Teaching and learning plan on introducing differential calculus

Teaching and learning plan on introducing differential calculus
Teaching & Learning Plans
Introduction to Calculus
Leaving Certificate Syllabus
The Teaching & Learning Plans
are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have
and also to knowledge which may be necessary in order to support them in
accessing this new topic.
Learning Outcomes outline what a student will be able to do, know and
understand having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/
or Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching
and learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing
the topic.
Lesson Interaction is set out under four sub-headings:
i.
Student Learning Tasks – Teacher Input: This section focuses on possible lines
of inquiry and gives details of the key student tasks and teacher questions which
move the lesson forward.
ii.
Student Activities – Possible Responses: Gives details of possible student
reactions and responses and possible misconceptions students may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to
support and scaffold student learning.
iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate
whether the goals/learning outcomes are being/have been achieved. This
evaluation will inform and direct the teaching and learning activities of the next
class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
2
Teaching & Learning Plans:
Introduction to Calculus
Aims
The aim of this series of lessons is to enable students to:
• understand what is meant by, and the difference between, average and
instantaneous rates of change
• recognise the need for differential calculus in terms of real-world problems
• understand the concept of the derivative of a function
• understand that differentiation (differential calculus) is used to calculate
instantaneous rates of change
• understand how to apply differentiation to calculate instantaneous rates of
change
Prior Knowledge
It is envisaged that, in advance of tackling this Teaching and Learning Plan, the
students will understand and be able to carry out operations in relation to:
• Functions
• Constant rates of change and calculating slopes from graphs
• Pattern analysis
• Describing graphs without formulae
• Distance, speed and time
• Indices
• Limits
• Tangents
Learning Outcomes
Having completed this Teaching and Learning Plan the students will be able to:
• describe rates of change in the real world
• use mathematical language to describe rates of change
• use the slope formula to calculate rates of change of linear functions
• use the slope formula to calculate average rates of change
• recognise that average rate of change and instantaneous rate of change
are identical for linear functions
• recognise that average rate of change and instantaneous rate of change
are not necessarily identical for non-linear functions
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Teaching & Learning Plan: Introduction to Calculus
• Recognise that the slope of the secant line between two points on a curve
is the average rate of change of the curve between those points
• Understand that the average rate of change over shorter intervals around a
point on a curve is a better estimate of the instantaneous rate of change at
that point
• Understand that the instantaneous rate of change is given by the average
rate of change over the shortest possible interval and that this is calculated
using the limit of the average rate of change as the interval approaches
zero.
• Recognise the notation associated with differentiation (e.g. slope, rate of
change, f’(x), dy/dx)
• Understand
• Understand that when you differentiate a function you generate a new
function (the slope function) which gives the slope of the original function
at any point
• Find the derivative by rule
Catering for Learner Diversity
In class, the needs of all students, whatever their level of ability level, are
equally important. In daily classroom teaching, teachers can cater for different
abilities by providing students with different activities and assignments graded
according to levels of difficulty so that students can work on exercises that
match their progress in learning. Less able students, may engage with the
activities in a relatively straightforward way while the more able students
should engage in more open-ended and challenging activities
In interacting with the whole class, teachers can make adjustments to meet
the needs of all of the students. For example, some students may engage with
some of the more challenging questions for example question number 12 in
Section A: Student Activity 1.
Apart from whole-class teaching, teachers can utilise pair and group work to
encourage peer interaction and to facilitate discussion. The use of different
grouping arrangements in these lessons should help ensure that the needs
of all students are met and that students are encouraged to articulate their
mathematics openly and to share their learning.
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Teaching & Learning Plan: Introduction to Calculus
Relationship to Leaving Certificate Syllabus
Sub-Topic
Students
learn
about
Learning outcomes
Students In addition students
working working at OL
at FL
should be able to
should be
able to
5.2 Calculus
Find first and second
derivatives of linear,
quadratic and cubic
functions by rule
Associate derivatives
with slopes and
tangent lines
Apply differentiation
to
•rates of change
•maxima and
minima
•curve sketching
In addition students
working at HL should
be able to
Differentiate linear and
quadratic functions from
first principles
Differentiate the
following functions
•polynomial
•exponential
•trigonometric
•rational powers
•inverse functions
•logarithms
Find the derivatives
of sums, differences,
products, quotients
and compositions of
functions of the above
form
Apply the differentiation
of above functions to
solve problems
Resources Required
Whiteboards, rulers, Geogebra, calculator.
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Teaching & Learning Plan: Introduction to Calculus
Lesson Interaction
Student Learning Tasks: Student Activities: Possible and Expected
Teacher Input
Responses
Teacher’s Supports and
Actions
Checking
Understanding
Teacher Reflections
Section A – Rates of Change
»» We are going to look
at rates of change.
Where have we
looked at rates of
change before?
• The slope of a line
•
• Lines have a constant rate of change
• Rate of change of a line can be found using ‘rise
over run’
»» Encourage students to
recall as much as they
can remember about
rates of change from
their Junior Certificate
learning.
»» What prior
knowledge do
the students
display?
»» Write all the answers
on the board.
• Investigating the change from a table
• Investigating the ‘change of the change’ from a
table
• If the first change is constant the pattern is linear
• If the second change (change of the change) is
constant then the pattern is quadratic.
• If both ‘change columns’ develop in the same
ratio, the pattern is exponential
Note: Remind students
that the money box
problem is a function of
Natural Numbers mapped
to Natural Numbers.
This will be important to
remember later in the
lesson as we can only
analyse a continuous
function using calculus.
• If the rate of change is positive the pattern is
increasing
• If the rate of change is negative the pattern is
decreasing
• The money box problem
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Student Activities: Possible
Teacher Input
and Expected Responses
»» Can you explain in
words what the slope
formula measures?
• How slanted a line is.
• How steep a line is.
Teacher’s Supports and Actions
Checking
Understanding
»» Draw a graph of a general line on the board
going through points (x1, y1) and (x2, y2)
Teacher Reflections
• How the ys are changing
as the xs change
»» Revise with the students that the formula
measures how the ys are changing as the xs
change
»» In pairs, write a
sentence on your
white board to
explain how the ys are
changing as the xs are
changing when the
slope of a line is 3.
• The ys increase by 3 units
every time the xs increase
by 1 unit.
»» D
raw a graph of y = 3x, x ∈ R on the board
as students write their sentence.
»» Can students
verbalise the
rate of change
when the slope
is 3?
»» Circulate to monitor progress. Facilitate
discussion if there are difficulties.
»» Ask a pair of students to call out the answer.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
»» In pairs, write a sentence
on your white board to
explain how the ys are
changing as the xs are
changing when the slope
of a line is -0.5.
Teacher’s Supports and Actions
Student Activities:
Possible and Expected
Responses
• The ys decrease by 0.5 »» Draw a graph of y = 4 - 0.5x on
units every time the xs
the board as students write their
increase by 1 unit.
sentence.
Checking Understanding
Teacher Reflections
»» Can students verbalise
the rate of change when
m = -0.5?
»» Circulate to monitor progress.
Facilitate discussion if there are
difficulties.
»» Ask a pair of students to call out the
answer.
»» Let’s give some context
to the xs and ys. Can
anyone remember what
the formula for speed is
in terms of distance and
time?
© Project Maths Development Team 2013
•
»» Write the formula on the board.
Note: The accurate formula for speed is:
There is no need to mention the
accurate formula here as students will
discover this formula themselves on
Student Activity 5.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
»» Do the following
•
question on your
whiteboards. A woman
drives along a 40km
straight stretch of
motorway in America
as part of her journey
along route 66. She puts
on her cruise control to
drive this section at a
constant speed. It takes
the woman 30 minutes
to drive to the end of this
stretch of motorway. At
what speed did she travel
along this road?
Teacher’s Supports and
Actions
Checking Understanding
»» Write the question on the
board.
»» Can students work with
the speed formula?
»» Observe what students
are writing. Assist them as
required.
»» Do they understand the
units of measurement
km/min?
»» Ask a student to come to
the board and write out the
answer.
»» Let’s represent this
journey on a graph.
Which variable is
dependent? Distance or
time?
• The distance depends on the
time.
»» Ask a student for the answer.
»» Which variable is
independent?
• Time, because that is going to
happen anyway.
»» Ask a student for the answer.
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Teacher Reflections
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible Teacher’s Supports and Actions
and Expected Responses
Checking Understanding
»» Which variable do we
• The independent variable »» Ask a student for the answer.
traditionally put on the
x-axis?
»» Now draw a graph to
represent the journey.
• Students draw a graph
»» Ask a student to draw their graph
to represent the journey on the
board.
Teacher Reflections
»» Do the students know
how to set up their axes?
»» Can students draw the
graph based on the
information given?
»» Can we relate the
formula for speed,
• They are both fractions.
• Distance is on the y-axis
to the slope formula,
• Time is on the x-axis
?
»» Can we use the slope
formula to find the
speed?
• Yes
»» Write on the board
»» Can students see that
slope and speed are the
same formula in this
example?
»» Help students understand this on
the graph drawn on the board.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Supports and
Actions
»» We can see that speed is
• How the distance changes as time
a rate of change in this
changes.
example. In the slope
formula we measure how
the ys change as the xs
change. What rate of
change are we measuring
when we find speed?
Checking Understanding
»» Write this on the board: »» Can students verbalise the
"Speed measures how
changing quantities in speed?
the distance changes as
time changes."
