DIFFERENTIATIONINTEGRATION

Coline Diver Paraparaumu CollegeMark McGuiness Victoria University

Content

• Why students take the differentiation and integration

• Progression and selection process• Student numbers / proportions• Teaching resources• Differentiation• Integration

Student Consideration

• Requirement for tertiary study, particularly engineering, physical sciences and flying/piloting

• Enjoyment of the “challenge” of the subject

• “Don’t know what else to take”

• “Want to be dux”

Teaching / Learning Resources

• Use of Graphic calculators is encouraged throughout the Mathematics Department (as early as possible)

• For both Differentiation and Integration these are usually used as a check

Calculus at Paraparaumu College• AS 91575 Trigonometric Methods

• AS 91578 Differentiation Methods

• AS 91577 Algebra of Complex Numbers

• AS 91579 Integration Methods

• AS 91573 Geometry of Conic Sections

A total of 24 Credits

Student Numbers / Proportion• Y13 Mathematics class:

3 of the 12 students elected to take Differentiation

• Y13 Calculus class:21 students were enrolled for both Differentiation and Integration19 of these attended the NCEA ExamThe number of students in the cohort is 210The number of students taking any form of Mathematics or Statistics at Level 3 is 89 (4 students did both MS and MC)

Progression

Y13Y12Y11

Mathematics

Math with Calculus

Calculus

Mathematics

Math with Statistics Statistics

Entry: Y12 Mathematics with Calculus

• 14+ credits at Level 1

• At least 12 credits from externally assessed standards

• Algebra required – ideally with at least a merit grade

• At teacher / HOD discretion

Entry: Y13 Calculus

• 14+ credits at Level 2

• Algebra (Merit grade) and Calculus are both required and ideally Graphing

• Students who do not meet this requirement can do Y13 Mathematics which gives them an option to do the Differentiation standard later in the year

• At HOD’s discretion

Differentiation

6 credits, assessed externallyDerivatives of power, exponential, logarithmic (base e) and trig functionsOptimisationEquations of normals (tangents at Level 2)Maxima, minima and points of inflectionRelated rates of changeDerivatives of parametric functionsChain, product and quotient rulesProperties of graphs (limits, differentiability, continuity, concavity)

No longer assessed• Differentiation from first principals

• Implicit differentiation

Achieved Level Exemplar (2013)

( )2tan 1y x= +

In 2013 a student achieved the standard by:

In Question 1� Correctly differentiating

� Finding the gradient of the tangent to the function

( ) ( )ln 3 at the point where 0xf x x e x= − =

In Question 2� Identifying 3 out of 5 “conditions” from the graph of a

function

AND in Question 3 answered another question demonstrating “limited knowledge of differentiation techniques”

Almost correct use of the quotientrule for an incorrectly written function,BUT demonstrates “limited knowledgeof differentiation techniques”

Excellence Level Exemplar (2013)From Question 1

Formulae for both curved surface areaand volume of cylinder on Formula Sheet.Identifies need to maximise volume havingwritten it in terms of r.Communicated solution, and used unitsSome knowledge of measurement required.

From Question 2

From Question 3

In 2013 a student could get Excellence if they scored a total of 21 – 24 for the three questions in the assessment

This means they had to get an Excellence grade for at least two of the questions and a Merit grade for the other

This score spread was the same for both Differentiation and Integration

Integration6 credits, assessed externally

• Integrating power, exponential (base e), trig and rational functions

• Reverse chain rule, trig formulae• Rates of change problems• Areas under or between graphs of functions (by integration)• Finding areas using numerical methods (rectangle, trapezium,

Simpson’s rule)• Differential equations of the forms y’=f(x) or y’’=f(x) for the

above functions, or where variables are separable (y’=ky) in applications such as growth and decay, inflation, Newton’s Law of Cooling and similar situations

No longer assessed• Volumes of revolution

NOTE:Areas under or between graphs has been moved from Level 2 to Level 3

Achieved Level Exemplar (2013)In 2013 a student achieved the standard by:In Question 1• Integrating

• Correctly integrating, but then giving an incorrect answer to:

( )2xe dxπ −∫

In Question 2:• Using the Trapezium Rule to find using values

given in a table

In Question 3:• Calculating an area:

( )2

1f x dx∫

Rationale for Achieved Grade

• Question 1: “the candidate has shown the ability to integrate some functions”

• Question 2: “the candidate has been able to use the Trapezium Rule”

• Question 3: “the candidate has shown the ability to integrate some functions”

Excellence Level Exemplar (2013)

Question 3

Examination (Differentiation and Integration)

• The exam is 3 hours

• Many students do three standards in this time (for 17 credits)

• Students have a comprehensive formula sheet

Differences between Senior Secondary and first year Tertiary• Small class size

• Positive relationships formed

• Availability of teacher

• Unscheduled “classes” (before school and most lunch times)

• Course content is similar

Finally• Thanks to WGC and VUW for the Calculus Scholarship

programme they coordinated

• MAX Math 153

Differentiation (3.6)Integration (3.7)Coline Diver, Paraparaumu College

Mark McGuinness, Victoria University

Entry and calculus• 16 NCEA level 3 AS credits

• mathematics not statistics in 2016

• MATH141 Calculus �

• mathematics or statistics in 2015

• all first year MATH �

• mathematics or statistics in 2016

• ENGR121 �

• NCEA differentiation, integration, trig or complex numbers: direct entry to MATH142

Calculus �

Otherwise MATH132

VUW Calculus• MATH132 Intro to Mathematical Thinking

• basic ideas of calculus

• MATH141 Calculus 1A

• main start point for math majors

• MATH142 Calculus 1B

• start point if good calculus background

• ENGR121 Engineering Maths Foundations

• ENGR122 Engineering Maths with Calculus

• MATH177 Probability & Decision Modelling

MATH141 sample: differentiation

rate of changeslope of f(x)

limit of secant line slopes

Lecture 5 of MATH141, week 2

What we do in the shadows… MATH141, week 5

MATH141 sample: integration

Final Exam:

Engineering Mathematicsrepackages MATH100• ENGR121 Engineering mathematics foundations

• serves all ENGR students; will serve COMP in the future

• has introduction to differentiation

• plus function, graphs, logic, probability intro

• ENGR122 Engineering mathematics with calculus

• Electronic and Computer System Engineering

• more differentiation, plus integration, vectors, matrix

• ENGR123 Engineering mathematics with logic and statistics

• Network or Software Engineering;COMP in future

Why offer ENGR math?

• recent growth in ENGR students

• passing math courses was a bottleneck

• existing math courses were not tailored to ENGR needs

• $

ENGR121 sample material

Differentiate

?

2014: a Pilot Year for ENGR math

• pass rates 80%

• more A’s than before

• more students say it’s their favourite course

• labs need to be better integrated with math

Success at VUW?• lifestyle can be a challenge

• lack of engagement:

• ~10% failure: no evidence of any work

• bimodal grade distribution

• good students do well reliably

• others teeter near failure

• various help is available