Transcript Projects for Mathematics: Inside and Outside of the Classroom

Projects in Mathematics
Adaptable For All Classes
Presented by: Marianne Ilmanen
Joe Henderson
Why Projects ?
• Projects give students a longer period of time
to solve a problem than is available in a class
period.
• Projects allow students to deepen their
understanding of math with hands-on,
problem-solving activities.
• Projects help to improve student reasoning,
problem-solving, communication and make math
connections to real world situations
• Projects
can be done in groups that
require students to contribute to a group
effort and to accept group consensus.
Sample Projects
• Projects that Incorporate Literacy into
Mathematics: Career Investigation
• In Class Group Projects: Review for Final Exam
Research Projects: Standardized Test
Question Analysis
• Real World Topics that Reinforce the curriculum:
Recipe for Math
Pyramid Power
Mapping Distance in a Plane
• Projects Adapted for Different Course Levels:
Function Families
Volume of a Box
•
Career Investigations Project
Project Description:
Students investigate a career that has
strong mathematics base from a list of
careers.
- Students are given a list of careers that have a
mathematics base, a set of questions to be
addressed and the project grading rubric.
- Students choose a career from the Career List.
- Students research their chosen
career using www.bridges.com at the
career center or at home.
Career Project Questions
1. What are the duties of the career?
2. What academic path is required for the
career?
3. What math classes do you need to take
during your education to prepare for this
career?
4. How is math important for this career?
5. Which schools have respected programs in
this field?
6. Are there jobs available in this career in
your area?
Career List
Actuary
Administrator
Bank financial manager
Bank loan officer
Business consultant
Cash flow analyst
Communications consultant
Computer aided designer
Computer network designer
Computer software designer
Computer technician
Construction
Consumer behavior analyst
Financial analyst
Corporate planner
Cost account
Customer service rep
Demographic analyst
Economic analyst
Engineer
Environmental forecaster
Factor analyst/medical research Insurance analyst
Factor analyst/ social systems
Human resources manager
Investment analyst
Industrial cost controller
Management consultant
Marketing consultant
Modeler of genetic systems
Pension fund controller
Policy change analyst
Portfolio manager
Product designer
Production planner
Program analyst
Research data analyst
Researcher
Resource analyst
Safety coordinator
Salary and benefits analyst Statistical consultant
Stock and bond analyst
Tax consultant
Taxation systems consultant
Teacher/professor
Urban planner
Veterinary Medicine
Agricultural Management
Biomedical Research
Petroleum Engineering
Construction Management
Independent Music Production Cartography
Aerospace Medicine
Astronomical Research
Meteorology
Food Management
Operations Research
Government Finance
Special Education
Public Accounting
Computer Consulting
Manufacturing Engineering
Architect
Structural Engineer
Navigator
Physicist
Seismologist
Surveyor
Product Manager
Statistician
Math Textbook Editor
Opinion Researcher
Actuary
Math TV Content Director
U.S. Navy Officer
Airplane Pilot
Helicopter Pilot
Casino Manager
Rubric
Final Exam Review
Project Description:
Students identify the main
topics that are covered on the final
exam, using post-it tabs.
Final Exam Review
Project
Directions
1. Provide a sheet of
butcher paper
for each general topic.
2. Title each sheet with a
topic.
You may add any key
questions that you
want addressed.
Linear Equations
X and Y intercepts
Slope
Slope-intercept form
Point-slope form
Standard form
Labeled Graph
3. Pre-select groups of 4 students
for each group.
4. Arrange your room into clusters
of four desks. Place a poster on
each table and assign a group to each.
WARNING
DO NOT let the
students pick their
own groups!
5. Give each group a different color
marker so that their contributions
to each poster can be identified.
6. Have the groups rotate to each poster at one
minute intervals.
7. At each stop, the group will provide the
definition of one item on the poster. They can
write out a definition or draw a picture with
labels.
Each group should have
a person assigned to
each of the following
job assignments:
 One writer/recorder
 Two journal researchers
 One person to check for accuracy
Example of Rotation
8. When the groups have rotated through
all of the stations, they are assigned
to check and edit the poster at their
last station.
9. Each student is given a
copy of the Review
Outline.
10. Each group presents their
posters while the other students
complete their individual outlines.
Standardized Test Question
Analysis Project
Purpose:
To allow students to assess their own
math knowledge, to determine if the
test is trying to confuse them in the
question or with the answers and to
discover if the answers contain any
“boobie traps”.
Preparation:
• Select a set of standardized test
questions, so that each student will have
a unique question.
• Prepare a large scale version of the
questions for use on the overhead
projector or for use in power point.
• Give each student one of the selected
standardized test questions and a copy
of the rubric.
Independent Student Work
Students answer a set of questions
about their multiple choice problem.
Students type their answers in
paragraph form…or make a poster.
Students present their answers in class
to the other students.
Activities and Questions
1. Draw a diagram or picture.
2. What math knowledge do you have to
have to answer the question?
3. How are they trying to confuse you
in the question?
4. Show your work and answer the math
question.
5. How are they trying to confuse
you with the answers?
6. Which answers are “boobie traps” ?
Rubric
Recipe for Math Project
Project Description:
Using the recipe for HERSHEY'S
"PERFECTLY CHOCOLATE"