Teacher Reflections
»» We will now look at
other examples of rates
of change. Working in
pairs, complete Section
A: Student Activity 1.
Take five minutes to read
it first without a pen in
your hand.
»» After 5 minutes ask the
students to complete the
table.
© Project Maths Development Team 2013
Students complete table.
Rates of Change
1 y's change as the x's
change
2 Distance travelled changes
as time changes
3 Height of water changes as
time changes
4 Length of metal rod
changes as temperature
changes
5 Number of bacteria
changes with time
6 Production costs change
with respect to the
quality of the product
manufactured
7 Height of a flower changes
with respect to time
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Independent Dependent
Variable
Variable
x
y
t
d
t
h
r
l
t
n
q
p
t
h
»» Distribute Section A:
Student Activity 1.
»» Observe what students
are writing. Assist them
as required.
»» Ask students for their
answers.
»» Do students understand the
concept of dependent and
independent variables?
»» Can students make the
connection between slope,
rate of change and the
examples in the activity?
»» Can students verbalise the
rates of change?
»» Can students think of another
example of a rate of change?
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
»» We have looked at rates
of change from first year.
Rates of change are all
around us and are very
important.
»» Offer some more examples
of rates of change like
“10c per text” or “85c per
minute for a phone call to a
landline”
»» Do students appreciate that
they are surrounded by rates
of change?
»» Now, let’s look at some
• Students complete the
of the vocabulary we
Word Bank.
use to describe rates
Word Bank
of change. Working in
slope
pairs, complete Section
rate of change
A: Student Activity 2.
increasing
Again, discuss the activity
fast
for five minutes first
steep
without pens in your
slower
hands and then fill in the
level
Word Bank.
no change
»» Distribute Section A: Activity »» Can students verbalise their
2
reasoning for applying
different rates of change to
»» Circulate to monitor
different parts of the graph?
progress. Facilitate discussion
if there are difficulties.
»» Are student using the correct
terms to describe rates of
»» By asking different groups
change?
create a class Word Bank on
a poster for the wall.
constant
decreasing
the rate of decrease slows down
Teacher Reflections
Note: Discuss and expand any
misconceptions regarding rates
of change here.
Note: If “the rate of decrease
slows down” arises as an
observation, this could be
expanded to informally discuss
the underlying concept of the
second derivative as the rate at
which the slope is changing. If
this arises keep the discussion
very informal.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Student Activities: Possible
Teacher Input
and Expected Responses
Teacher’s Supports and Actions
»» Now we are going
to look at Section A:
Student Activity 3 –
Part 1.
»» Distribute Section A: Activity 3 –
Part 1.
»» Describe the rate of
change in Question 1.
• Height changes as time
changes
• Students sketch two graphs.
»» Working in pairs,
work through this
activity. Take five
minutes to read it first
without a pen in your
hand.
»» After 5 minutes
ask the students to
commence writing.
• It takes container B longer
to fill because its radius is
wider than the radius in
container A.
It takes the inverted cone 8
seconds to fill as the volume
of a cone is one third the
volume of a cylinder.
Checking Understanding
Teacher Reflections
Note: The volume of water in both
»» Do students understand how to
containers increases at the same rate.
describe this rate of change?
»» Give students time to discuss what »» Do students understand that
is happening in the containers.
the rate of change of the
height of water, as a cylinder
»» Circulate to monitor progress.
fills, is a linear graph whereas
Facilitate discussion but allow
the rate of change of the
time for students to arrive at
height, as the inverted cone
what is happening.
fills, is curved?
»» Encourage students to see the link »» Do students understand that
between the radius not changing
that cylinder B fills slower
in a cylinder and the constant rate
because container B has a
of change in the height of the
bigger radius?
water level with time.
»» Can students represent these
»» Ask one of the pairs of students
situations on a graph?
to put their graphs on the board.
»» Use GeoGebra files to show the
graphs of the containers filling.
»» What vocabulary from
your Word Bank can
you use to describe
how the containers
are filling?
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»» Ask a number of students to
describe the graphs in words
using the vocabulary from their
Word Bank.
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»» Can students describe in words
how the cone is filling?
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Student Activities: Possible and Expected
Teacher Input
Responses
Teacher’s Supports and Checking Understanding
Actions
»» Now we are going
to look at Section A:
Student Activity 3 –
Part 2.
»» Distribute Section A:
Student Activity 3 –
Part 2.
»» Working in pairs,
work through this
activity.
• Students find that the slopes of [AB] and
[OA] are both m = 2. Students conclude
that the slope at any point on [OA] is 2
as a line has a constant rate of change.
Students then conclude that the slope at
the point C is 2 because it is one of the
points that lie on the line [OA] that has a
constant rate of change 2.
• Students find that the slope of [AB] is m =
0.5. Students conclude that the height of
water in the cylinder changes at a rate of
0.5 cm/sec. Students see that the height
of water changes at a rate of 0.5 cm/sec
at any time between 0 and 24 seconds
because a line has a constant rate of
change. Students then conclude that the
rate of change at the point C is 0.5 cm/sec
as C lies on the line whose constant rate
of change is 0.5 cm/sec.
»» Observe what
students are writing.
Assist them as
required.
»» Ask students to
explain to the class
their answers and
reasoning to Q1, Q2
and Q3.
Teacher Reflections
»» Can students use the
slope formula?
»» Can students relate the
slope to the rate of
change of the height
of the water level with
time?
»» Can students
understand that the
slope along a straight
line is always the same?
»» Can students
understand that the
slope along the curve is
changing?
• Students find that the slope of [AB] is
m = 1.03 and the slope of [OB] is m = 1.50.
Students conclude that the slopes of [AB]
and [OB] do not help them find the slope
at the point C as a curve does not have a
constant rate of change.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible Teacher’s Supports and
and Expected Responses
Actions
Checking Understanding
»» Recall the example of
finding the constant
speed of the car on cruise
control along a long
stretch of road.
»» R
emind students of the
question if they cannot recall
it.
»» Can students remember this
example?
»» Do you recall the formula •
for speed in terms of
distance and time?
»» Write the formula on the
board.
»» Do they know the general
formula for speed?
Teacher Reflections
Note: Again, the accurate
formula for speed is:
There is no need to mention
the accurate formula here
as students will discover this
formula themselves on Section
A: Student Activity 5.
»» Can you describe the
rate of change ‘speed’
in terms of distance and
time?
• Distance changes as time
changes
»» Write the wording on the
board.
»» Can they verbalise constant
speed as a rate of change in
terms of distance and time?
Note: There is no need to talk
about average or instantaneous
speed at this point. This will
be developed in the following
activities.
»» Let’s now look at Section
A: Student Activity 4.
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»» Distribute Section A: Student
Activity 4.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
»» Working in pairs, work
through this activity.
Take five minutes to read
it to yourselves without a
pen in your hand.
• Students draw a graph.
»» Circulate to monitor
progress. Facilitate
discussion if there are
difficulties.
»» Can students represent the
situation on a graph?
Teacher Reflections
»» Do they understand that the
graph is linear?
»» After 5 minutes ask
students to commence
writing.
• The train passes the two students
at a speed of 120 km/hr.
»» Can students use the speed
formula?
• The train also passes the teacher
at 120 km/hr because the train is
travelling at a constant speed.
»» Do students understand
that the train will pass
the teacher at the same
speed because it passes at a
constant rate of change?
»» Wrap up: We see from
our investigations that if
we know the slope of a
line, there is no difficulty
in getting the slope of
any other point on that
line.
»» Let’s look at another
example of speed. Look
at Section A: Student
Activity 5.
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»» Distribute Section A:
Student Activity 5.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Student Activities: Possible
Teacher Input
and Expected Responses
Teacher’s Supports and Actions
Checking
Understanding
»» Working in pairs
answer the first 3
questions. Take five
minutes to read it to
yourselves first.
• Usain Bolt’s speed for the
race is 10.44 m/s
»» Circulate to monitor progress. Facilitate
discussion if there are difficulties.
• His speed was not 10.44
m/s during the whole race
because we can see from
the graph that he was
slower at the beginning of
the race.
»» Engage students in a classroom
discussion about these three questions.
»» Do students
understand that the
rate of change is not
constant during the
whole race?
»» Write the words ‘Average Speed’ on the
board as a new term.
»» Can they see this
from the graph?
»» Introduce the correct formula for the
speed of a journey:
»» Do they therefore
understand that
10.44 m/s is a
representation of the
average speed for the
race?
• As he is not running at
the same speed during
the whole race, 10.44 m/s
represents his average
speed.
»» Working in pairs,
answer question 4.
• Students draw in a line on
the graph.
Note: Discuss and expand on any
misconceptions regarding these questions.
»» Observe what students are writing.
Assist them as required.
»» Ask a student to draw their graph on
the board.
»» Ask a student for the answer to part (ii)
Teacher Reflections
»» Can students see
that the slope of the
secant line between
the two end points
of this curve is the
same as Usain Bolt’s
average speed?
»» Ask a student to give their answer to
part (iii) and explain their reasoning.
ii m = 10.44
iii The slope of the secant line »» Engage the students in a discussion
is the same as the average
about secants and how they represent
rate of change.
the average rate of change between
two points.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
»» In conclusion, the
slope of the secant line
between two points is
the same as the average
rate of change between
two points.
