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

# FSMQ Level 2 Algebra and graphs scheme of work

Before starting this Higher (Level 2) FSMQ students should be able to:

• calculate with numbers (including large and very small numbers) by carrying out operations in the correct order, working both by hand and with a calculator
• calculate with fractions and decimals (including expressing one quantity as a fraction of another and converting between decimals and fractions)
• calculate with percentages
• round values to the nearest whole number, 10, 100, 1000, $\frac{1}{10}$ (0.1), $\frac{1}{100}$ (0.01) or to an appropriate number of decimal places or significant figures.
• substitute into formulae expressed in words and symbols

A suggested work scheme showing topic areas and methods to be covered is given below. This recommends 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. There is plenty of scope for varying the order and time allocation as many of the mathematical techniques can be introduced/revised in several different contexts. Although the topics are listed separately, it would often be beneficial to use a variety of skills within the same piece of work.

Some techniques should be introduced as soon as possible and used throughout the course.  These include:

• using a (scientific or graphic) calculator effectively and efficiently, including the use of memory facilities, functions and standard form for large and small values
• doing calculations without a calculator using written methods and mental techniques
• recording and presenting data in tables using an appropriate degree of accuracy and correct units (grouping data where appropriate)
• graph plotting by hand and using either computer software or a graphic calculator
• checking calculations using estimation, inverse operations and alternative methods.

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

 Topic area Content Nuffield resource The links below go to pages from which you can download the resources, some recently revised. Plot graphs of real data (5 hours) Plot accurate graphs of data pairs by hand and using either a graphic calculator or function plotting software, ensuring that the graphs are correctly scaled and labelled. Average data if necessary (mean, mode, median for discrete data). Use coordinates in all four quadrants where appropriate. Fit lines by eye (both straight and curved, as appropriate). Find the intercepts of linear and non-linear graphs with axes and where appropriate understand their physical significance. Use trace facilities with graphic calculators or function plotting software where possible. Identify errors in data by inspection of the data set and by graphical means. Recognise that measurements expressed to a given unit can have a maximum error of half a unit; using notation such as 300±50 to express errors. Road test    Use data from a road test on a sports car for practice in drawing and interpreting graphs. Optional use of spreadsheet. Errors  (Use the first part only)  Slide presentation showing errors in measurements and how errors accumulate in calculations involving measurements.  Accompanying notes and worksheets. Matching graphs and scenarios   Twelve pairs of cards for students to match.  One card in each pair shows a graph and the other gives a description of the real situation that the graph represents.  Slide presentation to aid discussion (same graphs with titles and labels). Linear graphs (3 hours) Calculate in appropriate units and understand the physical significance of the gradients of linear graphs.  Consider the intercepts of linear graphs with axes and where appropriate understand their physical significance. Hire a coach  Introduces the concepts of gradient and intercept for linear graphs using Excel. Standard Form (2 hourS) Convert between standard form and ordinary numbers. Use a calculator to perform calculations with numbers expressed in standard form. Large and small    Slide presentation and examples from real contexts to introduce, practise or revise standard form.

 Topic area Content Nuffield resource The links below go to pages from which you can download the resources, some recently revised. Graphs of functions (6 hours) Use functions to find data pairs: $y&space;=&space;mx&space;+&space;c$ $y&space;=&space;kx^2&space;+&space;c$ $y&space;=&space;ax^2&space;+&space;bx&space;+&space;c$   $y&space;=&space;ax^3&space;+&space;bx^2&space;+&space;cx&space;+&space;d$ $y&space;=&space;\frac&space;{k}{x}$ and functions with other variables such as $P&space;=&space;\frac&space;{k}{V}$ Use tables to display results and where appropriate stages of calculations. Use function notation such as $f&space;\left&space;(&space;x&space;\right&space;)&space;=&space;kx^2&space;+&space;c$ Look for patterns which data-fitting proportional ($y&space;=&space;mx$), linear ($y&space;=&space;mx&space;+&space;c$), and quadratic models of the form $y&space;=&space;kx^2$  have. Plot graphs of functions, using co-ordinates in all four quadrants. Consider the main features of direct and indirect proportional, linear and quadratic models and their differences. Predict the shapes of graphs of direct and inverse proportion, linear and quadratic functions from an algebraic statement. Linear graphs   Slide 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. Spreadsheet graphs   Interactive spreadsheet graphs introducing the shape and main features of proportional, linear, inverse proportional and quadratic graphs. Linear relationships    Example and exercise involving proportionality and other linear relationships in scientific contexts. Quadratic graphs   Slide presentation, notes and exercise on drawing quadratic graphs and using them to solve equations. Substitution into formulae including conversion of units (3 hours) Convert within and between metric and imperial systems, including inches, feet, yards, miles using conversion factors and the use of formulae such as  $L$ = 3.28$l$ for converting metres to feet. Substitute data into other formulae, using BIDMAS to find secondary data including formulae with multiples and fractions of linear terms, powers (positive and negative integers and fractions) and brackets. Hot water tank: Formulae  Slide presentation, notes and exercise. Students learn to substitute values into formulae, and to use a calculator to evaluate expressions. Non-linear graphs   Draw graphs from data and formulae then use them to solve problems in real contexts. Includes slide presentation. Re-arrange algebraic expressions (3 hours) Rearrange algebraic expressions by: i  collecting like terms, for example $na$, $nab$, $na^2$ ii expanding brackets, for example $2\left&space;(&space;x&space;+&space;y&space;\right&space;)$, $a\left&space;(&space;2a&space;+&space;3b&space;\right&space;)$,     $\frac&space;{1}{2}\left&space;(&space;4x^2&space;+&space;6x&space;\right&space;)$ iii extracting common factors from expressions like $\left&space;(&space;4x&space;+&space;6x^2&space;\right&space;)$ and $\left&space;(&space;16y&space;-&space;8xy&space;\right&space;)$ Algebraic expressions  Slide presentation, Information sheet, practice sheet and application to perimeters and areas.

