Introduction

There are four tasks for more able pupils at key stage 2:

• Before and after
• Turning triangles
• Maths diseases
• Number slide

Teachers can use these with individuals, pairs of pupils or small groups. These tasks can also be adapted to be used with the whole class.

Before and after

This task involves pupils exploring number sequences and completing a two-part task.

You may adapt the task for the whole class to work with the ideas at an appropriate level. For example, some pupils might work only on the first two elements (a and b) of Part 1, or work only with sequence A of Part 2.

Pupils are likely to take about two to three hours on the task in total, although this may vary. They are unlikely to complete the task in one sitting; revisiting the task after a break of hours or days will allow time for reflection and is to be advised.

Prior knowledge

Pupils working on this task will need to have prior knowledge or understanding of:

• sequences and multiples
• the fact that zero is a multiple of 3, as is -3, -6 and so on.

Assessment focus

• Recognise and describe number patterns … recognising patterns and using these to make predictions; make general statements, using words to describe a functional relationship, and test these (Ma2, 2b).
• Organise work and refine ways of recording (Ma2, 1f).
• Use notation, diagrams and symbols correctly within a given problem (Ma2, 1g).
• Present and interpret solutions in the context of a problem (Ma2, 1h).
• Communicate mathematically, including the use of precise mathematical language (Ma2, 1i).
• Search for patterns in results; develop logical thinking and explain reasoning (Ma2, 1k).
• Recognise, represent and interpret simple number relationships, constructing and using formulae in words then symbols (Ma2, 4d).

Resources

For each pupil or group of pupils:

• a copy of the photocopiable two-page pupil sheet
• paper and pencil
• optional – a spreadsheet or graphic calculator for Part 2.

For the teacher:

• solutions and what to look for.

What to do

• Tell the pupils that they will be exploring number sequences and that the task has two parts.
• Let them know that the first part asks them to look at a number sequence and to think about how it continues, working out the 500th number in the sequence without writing down all the terms. Then they will have to explore what numbers could have come earlier in a similar sequence. Since there is more than one possible answer pupils will need to consider how to find all possible answers.
• Tell them that in the second part of the task they are going to explore and compare three different number sequences.
• Let pupils explore ideas freely and test their understanding. Many pupils will, in the exploratory phase of the task, produce work that is unsystematic or unclear. After the exploratory phase, encourage them to present a solution that is logical and clear.
• Engage with the pupils to establish their mathematical reasoning and understanding. Pupil's explanations and justifications are important aspects of their mathematical working and are rarely fully expanded in written explanations, although this should be encouraged where the pupils are confident writers. For example, if pupils state there is only one solution to Part 1c, encourage them to explore further and if necessary give an example of how a number sequence can be worked backwards in different ways using the two possible rules.
  [Misunderstandings can be explored and corrected through discussion.]
• In Part 2, discussion may be needed to find out whether zero is a multiple of 3 (it is) and whether negatives can be multiples of 3 (they can).
• If pupils are not confident working with negative numbers, sequence C in Part 2 can be worked from larger starting numbers.

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Turning triangles

In this two-part task pupils explore the angles of polygons, triangles and other shapes.

• Part 1: pupils consider regular polygons as being made of isosceles triangles that fit together around a point. They work out a set of angles within a pentagon and decagon before investigating other polygons.
• Part 2: pupils investigate windmill patterns produced by fitting isosceles triangles together around a point. They are then encouraged to generalise rules for angles in windmill patterns enclosed by polygons.

This task is intended for more able pupils but can be adapted to be accessible to all. To adapt the task to an appropriate level for whole class work, for example, pupils could be asked to find only six of the twelve polygons made from isosceles triangles that fit together around a point. Making the task more concrete by using shapes made from plastic or card may also be helpful.

Pupils should spend two or three hours on the task, though this may vary. Although pupils may complete the task in one sitting - revisiting the task after a break of hours or days allows time for reflection and is to be advised.

Prior knowledge

Pupils working on this task need to know or understand:

• the angles of a triangle add up to 180°;
• the angles at a point add up to 360°;
• the angles on a straight line add up to 180°;
• isosceles triangle, pentagon, decagon and regular polygon;
• an equilateral triangle is a special case of an isosceles triangle.

Resources

For each pupil, or group of pupils:

• a copy of the photocopiable two-page task pupil sheet
• paper and pen
• if pupils prefer, a ruler to draw diagrams (optional).

For the teacher:

• solutions and what to look for.

What to do

This is not a traditional test and is designed to be worked collaboratively.

• Explain to pupils that the task has two parts. They will be exploring shapes and angles.

Part 1

• Explain to pupils that in Part 1 they are going to look at ways in which polygons can be made by fitting isosceles triangles around a point.
• Ensure that pupils read 1 (b) correctly. Errors result if pupils begin working with 10°.
• In Part 1 (c) pupils will find only eleven possible polygons if they do not consider an equilateral triangle to be an isosceles triangle. However, by drawing parallels with their knowledge that a square is a special rectangle, teachers can challenge this misunderstanding.

