9th grade public research lesson on January 7, 2007 at Las Cruces High School
Mathematics Unit Plan for Grade 9
For the lesson on January 7, 2007
At Las Cruces High School
Instructor: Mary Andrews
Unit Plan developed by: Mary Andrews, Connie Jaramillo, Douglas Lutz, Sandy Nesbitt
1. Title of the Unit: Discovering the meaning of slope.
2. Background Information:
Goal of the unit:
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This unit will provide students with understanding and interpreting the
meaning of slope given linear equations, real world problems, or data
set’s.
Students will be able to determine, identify, and evaluate slope and justify
their reasoning (conjectures).
3. Narrative Overview of Background Information:
This lesson study will focus on building student understanding of graphs, real
world problems, and linear equations. Based on CRT, criterion reference testing, it has
been shown that student perform poorly on problems related to linear equations, slope,
using real world problems. Our lesson study group wanted to focus on problem areas that
repeat themselves year after year. One area of weakness that our students have is
graphing and understanding slope. We wanted to create a unit that addressed their
understanding of slope, not just as a number in front of “x”, but to have a deep visual
understanding of how “x” changes, and to gain a deeper understanding of what rate of
change is.
This lesson study will focus on building higher mathematical thinking and
problem solving. Prior to this lesson student’s will have seen and understand the
coordinate plane, ordered pairs, plotting points, quadrants, and given a linear equation,
student can make a T chart.
4 . How this unit is related to the curriculum:
New Mexico standards and benchmarks expect students to represent and analyze
relationships using written and verbal expressions, tables, equations, and graphs, and
describe the connections those represent.
Strand: Algebra, Functions, and Graphs
Performance Standard: Students will understand algebraic concepts and applications
A. Represent and analyze mathematical situations and structures using algebraic
symbols.
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translate from verbal expression to algebraic formulae (e.g., 'Set up the equations
that represent the data in the following equation: John?s father is 23 years older
than John. John is 4 years older than his sister Jane. John's mother is 3 years
younger than John’s father. John’s mother is 9 times as old as Jane. How old are
John, Jane, John’s mother, and John’s father?')
given data in a table, construct a function that represents these data (linear only)
given a graph, construct a function that represents the graph (linear only)
B. Understand patterns, relations, functions, and graphs.
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Distinguish between the concept of a relation and a function.
Determine whether a relation defined by a graph, a set of ordered pairs, a table of
values, an equation, or a rule is a function.
Describe the concept of a graph of a function.
Translate among tabular, symbolic, and graphical representations of functions.
Determine the domain of independent variables and the range of dependent
variables defined by a graph, a set of ordered pairs, or a symbolic expression.
Identify the independent and dependent variables from an application problem
(e.g., height of a child).
Describe the concept of a graph of an equation.
Work with composition of functions (e.g., find f of g when f(x) = 2x - 3 and g(x)
= 3x - 2), and find the domain, range, intercepts, zeros, and local maxima or
minima of the final function.
C. Use mathematical models to represent and understand quantitative relationships.
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Model real-world phenomena using linear and quadratic equations and linear
inequalities (e.g., apply algebraic techniques to solve rate problems, work
problems, and percent mixture problems; solve problems that involve discounts,
markups, commissions, and profit and compute simple and compound interest;
apply quadratic equations to model throwing a baseball in the air ).
Use a variety of computational methods (e.g., mental arithmetic, paper and pencil,
technological tools).
Express the relationship between two variables using a table with a finite set of
values and graph the relationship.
Express the relationship between two variables using an equation and a graph:
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solve linear inequalities and equations in one variable
Generate an algebraic sentence to model real-life situations.
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D. Analyze changes in various contexts.
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Estimate the rate of change of a function or equation by finding the slope between
two points on the graph.
Evaluate the estimated rate of change in the context of the problem.
4. Instructional sequence for the unit:
Learning Activity
Teacher Response
1. Introduction: Two bottles
will be filled with colored
water.
Which bottle do you think
will empty first?
One bottle contains more
water than the other. The
bottles with the greater
amount of water will have
more holes than the bottle
with less water. Holes are
of equal size.
As the students are
watching the bottles drain,
they will be asked to
discuss everything they
observed.
Teacher will present
students with the following
problem:
Was your prediction
correct? Why or why not?
We anticipate that the
students will observe the
Teacher will begin draining following changes:
At first, one bottle’s height
the water in bottles while a
student times the draining in was greater than the other.
At a certain point the bottle
20 second intervals.
Students will tell the teacher with the greater height was
which bottle has more water the same as the other bottle
and then continued to have
at each interval.
the lesser amount.
Activity 2:
(Setting up the scenario:
When you fly, there isn’t
just one airline that can take
you where you need to go.
Name some of those
airlines…………Las
Cruces is getting ready to
build a Space Port that will
fly people in to space and
take them into orbit around
the earth. It takes
approximately 90 minutes
to orbit the earth. Just as
there are many airlines, this
Space Port will also have
many companies to take
people in to space. They
will be competing for our
business.)
