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Decide what maths is 'at the top of the tree' for this topic - think about conceptual understanding and what you want students to be able to do.  

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Decide what maths is 'at the top of the tree' for this topic - think about conceptual understanding and what you want students to be able to do.  

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Consider what gaps students might have - procedural understanding 

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Consider what gaps students might have - procedural understanding 

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Plan a lesson for your chosen area. If your area is large, pick a topic within it. 

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Plan a lesson for your chosen area. If your area is large, pick a topic within it. 

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The subject content of this specification matches that set out in the Department for Education's Mathematics GCSE subject content and assessment objectives document. This content is common to all exam boards.

The content has been organised into broad topic areas and given a reference as follows:

Number references start with N

Algebra references start with A

Ratio, proportion and rates of change references start with R

Geometry and measures references start with G

Probability references start with P

Statistics references start with S.

All content can be assessed on any of the three question papers. As such, some questions will draw together elements of maths from different topic areas.

The weighting of the topic areas has been prescribed by Ofqual and is common to all exam boards. The table below shows the approximate weightings of the topic areas for the overall tier of assessment, not for each individual question paper.

Topic Area

Foundation Tier (%)

Higher Tier (%)

Number

25

15

Algebra

20

30

Ratio

25

20

Geometry

15

20

Probability and statistics (combined)

15

15

The subject content, aims and learning outcomes, and assessment objectives sections of this specification set out the knowledge, skills and understanding common to all GCSE Mathematics exams.

Within this specification, the assessment will reflect the key concepts of the subject as articulated in the subject content and assessment objectives document.

In line with the requirements set by the Department for Education, the expectation is that:

all students will develop confidence and competence with the content identified in the "basic foundation content" column

all students will be assessed on the content identified by the "basic foundation content" and "additional foundation content" columns; more highly attaining students will develop confidence and competence with all of this content

only the more highly attaining students will be assessed on the content identified in the "higher content" column. The highest attaining students will develop confidence and competence with this content.

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

The content in the "basic foundation content" column and "additional foundation content" column can be assessed on Foundation tier question papers.

All content can be assessed on Higher tier question papers.

Notes are added to exemplify some of the specification references.

In addition to this subject content, students should be able to recall, select and apply mathematical formulae. See the Appendix for a list of the DfE prescribed formulae.

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The subject content of this specification matches that set out in the Department for Education's Mathematics GCSE subject content and assessment objectives document. This content is common to all exam boards.

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AQA Maths revision past exam papers for the new 1-9 GCSE Specification.

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Core concept 6.2 covers how the formulae for perimeter, area and volume are derived and connected, and the importance of reasoning mathematically to solve a range of problems.

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Pupils work out how much material is needed to make bunting to go round a garden.

Suitability

National Curriculum levels 5 to 8

Time

45 minutes to 1 hour

Resources

Paper, ruler and calculator

Representing

Choose a method to determine the amount of material.

Analysing

Work out how many flags should be made and how much material is required.

Communicating and reflecting

Communicate their findings effectively and check that the solution fits the problem.

You can help pupils to understand the task, for example, with a short video on bunting at a street party (eg: http://www.youtube.com/watch?v=wVYjCa77CAw ) with comments:

Bunting is the word for a series of flags, usually triangular, that you often see on occasions like street parties.

You are asked to work out how much material is needed to make bunting to go round someone's garden. You are given some facts and two steps to follow.

The task requires percentages, tessellations and areas and perimeters.

During the work, the following probing questions may be helpful:

Are you confident that your method will work out how much material Kim needs?

Are there different ways you could plan your lay-out to get the most flags out of each metre of material? Is yours the best way? How do you know?

Is there any other information that Kim needs from you?

Teacher notes about possible tessellations are given in the Annex at the end.

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Pupils work out how much material is needed to make bunting to go round a garden. Suitability National Curriculum levels 5 to 8 Time 45 minutes to 1 hour Resources Paper, ruler and calculator Representing Choose a method to determine the amount of material.

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Pupils use their knowledge of area of circles and rectangles to solve a problem.

Suitability

National Curriculum levels 6 to 8

Time

About 1 hour

Resources

Calculator, paper (may request squared, graph or plain), pair of compasses, a ruler

Representing

Break the problem down into smaller steps.

Analysing

Use logical reasoning, and make calculations.

Interpreting and evaluating

Consider appropriateness and accuracy.

Communicating and reflecting

Communicate their findings effectively.

You might set the scene by showing the slides on a whiteboard. If asked, clarify that the thickness of the pastry when re-rolled should be the same as originally; don't volunteer this information since it can form part of the assessment.

This task looks at a practical issue - the making of pies. You are asked to calculate the maximum number of pies Anna can make from a rectangle of pastry; note she has to cut whole circles for the pies.

