Tasks in Calculus: Results of a 9-Year Evolution

Tasks in Calculus: Results of a 9-Year Evolution

Geneviève Savard and Kathleen PineauÉcole de technologie supérieure

Montréal, Canada

@ École de technologie supérieure

All teachers and students have the same calculator and textbook and there is a common final exam1999

TI-92 Plus Harvard group text Calculus: Single Variable

2004 Voyage 200 Stewart’s Calculus, Concepts and Contexts

2006 Voyage 200 Home grown text

Read and understandthe problem

Translate tosketches, graphs, tables, ...

Problem solving

Establish a plan

Intermediate information?Strategy? Necessary tools?

Execute the plan

Calculate, solve equations, apply algorithms,…

Examine the solution

Validate resultInterpret result in context



Problem solving

Establish a plan


Execute the plan






Problem solving

Establish a plan


Execute the plan




A classic…

Find the equation of the line tangent to the graph of

at the point where

2

107

xy

5.x

Why?

Ex1: answering the question Why?

2D cross-section

Graded Class work Fall 2004, Textbook 2006

Objective: create the need for the mathematical tool

piece supportEquation of the parabola

Contact points between the conical support and the piece must be at x=5 and x= -5.

a) Angle?

b) Height of the cone?

2

10, 10 07

xy y

Ex1: Hands-on

Determine the dimensions of the cone (height, apothem and radius of base). Build one with cardboard.

New opportunity for validationConsolidation of geometrical notionsPreparation for optimisation problem 2D cross-section

piece support

New ideas...

Ex2: Evaluating understandingAt 13:00, a rainwater tank contains 500 litres of water. The rainwater is filling the tank at a rate of

where t is measured in hours from 13:00.

a)Calculate the area under the curve y=Q(t) for t ranging from 1 to 3. What does this value represent in this context?b)What will be the volume of water in the tank at 3:00 pm? Explain your reasoning.c)If the tank can only contain 1 250 litres of water, will it overflow? If so, at what time? Explain your reasoning. If it does not overflow, explain why in the context of our problem.d)What expression containing a definite integral makes it possible to calculate the average flow rate between 13:00 and 15:00? Calculate this average flow rate and interpret what it represents on the graph of y=Q(t).

Lh2

2500

1 2

tQ t

t

Final exam, summer 2006

A lot of sub-

questions

At 13:00, a rainwater tank contains 500 litres of water. The rainwater is filling the tank at a rate of


a)Calculate the area under the curve y=Q(t) for t ranging from 1 to 3. What does this value represent in this context?

Answer...

Represents the quantity of water (410.5 L) that was added to the tank between 2 and 4 pm.

Ex2: Starting with basics

3

21

2500410.5

1 2

tdt

t


Easy

Not so easy•Units, area linked to litres !!!•Frame of reference

Lh2

2500

1 2

tQ t

t

Ex2: Guiding students

At 13:00, a rainwater tank contains 500 litres of water. The rainwater is filling the tank at a rate of


b)What will be the volume of water in the tank at 3:00 pm? Explain your reasoning.

Answer... Initial quantity plus change...

2

20

2500500 1005.9 L

1 2

tdt

t

Final exam, summer 2006Purpose: setting up c)

Lh2

2500

1 2

tQ t

t

c) If the tank can only contain 1 250 litres of water, will it overflow? If so, at what time? Explain your reasoning.If it does not overflow, explain why in the context of our problem.

Answer...

There will be overflow at approximately 4:30 pm.

20

2500500

1 2

x tV x dt

t

Ex2: Making sense of notation

Final exam, summer 2006Difficult concept: variable upper limit

1250 0.354 3.479V x x or x

Ex2: Changing registers

d) What expression containing a definite integral makes it possible to calculate the average flow rate between 13:00 and 15:00? Calculate this rate and interpret what it represents on the graph of y=Q(t).

Answer...

The area under the curve is the same as that of the rectangle

2

20

2500

1 2 505.9 L/h = 252.95 L/h

2 0 2

tdt

t


Let f(t) be the function graphed below.

a) Evaluate g(12), g’(12), and g’’(12).Show how you go about getting your answer.

Ex3: Working with graphs

0

x

g x f t dtDefine

Quiz, fall 2007- TI prohibited !

2 4 6 8 10 12

-3

-2

-1

1

2

3

t

y=f(t)

Ex3: Adding context

b) Suppose that f(t) represents the flow of liquid entering or exiting a tank in L/min at time t. Suppose also that the initial volume of liquid in the tank is 60 litres. What are the minimum and maximum volume of liquid contained in the tank during the first 12 minutes?


t V

0 60

2 57

4 54

6 57

8 62

10 64

12 662 4 6 8 10 12

60

t

V

Ex3: Creating expertise

c) In this context, what is the average flow rate for the first 12 minutes?


2 4 6 8 10 12

-3

-2

-1

1

2

3

t

y=f(t)

Variations on the same themeEx2 with symbolic register Students guided with

questions Emphasis on contextual

interpretation: units, frame of reference, ...

Ex3 with graphical register Students unguided, they

choose the tools Emphasis on graphical

interpretation

CAS is technically unnecessary. However, it offers (in the teaching/learning process) the possibility of switching easily from one register to another.

Validation of results facilitated by working with different registers (constructing an expertise regarding what is useful about a particular register)

Ex4: Using CAS

The maximum level of liquid contained in a tank is 1.6 m. The tank’s shape is obtained by revolving around the y axis the region bounded by the curves

where x and y are measured in metres.

Your job is to evaluate the height of the liquid when the tank is half full.

ln(3 4), 0, 1.6, 0y x y y x

Graded homework, fall 2007

A change in register is necessary

a) Illustrate the problem: 2D cross section and 3D outline showing important values

? 01 xy


?


a) Illustrate the problem: 2D cross section and 3D outline showing important values continued

b) What is the maximum volume of liquid contained in the tank?

Maximum volume is 33.90 m3.

Students compare with the volume of a cone or a cylinder or...

Disk method


?

c) Graph volume function with respect to height of liquid.

d) From this graph, estimate the height of the liquid when the tank is half full.

1.3h

v(h)33.90

16.95

0.5

v

h

Judicious use of solver

e) Refine your estimate using the calculator

Numerical validation: v(1.29765) = v(1.6) / 2

Common sense validation: using the shape of the tank, the height of the liquid when the tank is half full should be more than 0.8 m (half the height of the tank.)


Conclusions More emphasis on

Context Interpretation of results (mathematical and contextual) Validation (facilitated by work in different registers)

Our intentions... Make (through the “imagery” of contexts) the mathematical

notations easier to understand and to piece together Create the need for mathematical tools in the student, thereby

acting on motivation and ability to choose the appropriate tool Generate a need for interpretation (that could lead to one of

validation) Work on communication skills

Read and interpret a varied set of problems Choose arguments Write in natural language interpretation of results

Develop mathematical control over the use of the tool

Documents

Tasks in Calculus: Results of a 9-Year Evolution