FSMQ Teacher Delivery Guide - Numerical Methods

Teacher Delivery Guide:Numerical methods

Content / Learners should be able to / Notes
Numerical Methods (NM)
Solving equations / NM1 / solve equations approximately by considering the change of sign.
NM2 / use a simple iterative method to solve equations approximately.
NM3 / recognise when these numerical methods may fail.
Gradients of tangents / NM4 / use a chord to estimate gradient of a tangent to a curve at a point.
NM5 / recognise how to improve an estimate for the gradient of a curve at a point.
Area under a curve / NM6 / use rectangular strips to estimate the area between a curve and the x-axis.
NM7 / use trapezium rule to estimate the area between a curve and the x-axis. / Formula will be provided.
NM8 / recognise whether an estimate would be an over or under estimate, and understand how to calculate an improved estimate.
Applications of Numerical methods / NM9 / apply numerical methods in context where appropriate. / e.g. determine the velocity from displacement-time curve.

NM1-NM3: Solving Equations

General approach

It is important to stress that numerical methods should only be used once algebraic routes have been tried or considered. The introduction of numerical methods at this level allows learners to be able to investigate problems beyond the range of their current algebraic skills, for example solving cubic or higher order equations which do not factorise..

The link between algebraic and graphical methods should be made clear and students should be encouraged to always start with a sketch of the function. This will help visualise why numerical methods are not always successful and the reasons why failure may happen.

The key conceptin the change of sign method is that when the y-axis values change from negative to positive or the reverse, there will be a root if the curve is continuous.The algorithms are decimal search and interval bisection.

In the decimal searchalgorithm, it is important to look methodically at all numbers of a certain type within an interval, note where a sign change occurs and then repeat or iterate on a magnification of the interval. This is ready done using the table function on scientific calculators, choosing appropriate start and end limits with the step to increment.

Interval Bisection cuts the interval in half and looks for which half has a change of sign. Then bisect that interval and repeat. This technique requires less individual calculations which may be useful depending on the software used and the nature of curve under investigation.

Failure to find the root occurs when there is a repeated root, there are two closely positioned roots within an interval or if the curve is discontinuous.

Simple iterative methods include the notion that and . We can illustrate the concept graphically either using a staircase or a cobweb diagram. These plot against and then trace an initial value either converging to or diverging away from the root. Use the iterative formula. At this level assessment questions would require students to comment on the success or failure of a given iterative formula, however investigating the results from different rearrangements would help emphasis why this algorithm may not always succeed in finding the required root.

The solve function on scientific and graphical calculators uses the Newton-Raphson algorithm. This algorithm is beyond the scope of the FSMQ, but might be an interesting progression for further reading.

Prior knowledge

GCSE (9-1)

4.01 Approximation and Estimation

Students need to be able to express their answers to an appropriate level of accuracy, rounded correctly. They should be able to identify the upper and lower bounds of their estimation.

6.02 Algebraic formula

Students need to be confident with substituting numerical values into formulae and expressions, and with manipulating algebraic expressions to change the subject of a formula

6.03 Algebraic equations

GCSE maths focuses mainly on linear and quadratic equations, but does introduce the concept of finding approximate solutions using graphs and systematic iterations.

6.06 Sequences

Simple iterations require a knowledge of subscript notation for term to term rules.

7.01 Graphs of equations and functions

Solving equations using numerical methods requires students to link functions with their graphical representation.

Misconceptions

Students often miss the importance of a sketch to identify appropriate starting points for finding roots. Whilst it is common for exam questions to provide initial intervals, students would benefit from identifying these intervals themselves using graphing software.

It is important for students to recognise the limitations of these numerical methods. The failure to identify a root within a required interval should not lead students to immediately conclude that no root is present within that interval.

Change of sign failure may be due to there being two roots within an interval resulting in no change of sign seen. Iterative methods may converge of an alternate root, it will also only find one of the roots if two or more are close together.