You are given the dimensions of the pastry and are told Anna can roll the pastry, then re-roll the left over once only.

The task assesses geometric understanding, with a focus on circles.

During the task, the following probing questions may be helpful:

Can Anna use all of the pastry in the first rolling? Why not?

She wants to make as many pies as possible. What should she think about when rolling out the leftovers?

When Anna uses the leftover pastry, what size rectangle should she make? Why?

How certain are you that the number you have found is the maximum possible?

The following values may be helpful; they are given to two decimal places to help check pupils' rounding skills.

Total area per pie = (25π = 78.54 cm ²) + (9π = 28.27 cm ²) = (34π = 106.81 cm ²)

Assuming 12 pies cut from fist rectangle, remaining area = 518.23 cm ²

Theoretical maximum number of pies: 16 (1800 ÷ 34π = 16.85)

Actual maximum number of pies: 15

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Pupils use their knowledge of area of circles and rectangles to solve a problem. Suitability National Curriculum levels 6 to 8 Time About 1 hour Resources Calculator, paper (may request squared, graph or plain), pair of compasses, a ruler Representing Break the problem down into smaller steps.

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Pupils work out how adventurers can maximise the area in which they can dig for gold given a fixed perimeter.

Suitability

National Curriculum levels 4 to 6 (can be extended up to level 8)

Time

30 minutes to 1 hour

Resources

Ruler, pencil, calculator and 1 cm squared paper. It may also be helpful to make available some graph paper.

Representing

Draw various rectangles with a given perimeters.

Analysing

Find areas of rectangles.

Interpreting and evaluating

Consider different rectangles while trying to maximise the area. Find that a square is the best. Go on to find that, if the adventurers work together, they can get a bigger area each.

Communicating

Explain how they know this is the best arrangement and that working together is best.

Check that pupils understand the context, for example with questions such as:

How do you think the adventurers would mark out their plot of land?

How should they share a plot that two of them had marked out?

The adventurers need to understand your thinking, so show your findings in an organised way.

Pupils can tackle this task in different ways, but they might be expected to:

draw rectangles and find their areas

collect results in a table or ordered list

find a pattern in their data

The work could be extended to:

finding patterns in their results and finding an nth term

drawing a graph to clarify their results

finding the areas of other shapes

This would take the content to National Curriculum levels 6 - 8.

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Pupils work out how adventurers can maximise the area in which they can dig for gold given a fixed perimeter. Suitability National Curriculum levels 4 to 6 (can be extended up to level 8) Time 30 minutes to 1 hour Resources Ruler, pencil, calculator and 1 cm squared paper.

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What can you say about these two shapes?

What is the area of each one? What is the perimeter of each one?

What can you say about the shapes below?

Can you draw a shape in which the area is numerically equal to its perimeter? And another?
Can you draw a shape in which the perimeter is numerically twice the area?
Can you draw a shape in which the area is numerically twice the perimeter?
Can you make the area of your shape go up but the perimeter go down?
Can you make the perimeter of your shape go up but the area go down?

Can you draw some shapes that have the same area but different perimeters?
Can you draw some shapes that have the same perimeter but different areas?

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What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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I have started working on finding the area of 2D shapes with year 9 and I wanted them to see the connections with the area of a rectangle. Firstly because they are most confident with that shape but mostly because when given a formula they don't remember the considerations that need to be made. For example; if they learn area of a triangle is 1/2 x b x h they don't remember that you can't just take 2 of the lengths as 'b' and 'h' unless they are the two lengths connected by a right angle.

Show a class an equilateral triangle with sides 6cm and if they calculate 1/2 x 6 x 6 then its time to reinforce the idea of a triangle being half a rectangle.

I plan to show my class the following images and ask them to find which shapes are half the rectangle, which are more and which are less.

Then i plan to get students to try and derive the areas of triangles, parallelograms, kites and ultimately trapeziums from the rectangle. I have used a rectangle that i cut and reform to show this too.

I have found with compound area involving rectangles that i give the same problem but split it differently each time. I ask the class to find the area of each and if they notice they are all the same shape each time then see if they can prove it 3 different ways.
shaded shapes
UPDATE!

I was pleased that students imagined pieces moving or cut the shapes up in order to see if the shaded part was more, less or equal to a half.

We then derived the areas of other 2D shapes from the area of a rectangle.

UPDATE!

I made another similar task

Students have to group the shapes so that the shapes in each group have the same area as one another.

UPDATE:

I've been rethinking some questions for 2D shapes. I want Ss to consider whether they should split or form a rectangle to find the area of compound shapes

combining two shapes for the same area

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I have started working on finding the area of 2D shapes with year 9 and I wanted them to see the connections with the area of a rectangle. Firstly because they are most confident with that shape but mostly because when given a formula they don't remember the considerations that need to be made.