 Topic area Content Nuffield resource The links below go to pages from which you can download the resources, some recently revised. Areas under graphs (4 hours) Calculate estimates of areas under graphs and understand their physical significance (if any) using the formula for the area of a trapezium (and triangle if necessary). Speed and distance Slides to introduce area under a speed-time graph accompanied by examples for students to try. Optional use of spreadsheet/graphic calculator. Rearrange formulae (4 hours) Include examples such as: · to give $u$ if $v&space;=&space;u&space;+&space;10t$ · to give $I$ if $P&space;=&space;1000I^2$  or $P&space;=&space;I^2R$ · to give $P$ if $A&space;=&space;P&space;+&space;\frac&space;{PRT}{100}$ Goldfish bowl: rearrange formulae  Students practise rearranging formulae. There is a worksheet for individual work, or a set of cards for group work to develop this skill. Line and curve fitting  (6 hours) Recognise the main features of the data and graphs of: · direct proportional ( $y&space;=&space;mx$) and linear ($y&space;=&space;mx&space;+&space;c$ ) models and be aware of their differences · quadratic models of the form $y&space;=&space;kx^2$ Recognise the graphs of inverse proportional models Use the gradient and intercept of a straight line fitted to data to find an algebraic statement for it. Understand when 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. Find a function to fit data using substitution of values into a given expression for the model $\left&space;(&space;y&space;=&space;mx&space;+&space;c,&space;~&space;y&space;=&space;kx^2&space;+&space;c,&space;~&space;y&space;=&space;\frac&space;{k}{x}&space;\right&space;)$ 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. Modelling a test drive   Includes data from a road test on a sports car.  Worksheet giving practice in fitting linear and quadratic functions. Optional use of spreadsheet. Boyle’s Law (assignment)  Data sheet giving pressure and volume of a fixed mass of gas, an assignment and sample examination question based on this experimental data. Shoot (assignment)   Students investigate the distance travelled by an object rolling down an inclined plane using graphical and algebraic techniques. Includes a guidance sheet. Experiments   List of seven experiments that generate linear and non-linear data. Students find appropriate algebraic models.

 Topic area Content Nuffield resource The links below go to pages from which you can download the resources, some recently revised. Solve equations  (5 hours) Using graphs · to determine a value of $a$ when you know $f(a)$ · to solve equations involving functions of the form $y&space;=&space;mx&space;+&space;c$,   $y&space;=&space;kx^2&space;+&space;c$,   $y&space;=&space;kx^3$ , $y&space;=&space;\frac{k}{x}$, and other functions useful to students’ other work. Using algebra · to form and solve exactly equations where the unknown appears in only one term including where the unknown is squared, such as $2x^2+&space;14&space;=&space;20&space;\Rightarrow&space;x&space;=&space;\sqrt3&space;~or&space;-&space;\sqrt3$ · to form and solve exactly equations where the unknown appears in two terms, each of the same power such as $4x&space;-&space;2&space;=&space;2x&space;+&space;8,~&space;3x^2&space;+&space;4&space;=&space;20&space;-&space;x^2$ Quadratic factors and graphs  Twelve sets of cards. Each set has a quadratic graph and its equation in factorised form and expanded form. Students match each graph with one or both of its equations. Can also be used later for practice in factorising quadratics. Solve simultaneous equations (5 hours) Find the approximate solution of linear simultaneous equations by finding the point of intersection of two straight lines. Form and solve pairs of linear simultaneous equations by an algebraic method and interpret solutions geometrically when appropriate. Plumbers’ prices  Introduction to the graphical solution of simultaneous equations using Excel in real contexts. Can be used as a follow-up to 'Hire a coach'. Circuit boards (assignment)   Students investigate the cost efficiency of two machines using graphical and algebraic techniques. Includes a guidance sheet. Solve quadratic equations  (6 hours) Form and solve quadratic equations of the form $ax^2&space;+&space;bx&space;+&space;c&space;=&space;0$ by · factorising with $a&space;=&space;1$ · using the formula $x&space;=&space;\frac&space;{-b\pm&space;\sqrt&space;{b^2&space;-&space;4ac}}{2a}$ Factor cards      Nearly 100 pairs of cards showing a wide variety of quadratic expressions and their factors.  Use in a pairing activity to give students practice in expanding brackets or factorising. Quadratic formula  Two examples introducing the quadratic formula and a set of similar equations to solve. Road tunnel     Students solve a problem involving a road tunnel by finding solutions of quadratic equations using a graph drawn in Excel and then using the quadratic formula. Includes a range of other problems. Revision (8 hours) Revise topics and try past papers. Discuss data sheet – make up and try questions based on it.

Page last updated on 20 February 2012