Two companies have built
spaceships that will take the
common person in to space
and orbit the earth. Bowing
will charge $500 plus $50
for each orbit. Lochheed
will waive the initial fee.
However, Lochheed will
charge $150 per orbit. Your
task will be to decide which
the better company is.
Students will be given a
handout of the problem and
read along as the teacher
reads the problem.
In pairs, students will be
asked to make table of the
number of orbits from 1 to
9, and the cost. One student
from each pair will make a
table for Bowing and the
other student will make a
table for Lochheed.
After students have
completed their tables,
students will be given graph
paper and be asked to graph
their data.
A few select students will
be asked to transfer their
graphs to an overhead
transparency.
As the teacher is walking
around, she will answer
anticipated questions and
question students about
their data.
Again, each student will
graph their own data.
While the select students
are transferring their graphs
to transparencies. Teacher
will bring the rest of the
class together and go over
the table.
The purpose for this activity
will be to allow students to
correct their graphs.
Teacher will present two
overhead graphs and ask the
following questions:
1. Point to y intercept and
have students explain
meaning.
The purpose of this question
is to have students
recognize the y intercept
and its meaning.
2. As students how the cost
will increase from 1 orbit to
2, 2 orbits to 3, and so on.
The teacher will do the
same thing for both graphs.
The purpose of these
questions is to have the
students notice the constant
change in cost and lead
them into calling it the
slope. Once students
recognize the y intercept
and slope, students will be
asked to write the equation
of the line.
The teacher will ask the
following question.
Which company is the
better choice and why.
Students will use the
think/pair/share method.
Take 2 minutes to write
your response on the given
paper. Then discuss this
question with your partner
and be ready to share with
the class.
Next, the teacher will ask
the students to answer the
following question: Is there
a time when it doesn’t
matter which company you
choose?
The teacher will then put
both graphs together and
ask the following questions:
How can we tell when the
two companies are the same
cost? How much is it at 5
orbits for Bowing and
Lochheed?
Students will be asked to
raise their hands if they
choose Bowing, and again
raise their hand if they
choose Lochheed. One
student for each company
will be chosen to explain
their reasoning.
Teacher will choose
students to explain their
reasoning.
Students should notice that
at 5 orbits the cost is the
same. The teacher will ask
the students: How can we
show that at 5 orbits the
cost is the same? We
anticipate the students will
say, “put the two graphs
together.”
Point out on graph.
At two orbits, which is the
better company? How much
is Lochheed? How much is
Bowing? What is the
difference?
Again, at 4 orbits? Which
The purpose for this
activity: We are hoping that
students will begin to notice
the slopes of the two
graphs. They should begin
to see that past 5 orbits,
Lochheed is increasing at a
is the better company?
How much is each? What
is the difference?
Teacher will make the
following summation:
From 1 to 4 orbits,
Loohheed is the cheaper
company.
Ask the same questions for
6 orbits and 8 orbits and
make the summation for
Bowing being the cheaper
company.
The teacher will show on a
graphing calculator the
equation y = x. Students
will discuss the slope and
steepness. They will
determine the table verbally
and then be shown the table
on the overhead
screen/graphing calculator.
The following questions
will be asked; As x
increases by 1, how does
the y increase? They will
discuss the rate of change in
x.
Then students will be asked
to guess what the line y =
2x will look like. Students
will be asked to share their
guess with their partner.
Teacher will then show
rate that is 3 times greater
than Bowing. This is when
the steepness of the slope
will begin to be discussed.
graph after guesses and look
at the table and discuss rate
of change.
Students will be asked to
guess the graph of y = ½ x
and discuss their guess, then
they will be asked to look at
the table.
Conclusion: The teacher
will once again refer to the
slope of the activity and
have the students see the
relationship between the
steepness of the slope and
how fast the graph
increases.
We want the students to
recognize how the steepness
of the slope is directly
related to the slant of the
line. As the change in y
increases over the change in
x.
5. Evaluation
6. Appendix
An Introduction to the Profound Potential of Connected Algebra Activities:
Issues of Representation, Engagement and Pedagogy (ED489551)
2004-07-00
Author(s):Hegedus, Stephen J.; Kaput,
Pub Date:
James J.
