Problem Solving and GCSE

Assessment: Fit for purpose?

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The new GCSE specifications

GCSE specifications in mathematics should provide a broad, coherent, satisfying and worthwhile course of study.

They should encourage students to develop confidence in, and a positive attitude towards, mathematics and to recognise the importance of mathematics in their own lives and to society.

They should also provide a strong mathematical foundation for students who go on to study mathematics at a higher level post-16.

Subject aims and learning outcomes

GCSE specifications in mathematics should enable students to:

• develop fluent knowledge, skills and understanding of mathematical methods and concepts

• acquire, select and apply mathematical techniques to solve problems

• reason mathematically, make deductions and inferences and draw conclusions

• comprehend, interpret and communicate mathematical information in a variety of forms appropriate to the information and context.

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modelling and problem solving are emphasised

• “Students should be aware that mathematics can be used to develop models of real situations and that these models may be more or less effective depending on how the situation has been simplified and the assumptions that have been made.”

• “Students can be said to have confidence and competence with mathematical content when they can apply it flexibly to solve problems.”

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AO1: Use and apply standard techniques

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AO1 Weighting

Students should be able to:

•accurately recall facts, terminology and definitions

•use and interpret notation correctly

•accurately carry out routine procedures or set tasks requiring multi-step solutions

40%

(Higher)

50%

(Foundation)

Short tasks:

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AO2: Reason, interpret and communicate mathematically

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AO2 Weighting

• make deductions, inferences and draw conclusions from mathematical information

• construct chains of reasoning to achieve a given result

• interpret and communicate information accurately

• present arguments and proofs

• assess the validity of an argument and critically evaluate a given way of presenting information

30%

(Higher)

25%

(Foundation)

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1. Do they have to

stop for petrol?

Explain your

reasoning.

2. Suppose they

decide to stop for ten

minutes.

At what time will

they reach London?

Construct chains of reasoning

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The following real data shows how the

percentages of cars involved in traffic

accidents is related to the speed at

which they were driving.

(a) Max claims that the graph shows that

almost two thirds of accidents occur to

cars travelling below 40mph.

(b) Max concludes that it is safer to drive

at over 40 mph than to travel below

40mph.

Is Max right in drawing each of these

conclusions?

Explain why or why not.

Interpret and communicate information accurately

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Assess the validity of an argument

Which diagram is best for proving Pythagoras’ theorem? Explain why.

AO3: Solve problems within mathematics and in other contexts

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AO3 Weighting

• translate problems in mathematical or non-mathematical contexts into a process or a series of mathematical processes

• make and use connections between different parts of mathematics

• interpret results in the context of the given problem

• evaluate methods used and results obtained

• evaluate solutions to identify how they may have been affected by assumptions made.

30%

(Higher)

25%

(Foundation)

Modelling

1. Last Sunday an accident caused a traffic jam 12 miles long on a two lane motorway. How many cars do you think were in the traffic jam? Explain your thinking and show all your calculations. Write down any assumptions you make. (Note: 5 miles is approximately equal to 8 kilometres)

2. When the accident was cleared, the cars drove away from the front, one car every two seconds. Estimate how long it took before the last car moved.

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Modelling

1. Mrs. Grundy is planning to sell her home-made cola. These pictures show the top and side views of the type of bottle she plans to use. They are drawn accurately, full size.

2. Calculate the volume of soda that is now in the bottle, in cubic centimetres. Do this as accurately as you can. Show your method clearly. State any formulae that you use.

3. Do you think that your calculation for the volume is too large or too small? Explain why you think this.

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Modelling

1. Sue and Terry are making dogs and bears for the next charity fair. They want to make as much money as possible for the charity.

2. They only have time to make a total of 18 toys. They only have £60 to spend on materials. The materials to make each dog cost £3 and the materials for each bear cost £4. They sell each dog for £8 and each bear for £10

3. How many of each toy should they make?

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