The French prime minister, Edouard Philippe, has met opposition party leaders as nationwide protests continue to spread across France, with students blockading about 100 schools.

After thousands of masked protesters fought running battles with police and burned cars, buildings and barricades in the most affluent areas of Paris on Saturday, the gilets jaunes – or yellow vests – citizens’ protest movement continued on Monday with peaceful anti-government demonstrations at barricades on roads and at fuel depots across France.

High-school students – who have been protesting against changes to colleges and the university system – also seized on the mood of protest and stepped up their blockades.

About 100 high schools were fully or partially blockaded around the country, including in the southern city of Toulouse and in Créteil in the Paris area.

Seven teenagers were arrested after riot police were called to the Jean-Pierre Timbaud high school in Aubervilliers in the northern Paris suburbs where a car was overturned and bins were set alight.

A car burns outside the Jean-Pierre Timbaud high school in Aubervilliers. Photograph: Thomas Samson/AFP/Getty Images

French authorities traditionally fear high school students joining protests because their demonstrations often spread fast. An official at the education authority in Créteil said: “Pretexts are clearly being used to commit urban violence.”

An 80-year-old woman died in Marseille on Monday after she was hit by a police teargas grenade on Saturday. She was inside her apartment and had opened her windows to close her shutters when she was hit by the grenade as police confronted protesters. She died in hospital after surgery.

Meanwhile, 11 fuel depots across France were shut down after being blockaded by protesters. More than 70 petrol stations had run out of fuel and restrictions were in place in Brittany on how much fuel motorists could purchase.

An opinion poll for Harris Interactive taken after Saturday’s violence in Paris found 72% of French people still supported the protest movement that began last month in response to a rise in environmental taxes on fuel and has morphed into opposition to the government of the centrist president, Emmanuel Macron, amid a sense that the tax system is unfair and favours the rich.

Police arrest a high school student during a demonstration in Bordeaux. Photograph: Nicolas Tucat/AFP/Getty Images

The prime minister is expected to meet protesters’ representatives on Tuesday, but the grassroots movement, which emerged on social media, has no leadership nor a defined structure. One Paris representative said he had received death threats warning him not to meet the government.

Jacline Mouraud, one of the protest movement’s prime instigators, said scrapping the fuel tax was a “prerequisite for any discussion” with the government.

Macron’s challenge in calming the widespread anger against the government is complicated by his own desire not to yield to street protests, which repeatedly forced his predecessors into U-turns.

“Thinking that, as we have always done for 30 years, that you make a little symbolic gesture and then we sweep the dust under the carpet, that doesn’t resolve the fundamental, structural problem,” the government spokesman Benjamin Griveaux said, implying the government would not make major concessions.

Most opposition politicians urged the government to abandon a planned tax on fuel that was to begin in January; the only exception was the Green party, which said instead that tax must be fairer.

Laurent Wauquiez, the hardline leader of the rightwing Les Républicains party, said Macron should call a referendum on carbon tax “to let the people’s voice be heard”, but did not specify exactly what form a referendum would take.

The economy minister, Bruno Le Maire, said the solution for tackling the low purchasing power of struggling families lay in reducing the tax burden in France, which is among the highest in Europe.

“We must speed up the reduction of taxes,” he said. “But for that we must speed up the decrease in public spending.”

Saturday’s violence in Paris has alarmed the French business community, which claims it has already lost billions of euros, as representatives met the economy minister.

Hotel unions said bookings were down 15%. Retailers in Paris and other cities in which there were violent protests on Saturday, such as Toulouse, said they had suffered major losses in the crucial run-up to Christmas.