»» Write the conclusion on the
board.
»» In pairs, have a discussion • Any discussion is good – it
about question 5.
is not necessary for the
students to come up with
the correct strategy as this
will be developed in the
next worksheet.
»» Facilitate and encourage any
discussion and ideas.
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Teacher Reflections
• student answer/response
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Teaching & Learning Plan: Introduction to Calculus
Teacher’s Supports and Actions
Student Learning Tasks: Teacher Student Activities:
Input
Possible and
Expected Responses
»» So far we have investigated a
few different situations where
we see rates of change. The
cylindrical containers and the
passing train had a constant
rate of change whereas the
cone container and Usian Bolt’s
race did not have a constant
rate of change.
»» The situations that had a
constant rate of change were
represented as a line whereas
the situations without a
constant rate of change were
represented as a curve.
Checking
Understanding
Teacher Reflections
»» Write a comparison table on the board. »» Are students able
to recall what they
Filling cylindrical
Filling an invertd cone
have explored about
containers
container
rates of change this
Train passing
Usain Bolt's race
far?
• Linear Graph
• Curved Graph
•
No problem
•
Unable to find the
finding the
exact slope at a single
slope at any
point on the line
single point on •
Best we can do is find
the line
an average rate of
change
»» Do students
recognise that there
is a difficulty in
finding the slope at
a single point on a
curve?
»» On a linear graph we had no
problem finding the slope of
any point on the line whereas
on a curved graph we were
unable to find the exact slope
at a single point on the curve;
the best we could do was find
an average rate of change.
»» Now we are going to look at
Section A: Student Activity 6 –
part 1 to see if it is possible to
get the slope at a single point
on a curve.
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»» Distribute Section A: Student Activity
6 – part 1.
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19
Teaching & Learning Plan: Introduction to Calculus
Student Learning
Student Activities: Possible and Expected
Tasks: Teacher Input Responses
Teacher’s Supports and
Actions
»» Work through this • Students write their answers
Activity in pairs.
1. Victoria’s average speed is 0.5km/min.
Take five minutes 2. Students draw in secants.
to read it to
yourselves first.
»» Circulate to monitor
»» Do students
progress. Facilitate discussion
understand that
if there are difficulties.
the slope of the
secant between two
»» Encourage students to
points is the same
discuss this activity.
as the average rate
of change between
»» Ask a student to put the
two points?
graph on the board.
»» Have students
»» Ask another student to put
recognised that the
the table on the board.
slope of the secant
closest to point A
»» Ask a student to verbally
will be the best
give their answer to
estimate for the rate
question 4 and explain their
of change at the
reasoning.
exact point A?
3. Students fill in the table.
Slope of Secant [AB]
0.60 Average speed between
Slope of Secant [AC]
0.55 Average speed between
A and B = 0.6 km/min
A and C = 0.55 km/min
Slope of Secant [AD]
0.45 Average speed between
A and D = 0.45 km/min
Slope of Secant [AE]
0.40 Average speed between
A and E = 0.4 km/min
Checking
Understanding
Teacher Reflections
»» Ask a number of students to
verbally answer question 5
and explain their reasoning.
Encourage a classroom
discussion on this question.
4. The slope of secant AE will be the best
estimate of the slope at the point A.
5. By looking at the slopes of secants
nearer to the point A. The slope of the
secant closest to point A will be the best
estimate.
© Project Maths Development Team 2013
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20
Teaching & Learning Plan: Introduction to Calculus
Student Learning
Student Activities: Possible and
Tasks: Teacher Input Expected Responses
Teacher’s Supports and Actions
Checking
Understanding
»» Now let’s look at
• Students fill in the table.
Section A: Student
The interval of x values for secant AB= 6
Activity 6 – part 2.
5
»» Observe what students are writing. Assist them
as required.
»» Ask a student to write up the table on the
board.
»» Encourage a class discussion on question 2.
»» Do students
understand
how to find
the interval
of x values?
»» Remind them of their study on limits and how
we can use the limit as the interval approaches
0 to make the interval as close as possible to
zero without actually becoming 0.
»» Introduce students to the variable ‘h’ or ‘∆x’ as
how we describe this interval of x values that
we would like to make very small.
»» Therefore the slope of the secant closest to the
point A will be found using
»» Do the
students
recall their
study of
limits?
The interval of x values for secant AC=
The interval of x values for secant AD=
The interval of x values for secant AE=
The interval of x values for secant AF=
The interval of x values for secant AG=
»» Is there any way
we can make
the interval so
small that it is
practically zero?
3
2
Teacher Reflections
1
0.5
• Students discuss the smallest
possible interval they can think
of.
• Let the interval be the smallest
number in the world.
• Use a limit and let the interval
approach 0.
»» Draw a graph on the board to help explain this,
see Appendix 1.
»» Show the GeoGebra file of Angry Birds to
reinforce this.
© Project Maths Development Team 2013
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21
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
»» We’ve seen how rates of
change appear in lots of
different contexts. Let’s
look at one more such
context.
»» In Angry Birds what is the
aim of the game?
»» How do you do this?
What is your strategy?
»» What shape does the
flight path of the angry
bird make?
Teacher’s Supports and Actions
»» Open up the GeoGebra file Angry-Birdsand-Calculus.ggb. Make sure 'Show
Background' is clicked.
Note: there is a static diagram of the Angry
Birds file in the Appendix of the T&L plan.
• To kill the pigs using the
least number of angry
birds.
• Launch the angry birds
at different angles and
different speeds to hit the
target.
• A quadratic.
»» Fly the angry bird again this time with
'Show Flight Path' clicked and 'Show
Background' unclicked.
• Measure it.
• Construct a secant.
• Measure the slope at the
point.
»» Show the point on the graph by clicking
'Show Point'.
»» Remind students that they faced a similar
problem with Victoria Pendelton.
»» Show the secant on the graph by clicking
on 'Show Secant'.
© Project Maths Development Team 2013
Teacher Reflections
»» Demonstrate the game by flying the angry
bird across the screen using the slider Fly.
»» Demonstrate that the bird’s flight path is
part of a quadratic function by clicking on
'Show Full Quadratic'.
»» How might we calculate
the rate of change of
the angry bird at a given
point during its flight?
Checking
Understanding
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KEY: » next step
»» Do students
recognise
the flight
path as being
quadratic in
shape?
»» Can students
apply a similar
approach to
that introduced
with Victoria
Pendelton
to suggest a
solution?
• student answer/response
22
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and Actions
»» How might we modify our
secant so that it would
provide a better estimate
of the rate of change
of the angry bird at the
point?
• Construct a secant using a
point closer to the point of
interest.
»» Drag the purple point towards
»» Do students understand that
the blue point as a demonstration
the closer the two points
of the idea of a better secant.
on the function are, the
better the secant estimates
»» Show the slope of the secant
the slope at the point of
and average rate of change
interest?
calculation by clicking on 'Show
Secant Workings'.
»» How might we get the
actual rate of change at
the point of interest – as
opposed to an estimate?
• Drag the purple point until
it is directly on top of the
blue point.
»» Drag the purple point over
the blue point such that the
secant disappears and the slope
and average rate of change
calculations are undefined.
»» Does this approach
work for getting the
instantaneous rate of
change? Why doesn’t it
work?
• No.
»» Discuss the secant workings and
• The two points are on top of
highlight division by zero.
each other so we no longer
have a secant and so cannot
get the slope.
• When we go to calculate the
slope we end up dividing by
zero which is undefined.
»» Do students recognise that
the approach breaks down
due to division by zero?
»» We would like to know
what happens when the
purple point approaches
the blue point so that the
two points are ever closer
but never directly on top
of each other. Is there a
tool in mathematics which
allows us to investigate
such an occurrence?
• Limits.
»» Do students understand that
we could use a limit to find
the slope at a point?
© Project Maths Development Team 2013
»» Remind students of various
examples of limits e.g. the
introduction to .
e
www.projectmaths.ie
Checking Understanding
Teacher Reflections
»» Do students understand the
closest the two points can be
is when one sits directly on
top of the other?
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Teaching & Learning Plan: Introduction to Calculus
Teacher’s Supports and Actions
Student Learning
Student Activities:
Tasks: Teacher Input Possible and Expected
Responses
Checking Understanding
Teacher Reflections
»» What limit do we
want to find?
• The limit of the
slope formula
between the
two points as the
distance between
the two points gets
smaller and smaller.
»» Write up a semi-mathematical expression for »» Do students understand
this limit e.g. lim (slope) as distance between
that we are looking at the
points gets smaller.
limit of the slope formula
as the distance between the
two points gets smaller?
»» This expression
is messy. Could
we use some
variables to write
it in a more
mathematical
form?
• Yes, using the slope
formula.
»» Re-write this expression using the standard
slope formula
»» The choice of
the second point
depends on the
location of the
first point. For
this reason, could
we re-write the
denominator
in terms of the
location of the
point of interest?
as distance between points gets smaller.
• Yes, x2 - x1.
»» Suggest that we could use the horizontal
distance between the two points as a
measure of the distance between the two
points.
»» Do students understand
that the horizontal distance
is a good measure of how
close the two points on the
»» Re-write the expression in the following form
function are?
»» Demonstrate that this distance is just x2 - x1
and suggest that we re-name this distance h.
encouraging students to discuss how we
might write the limit (as distance between
points gets smaller) of the expression
mathematically.