Part 2

• Explain to pupils that in Part 2 isosceles triangles are arranged differently, to produce a windmill pattern. Pupils use what they have learned in Part 1 and what they know about angles to work out the angles in the triangles. They then consider the polygon that can be drawn by joining the exterior points of the windmill. They work out the angles and try to generalise what [rules] they have discovered for any such shape.
• Encourage pupils to communicate effectively, both orally and in writing.
  [Engage with the pupils to establish their mathematical reasoning and understanding. Discussion can help pupils explore and correct misunderstandings.]
• Encourage them to explain and justify their mathematical working, especially when the pupils are confident writers.
• Pupils should be free to decide whether they want to draw diagrams to help them with this task. Many will want to do so, particularly in the early stage of trying to visualize a problem. However, drawing is not compulsory. Some pupils may need reassurance that drawings are not necessary. Encourage able pupils to calculate angles without drawing.
• Part 2 (b) is more challenging than the earlier work on the task. Pupils may need further guidance and discussion.
• Part 2 (c) invites exploration of angles in a complicated context. Even very able pupils may want to revert to drawing or using cut out triangles to help them. Once pupils have understood the activity they should be encouraged to use visualisation rather than drawing.
• Some able pupils should be encouraged, where appropriate, to work algebraically.
• When a pupil has a complete and well presented numerical calculation for part 2 (b), you can suggest that they substitute letters for numbers in their working. This will help with Part 2 (c).

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Maths diseases

This task involves pupils exploring percentage increases and decreases. The task has three parts.

Part 1: pupils explore the effect of decreasing numbers by 20%.
Part 2: pupils investigate the percentage increase that will undo a percentage decrease of 20%.
Part 3: pupils investigate the effect of combining two percentage increases.

The context of the task is to investigate tablets that may cure an imaginary disease. Teachers who trialled the task recommended emphasising the imaginary nature of the disease when introducing the task, remaining sensitive to different pupils’ needs.

You may adapt the task for the whole class to work with the ideas at an appropriate level. For example some pupils might begin by looking at percentage decreases of 50% before considering decreases of 20%. You might decide that some pupils should not proceed beyond part 2.

You may choose to ask pupils to record their answers to part 1 and the first question in part 2 on the sheets provided, then to use separate paper for subsequent work. Alternatively you may choose to ask pupils to use separate paper throughout.

This task is likely to take between one and two hours. However the time spent working is likely to vary considerably according to the ability level of the pupil. Although pupils may complete the task in one sitting, revisiting the task after a break of hours or days allows time for reflection and is to be advised.

Suggestions for optional open-ended extension work are also provided.

Prior knowledge

Pupils working on this task need to have prior knowledge of:

• understanding unit fractions and fractions that are several parts of the whole
• finding percentages of whole numbers

• recognising the equivalence between fractions and percentages.

Assessment focus

• Understand that 'percentage' means the 'number of parts per 100' and that it can be used for comparisons; find percentages of whole number quantities, using a calculator where appropriate (Ma2, 2f).
• Organise work and refine ways of recording (Ma2, 1f).
• Use notation diagrams and symbols correctly within a given problem (Ma2, 1g).
• Present and interpret solutions in the context of a problem (Ma2, 1h).
• Communicate mathematically, including the use of precise mathematical language (Ma2, 1i).
• Understand and investigate general statements (Ma2, 1j).
• Recognise and represent simple number relationships (Ma2, 4d).

Resources

Each pupil or group of pupils will need:

• a copy of the three photocopiable pupil sheets
• paper and pencil
• a calculator.

For the teacher:

• solutions and what to look for

What to do

This is not a traditional test and is designed to be collaborative.

• Tell the pupils that they will be investigating the effects of an imaginary disease called
  minus-twentypercentitis.
• Tell them that the task has more than one part.
• Tell them that they may use a calculator throughout if they wish.

• Give the pupils the part 1 sheet.
• Explain that in the first part of the task they will be exploring what happens when the answers to calculations are decreased by 20%.
• Use the example provided to discuss the disease minus-twentypercentitis with the pupils.
• Engage with the pupils to establish their mathematical reasoning and understanding.

• Before giving pupils the part 2 sheet, you may wish to give them the opportunity to predict a cure for minus-twentypercentitis, ie the percentage increase that will undo a 20% decrease.
• Tell the pupils that in the second part of the task they will be investigating a possible cure for minus-twentypercentitis.
• If pupils are not already recording their answers on separate paper, encourage them to do so from question (b) onwards, and for the remainder of the task.

• Give the pupils the part 3 sheet.
• Tell the pupils that in the third part of the task they will be investigating what happens when two tablets are taken instead of a single plus25% tablet.
• Discuss the example provided, ensuring that pupils understand that the two tablets are taken one after the other (ie the effects of each tablet are independent).
• Encourage the pupils to explain and justify their mathematical working.

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Number slide