Progression

A Level Maths

1.02 Algebra and functions

One of the reasons for these numerical methods to fail in more complex functions is the presence of asymptotes. The another reason for these methods to fail to find a required root is where two or more roots are close together. Curve sketching helps identify if either of these situations applies in specific cases.

1.09 Numerical Methods

The stage 2 content for A Level Maths develops change of sign and iterative methods and also introduces Newton-Raphson.

A Level Further Maths

Alternate numerical methods may be covered in the optional strands of different Further Maths specifications.

NM4-NM5: Gradients of Tangents

General approach

This section could be used to develop the GCSE work on estimating gradients as a stepping stone towards formal differentiation, section CA1-CA7.

No specific method is defined, either a one sided (forward or backwards difference) or a central difference may be used as appropriate

Prior knowledge

GCSE (9-1)

4.01 Approximation and Estimation

Students need to be able to express their answers to an appropriate level of accuracy, rounded correctly.

7.02 Straight line graphs

This section revisits the general equation of a straight line and the formula to find the gradient of a line.

7.04 Interpreting graphs

This section introduces the concept of graphical representation of rates of change and estimating the gradient of real life graphs.

Misconceptions

Students often make mistakes with signs when substituting values to find the gradient between two points. Reference to the curve to ensure the direction of slope makes sense.

Over specification of decimal accuracy should be avoided when undertaking any estimation work.

Progression

A Level Maths

1.02 Algebra and functions

One of the reasons for these numerical methods to fail in more complex functions is taking points on either side of an asymptotes.

1.07 Differentiation

The numerical methods for estimating gradient can be developed further to greater degrees of accuracy, leading to an understanding of general differentiation from first principles

A Level Further Maths

Numerical differentiation is one area that may be developed in Further Maths optional strands.

Area under a curve

General approach

This section could be used to develop the GCSE work on estimating areas under graphs before studying formal analytical integration, section CA8-C13.

Prior knowledge

GCSE (9-1)

4.01 Approximation and Estimation

Students need to be able to express their answers to an appropriate level of accuracy, rounded correctly.

6.02 Algebraic formula

Students need to be confident substituting values into given formulae to find y values, and using the trapezium rule.

7.04 Interpreting graphs

This section introduces the concept of graphical representation of real life scenarios and interpreting the area under the curve in context.

10.03 Area calculations

A general understanding of calculating area of plane shapes is needed for this topic. This progresses the idea of finding the area of a single trapezium, using algebraic manipulation to develop the trapezium rule for the sum of a series of strips under a curve.

Misconceptions

Students need to refer their calculations back to the original curve to determine when the method has produced an over or underestimate.

Care is needed not to over specify the quoted answer, especially when the values of have been rounded.

Progression

A Level Maths

1.09 Numerical Methods

The stage 2 content for A Level Maths develops numerical integration and its use to solve problems for which analytical techniques are not yet understood.

A Level Further Maths

Numerical integration is one area that may be developed in Further Maths optional strands.

Thinking Contextually

The section on numerical methods develops students’ understanding of the theories behind the corresponding analytical techniques and provides a method for finding estimated solutions for problems beyond their current analytical knowledge.

The section brings together algebra and coordinate geometry, so the use of graphing software will help students visualise the theory and demonstrate the process of improving estimates.