© Project Maths Development Team 2013
»» Do students recognise
that this expression is not
written mathematically?
www.projectmaths.ie
»» Can students write the limit
as h → 0?
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24
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Checking Understanding
Actions
»» Demonstrate how y1 may be »» Do students understand
where the new notation has
written as f (a) and how y2
come from?
may be written as f (a+h).
»» Show this representation in
the Geogebra file by clicking
on 'Show Labels'.
»» This is Calculus.
Calculus is the branch
of mathematics that
allows us to calculate
instantaneous rates of
change.
»» Write out the full definition
of the derivative
Teacher Reflections
»» Do students understand that
we are simply looking at the
limit of the standard slope
formula re-written in terms
of a single point?
»» Do students understand
that calculus provides the
instantaneous rate of change
as opposed to the slope
»» Show students the definition
of the secant which only
of the instantaneous rate
provides an estimate?
of change in GeoGebra by
clicking on 'Show Calculus'.
»» Do students understand
that the error on the rate
»» Discuss that the closest
of change provided by the
secant offers a very good
secant can be significant
approximation for the
when carrying out rate-ofinstantaneous rate of
change calculations in the
change but that even such
real world?
a small error can have
consequences for real-world
calculations.
© Project Maths Development Team 2013
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
»» We have seen that
a secant provides
a geometrical
representation of
average rate of
change. There is
also a geometrical
representation of
instantaneous rate
of change. Could you
suggest what that is?
• It’s where the secant only
touches the function at
one point
»» On the Geogebra file, move
the purple point away from
the blue point then bring
them back together. Ask
students if they can identify
the geometrical relationship
which the secant is
approaching?
»» Do students recognise that
the secant is getting closer
to being a tangent to the
curve as the points get closer
together?
»» What do you notice
about the slope of the
tangent to the curve?
• It is the same as the
instantaneous rate of
change at that point
© Project Maths Development Team 2013
• It’s the tangent to the
curve at that point
www.projectmaths.ie
Teacher Reflections
»» Calculate the slope of the
»» Do students understand that
tangent by clicking on 'Show
the slope of the tangent
Tangent'.
to the function is the
instantaneous rate of change
at the point of contact?
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26
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible Teacher’s Supports and Actions
and Expected Responses
Checking Understanding
Section B – Rules of Differentiation
Teacher Reflections
»» We now see the need for
being able to calculate
the instantaneous rate
of change. We also see
that we can calculate
instantaneous rate of
change at a given point
using
»» This approach may be
extended to calculate a
general expression which
tells us the instantaneous
rate of change at any point
(x) along the function.
»» Write up the generalised limit
and discuss the differences
between this limit and
»» Do students understand
the difference between
the two limits presented?
»» Can students calculate
derivative functions by
first principles?
»» Proceed to apply the limit
»» Do students understand
what the derivative
function means?
to calculate derivative
functions of linear
functions and quadratic
functions (Differentiation
by First Principles - H.L.
only).
© Project Maths Development Team 2013
»» Can students correctly
apply the derivative
function to calculate
instantaneous rates of
change?
www.projectmaths.ie
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Teacher
Input
Student Activities:
Possible and Expected
Responses
Teacher’s Supports and
Actions
Teacher Reflections
»» We will now investigate how
to work out the instantaneous
rate of change practically – such
that we can perform these
calculations efficiently. This will
allow us to use calculus to solve
real-world problems.
»» We will do this by looking at how
the slope of various functions
changes from one location to
the next on a function and see if
there is a pattern to this change.
Identifying a pattern in the
slopes would allow us to predict
slope (and instantaneous rate of
change) at any point.
»» We will start by examining linear
functions.
»» For each graph A-D in Section B: • The slope is zero.
»» Distribute Section B :
Student Activity 1, can you write • The slope is the same
Student Activity 1 to
down their slopes at each point
everywhere. We are
students.
given in the table?
looking at a straight line. »» Circulate to check if
• A horizontal line
students understand the
has a slope of zero
task.
everywhere.
»» Circulate to check/reinforce
• The rise is zero –
students’ understanding
therefore the slope is
of slope of linear functions
zero.
through questioning – e.g.
Do you need to calculate
the slope at each point?
Why?
© Project Maths Development Team 2013
www.projectmaths.ie
Checking Understanding
»» Do students understand that
slope means rate of change?
»» Do students understand that
the slope of a horizontal line
is zero?
»» Can students apply their
Junior Cert. understanding
of slopes of linear functions?
»» Do students recognise that
the slope of a linear function
is the same at all points?
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Student Activities: Possible
Teacher Input
and Expected Responses
Teacher’s Supports and Actions
Checking Understanding
»» If I came up with a
new function e.g. q
(x) = 95, could you tell
me about the slope of
that function just by
looking at the form of
the function?
»» Ask students what type of
function is represented by
f (x) = c?
»» Do students recognise
that the function f (x) = c
represents a line parallel to
the x-axis?
• The slope is 95 (incorrect).
• The slope is zero.
• The slope is zero
everywhere.
• That’s the equation of a
horizontal line – therefore
it has zero slope.
»» Suggest to students that they
sketch this function?
Teacher Reflections
»» Can students read the slope
from the function form?
»» Ask students if they could have
predicted the shape of the graph
without sketching the function?
»» If you were given any
horizontal line could
you explain how to
find the slope at any
point.
• It’s just zero.
• Its slope is zero
everywhere.
»» Can you explain why
the rate of change
of a function f (x) = c
should be zero?
• Rate of change means
slope and we’ve seen
slope is zero.
»» Can students generalise
the pattern they have just
discovered?
»» Can students explain in
words how to predict the
slope of a horizontal line?
• The function remains
steady at the same value
all the time so it doesn’t
change.
• The y-value is constant so
there is no change.
© Project Maths Development Team 2013
»» Encourage students to generalise
what they have just investigated.
www.projectmaths.ie
»» Remind students that rate of
»» Do students understand that
change and slope are the same
rate of change is equivalent
thing.
to slope?
»» Encourage students to discuss
what rate of change means.
»» Do students understand that
»» Encourage students to explain
a horizontal line means no
what a function would need to
change in the function and
do to have a zero rate of change.
that this means a zero rate
»» Encourage students to sketch out
of change?
an example of a graph where the
rate of change is zero.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible and Teacher’s Supports and Actions
Expected Responses
»» We have looked at a
really simple example
of a linear function – a
horizontal line. We will
now look at other types
of linear function.
»» For each graph A-F in
Section B – Student
Activity 2, can you write
down its slope at each of
the points indicated in
the table?
Checking Understanding
»» Distribute Section B: Student
Activity 2 to students.
• The slope is given by
• The slope is the same at all
points along each graph –
we’re dealing with a straight
line.
• We only need to calculate the
slope at one point along the
line.
»» Circulate to check that students
can calculate slope.
»» Circulate to check that students
are calculating (or predicting) the
slope at all points on each line.
»» Encourage students to calculate
slope in different ways (formula,
graph etc.).
»» Sketch the graph on the board or
graph it using GeoGebra.
Teacher Reflections
»» Can students apply
to calculate slope?
»» Can students read slope
directly from the graph?
»» Do students understand
that slope is the same at
all points on a straight
line?
»» If you were presented
• Extend the graph
with a straight-line graph • Calculate the slope at any
in the domain -4 < x < 4
point – this is the slope at
and you were asked to
every point since we have a
determine the slope of
straight line
the line when x = 15 how
would you do it?
»» Do students understand
that knowing the
slope at one point on a
straight line means that
they know the slope at
all points on the line?
»» If you were presented
with a new linear
function f (x) = 13.5x + 5
could you describe what
its slope is for all values
of x?
»» Can students read slope
directly from a function
without the need for a
graph?
© Project Maths Development Team 2013
• It’s slope is 13.5
»» Make links to the function form
• Its slope is the same
of the equation of a straight line
everywhere i.e.13.5
y = mx + c.
• Its slope does not change since »» Encourage students to sketch
we have a straight line.
out the function – ask them how
• The slope is just the
they are doing so (using their
co‑efficient of x.
knowledge of slope and intercept).
www.projectmaths.ie
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30
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
»» Can you generalise the
pattern you have just
discovered? Suppose you
were given a function
a(x) = 7x or b(x) = -8x or
g(x) = nx, how would you
calculate its slope?
Student Activities: Possible
and Expected Responses
• Take the co-efficient of x.
Teacher’s Supports and
Actions
Checking Understanding
»» Encourage students to look
back through their table
together and explain how
their slope was related to
the original function each
time.
»» Can students apply their
learning to find the slope of
any linear function f(x) = nx?
»» Suppose we modified the • The same way as before.
»» Encourage students to
previous three functions
sketch the new graph with
so that they now read
the corresponding old graph
• Slope is not affected by
a(x) = 7x + 2 or
and compare their slopes.
the number added on –
b(x) = -8x + 5 or
only by the co-efficient of
f(x) = nx + c, how would
»» Make links with shifting and
x.
you calculate their
scaling of functions.
slopes?
• Each graph is just a shifted
»» Discuss the functional
version of the previous
form of a linear function
graphs so the slope is
(f(x) = mx + c) and what
unaffected.
each term in the function
represents.
»» Make the link between this
approach and that used for
functions which produce
horizontal lines. Does our
current approach work for
those functions? Yes the
previous horizontal line
functions may be written as
y = 0x + c).
