Uplift Education: North Hills Preparatory
2014-2015
Pratibha Sinha: IB Math Studies
IB Unit: Introduction to Differential Calculus - Section 6
IB Framework
Connection to Unit Question:
Connection to Learner Profile Attribute(s):
Thinkers They exercise initiative in applying thinking skills
Unit Question: How can we determine the rate of change
mathematically and interpret its contextual meaning to make
decisions in real life situations?
-
Scholars will use symbolic calculus to represent rate of
change.
Scholars will connect their prior knowledge of slope to
understand the instantaneous rate of change.
Scholars will model real life situations with gradient
functions
Scholars will relate the derivatives to the optimization
process
critically and creatively to recognize and approach complex
problems, and make reasoned, ethical decisions.
Knowledgeable They extend their previous learning of slope
to understand the rate of change in calculus.
Lesson Vision
Objective(s)
 SWBAT
Apply differential calculus to find optimal solutions in
real life situations
Key Points and Vocabulary
Key Points
 rate of change
 Positive and negative gradient
 Critical points in a function
 optimization
Vocabulary:
 gradient
 local maximum
 local minimum
 point of horizontal inflection
 rate of change
 absolute minimum
 absolute maximum
 sign diagram
 derivative
 end points
 constraints
DP Content Objectives: (from DP Guide)
Standard 7.6
optimization problems
(maximizing profit, minimizing cost, maximizing
volume for a given surface area) In examinations,
questions on kinematics will not be set
ELPS Alignment:
(1) The Student is expected to develop an awareness of his or
her own learning processes in all content areas:
(A) use prior knowledge and experiences to
understand meanings in English;
(C) use strategic learning techniques such as concept
mapping, drawing, memorizing, comparing, contrasting,
and reviewing to acquire basic and grade-level
vocabulary;
(F) use accessible language and learn new and
essential language in the process;
(2) The ELL listens to a variety of speakers including
teachers, peers, and electronic media to gain an increasing
level of comprehension of newly acquired language in all
content areas:
(E) use visual, contextual, and linguistic support to
enhance and confirm understanding of increasingly
complex and elaborated spoken language;
(3) The ELL speaks in a variety of modes for a variety of
purposes with an awareness of different language registers
(formal/informal) using vocabulary with increasing fluency
and accuracy in language arts and all content areas:
(E) share information in cooperative learning
interactions;
( (G) express opinions, ideas, and feelings ranging from
communicating single words and short phrases to
participating in extended discussions on a variety of
social and grade-appropriate academic topics;
(4) The ELL reads a variety of texts for a variety of purposes
with an increasing level of comprehension in all content areas:
(C) develop basic sight vocabulary, derive meaning of
environmental print, and comprehend English
vocabulary and language structures used routinely in
written classroom materials;
(5) The ELL writes in a variety of forms with increasing
accuracy to effectively address a specific purpose and
audience in all content areas:
(B) write using newly acquired basic vocabulary and
content-based grade-level vocabulary;
Assessment:
Problem:
Accommodations/Modifications for Special Populations:
All class notes, power points, and handouts available to scholars online on our class webpage at
www.northhillsprep.org
Visual as well as auditory presentation of information at all times.
Opportunity for oral clarification or verbal delivery.
Vocabulary definitions and clarifications as we read/real time.
Movement and rotating partner collaboration.
Preferential seating and active teacher monitoring.
Teaching Plan
Do Now
Time: ___10_ minutes
Hook
Time: _5___ minutes
Quiz – Rate of change problems and identifying maximum and minimum (IB test
bank)
Show scholars the map of the middle earth from ‘Lords of the Rings’ and discuss how
Frodo’s journey was optimized for him to be successful.
Discuss the all the possible options he had to complete the given task and how he
made his decisions to be successful to bring scholars into the lesson 
Intro to New Material (I Do)
Time: __20__ minutes
Hands on ActivityScholars will work in partners.
Hand out a grid paper to each of the pairs and ask them to draw four congruent
squares in each corner of the paper (the size of the four corners has to be given
by the teacher). Using the scissors and tape, they will then cut out the squares to
create an open topped box.
Then they are asked to record the width, length,and height of the box along with
its volume.
All groups record the data on the board and then they draw a graph to find the
maximum volume and compare the size of the cut out corner.
Connect this activity with the optimization process in calculus.
Optimization
Powerpoint presentation on optimization
-
Go over the problem solving method
Draw a clear diagram of the situation
Construct a formula with a variable to be optimized
Find the derivative and find the values of x where it is zero
Show the sign diagram using the domain and then test for a maximum or
minimum
Checks for Understanding, Exemplar Student Responses, Potential Misconceptions
What is optimization?
It is the process of finding the optimum solution using the maximum or minimum
Does the maximum or minimum value always occur when the derivative is zero?
