The rocket launches come after an IDF tank struck a Hamas facility in Gaza on Tuesday, reportedly killing two members of Hamas' military wing's al-Nukhba elite unit and wounding six others. Israel deployed its Iron Dome system and intercepted 11 launches, though several civilians were injured by shrapnel.
The IDF confirmed they responded by firing at 140 targets in Gaza.
The BBC has been facing a backlash by the Israeli Foreign Ministry on Thursday morning over a story headline that was interpreted as "anti-Israel" and biased.
At least three Palestinians died - a pregnant woman, her one-year-old daughter and a Hamas militant, according to the Gaza Health Ministry.
Over the past four months, 163 Palestinians have been killed by Israeli fire, including at least 120 protesters, according to the Gaza Health Ministry and a local rights group.
The cross-border violence comes soon after a delegation of senior Hamas officials visited Gaza to discuss a ceasefire agreement with Israel proposed by Egypt and the UN.
In addition, the Israeli Air Force fired three missiles at a concrete factory, and three other missiles into a nearby area, northwest of Gaza city.
The flare-up came after officials on both sides had talked about potential progress in an effort by the United Nations and Egypt to broker a truce to end months of violence and alleviate deepening humanitarian and economic hardship in the Gaza Strip.More news: Sam Curran leads England's fightback against India
In a separate statement, the Israeli Air Force (IAF) said its fighter jets targeted 12 terror sites in the Gaza Strip.
In Israel, Barzilai Medical Center in Ashkelon said that it treated 23 injuries overnight from the rocket attacks, the majority from anxiety or orthopedic injuries.
Israeli media said officials were considering evacuating residents from areas near the Gaza border.
Israel has been battling large fires caused by kites and balloons rigged with incendiary devices or burning rags launched from Gaza that have destroyed forests, burned crops and killed wildlife and livestock.
Israel's defence chief said last month that Gaza's only commercial border crossing, Keren Shalom, would reopen if calm persisted.
The IAF also struck a vehicle in the northern Gaza Strip from which they claim had been used to launch rockets.
"This is the only barrier of its kind in the world, which will in effect block the possibility of infiltration of Israel from Gaza via the sea", Defense Minister Avigdor Liberman stated at the end of May.
US Envoy Jason Greenblatt who has been involved in peace negotiations between Israel and Palestine, pointed a finger squarely at Hamas for the escalation of violence. Since then, the group has fought three wars with Israel, most recently in 2014, which has seen deaths and injuries of thousands of civilians and has left much of Gaza is ruin. "If current escalation however is not contained immediately, the situation can rapidly deteriorate with devastating consequences for all people".