© Project Maths Development Team 2013
www.projectmaths.ie
Teacher Reflections
»» Do students understand the
difference between slope
and intercept?
»» Do students understand
what each term in a linear
function represents?
»» Do students understand
that slope is unaffected by
adding a constant amount to
a function?
»» Can students apply their
learning to find the slope
of any linear function
f(x) = nx + c?
»» Can students describe
verbally how to calculate the
slope of a linear function?
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31
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Teacher
Input
Student Activities: Possible Teacher’s Supports and
and Expected Responses
Actions
Checking Understanding
»» Many real-life situations are
modelled by quadratic functions
e.g. a projectile falling through
the air. One way of determining
how quickly a projectile falls is to
calculate its rate of change as it falls.
»» Given the quadratic function
f(x) = x2, if you were told that the
slope of the function at the point
x = -3 is -6, could you say what the
slopes of the function are at all
other points?
Teacher Reflections
• No.
• The slope is not the same
at all points on a curve.
• The slope changes as you
move along a curve.
• Because of symmetry I can
say that the slope at x = 3
is 6.
»» Given that we are dealing with a
• The slope of a tangent to
curve what do we mean by the slope
the curve at that point.
at a point?
»» We have now identified a problem
with rates of change for quadratic
functions – they are not the same
so knowing the rate of change at
one point does not mean we know
the rate of change at any point. Do
you think it’s the same for other
functions that are non-linear?
© Project Maths Development Team 2013
»» Display the function
»» Do students appreciate
f(x) = x2 on the board using
that the rate of change is
a sketch or GeoGebra.
different at all points on a
»» Move around the curve and
curved graph?
discuss slope as you go.
»» Using a sketch or GeoGebra
add a tangent to the curve
at one point. Move the
tangent around the curve
and ask what is happening
to the slope as we do so.
Does the slope remain the
same throughout?
»» Calculate the slope of the
tangent at a couple of
points along the curve.
• Yes.
»» Sketch different examples
• Any function whose graph
of non-linear functions on
is curved will not have
the board.
the same rate of change
everywhere.
www.projectmaths.ie
»» Do students understand that
when we say slope at a point
on a curve what we mean is
the slope of a tangent to the
curve at that point?
»» Do students understand
what a tangent is?
»» Can students calculate the
slope of different tangents
from the graph?
»» Do students understand
that non-linear functions do
not have a constant rate of
change?
32
KEY: » next step
• student answer/response
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Teacher
Input
Student Activities: Possible
and Expected Responses
»» Can we predict what the rate
of change (slope) of a quadratic
function will be at any point?
• Maybe there is a pattern
to how the slopes change.
Teacher’s Supports and
Actions
Teacher Reflections
»» Ask all students to complete
Section B: Student Activity 3(a)
only – for the function f(x) = x2.
»» Distribute Section B:
Student Activity 3 to
students.
»» Explain how to complete the
task
• Measure the slope of the
tangents at each point on
the graph(A-G).
• Fill in these slopes in the
table.
• Complete the Change
column in the table.
• Graph the slopes as a
function of x on the graph
paper provided.
• Investigate if there is a
pattern in the slope values
using the table and the
graph.
• Write down the pattern of
the slopes.
»» Support activity with
GeoGebra file if possible.
© Project Maths Development Team 2013
www.projectmaths.ie
Checking Understanding
»» Circulate to ensure
students understand the
task and can read slopes
from the graph.
»» Encourage students
to use rulers to help
measure the slopes.
»» Remind students that
each tangent is a straight
line so you can measure
its slope at any point on
the tangent.
»» Can students calculate
the slope of a tangent
accurately?
»» Can students complete
the Change column in the
table and recognise what
the values represent in
terms of the pattern of
the slopes?
»» Can students complete
the graph of f '(x) and
recognise the resulting
shape?
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33
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Teacher
Input
Teacher’s Supports and Actions
Student Activities:
Possible and Expected
Responses
Checking Understanding
»» Are the slope values you
measured predictable in any
way?
• Yes – they form a
linear pattern.
• Yes – they all lie on
the same straight
line.
»» Encourage students to discuss the
change column in the table and
the shape of the graph.
»» Do students recognise that the
slopes follow a linear pattern?
»» Do students understand
what a constant change and
a straight-line graph means
in terms of the underlying
pattern?
• Extend my graph and »» Encourage students to discuss
measure the slope of
different ways in which this could
the tangent at that
be done? Are some approaches
point.
better than others? Explain.
• Use my slope pattern »» Demonstrate to students that
to predict what the
their prediction is correct by
slope will be at each
measuring the slope of a tangent
point.
at x = 57 using GeoGebra.
»» Do students recognise that the
pattern they have discovered
allows them to predict the
slope of their function f(x) = x2
at any point along the curve?
»» For the function f(x) = x2, could
you tell me what the slope of
that function is when x = 8? How
about when x = 57? How might
you do it?
»» The slopes of the function
f(x) = x2 themselves form a
pattern. The slopes form a linear
pattern and can be represented
by a linear function. We call
this function the slope function,
the differential function or
simply the derivative. There are
various different notations used
to denote the slope function,
including f '(x) and
.
»» Write up new language and
notation on the board and
encourage students to record it.
»» Link the new language and
notation to the table and graph
»» Stress the fact that we have
created a new function and that
this allows us to calculate slopes
at any point.
Teacher Reflections
»» Do students recognise that the
slopes themselves make up a
function?
»» Do students appreciate that
the slope function tells us
what the slope of our original
function is at any point?
»» Do students understand the
language and associated
notation used to describe the
slope function?
»» This slope function allows us to
calculate the slope of our original
function at a given point.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning
Tasks: Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and Actions
Checking Understanding
»» If you were given a • We would need the slope »» Give different groups of students
different quadratic
functions for each of those
one of the three functions
function – e.g.
functions.
(f(x) = 2x2, f(x) = 3x2 or f(x) = 4x2)
2
f(x) = 2x ,
to complete in the same way as
2
f(x) = 3x or
• We would just multiply
the function f(x) = x2 (see Section
2
f(x) = 4x , could
our previous slope
B: Student Activity 3(b), 3(c) and
you predict what
function by the
3(d)).
their slopes would
co‑efficient of x2.
be when x = -20?
»» Circulate to ensure students are on
What would you
• We could graph our
task.
need? How might
function and measure the
you find this out?
slope of the tangent at
»» Ask groups of students to come
that point.
to the board and describe their
results.
• We could graph a piece
of our function and use
»» Encourage different groups who
the slopes of the tangents
investigated the same function
to work out our slope
to agree/disagree with the results
function.
presented at the board.
»» Encourage groups who
investigated a different function to
listen and to compare and contrast
what’s presented at the board to
their own findings.
»» Do students understand
that the new functions are
different in shape to the
function f(x) = 2x2 and so will
have different slopes?
Teacher Reflections
»» Do students understand that
each new function will have
its own slope function which
is needed if we want to
calculate their slopes?
»» Do students recognise
similarities and differences
between their own work
and that of their peers?
»» Can students use the
different slope functions to
predict what the slope at any
point on a given quadratic
will be?
»» Ask students from the groups who
are not presenting to predict what
the slope of the function being
discussed function will be at a
specific point.
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35
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
»» Can we summarise
• The slope functions
the work the class as
for each quadratic are
a whole has done? Is
different.
there anything common
across all of the functions • The slope functions of
investigated?
each quadratic is a linear
function.
Teacher’s Supports and
Actions
Checking Understanding
»» On one side of the board,
summarise the work
completed in a table
consisting of the functions
investigated and their slope
functions.
»» Do students understand
why each function’s slope
function is different?
»» Write up the main findings
on the board and encourage
students to make a note of
these.
»» When we investigated
• No, because the slope is
linear functions we
not constant across the
discovered that we
function – it changes from
don’t actually need a
place to place.
graph to determine
what the slope is at any
point – we can read the
slope directly from the
function itself. Can we do
the same for a quadratic?
»» Bring student’s attention
to the summary table on
the board and in Section B:
Student Activity 3(e).
Teacher Reflections
»» Do students recognise that
the slope functions of all
our quadratic functions are
linear functions?
»» Do students recognise the
relationship between the
original function and the
slope function?
»» Ask them if they can identify
a pattern which relates
the slope function to the
original function?
»» Ask them to discuss this
pattern and to describe how
they might use the original
function to find the slope
function.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
»» If you were given the
function f(x) = 5x2 could
you write down its slope
function? Could you
predict what the slope of
this function would be
when x = 12? How about
when x = -4? How did
you do it?
• The slope function is
f '(x) = 10x.
»» Encourage students to
discuss their approach.
• We got it by multiplying
the co-efficient of x2 by
the power and reducing
the power by 1.
»» Encourage students to
compare results and discuss
any differences.
»» Can students recognise
the relationship between
a function and its slope
function and use this to
write down the slope
function of f(x) = 5x2?
»» Given the general
quadratic function
f(x) = ax2 could you write
down its slope function?
© Project Maths Development Team 2013
Teacher Reflections
»» Ask students how they might »» Can students use the slope
• The slope of the function
check if their predictions are
function to predict the slope
when x = 12 is 120 and the
correct.
of the function at the points
slope when x = -4 is -40.
given?
»» Using GeoGebra confirm
students’ results by
»» Do students understand
measuring the slope of the
that they can determine the
tangent at each point.
slope function of a quadratic
simply by inspecting the
»» Ask students to write
original function?
down their own quadratic
functions and get their
partner to work out their
slope functions. Swap and
discuss answers.
