Nuffield Mathematics teaching resources are for use in secondary and further education

FSMQ Level 3 (pilot) Algebra scheme of work

 

This Advanced (Level 3) unit is NOT a free-standing qualification in the AQA pilot and no separate FSMQ certificate is available for it.

Also note that the AQA assessment of this unit is by examination only.

Before starting this unit learners should be able to:

  • plot by hand accurate graphs of paired variable data & linear & simple quadratic functions (including the type y = ax2 + bx + c) in all four quadrants
  • recognise and predict the general shapes of graphs of direct proportion, linear and quadratic functions (including the type y = kx2 + c)
  • fit linear functions to model data (using gradient and intercept)
  • rearrange basic algebraic expressions by collecting like terms, expanding brackets and extracting common factors
  • solve basic equations by exact methods including pairs of linear simultaneous equations
  • use power notation (including positive and negative integers and fractions)
  • solve quadratic equations by factorising and by using at least one of: graphics calculator, the formula \frac {- b \pm \sqrt{b^2 - 4ac}}{2a} (must be memorised), completing the square

A suggested work scheme for this unit is given below.  It includes some revision of the above as well as the other topics and methods to be covered. 

The following techniques should be introduced as soon as possible and used throughout the course:

  • using a calculator effectively and efficiently, including the use of memory and function facilities (recording the working as well as the result)
  • doing calculations without a calculator using written methods and mental techniques
  • graph plotting using computer software or a graphical calculator and using trace and zoom facilities to find significant features such as turning points and points of intersection
  • checking calculations using estimation, inverse operations and different methods.

Note that the AQA assessment of this core unit is by examination only and you should disregard any references to Coursework Portfolio requirements in the assignments listed below which have not yet been updated.  These have been included for possible use as classroom activities but will not form part of the AQA assessment of this core unit.

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.
(originally written for the 'Working with algebraic and graphical techniques' FSMQ)

 

Linear functions

(4 hours)

 

Revise the main features of graphs of direct proportional (y = mx) and linear (y = mx + c) functions.  Fit such functions to  real data using gradients and intercepts.

Understand whether it is appropriate or not to use a particular function to model data by consideration of intercepts, long term behaviour, and so on in real world terms.

Solve linear simultaneous equations using graphical and algebraic methods.

Linear graphs  
Presentation and activity to introduce linear graphs.

Graphs of functions in Excel  
This activity shows students how to draw graphs of algebraic functions in Excel.

Interactive graphs  
Uses interactive spreadsheet graphs to introduce the shape and main features of proportional, linear, quadratic and power graphs. (Can be split into 3 separate parts.)

Graphic calculators  
Presentation introducing students to the CASIO fx-7400G PLUS calculator.

Using the CASIO fx-7400G PLUS   
Notes on how to use this calculator - includes how to draw the graph of a function, investigate how well a model fits data and how to find a model.

Graphic calculator – Equations   
A variety of equations to provide practice in using a graphic calculator. 

Car bonnet   
Students are asked to consider linear approximations to temperature data.

Match linear functions and graphs   
Twelve sets of cards, each containing a linear graph, its equation and the real situation it represents (students match).

Simultaneous equations on a graphic calculator 
Instructions for using the CASIO fx-7400G PLUS calculator to solve simultaneous equations.

Quadratic functions
(8 hours)

Draw graphs of quadratic functions of the form:

· y = ax2 + bx + c

· y = (rx s )(x – t)

· y = m(x + n)2 + p

Relate the shape, orientation and position of the graph to the constants and zeros of the function f(x) to roots of the equation f(x) = 0. 

Fit quadratic functions to real data.

Revise solving quadratic equations by:
· factorising
· using the formula \frac {- b \pm \sqrt{b^2 - 4ac}}{2a}

Rearrange any quadratic function into the forms y = ax2 + bx + c and y = a(x + b)2 + c

Find maximum and minimum points of quadratics by completing the square.

Test run  
Students interpret a speed-time graph and fit both linear and quadratic models.  The performance data is also given in an Excel spreadsheet for comparison with models.

Model the path of a golf ball   
Students consider linear and quadratic models for the path of a golf ball.

Broadband A, B, C   
Instructions showing how to use Excel, a graphic calculator & algebra to find a quadratic model for the growth in broadband connections in recent years.

Presentation shows the algebra version.

Two on a line and three on a parabola   
Spreadsheets giving a linear or quadratic function passing through particular points.

Factor cards   
Nearly 100 pairs of cards showing a wide variety of quadratic expressions and their factors. Pairing will give students practice in expanding brackets or factorising.

Water flow  
Includes data about the velocity of water as it flows along an open channel and sample examination question. Data could also be used to give practice in finding a quadratic model.

Completing the square   
Presentation shows how to complete the square and use this form to sketch graphs.  Card-matching activity using a selection from 24 sets each of three cards showing a quadratic graph, the corresponding function, and its completed square form.

 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

Gradients of curves, maxima and minima

(5 hours)

 

Calculate and understand gradient at a point on a graph using tangents drawn by hand (and also using zoom and trace facilities on a graphic calculator or computer if possible).

Use and understand the correct units for rates of change.

Interpret and understand gradients in terms of their physical significance.

Identify trends of changing gradients and their significance both for known functions and curves drawn to fit data.

Find local maximum and minimum points and understand their significance in terms of the real situation.

