# What you MUST teach in KS4

Discussion in 'Mathematics' started by Subjectknowledge, Nov 6, 2007.

1. Has anyone else looked at the new statutory orders for the national curriculum? The links are
http://curriculum.qca.org.uk/subjects/mathematics/keystag...
http://curriculum.qca.org.uk/subjects/mathematics/keystag...

This tells us what schools are statutorily required to teach from next September.

Below I have reproduced the entire 'range and content' statement for KS4 number and algebra (NB starred items have explanatory notes, which I have also copied below see explanatory notes).

The 'range and content' is incredibly vague, but it does include a statement that students in KS4 are required to be taught 'mathematics that includes' representations and calculations with real numbers(in contrast, at KS3, pupils have to be taught calculations with rational numbers); simultaneous equations, including cases where one is a quadratic and transformations of function etc.

The range and content statement does not say that pupils who finish KS3 at level 3, 4 or 5 are exemptfrom the requirement that they should be taught the higher level material.

I've had a think about representations of real numbers: it obviously means at least two representations have to be taught: Since we can assume they know about rational numbers from KS3, we could represent real numbers as equivalence classes of Cauchy sequences of rational numbers; or maybe via Dedekind cuts; or perhaps as continued fractions; or as infinite decimals. I am undecided about whether to demonstrate that pi and e are transcendental, or whether pupils need to know that, although rational numbers are dense among the reals, they are countably infinite, whereas Cantor's diagonalisation argumnet shows that the reals between 0 and 1 are uncountably infinite.

I look forward to applications: 'If apples cost e^Sqrt(2 pi) pence per kg, how many kg can I but for sqrt(17)^sqrt(17) pence?

Of course, the statement provides freedom to teach other things, like what is on the Foundation level syllabus, but (statutory) 'orders is orders'!

Range and Content

This section outlines the breadth of the subject on which teachers should draw when teaching the key concepts and key processes.

The study of mathematics should enable students to apply their knowledge, skills and understanding to relevant real-world situations*.
The study of mathematics should include:

3.1 Number and algebra
a) real numbers, their properties and their different representations

b) rules of arithmetic* applied to calculations and manipulations with real numbers*, including standard index form and surds

c) proportional reasoning, direct and inverse proportion*, proportional change and exponential growth

d) upper and lower bounds

e) linear, quadratic and other expressions and equations*

f) graphs of exponential and trigonometric functions

g) transformation of functions

h) graphs of simple loci

Explanatory notes
Relevant real-world situations: Mathematical skills are required in many workplace settings, for example understanding relationships between variables in stock control (food processing) or calculating and monitoring quantifiable variables of a hotel?s performance (tourism).

Rules of arithmetic: This includes knowledge of operations and inverse operations and how calculators use precedence. Students should understand that not all calculators use algebraic logic and may give different answers for calculations such as 1 + 2 x 3.

Calculations and manipulations with real numbers: This includes using mental and written methods to make sense of everyday situations such as temperature, altitude, financial statements and transactions.

Proportion: This includes percentages and applying concepts of ratio and proportion to contexts such as value for money, scales, plans and maps, cooking and statistical information (eg 9 out of 10 people prefer?).

Linear, quadratic and other expressions and equations: This includes relationships between solutions found using algebraic or graphical representations and trial and improvement methods. Simultaneous equations should include one linear and one quadratic equation.

2. Has anyone else looked at the new statutory orders for the national curriculum? The links are
http://curriculum.qca.org.uk/subjects/mathematics/keystag...
http://curriculum.qca.org.uk/subjects/mathematics/keystag...

This tells us what schools are statutorily required to teach from next September.

Below I have reproduced the entire 'range and content' statement for KS4 number and algebra (NB starred items have explanatory notes, which I have also copied below see explanatory notes).

The 'range and content' is incredibly vague, but it does include a statement that students in KS4 are required to be taught 'mathematics that includes' representations and calculations with real numbers(in contrast, at KS3, pupils have to be taught calculations with rational numbers); simultaneous equations, including cases where one is a quadratic and transformations of function etc.

The range and content statement does not say that pupils who finish KS3 at level 3, 4 or 5 are exemptfrom the requirement that they should be taught the higher level material.

I've had a think about representations of real numbers: it obviously means at least two representations have to be taught: Since we can assume they know about rational numbers from KS3, we could represent real numbers as equivalence classes of Cauchy sequences of rational numbers; or maybe via Dedekind cuts; or perhaps as continued fractions; or as infinite decimals. I am undecided about whether to demonstrate that pi and e are transcendental, or whether pupils need to know that, although rational numbers are dense among the reals, they are countably infinite, whereas Cantor's diagonalisation argumnet shows that the reals between 0 and 1 are uncountably infinite.

I look forward to applications: 'If apples cost e^Sqrt(2 pi) pence per kg, how many kg can I but for sqrt(17)^sqrt(17) pence?

Of course, the statement provides freedom to teach other things, like what is on the Foundation level syllabus, but (statutory) 'orders is orders'!

Range and Content

This section outlines the breadth of the subject on which teachers should draw when teaching the key concepts and key processes.

The study of mathematics should enable students to apply their knowledge, skills and understanding to relevant real-world situations*.
The study of mathematics should include:

3.1 Number and algebra
a) real numbers, their properties and their different representations

b) rules of arithmetic* applied to calculations and manipulations with real numbers*, including standard index form and surds

c) proportional reasoning, direct and inverse proportion*, proportional change and exponential growth

d) upper and lower bounds

e) linear, quadratic and other expressions and equations*

f) graphs of exponential and trigonometric functions

g) transformation of functions

h) graphs of simple loci

Explanatory notes
Relevant real-world situations: Mathematical skills are required in many workplace settings, for example understanding relationships between variables in stock control (food processing) or calculating and monitoring quantifiable variables of a hotel?s performance (tourism).

Rules of arithmetic: This includes knowledge of operations and inverse operations and how calculators use precedence. Students should understand that not all calculators use algebraic logic and may give different answers for calculations such as 1 + 2 x 3.

Calculations and manipulations with real numbers: This includes using mental and written methods to make sense of everyday situations such as temperature, altitude, financial statements and transactions.

Proportion: This includes percentages and applying concepts of ratio and proportion to contexts such as value for money, scales, plans and maps, cooking and statistical information (eg 9 out of 10 people prefer?).

Linear, quadratic and other expressions and equations: This includes relationships between solutions found using algebraic or graphical representations and trial and improvement methods. Simultaneous equations should include one linear and one quadratic equation.

3. I never read that tosh. I'm just waiting for the new GCSE specs and I'll teach that.