»»
f '(x) = 2ax
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»» Write a few possible options
on the board and ask
students to determine which
is correct
»» Can students apply their
approach to finding a slope
function to a more general
example?
»» Encourage students to
explain to each other why
they chose their answer
»» Are all students comfortable
using calculus language and
notation?
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible Teacher’s Supports and Actions
and Expected Responses
»» Several real-world situations,
including the volume of
three-dimensional objects
like cuboids, spheres and
cones, are modelled by cubic
functions. Accordingly we
would also like to be able to
calculate the rate of change
at any point on a cubic
function.
Checking Understanding
»» Sketch a cubic function (or graph »» Do students recognise that the
rate of change of a cubic function
it with GeoGebra) so students
changes from one point to the next?
understand what it looks like.
»» Do students recognise that when
they are asked to find the rate
of change at a point on the cubic
function they are faced with the
same problem as they were met
with when investigating quadratic
functions?
»» Given a cubic function,
can you determine how to
calculate its slope at any
point? How might you go
about this?
• See if the slopes of the cubic
function form a pattern?
• Construct a tangent at that
point and measure its slope.
• Measure the slopes of
different tangents along
the cubic function and see if
there’s an obvious pattern.
• Find the slope function for
the cubic function.
»» Distribute Section B: Student Activity
4 to students.
»» Explain to students that the slopes
of the tangents at each point are
already calculated (they are very
difficult to read from the graph).
»» Ask students to complete the tables
for f(x) = x3 and f(x) = 2x3 (Section B:
Student Activity 4(a) and 4(b)), and
to see if they can identify a pattern
among the slopes.
»» Ask students to complete the graphs
for f(x) = x3 and f(x) = 2x3, and to see
if they can identify a pattern in the
graph of the slopes.
»» Circulate to see if students
understand the task.
»» Can students read the slopes at each
point correctly?
»» Can students complete the table and
use it to determine the pattern of
the slopes?
»» Do students recognise from the
shape of the graph that it represents
a quadratic function?
»» Can students link the pattern in the
change columns of the table to the
shape of the graph?
»» Could you calculate what
the slope of the function
f(x) = x3 is at the point on
the function when x = 16?
How would you do it?
• Use the slope function for
f(x) = x3.
• Extend my graph, construct
a tangent at that point and
measure its slope.
»» Remind students that they have
completed a similar task when
investigating quadratics.
»» Ask students to discuss the
advantages/disadvantages of the
different approaches.
»» Do students understand that every
cubic function has its own slope
function and that this may be used
to calculate its slope at any point?
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Teacher Reflections
38
Teaching & Learning Plan: Introduction to Calculus
Student Learning
Student Activities: Possible Teacher’s Supports and Actions
Tasks: Teacher Input and Expected Responses
»» Can we summarise • Each cubic function has
what we’ve
its own slope function
• The slope function of
discovered
all cubic functions is a
about the slope
quadratic function.
function of a cubic
function?
»» Could we write
down the slope
function of a
cubic without the
need for a table
or graph? How
would you do so?
• Multiply the coefficient
by the power and
decrease the power by 1.
»» Given the function • -2.5 x (4)3 = -160
r(x) = -2.5x3, could
(incorrect)
you calculate its
• Find the slope function
slope when x = 4?
and substitute x = 4 into
How would you
it.
do so?
• Plug x = 4 into
r '(x) = -7.5x2 to get
r '(4) = -120
© Project Maths Development Team 2013
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Checking Understanding
»» Write each of the original
»» Do students recognise that the
functions on the board with their
derivative of a cubic function is a
slope function beside them.
quadratic function?
»» Ask students why each cubic
»» Do students appreciate that each
function has a different slope
cubic function has its own slope
function?
function and why this is so?
Teacher Reflections
»» Highlight the summary table for »» Can students identify the general
cubic functions on the board.
rule for finding the slope function of
»» Direct students to their own copy
a cubic function?
of the summary table in Section
B: Student Activity 4(c).
»» Include some additional
functions and their slope
functions in the table if required.
»» Ask students to discuss their
ideas with their partners and to
provide a written explanation of
how they would find the slope
function in their Activity sheet.
»» Demonstrate the correct solution
by constructing the function and
tangent in GeoGebra.
»» Stress how our ability to find the
slope function directly from the
original function reduces our
workload significantly.
»» Encourage students to make up
their own cubic functions, swap
over with their partner and
find the corresponding slope
functions.
»» Do students understand that they
need to find the slope function first?
»» Can students find the slope function
by inspecting the original function.
»» Do students recognise the type of
function they need to create?
»» Can students determine the slope
functions correctly?
»» Can students identify correct and
incorrect answers and explain their
reasoning?
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39
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Teacher’s Supports and Actions
Checking
Understanding
»» We have now discovered • We applied the same
a quick rule for writing
approach when finding
down the slope function
the slope function of a
of any cubic function
quadratic function.
3
f(x) = ax . Does this rule/
approach look familiar in
any way?
»» Ask students to think back to our
previous activity on quadratics.
»» Display the summary tables for both
quadratics and cubics on the board.
»» Do students recognise
that they applied
the same approach
to reading the slope
function from both a
quadratic and cubic
function?
»» Quadratic and cubic
functions are all part
of the same family
of functions known
as polynomials. Let’s
look at a higherorder polynomial
- k(x) = 5x4. Based on
your investigation of
quadratics and cubics so
far, could you suggest
what its slope function
is? How did you do it?
• Graph it and investigate
the pattern of the slopes.
• Do the same as we did
with quadratics and cubics
– multiply by the power
and decrease the power
by 1.
• k '(x) = 20x3.
»» Encourage students to use inspection of
the function to try to get to the slope
function.
»» Confirm their answer by graphing the
derivative of the function using GeoGebra
or by choosing a point on the function
and measuring its slope at that point.
»» Do students
understand what a
polynomial is?
»» Can students extend
what they have
discovered for
quadratics and cubics
to higher-order
polynomials?
»» Linear functions are
also members of the
polynomial family. Can
we find the slope of a
linear function using the
same rule we used for
quadratics and cubics?
• No – it won’t work
(incorrect).
• Yes – it will work for some
linear functions if we
write in the power of x
as x1 but it doesn’t work
for the functions which
represent horizontal lines
(incorrect).
• Yes it works for all
functions if we write
f(x) = 3 as f(x) = 3x0.
»» Distribute Section B: Student Activity 5.
»» Ask students to complete the table by
filling in their results from Section B:
Student Activities 1 – 4.
»» Encourage students to discuss if they
think the same rule applies across all the
polynomial functions we investigated.
»» Highlight that for the rule to work the
polynomial must be written in the form
f(x) = axn.
»» Highlight the presence of this rule in the
maths tables (Page 25).
»» Do students recognise
that x may be written
as x1 and that 1 may
be written as x0?
»» Do students
appreciate that the
same simple rule may
be applied to find the
slope function of any
function of the form
f(x) = axn?
© Project Maths Development Team 2013
Student Activities: Possible
and Expected Responses
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Teacher Reflections
40
Teaching & Learning Plan: Introduction to Calculus
Teacher’s Supports and Actions
Student Learning Tasks: Student Activities:
Teacher Input
Possible and
Expected Responses
Checking Understanding
»» Given any polynomial
could you now
write down its slope
function?
»» Ask students to complete the table
in Section B: Student Activity 5 Q2 &
Q3.
»» Circulate to check that students are
completing the task correctly.
»» Ask students how they would find
the slope of a given function at a
specific point e.g. What is the slope
of f(x) = 0.5x2 at the point on the
function when x = 7?
»» Check if students are re-writing the
functions into the required form
before applying their rule.
»» Encourage students to make up
additional questions, to swap these
over and to try them out.
»» Can students apply their rule
to correctly find the slope
function?
»» Do students recognise when
a function must be re-written
in the correct form before
differentiating by rule?
»» Can students use their
knowledge of indices to re‑write
functions in the correct form?
»» Can students write down the
general rule using the same
mathematical notation as
presented in the maths tables?
»» Do students understand what a
polynomial function is?
»» Write the quadratic f(x) = x2 - x - 6
on the board. Explain how it may be
viewed as a combination of three
separate functions added together.
»» Do students recognise that this
function is formed by adding
three separate functions?
• Yes.
»» We have investigated
various functions of
the form f(x) = axn
and have discovered
a reliable and
quick approach to
determining their
slope functions.
»» We now want to
do the same for
polynomials which are
a combination of a
quadratic, a linear and
a horizontal line.
© Project Maths Development Team 2013
Teacher Reflections
»» Can students apply their
»» Write each part out as a separate
knowledge of differentiation
function.
2
to write down the correct slope
»» g(x) = x
function for each part?
»» h(x) = -x and
»» k(x) = -6 and ask students to write
down the slope function of each one.
»» Write each slope function beside its
original function.
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks: Student Activities: Possible Teacher’s Supports and Actions
Teacher Input
and Expected Responses
Checking Understanding
»» Given a function
f(x) = x2 - x - 6, how
could we determine
how to find its slope
function?
»» Can students complete the
table and graph, and use
them to write down the
correct slope function?
• Graph it.
• Construct tangents to the
function and measure
their slopes.
