And, in a move that has angered traditionalists, the amount of time teenagers spend being tested on Shakespeare is to be cut.
The proposals are part of the first revamp of the national tests since they were introduced a decade ago and come as ministers confirm challenging primary targets for 2004.
Plans to ask more demanding questions are a response to criticism of widespread "teaching to the test". Government advisers hope that the inclusion of more reasoning and investigation questions in exams will make teachers focus more on these activities in class. The Qualifications and Curriculum Authority proposals are not, however, an admission that the tests are too easy - key stage 2 maths scores actually fell last year.
New questions in the 2003 maths paper will examine whether pupils can apply mathematical reasoning in a practical context.
For instance, rather than require pupils to work out a percentage, a question might ask: "If all pupils at a school voted on the colour of a new sweatshirt and 40 per cent voted for green, half voted red and 32 voted blue, how many children voted?"
Jackie Bawden, QCA manager of assessment, said: "Some of the questions will be a little harder but that is necessary if we are going to support numeracy strategies."
In science, the key stage 2 test will be extended from 35 to 45 minutes to allow more investigative work. At present the test is dominated by questions that simply ask pupils to retrieve facts.
Harder questions could put the new 2004 targets in jeopardy. By then 85 per cent of 11-year-olds should reach level 4 in English and maths and 35 per cent should reach level 5.
Sheila Dainton, Association of Teachers and Lecturers policy officer, said:
"These questions require a different sort of thinking which has to be good. We don't want to produce children who are full of facts but cannot apply what they know. The changes may cause a dip in results but ministers will have to live with it if they want better learning."
Professor Dylan Wiliam, of London University's King's College, warned that questions set in context could disadvantage working-class pupils and girls.
But a QCA spokeswoman said the proposed changes had undergone extensive trials to ensure that standards remain consistent. This means that when questions are harder, grade boundaries are adjusted.
The Shakespeare test for 14-year-olds is due to be shortened from an hour to 45 minutes to "better reflect the amount of time spent on Shakespeare in key stage 3". Questions in the reading test will be more focused. Pupils will give shorter answers, making it easier to show where marks are earned.
Ruth Moore, president of the National Association for the Teaching of English, said the moves could threaten extended writing.
"English is a complex subject. Trying to reduce it to a series of key tasks will not lead to deeper understanding," she said.
Nick Seaton, from the Campaign for Real Education, claimed the Shakespeare proposal downgraded an important part of Britain's cultural heritage.
Ministers are now considering the QCA proposals.