Tin can   
Students design a tin can, using algebraic and graphical techniques.  Optional use of the internet.

Maximum and minimum problems  
Presentation and practice questions using a spreadsheet or graphic calculator to solve problems involving maximum and minimum values. 

Power functions and inverse functions

(5 hours)

Draw graphs of functions of powers of x including y = kx^n where n is a positive integer, y = kx^-^1 = \frac{k}{x}  , y = kx^-^2 = \frac{k}{x^2} ,  and y = kx^\frac{1}{2} = \sqrt x
Learn the general shape and position of such functions.

Find the graph of an inverse function using reflection in the line y = x.

Solve polynomial equations of the form axn = b .

Interactive graphs    
See above.

Growth and decay

(8 hours)

Draw graphs of exponential functions of the form y = ka^m^x and y = ke^m^x (m positive or negative) and understand ideas of exponential growth and decay.

Draw graphs of natural logarithmic functions of the form y = a~\textup{ln}(bx) and understand the logarithmic function as the inverse of the exponential function.  Understand how logarithms can be used to represent numbers.

Solve exponential equations of the form  A~\textup{exp}(mx + c) = k)

Learn and use the laws of logarithms

· \textup{log}(ab) = \textup{log}~a + \textup{log}~b

· \textup{log}(\frac{a}{b}) = \textup{log}~a - \textup{log}~b

· \textup{log}(a^n) = n~ \textup{log}~a

Convert equations involving powers to logarithmic form,
for instance y = ka^m^x  gives \textup{log}~y = \textup{log}~k + mx\textup{log}~a .

Use natural logarithms to solve equations such as a^x = b .

Growth and decay    
Presentation that uses compound interest and radioactive decay to introduce exponential growth and decay.

Population growth   
Students use a given exponential function to model population data, then consider predictions made by the model.

Calculator table   
Students use the calculator’s table function to complete tables for population models then draw and use the corresponding graphs.

Ozone hole    
Data concerning depletion of ozone levels and the increase in the area of the Antarctic ozone hole over the last twenty years. Students investigate possible linear, quadratic and exponential models. Optional use of spreadsheet.

Climate prediction A and B 
Students use an Excel spreadsheet and/or graphic calculator to find polynomial functions to model temperature change and compare with exponential models.

Cup of coffee  
Data Sheet gives the amount of caffeine remaining in the bodies of a group of people at intervals of 1 hour after they have drunk a cup of coffee or cola.  Students are asked to model the data
(exponential and linear functions).

 

Topic area

Content

Nuffield resources
The links below go to pages from which you can download the resources, some recently revised.

Transformations of graphs

(6 hours)

Use the following to transform graphs of basic functions:

· translation of y = \textup{f}(x)  by vector \begin{bmatrix} 0 \\ a \end{bmatrix} to give y = \textup{f}(x)+ a

· translation of y = \textup{f}(x) by vector \begin{bmatrix} - a \\ 0 \end{bmatrix} to give y = \textup{f}(x + a)

· stretch of y = \textup{f}(x) scale factor a, invariant line x = 0 to give y = a\textup{f}(x)

· stretch of y = \textup{f}(x) scale factor \frac{1}{a} , invariant line y = 0 to give y = \textup{f}(ax)

Describe geometric transformations fully.

Use transformations to fit a function to data.

Sea defence wall (assignment) 
Two versions of an assignment in which students find functions to model the outline of a sea defence wall. The first version encourages students to work independently, the second is more structured for less able students.

 

Trigonometric functions

(10 hours)

Draw graphs of 

· y = A~\textup{sin}(mx + c)

· y = A~\textup{cos}(mx + c)

Learn the general shape and position of trigonometric functions and use the terms amplitude, frequency and period correctly.

Fit trigonometric functions to real data.

Solve trigonometric equations of the form y = A~\textup{sin}(mx + c) = k and y = A~\textup{cos}(mx + c) = k .

Coughs and sneezes  
Includes data about the way in which an outbreak of the common cold spreads. Students are asked to model the data using trigonometric and polynomial functions.

SARS A and B (assignments)  
Data set giving the number of deaths from SARS. Students choose, draw and evaluate functions to model the data.

Sunrise and sunset times (assignment)  
Students find and evaluate trigonometric functions to model how the amount of daylight varies with the day of the year.  Includes data for Adelaide, Brisbane and London.

Tides (assignment)  
Data set giving the water depth each hour during a day. Students choose, draw and evaluate functions to model the data.

 

Topic area

Content

Nuffield resource

Linearising data

(6 hours)

Determine parameters of non-linear laws by plotting appropriate linear graphs in applications of the cases below:

· y = ax2 + b by plotting y against x2

· y = axb and y = ax using natural logarithms

Earthquakes - Log graphs  
Examples involving earthquakes and planetary motion which can be used to introduce log graphs. Ideas of experiments and other situations which can be used for practice. Includes logs to base 10.

Gas guzzlers  
Slide presentation and activity in which students use a log graph to find an exponential function to model real data.

Smoke strata  
Includes data about the height of smoke layers due to a fire in a tall building and sample examination question. Data could also be used to give practice in linearising data.

Earthquakes (ln version only)   
Activity that uses natural logarithms to find an equation connecting the energy released by an earthquake and its Richter value. 

Revision (8 hours)

Revise topics. Work through revision questions and practice papers.

Discuss the data sheet - make up and work through questions based on it.

 

 

 

Page last updated on 25 January 2012