• Find the slope function
»» Having worked out
of each part and add
the slope function by
them together to get the
examining tangents to
complete slope function.
the function – could
you suggest a more
efficient approach?
• Yes.
»» Can you use this
knowledge to find
the slope functions of
different polynomials?
»» Distribute Section B: Student
Activity 6 to students.
»» Ask students to complete Part
(a) [and (b) if time permits
or give different quadratics
to different groups] in a
similar way to the activities on
quadratic and cubic graphs.
»» Ask students to relate their
slope function to what we have
already written on the board?
Can they spot any similarities?
»» Link the various parts of
the slope function to the
individual parts on the board by
highlighting or using arrows.
»» Ask students to complete
Section B: Student Activity 6(c).
»» Circulate to check student’s
work.
Teacher Reflections
»» Do students recognise that
the slope function is simply
the slope functions of the
individual parts added
together?
»» Do students understand how
to find the slope function of
any co-polynomial function?
»» Can students apply their
knowledge to find the slope
function of various copolynomial functions?
»» Ask students to write down
the rule for a general function
formed by adding different
functions together.
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42
Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
»» Can you complete
the Tarsia in Section
B: Student Activity
7 by matching the
correct function with
its corresponding slope
function?
• Yes.
»» Distribute Section B: Student »» Can students apply their
Activity 7 to students.
knowledge of differentiation
by rule to match up the
»» Ask students to complete
correct functions?
activity in groups.
»» Are students comfortable
»» Circulate to discuss any
with the different notation
problems students are
used?
having with the activity.
»» Do students recognise that
the function must be written
in a specific form if their rule
is to be applied?
Teacher Reflections
»» Can students use their
knowledge of indices to
correctly re-write each
function into the required
form?
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Teaching & Learning Plan: Introduction to Calculus
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Supports and
Actions
Checking Understanding
»» Can you write down a
list of the major learning
outcomes of this lesson?
• Rate of change and slope
are equivalent.
»» Circulate to question
students as to the main
points of the lesson.
»» Can students pick out the
important learning outcomes
from the lesson?
• The slope function of
a function allows us to
predict the slope of the
function at any point.
• The slope function of a
horizontal line is zero.
• The slope function of
a linear function is a
constant value.
»» Can you identify strengths/
weaknesses of each
approach?
• The slope function of a
quadratic function is a
linear function.
»» If you were asked to find the
rate of change of a function
at a particular point on that
function how would you do
it?
• Slope functions may be
determined by examining
the pattern of the slopes
of tangents to the
function.
• Slope functions may be
determined using a simple
rule.
© Project Maths Development Team 2013
»» Why is the slope function
important?
»» Can you describe two
different ways of finding the
slope function?
• The slope function of
a cubic function is a
quadratic function.
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Teacher Reflections
»» For any quadratic function
what shape is its slope
function?
»» Encourage students to share
their summaries.
»» Write up the main points on
the board.
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Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 1
Read the following examples of rates of change.
measures the rate at which the ys change as the
The slope of a line
xs change.
The speed of a vehicle measures the rate at which the distance travelled (d)
changes as time (t) changes.
We might be interested in the rate at which the height of water in a container
(h) changes as time (t) changes.
An engineer might be interested in the rate at which the length of a metal
rod (l) changes as temperature (r) changes.
A microbiologist might be interested in the rate at which the number of
bacteria (n) on a piece of cheese changes with time (t).
An economist could be interested in the rate at which production costs (p)
change with respect to the quantity of the product manufactured (q).
A gardener could be interested in the rate at which the height of a flower
changes with respect to time.
From the above examples, fill in the following table.
Rates of Change
1
ys change as the xs change
2
distance travelled changes as time changes
Independent
Variable
x
Dependant
Variable
y
t
d
3
4
5
6
7
Can you think of another rate of change that we might be interested in?
8
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45
Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 2
Looking at the graph above, discuss in groups how the depth of water
changes with time as Isabelle takes her bath.
Create a word bank of terms that were used during your discussion.
Word Bank
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46
Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 3, part 1
1. (i) Two cylindrical containers, A and B, are filled with water. The volume of
water increases at the same rate in both and the height of both containers is
12cm. Sketch a graph to show the rate at which the height of the water level
changes with time for both containers. Put both containers on one graph.
Container A is full after 6 seconds and container B is full after 24 seconds.
(ii) Why does it take container B longer to fill?
2. (i) Water flows into a vessel in the shape of an inverted cone as shown below.
The volume of water increases at the same rate as for the two cylinders
above. The vessel has the same height and radius as container B. How long
will it take to fill the vessel?
(ii) As water is poured into the vessel, sketch a rough graph to show how the
height of the water level changes with time.
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47
Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 3, part 2
3. Graph of cylinder A being filled
(i)Using
or rise over run, find the slope of [AB]
(ii)Using
or rise over run, find the slope of [OB]
(iii)What is the slope of the line at any point on the line segment [OB]?
Explain your reasoning.
(iv)What is the slope of the line at the point C? Explain your reasoning.
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48
Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 3, part 2
4. Graph of cylinder B being filled
(i)Using
or rise over run, find the slope of [AB]
(ii) At what rate is the height of the water rising at any time between 0 and 24
seconds? Explain your reasoning.
(iii)At what rate is the height of the water rising at point C (after 16 seconds)?
Explain your reasoning.
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49
Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 3, part 2
5. Graph of cylinder B being filled
(i)Using
or rise over run, find the slope of [AB].
(ii)Using
, find the slope of the line [OB], correct to two decimal places.
(iii)Can you use the slope of [AB] or the slope of [OB] to find the slope of the
graph at the at the point C? Give a reason for your answer.
© Project Maths Development Team 2013
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50
Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 4
Some Transition Year students decide to carry out an experiment on constant
speed. They have a class discussion on where they might see a model for constant
speed. They decide that if they go to a train station and choose a train that is not
scheduled to stop there, that the train will most likely pass them at a constant
speed. Two students from the class arrange to stand 100 metres apart at either
end of the platform and time the train between these two positions.
1. The two students stand 100 metres apart
and discover that it takes 3 seconds for
the front of the train to travel between
the two positions. Draw a graph to
represent how the distance changes
with time during these 3 seconds. Let
the position of the first student be at
the origin of the graph and put the
independent variable on the horizontal
axis.
2. At what speed does the train pass the two students in km/hour?
3. The teacher was standing half way between the students during the
experiment to supervise. At what speed did the train pass the teacher?
Give a reason for your answer.
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51
Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 5
In the 2009 World Championships in Berlin, Usain Bolt set the World Record for
the Men’s 100m sprint, running it in 9.58 seconds. Below is a table of Usain Bolt’s
split times every 10 metres during the race.
Distance (m)
0
10
20
30
40
50
60
70
80
90
100
Time (s)
0
1.89
2.88
3.78
4.64
5.47
6.29
7.10
7.92
8.75
9.58
1. How fast do you think Usain Bolt ran during the race?
Give your answer correct to 2 decimal places in m/sec.
2. Do you think he ran at this speed throughout the whole race?
Give two reasons for your answer.
3. What do you think your answer for Question 1 represents?
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Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 5
4. (i) Using a ruler, join the points (0,0) and (9.58,100) on the graph below.
(ii) Find the slope of this line.
(iii)The line that joins (0,0) to (9.58,100) has a special name. It is called a secant
line to the above curve. What observation can you make about the slope of
this secant line?
5. How do you think we could calculate Usain’s speed at precisely 1 second into
the race?
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Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 6, part 1
Below is a distance-time graph of the first ten minutes of a warm-up cycle by
Olympic Gold medallist Victoria Pendleton.
1. Over these 10 minutes, what is Victoria Pendleton’s average speed in km/min?
2. The coach wants to know what her speed is at exactly 3 minutes during this
warm-up. To help answer this question do the following:
(i) Using your ruler, draw in the secants [AB], [AC], [AD], [AE].
(ii) Fill in the following table. Answers correct to 2 decimal places.
Slope of Secant [AB]=
Average speed between A and B =
Slope of Secant [AC]=
Average speed between A and C =
Slope of Secant [AD]=
Average speed between A and D =
Slope of Secant [AE]=
Average speed between A and E =
3. The slope of which secant is the nearest estimate to Victoria’s speed after
exactly 3 minutes?
4. How might you find a better estimate for Victoria’s speed after 3 minutes?
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Teaching & Learning Plan: Introduction to Calculus
Section A: Student Activity 6, part 2
Below is a distance-time graph of the first ten minutes of a warm-up cycle by
Olympic Gold medallist Victoria Pendleton.
1. Average rates of change are found using
on a curve.
The denominator x2 - x1 tells us the length of the interval of x values or in
other words the length of the ‘run’ when using ‘rise over run’ to find the
slope.
Fill in the following table.
The interval of x values for secant [AB] =
6
The interval of x values for secant [AC] =
The interval of x values for secant [AD] =
The interval of x values for secant [AE] =
The interval of x values for secant [AG] =
2. What interval of x values would give us the secant whose slope is closest to
the tangent at the point A?
