and the army said two rockets were apparently intercepted above the city by an anti-missile battery.
Military spokesman Lieutenant Colonel Peter Lerner said the military's aim was to take a "substantial toll" on Hamas and to deplete its rocket capabilities. He said the army would gradually ramp up its strikes on Gaza.
He added: "The organisation is going to pay for its aggression. The country is not willing for this situation to continue."
About 2,000 people attended a funeral for eight Palestinians - including at least one militant, four adults and two children - who were killed on Wednesday.
In that attack, an airstrike flattened the home of a Hamas militant in the southern Gaza town of Khan Younis. Israel's military said it had called the home shortly before the airstrike to warn civilians to leave.
Palestinian president Mahmoud Abbas has called on the international community and the UN to "provide international protection for our people". He said Hamas leaders in Gaza want to restore calm and added: "Everyone I've talked to expressed his willingness to restore the truce and stop the escalation."
Tensions have been rising since the kidnapping of three Israeli teenagers in the West Bank on June 12.
The situation deteriorated last week after their bodies were found, followed a day later by the abduction of Palestinian teenager in Jerusalem. He was later found burned to death in what Palestinians believe was a revenge attack. Six Jewish Israelis have been arrested.
A US official involved in the investigation yesterday said the three Israeli teenagers were shot at least 10 times with a silenced gun in what appeared to be premeditated killings. The disclosure clashed with speculation by some Israeli and Palestinian commentators that the captors intended to take hostages for a prisoner exchange but panicked and shot them.