Rockets could be seen flying through the sky over the coastal city of Ashkelon, located about 12 kilometres from the Gaza Strip.
Various Gazan militant groups were firing barrages of rockets from the Gaza Strip into Israel on Saturday in response to the Palestinian casualties killed on Friday.
Meanwhile, the European Union on Saturday called for rocket fire from Gaza into Israel to "stop immediately" and threw its backing behind efforts by Egypt and the United Nations to calm the situation.
Two other Palestinians protesting against Israel near the border were also killed by armed forces.
Israeli fire killed three Palestinians, including a mother and her baby daughter, Gaza officials say.
Israeli Prime Minister Benjamin Netanyahu is due to meet with the security cabinet on Sunday to discuss Israel's continuing response.
COGAT, the Israeli defense body responsible for Palestinian civilian affairs, also said it was closing the fishing zone off Gaza's coast altogether and sealing Israel's two land crossings with Gaza.
The army said earlier it was only targeting military sites in Gaza.
On Friday, Israel said two soldiers were wounded by gunshots from the Hamas-controlled territory.
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"While children in Israel ran for shelter from incoming rocket fire from Gaza, these children in Gaza graduated from kindergarten", the tweet added.
The flare-up comes after Israel reported Friday that two of its soldiers were lightly wounded in a shooting incident along its border with Gaza.
But the Israeli military spokesman, Lieutenant-Colonel Jonathan Conricus, warned his country is ready to escalate the attacks to protect the border. Another two Palestinians were killed during weekly border protests.
Three people, including an elderly woman, were wounded Saturday by rocket fire, and dozens more were treated for shock.
Both Hamas and especially Islamic Jihad have vowed to violently disrupt Israel's hosting of the Eurovision Song Content in Tel Aviv scheduled to begin in less than two weeks.
In response the IDF launched air and artillery strikes against 30 Gaza sites belonging to Hamas, a militant group that controls the Gaza Strip, and against groups including Islamic Jihad.
Rocket fire and airstrikes continue to happen regularly, despite a supposed cease fire that was reached between the sides following a 50-day deadly war in 2014.
By nightfall, the army said militants had fired well over 200 rockets into Israel.
Palestinians have participated in regular demonstrations and clashes along the Gaza border for more than a year, calling on Israel to ease its crippling blockade of the enclave.
On Thursday, a Hamas delegation led by the group's Gaza chief Yahya Sinwar traveled to Cairo for talks with Egyptian officials on a truce with Israel, Hamas officials said.