CHOCOLATE CHIP COOKIES,
students will find algebraic
relationships of the ingredients.
Student Investigations
1. Students write an equation relating the
number of cups of brown sugar (S) to the
number of cups of flour (F ) in the cookie
recipe. Using their equation, students
determine the amount of flour or brown sugar
would be needed in different situations:
For a double recipe of cookies
For a recipe that increases the amount
of flour
For a recipe that increases the amount
of brown sugar
Using a Graph
2. Students graph the
equation relating the
amount of brown sugar
to the amount of flour
and use the graph to
answer these questions:
- How much brown sugar will be needed if 10 cups
of flour are used.
- How much flour will be needed if 6 cups of brown
sugar are used.
Making Connections
Students are asked to re-write the
cookie recipe for a bakery that
makes 24 dozen cookies in each
batch.
Pyramid Project
Project Description:
Students will make a scale model of the
Great Pyramid of Khufu. From the scale
model, they will determine estimates of the
actual measures.
Model
1. Using the pattern
provided, each student
creates a scale model of
the Great Pyramid
2. Given the scale of the
model, students determine
the actual measurements
of the Great Pyramid.
3. Students determine the
slope of each face of the
Great Pyramid from the
measurements.
Making Connections
4. Students draw a representation
of the Great Pyramid on a
coordinate System, determine
the coordinates of the corners
and the slope of the edge of
the face.
5. Students construct their own
scale models of the other two
pyramids of the Giza Plateau,
Khafre and Menkaure.
Distance on a Plane
Project Description:
Using maps of Sacramento and
Long Beach, students investigate
different ways to determine the
distances between different
landmarks.
Sacramento
Students investigate the
drivable distance and direct
distance between different
buildings
Long Beach
Students
use a map
of Long Beach
with a
coordinate grid
to investigate
relationships
of lines that
connect
schools.
Function Families
Project Description:
Function Families are sets of functions
with similar properties. The functions
used are those studied in each course and
enhance the concepts that the student
has learned.
Functions:
Course:
Linear Equations
Algebra
Quadratic Equations
Int Algebra
Advanced Math Various Functions
Linear Function Families
The purpose of the project is to investigate the properties of linear
equations and their graphs.Students are expected to prepare a
report for 10 different equations that contains the following data.
1) The equation in slope-intercept form
2) The equation in standard form
3) An x-y chart of at least 5 coordinate pairs
4) The graph on a coordinate system
5) The Domain
6) The Range
7) The x-intercept
8) The y-intercept
9) The slope
10) A description of the line that includes its direction, path and
how it compares to the graph of y = x
Sample Page for Linear Equations
X
X
Y
Y
-2
-2
1) The given equation, y = 2x + 4, is in slope-intercept form
0
0
2) The equation in standard form is 2x – y = - 4
-1
-1
2
2
y = 2x + 4
3) An x-y chart of coordinate pairs for this equation
4) The graph on a coordinate system
5) The Domain for this equation is x = { all real numbers}
6) The Range for this equation is y = { all real numbers}
7) The x-intercept is at ( -2, 0 )
8) The y-intercept is at ( 0, 3 )
9) The slope is m = 2
10) The graph is a line that is slanted up to the right. The path between any two points
is one space to the right and two spaces up. The line is steeper than the line y = x
Quadratic Equation Families
The purpose of the project is to investigate the properties of
quadratic equations and their graphs.Students are expected to
prepare a report for 10 different equations that contains the
following data.
1) The equation in general form
2) The equation in vertex form
3) The equation in factored form
4) The graph on a coordinate system
5) The Domain
6) The Range
7) The x-intercepts, if any
8) The y-intercept, if it exists
9) The location of its vertex, and identify it as a maximum or minimum
10) A description of how the parabola compares to the graph of y = x2
that includes its direction, width, location of the vertex and the
number of real roots.
Sample Page for Quadratic Equations
y = x2 + 2x  15
1) The given equation is in general form:: y = x2 + 2x  15
2) The equation in vertex form:
y = ( x – 1 ) 2 – 16
3) The equation in factored form: y = (x + 5) ( x  3)
4) The graph on a coordinate system
5) The Domain: x = { all real numbers }
6) The Range: y ≥ -16
7) The x-intercepts are at ( - 5, 0 ) and ( 3, 0 )
8) The y-intercept is at ( 0, -15 )
9) The vertex is at ( -1, -16 ). The vertex is the minimum value of y.
10) The parabola is the same size and opens upward the same as y = x2 . The vertex has
been moved to the point ( -1, -16 ). It has two real roots.
Function Families
The purpose of the project is to investigate the properties of a
variety of functions and their graphs.Students are expected to
prepare a report for 10 different equations that contains the
following data.
a) The name of the function
b) The format of the equation of the function
c) An example of the function
1)
2)
3)
4)
5)
6)
7)
8)
The equation
The graph on a coordinate system
The Domain, including any undefined values, if they occur
The Range
The roots (if, any exist)
The y-intercept
The intervals where it increases, decreases or is constant
The inverse function, and any restrictions that may apply so
that it is a function
Functions Used
1) The Constant Function: y = k
2) The Linear Function: y = mx + b or ax + by = c
3) The Quadratic Function: y = ax2 + bx + c or y = a(x  x1)(x  x2)
4) The Cubic Function: y = ax3 + bx2 + cx + d
or y = a(x  x1)(x  x2)(x  x3)
5) The Radical (Square Root) Function: y  a x  k  h
6) The Cubic Root Function: y  a 3 x  k  h
7) The Greatest Integer (Step) Function: y = a [[ x  k ]] + h
8) The Absolute Value Function: y = a | x  k | + h
f (x),for x  k 
y

