Download differentiation integration - School of Mathematics and Statistics

DIFFERENTIATION
INTEGRATION
Coline Diver
Mark McGuiness
Paraparaumu College
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
Integration
19 of these attended the NCEA Exam
The number of students in the cohort is 210
The number of students taking any form of Mathematics
or Statistics at Level 3 is 89 (4 students did both MS
and MC)
Progression
Y11
Y12
Y13
Calculus
Math with
Calculus
Mathematics
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 externally
Derivatives of power, exponential, logarithmic (base e) and
trig functions
Optimisation
Equations of normals (tangents at Level 2)
Maxima, minima and points of inflection
Related rates of change
Derivatives of parametric functions
Chain, product and quotient rules
Properties of graphs (limits, differentiability, continuity,
concavity)
No longer assessed
• Differentiation from first principals
• Implicit differentiation
Achieved Level Exemplar (2013)
In 2013 a student achieved the standard by:
In Question 1
2
y
=
tan
x
+1
Correctly differentiating
(
)
Finding the gradient of the tangent to the function
f ( x ) = ln ( 3 x − e x ) at the point where x = 0
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 quotient
rule for an incorrectly written function,
BUT demonstrates “limited knowledge
of differentiation techniques”
Excellence Level Exemplar (2013)
From Question 1
Formulae for both curved surface area
and volume of cylinder on Formula Sheet.
Identifies need to maximise volume having
written it in terms of r.
Communicated solution, and used units
Some 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
Integration
6 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
π − e2 x dx
∫(
)
• Correctly integrating, but then giving an incorrect answer
to:
In Question 2:
• Using the Trapezium Rule to find
given in a table
In Question 3:
• Calculating an area:
∫ f ( x ) dx using values
2
1
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
Otherwise
mathematics not statistics in 2016 MATH132
•
•
mathematics or statistics in 2015
•
•
all first year MATH
mathematics or statistics in 2016
•
•
MATH141 Calculus
ENGR121
NCEA differentiation, integration, trig or
complex numbers: direct entry to MATH142
Calculus
VUW Calculus
•
MATH132 Intro to Mathematical Thinking
•
•
MATH141 Calculus 1A
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•
main start point for math majors
MATH142 Calculus 1B
•
•
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basic ideas of calculus
start point if good calculus background
ENGR121 Engineering Maths Foundations
ENGR122 Engineering Maths with Calculus
MATH177 Probability & Decision Modelling
MATH141
sample: differentiation
rate of change
slope 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 Mathematics
repackages MATH100
• ENGR121 Engineering mathematics foundations
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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
•
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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?
•
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lifestyle can be a challenge
lack of engagement:
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bimodal grade distribution
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~10% failure: no evidence of any work
good students do well reliably
others teeter near failure
various help is available