Resources

Title / Organisation / Description / Ref
Intermediate Value Theorem. / Geogebra / Interactive computer aid. The interval can be moved to enclose the root and the sign change can be observed. Could be used as part of a starter or plenary or in a computing session linked with this topic. / NM1
Intermediate Value Theorem / Desmos / Visual aid. The interval can be moved close to the root. / NM1
Continuity - Intermediate Value Theorem Example 2 / Brian Veitch / Quick 5 minute video - simply explained with good visual aids. Could be used as starter or plenary. / NM1
MEI Numerical Methods Coursework – Decimal Search 1 / Diana Timofte / 13 minute video on the decimal search method, using Excel. Clear presentation. Lots of Excel advice. / NM1
Change of Sign: Decimal Search. / Geogebra / Demonstration of decimal search. / NM1
Interval Bisection. / Geogebra / Bisection demonstrated through a diagram; also gives table of iterates. Good plenary. / NM1
Interval Bisection Method / Wolfram Alpha / Online tool which produces Bisection iterates. User inputs f(x) and interval. / NM1
The intermediate value theorem / College of Arts and Sciences / Nice document with problems for students to solve. / NM1
Solve Me! / Nrich / Activity for students to do, finding the roots of a polynomial. / NM1
A cubic has one real root—can we find an approximation to it? / Underground Maths / Cambridge Colleges Examination for Entrance Scholarships and Exhibitions, Calculus (Group 3), 1952, Q2.
Interesting link between curve sketching, turning points and roots. / NM1
Iterations – Roots between / Mr Barton Maths / Tarsia puzzle matching functions with a range for the root. This puzzle was designed for legacy Core 3 Numerical methods, it will need slight changes to remove functions involving ln and trig functions (radians). / NM1
Overview of Change of sign with illustration. / Arizona State University / General result with specific example and diagram. / NM1 and NM3
Root Hunter / Nrich / Includes a further stretch activity. (Note that the trig example solution gives the answer in radians, which is not included in the FSMQ). / NM1 and NM3
PowerPoint which includes Decimal Search / Exeter College / PowerPoint which includes the decimal search method and fixed point iteration. The set of slides also includes Newton Raphson, beyond the scope of FSMQ. Very nice and comprehensive document with good diagrams. First 15 slides appropriate for FSMQ, covering many aspects of root finding numerical methods. Good resource for flipped learning. / NM1, NM2 and NM3
Iteration / CIMT / Includes some real world scenarios linked with the maths of root finding. Covers the FSMQ content and then onto Newton Raphson (A Level). / NM1, NM2 and NM3
Solve Me! / Nrich / Investigate solution of equation / NM1 and NM2
Iteration / Exam Solutions / 8 minute demonstration of find roots using rearrangement and iteration. First of two videos. / NM2
Iteration: How it works / Exam Solutions / 12 minute demonstration of how the rearrangement and iteration method works using graphs. Second of two videos. / NM2
Numerical methods: Solving by iteration / Geogebra / Demonstration of cobweb/staircase diagrams. / NM2
Rearrange into iterative form / Mr Barton Maths / Tarsia puzzle matching function with / NM2
Decimal Search Failure / Geogebra / Nice graphical example clearly showing a function where the decimal search may appear to indicate a solution when in fact none exists. / NM3
Zooming in / Underground Maths / Group activity to explore what happens to the graph as students change the scale of the axis. / NM4
Numerical Differentiation / Southampton University / Notes and examples on forward, backwards and central difference approximations for gradient. / NM4
Forward, Backward, and Central Difference Method / Alexander Maltagliati / Youtube video demonstration of each method (13 min). / NM4 and NM5
Is the Serpentine Lake really 40 acres? / Underground Maths / Investigation, using numerical methods, to evaluate the claim that the Serpentine covers an area of 40 acres. / NM6 and NM8
Approximating Area under a curve with rectangles / University of Notre Dame / Investigate using left, right or midpoint approximations. / NM6 and NM8
The trapezium rule / Geogebra / Demonstration of trapezium rule, using a slider to vary the number of trapezia and investigate the effect on the accuracy of the result. / NM7
Trapezium rule for radical function / Geogebra / Demonstration of trapezium rule, using a slider to vary the number of trapezia and investigate the effect on the accuracy of the result. / NM7
Underneath the arches / Underground Maths / Students are asked to decide whether using the trapezium rule would give an under or over estimate of the area between the curve and the x-axis for the sketch graphs from gradient match. Rather than focusing on the numerical aspect of using the trapezium rule, in this resource students are only given sketch graphs and are prompted to think about geometric features of the curves. / NM8