9) The Piecewise Function:


g(x),for
x

k


10) The Rational Function:
y
f (x)
g(x)
Sample Page for Functions Family
The Quadratic Function
The Standard Format of the function is y = ax2 + bx + c or
y = a(x  x1)(x  x2)
Example: y = x2  16
or y = (x + 4) ( x  4)
Domain: all real numbers
Range: y  -16
Roots: x = -4 and x = 4
Y-intercept: ( 0, -16)
The function increases for x > 0, and decreases for x < 0
The inverse function is y = x +16 , for x  0
Volume of a Box
Project Description:
Students in Geometry, Precalculus and
Calculus investigate the volume of a box
made by cutting out congruent squares
from each corner of a sheet of
cardboard and folding up the sides.
Course:
Project:
Geometry
Volume and Surface Area
Precalculus Volume as a function
Calculus
Minimize the volume function
Volume and Surface Area
Project Description:
x
8-2x
10-2x
Students make a box and calculate the
volume and the surface area of the box.
Then they compare the size of the square
that was cut from the cardboard to the
surface area and the volume of eight
different boxes.
Volume as a Function
Project Description:
x
V(x)=x(10-2x)(8-2x)
8-2x
10-2x
Students make a box and
calculate the volume.
They also write the function
V(x) to determine the volume, with x as
the measure of the side of the squares.
Using the graph of V(x), the students
analyze the function
Volume Function and Derivative
Project Description:
x
8-2x
V(x)=x(10-2x)(8-2x)
V(x)=4x3- 36x2+80x
V(x)=12x2 – 72x + 80
Students make a box and calculate the
volume. They also write the volume as the
function V(x). They find the derivative
V ’(x) to determine the critical values of
x and the maximum volume.
10-2x