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 1
Linear Functions and their Slope Functions
A
B
C
D
Slope (Rate of change) when x is:
Function
-3
-2
-1
0
1
Slope Function
2
3
f (x) = 3
f '(x) =
k (x) = -1
k '(x) =
g (x) = 1
g '(x) =
p (x) = -3
p '(x) =
q (x) = 95
q '(x) =
f (x) = c
f '(x) =
Complete the following: The rate of change of a constant is
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 2
Linear Functions and their Slope Functions
A
B
C
D
E
F
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 2
Linear Functions and their Slope Functions
Slope (Rate of change) when x is:
Function
-3
-2
-1
0
1
Slope Function
2
3
f '(x) =
f (x) = x
w '(x) =
w (x) = x + 2
r '(x) =
r (x) = 3⁄2 x
a '(x) =
a (x) = 3⁄2 x + 3
g (x) = -x
p (x) = -x + 3
n (x) =
q (x) =
f '(x) =
f (x) = nx
f '(x) =
f (x) = nx + c
Complete the following:
The rate of change of a function f (x) = nx is ______________________________________________________________
The rate of change of a function f (x) = nx + c is ___________________________________________________________
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 3, A
Quadratic Functions and their associated Slope Functions f(x) = x2
1 Using the appropriate GeoGebra file or the graph on the next page complete
the table by filling in the slopes of the tangents to the function at points A –
G.
2 Investigate the pattern of the slopes by completing the Change column in the
table.
3 Graph the slopes of the tangent (as a function of x) in the space provided.
4 Using your pattern-analysis skills, write down the slope function ( f '(x)).
Point
x
A
-3
B
-2
C
-1
D
0
E
1
F
2
G
3
Slope of Tangent f '(x)
Change of f '(x)
The slope function f '(x) is
Slope of f ’(x):
y-intercept of f ’(x):
Equation of f ’(x):
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 3, A
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 3, B
Quadratic Functions and their associated Slope Functions f(x) = 2x2
1 Using the appropriate GeoGebra file or the graph on the next page complete
the table by filling in the slopes of the tangents to the function at points A –
G.
2 Investigate the pattern of the slopes by completing the Change column in the
table.
3 Graph the slopes of the tangent (as a function of x) in the space provided.
4 Using your pattern-analysis skills, write down the slope function ( f '(x)).
Point
x
A
-3
B
-2
C
-1
D
0
E
1
F
2
G
3
Slope of Tangent f '(x)
Change of f '(x)
The slope function f '(x) is
Slope of f ’(x):
y-intercept of f ’(x):
Equation of f ’(x):
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 3, B
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 3, C
Quadratic Functions and their associated Slope Functions f(x) = 3x2
1 Using the appropriate GeoGebra file or the graph on the next page complete
the table by filling in the slopes of the tangents to the function at points A –
G.
2 Investigate the pattern of the slopes by completing the Change column in the
table.
3 Graph the slopes of the tangent (as a function of x) in the space provided.
4 Using your pattern-analysis skills, write down the slope function ( f '(x)).
Point
x
A
-3
B
-2
C
-1
D
0
E
1
F
2
G
3
Slope of Tangent f '(x)
Change of f '(x)
The slope function f '(x) is
Slope of f ’(x):
y-intercept of f ’(x):
Equation of f ’(x):
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 3, C
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 3, D
Quadratic Functions and their associated Slope Functions f(x) = 4x2
1 Using the appropriate GeoGebra file or the graph on the next page complete
the table by filling in the slopes of the tangents to the function at points A –
G.
2 Investigate the pattern of the slopes by completing the Change column in the
table.
3 Graph the slopes of the tangent (as a function of x) in the space provided.
4 Using your pattern-analysis skills, write down the slope function ( f '(x)).
Point
x
A
-3
B
-2
C
-1
D
0
E
1
F
2
G
3
Slope of Tangent f '(x)
Change of f '(x)
The slope function f '(x) is
Slope of f ’(x):
y-intercept of f ’(x):
Equation of f ’(x):
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 3, D
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 3, E
Quadratic Functions and their associated Slope Functions
Conclusion
1 Complete the table below and explain your findings.
Function f (x)
Slope Function f '(x)
f (x) = x2
f (x) = 2x2
f (x) = 3x2
f (x) = 4x2
f (x) = 5x2
f (x) = ax2
Explanation (in words).
To find the derivative (slope function) of a quadratic function…
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 4, A
Cubic Functions and their associated Slope Functions f(x) = x3
1 Using the appropriate GeoGebra file or the graph on the next page complete
the table by filling in the slopes of the tangents to the function at points A–G.
2. Investigate the pattern of the slopes by completing the Change column in the
table.
3. Graph the slopes of the tangent (as a function of x) in the space provided.
4. Using your pattern-analysis skills, write down the slope function ( f '(x)).
Point
x
A
-3
B
-2
C
-1
D
0
E
1
F
2
G
3
Slope of Tangent f '(x)
Change of f '(x)
Change of Change of f '(x)
The slope function f '(x) is
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 4, A
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 4, B
Cubic Functions and their associated Slope Functions f(x) = 2x3
1 Using the appropriate GeoGebra file or the graph on the next page complete
the table by filling in the slopes of the tangents to the function at points A–G.
2. Investigate the pattern of the slopes by completing the Change column in the
table.
3. Graph the slopes of the tangent (as a function of x) in the space provided.
4. Using your pattern-analysis skills, write down the slope function ( f '(x)).
Point
x
A
-3
B
-2
C
-1
D
0
E
1
F
2
G
3
Slope of Tangent f '(x)
Change of f '(x)
Change of Change of f '(x)
The slope function f '(x) is
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 4, B
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 4, C
Cubic Functions and their associated slope functions
Conclusion
1 Complete the table below and explain your findings.
Function f (x)
Slope Function f '(x)
f (x) = x3
f (x) = 2x3
f (x) = ax3
Explanation (in words).
To find the derivative (slope function) of a cubic function…
Have you seen this approach to finding the slope function before? Explain
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 5
Finding the slope function of a polynomial
1 Based on the results of activities Section B : Student Activities 1-4, complete
the following table:
Function f (x)
Slope Function f '(x)
f (x) = 3
f '(x) =
k (x) = 1
k '(x) =
p (x) = -1
p '(x) =
d (x) = - 3
d '(x) =
f (x) = x
f '(x) =
g (x) = 3⁄2 x
g '(x) =
h (x) = -x
h '(x) =
f (x) = x2
f '(x) =
a(x) = 2x2
a '(x) =
f (x) = 3x2
f '(x) =
g (x) = 4x2
g '(x) =
k (x) = x3
k '(x) =
p (x) = 2x3
p '(x) =
2. Can you summarise how you might find the slope function of any given
polynomial function?
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 5
Finding the slope function of a polynomial
3. Using this approach write down the slope functions of the following functions
Function f (x)
Slope Function f '(x)
f (x) = 5x
f '(x) =
g (x) = 4x3
g '(x) =
h (x) = -9x
h '(x) =
k (x) = -6x2
k '(x) =
f (x) = 0.5x2
f '(x) =
w (x) = 5x4
w '(x) =
g (x) = -3x7
g '(x) =
r (x) = x -2
r '(x) =
k(x) = x -1
k '(x) =
f (x) = 2x0
f '(x) =
z (x) = 1⁄x
z '(x) =
a (x) = √x
a '(x) =
g (x) = 2⁄x3
g '(x) =
f (x) = axn
f '(x) =
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 6, A
Co-polynomial functions and their associated Slope Functions
f(x) = x2 - x -6
1 Using the appropriate GeoGebra file or the graph on the next page complete
the table by filling in the slopes of the tangents to the function at points A–G.
2. Investigate the pattern of the slopes by completing the Change column in the
table.
3. Graph the slopes of the tangent (as a function of x) in the space provided.
4. Using your pattern-analysis skills, write down the slope function ( f '(x)).
Point
x
A
-3
B
-2
C
-1
D
0
E
1
F
2
G
3
Slope of Tangent f '(x)
Change of f '(x)
The slope function f '(x) is
Slope of f ’(x):
y-intercept of f ’(x):
Equation of f ’(x):
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 6, A
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 6, B
Co-polynomial functions and their associated Slope Functions
f(x) = x2 + 3x + 4
1 Using the appropriate GeoGebra file or the graph on the next page complete
the table by filling in the slopes of the tangents to the function at points A–G.
2. Investigate the pattern of the slopes by completing the Change column in the
table.
3. Graph the slopes of the tangent (as a function of x) in the space provided.
4. Using your pattern-analysis skills, write down the slope function ( f '(x)).
Point
x
A
-3
B
-2
C
-1
D
0
E
1
F
2
G
3
Slope of Tangent f '(x)
Change of f '(x)
The slope function f '(x) is
Slope of f ’(x):
y-intercept of f ’(x):
Equation of f ’(x):
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 6, B
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 6, C
Co-polynomial functions and their associated Slope Functions
Conclusion
1 Complete the table below and explain your findings.
Function f (x)
Slope Function f '(x)
f (x) = x - x - 6
2
f (x) = x2 + 3x + 4
f (x) = x2 - 4x - 5
f (x) = 3x2 - 5x
f (x) = 2x2 + 3x - 2
f (x) = ax2 + bx + c
Explanation (in words)
To find the derivative (slope function) of any function of the form
f(x) = g(x) + h(x) + k(x)...
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 7
Summary of how to find slope functions Tarsia
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Teaching & Learning Plan: Introduction to Calculus
Section B: Student Activity 7
Summary of how to find slope functions Tarsia
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Teaching & Learning Plan: Introduction to Calculus
Appendix 1
Introducing h as the interval of x values that approach 0
Slope of Secant =
Average Rate of Change =
Continous Rate of Change =
Slope of Tangent =
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