No, it can also occur at the end points of the domain
How can you test optimal solutions?
Using sign diagram test or graphical test
How do you find the derivative of a function?
By using the power rule of differentiation
How do you know if a function is increasing in a given interval?
If the gradient of the tangent is positive, the function is increasing
Why is the function increasing?
Because the f(b) is more than f(a) where a and b are x values and a<b
How can you identify the decreasing functions using the gradient of tangent lines?
If the derivative value in the given domain is negative, the function is decreasing
Explain why the function is decreasing using the x and y values.
If f(b) is less than f(a); where a and b are any values in the domain and a is less than b
What does the sign diagram represent?
The sign diagram shows the critical points and how the gradient of the tangent lines
vary in the various intervals in between.
What are the critical values or stationary points?
Critical values represent the points on a function where the gradient function or the
slope of the tangent lines is zero.
Misconception:
Always make sure to keep the order same for x and y coordinates when finding the
gradient.
Intervals on the domain can be represented as an inequality or by using the interval
notation.
In the interval notation, both the value inside the parenthesis are representing xvalues or the domain of the function.
In general, the rate of change of one variable with respect to another is the gradient
function.
Guided Practice (We Do)
Time: 20____ minutes
In Class Problems
Scholars will work on the DP exam style practice problems in the powerpoint with the
help of the teacher and their peers.
They will use the information presented to them to solve the given problems together
a group. We will have a discussion on the reasoning, rationale, and a way to crosscheck our answers as we go over these problems.
Topics for lesson 6 are:
-
Using optimization to find desired solutions
Checks for Understanding, Exemplar Student Responses, Potential Misconceptions
What are the critical values? How do you find them?
The critical values are the points in a function where the gradient of the tangent lines
is zero. We can find them by making the gradient function equal to zero and then
solving for x.
Can there be more than one critical value in a function?
Yes, there can be more than one critical value in a given function. The values are
separated by a solid line if the value of x exist and with a dotted line if the function is
undefined at those x-values.
What do the signs represent on the sign diagram?
The signs on the sign diagrams represent the value of the gradient function in that
interval.
How do you know if the function has a minimum point at a given critical point?
If the sign of the gradient is negative on the left and positive on the right, the critical
point represents a minimum.
How do you identify a maximum point from a sign diagram?
If the sign of the gradient is positive on the left and negative on the right, the critical
point in between represents a maximum.
How do you find the signs of the gradient function on the sign diagram?
By substituting any x-value in the given interval, we can determine the sign of the
function
How can you use the GDC in optimization problems?
- GDC can help you see the graph of the function and identify
maximum/minimum
- GDC can help you find the roots/x-intercepts/zeros of the first derivative
MisconceptionsEmphasize that the derivative of a function is also a function and can be graphed
Know that the variables in a function can change representing different quantities; so
the notation of the derivative function may change too based on the situations it is
used.
Independent Practice (You Do)
Time: __20__ minutes
Ask the scholars to work on the powerpoint problems in class.
Checks for Understanding, Exemplar Student Responses, Potential Misconceptions
As the scholars work on the independent practice questions, ask them the following
questionsWhat happens to the powers of the variables when you take the derivative of a
function?
The power of the variable goes down by 1
How do you take the derivative of a sum or difference?
The rule states that the derivative of a sum (or a difference) is equal to the sum (or the
difference) of the derivatives.
If v is the velocity of an object, what does dv/dt represent?
The dv/dt represents the rate of change of velocity with time which is equivalent to
the acceleration.
How would you decide by considering dv/dt whether the water was entering or
leaving the container?
If the value of the gradient is negative, the water is leaving and if the value of the
gradient is positive, the water is entering the container.
How can you double check your work?
By using the graphing function on the GDC
Misconceptions:
Make sure that scholars understand that
- the derivative is a function
- it changes values as we move on the graph
- it represents the slope at a given point on a graph
- the derivatives of curves vary along the shape
- the gradient of the tangent line can be used to determine if a function is
increasing or decreasing
-
Closure and Exit Ticket
Time: __10__ minutes
check your answer by graphing the function on the GDC
substitute the value of x in the gradient function to see if the slope at that
point is positive or negative
positive slope-means increasing function
negative slope-means decreasing function
zero slope means maximum or minimum
the quadratic function has a negative coefficient shows a maximum point
you can only use differentiation in functions with one variable
Choosing which variable to eliminate is an important skill. A bad choice will
make the function more complicated
Closure: Summarize what you learned and decide the learner profile attribute that you
exhibited today
TOK moments
 Is optimization unique to mathematics?
 How does mathematics fit into the scientific method?
 Does mathematics have a prescribed method of its own?
 Is mathematics a science?
Exit ticket
Practice problem from the IB test bank
Homework/Extension
Scholars will take a quiz on optimization.
Suggested homework: Topic 7 practice – all questions