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

FSMQ Level 3 (legacy) Working with algebraic and graphical techniques scheme of work

This FSMQ requires a total of 60 guided learning hours that could be used in a variety of ways, such as 2 hours per week for 30 weeks, 4 hours per week for 15 weeks, or 5 hours per week for 12 weeks.  Before starting this course learners should be able to:

  • plot by hand accurate graphs of paired variable data and linear and simple quadratic functions (including the type y = ax2 + bx + c) in all 4 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 using the formula \frac {- b \pm \sqrt{b^2 - 4ac}}{2a} (must be memorised).

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, but note that you will also need to allow time for students to complete an AQA Coursework Portfolio. The Coursework Portfolio requirements are listed on page 56 of the AQA Advanced FSMQ specification and also on page 101 of the AS Use of Mathematics specification. The assignments below provide some examples of the sort of work your students could include in their portfolios, but if possible you should use work that is more relevant to their other studies or interests.. 

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.

Topic area

Content

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

 

Linear functions

(3 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 (etc.) in real world terms.

Use error bounds to consider a range of possible functions to model data.

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 three 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, how to 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 – for students to match.

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

 

Topic area

Content

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

Quadratic functions
(7 hours)

Draw graphs of quadratic functions of the form:

· y = ax2 + bx + c

· y = (rx s )(x – t)

· y = m(x + n)2 + p

relating the shape, orientation and position of the graph to the constants and relating 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 and 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 that passes 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 for portfolio requirements or form the basis for an assignment.

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 3 cards showing a quadratic graph, the corresponding function and its completed square form.

Gradients of curves, maxima and minima

(3 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.

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. 

 

Topic area

Content

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

Power functions and inverse functions

(3 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 growth and decay.

Draw graphs of natural logarithmic functions of the form y = a ln(bx) and understand the logarithmic function as the inverse of the exponential function.

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

to convert equations involving powers to logarithmic form and solve them (using both base 10 and natural logarithms).

Growth and decay   
Presentation using 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

(4 hours)

Use:
· translation of y = f(x) parallel to the y axis to give y = f(x) + a

· translation of y = f(x) parallel to the x axis to give y = f(x + a)

· stretch of y = f(x)  parallel to the y axis to give y = af(x)

· stretch of y = f(x)  parallel to the x axis to give y = f(ax)

 

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

(8 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, wavelength, period and phase shift 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 resources
The links below go to pages from which you can download the resources, some recently revised.

Linearising data

(6 hours)

Determine parameters of non-linear laws by plotting appropriate linear graphs, for example:

· y = ax^2 + b
by plotting y against x^2

· y = \frac{a}{x} + b
by plotting y against \frac{1}{x}

· y = ax^3 + b
by plotting y against x^3

· y = a~\textup{sin}(x) + b
by plotting y against \textup{sin}(x)

· y = ax^b and y = a^x
using base 10 or natural logarithms

Log graphs - earthquakes  
Examples (involving earthquakes and planetary motion) that can be used to introduce log graphs.  Ideas of experiments and other situations that 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 & log version)  
Activity that uses natural or base 10 logarithms to find an equation connecting the energy released by an earthquake and its Richter value. 

Revision (6 hours)

 

 

 

 

Page last updated on 25 January 2012