Pub Type(s): Collected Works Proceedings; Reports Source: International Group for the
Psychology of Mathematics
Descriptive;
Education, 28th, Bergen,
Speeches/Meeting Papers
Norway, July 14-18, 2004
Peer-Reviewed: No
Descriptors:
Educational Technology; Algebra; Teaching Methods; Mathematics
Instruction; Computer Assisted Instruction; Instructional Effectiveness; Class Activities
Abstract:
We present two vignettes of classroom episodes that exemplify new activity structures for
introducing core algebra ideas such as linear functions, slope as rate and parametric
variation within a new educational technology environment that combines two kinds of
classroom technology affordances, one based in dynamic representation and the other
based in connectivi Note:The following two links are not-applicable for text-based
browsers or screen-reading software. Show Full Abstract
How "Focusing Phenomena" in the Instructional Environment Support
Individual Students' Generalizations. (EJ663477)
2003-00-00
Author(s):Lobato, Joanne; Ellis, Amy
Pub Date:
Burns; Munoz, Ricardo
Pub Type(s): Journal Articles; Reports Research
Source: Mathematical Thinking and
Learning, v5 n1 p1-36 2003
Peer-Reviewed: N/A
Descriptors:
Classroom Environment; Educational Change; Generalization; Mathematics
Instruction; Research and Instruction Units; Secondary Education; Teacher Student
Relationship
Abstract:
Investigates a way of connecting the classroom instructional environment with individual
students' generalization. Advances focusing phenomena, regularities in the ways in which
teachers, students, artifacts, and curricular materials act to direct attention toward certain
mathematical properties over others. Conducts an empirical study on slope and linear
functi Note:The following two links are not-applicable for text-based browsers or screenreading software. Show Full Abstract
Exploring the Phenomenon of Classroom Connectivity. (ED471755)
2002-00-00
Author(s):Hegedus, Stephen; Kaput, James Pub Date:
J.
Pub Type(s): Reports - Research;
Speeches/Meeting Papers
Source: N/A
Peer-Reviewed: N/A
Descriptors:
Algebra; Computer Uses in Education; Innovation; Mathematics Education; Secondary
Education
Abstract:
We describe highly generative and affectively powerful classroom activity structures that
are made possible by applying new levels of connectivity across diverse hardware
platforms. Based on teaching experiments involving core topics in basic algebra (slopeas-rate, linear functions, simultaneous conditions), we examine 3 kinds of activity
structures exploiting Note:The following two links are not-applicable for text-based
browsers or screen-reading software. Show Full Abstract
Making Sense of Slope. (EJ604110)
Author(s):Crawford, Ann R.; Scott,
William E.
Source: Mathematics Teacher, v93 n2
p114-18 Feb 2000
Author(s):Hegedus, Stephen; Kaput, James
J.
Source: N/A
2000-00-00
Guides - Classroom Teacher; Journal Articles
Peer-Reviewed: N/A
2002-00-00
Pub Date:
Pub Type(s): Reports - Research;
Speeches/Meeting Papers
Peer-Reviewed: N/A
Pub Date:
Pub Type(s):
Descriptors:
Algebra; Computer Uses in Education; Innovation; Mathematics Education; Secondary
Education
Abstract:
We describe highly generative and affectively powerful classroom activity structures that
are made possible by applying new levels of connectivity across diverse hardware
platforms. Based on teaching experiments involving core topics in basic algebra (slopeas-rate, linear functions, simultaneous conditions), we examine 3 kinds Note:The
following two links are not-applicable for text-based browsers or screen-reading
software. Show Full Abstract
Liquid Assets: Increasing Students' Mathematical Capital. (EJ666994)
2000-00-00
Author(s):Winter, Mary Jean; Carlson,
Pub Date:
Ronald J.
Pub Type(s): Guides - Classroom Teacher; Journal Articles
Source: Mathematics Teacher, v93 n3
p172-78 Mar 2000
Peer-Reviewed: N/A
Descriptors:
Algebra; Concept Formation; Group Activities; Learning Processes; Mathematical
Applications; Mathematics Instruction; Secondary Education; Teaching Methods
Abstract:
Describes a laboratory-type activity, liquid assets, used to illustrate, develop, or reinforce
central concepts in first-year algebra. These include linear function, slope, intercept, and
dependent and independent variables. Presents a group activity for collecting data,
transition from group to individual activity in plotting Note:The following two links are
not-applicable for text-based browsers or screen-reading software. Show Full Abstract
Activities. (EJ480188)
Author(s):Anderson, Edwin D.; Nelson,
Jim
Source: Mathematics Teacher, v87 n1
p27-30,37-41 Jan 1994
1994-00-00
Guides - Classroom Learner; Guides Classroom - Teacher;
Journal Articles
Peer-Reviewed: N/A
Pub Date:
Pub Type(s):
Descriptors:
Algebra; Concept Formation; Discovery Learning; Instructional Materials; Intermediate
Grades; Learning Activities; Manipulative Materials; Mathematical
Concepts; Mathematics Education; Mathematics Instruction; Measurement; Secondary
Education; Secondary School Mathematics; Worksheets
Abstract:
Presents a series of 5 reproducible worksheets for grade levels 5-10 that provide hands-on
activities to help develop the concept of slope. (MDH)
Activities: The Functions of a Toy Balloon. (EJ500126)
1994-00-00
Author(s):Coes, Loring, III
Pub Date:
Source: Mathematics Teacher, v87 n8
Pub Type(s): Guides - Classroom p619-22,628-29 Nov 1994
Teacher; Journal Articles
Peer-Reviewed: N/A
Descriptors:
Algebra; Functions (Mathematics); Geometric Concepts; Geometry; Learning
Activities; Lesson Plans; Manipulative Materials; Mathematics Instruction; Secondary
Education; Volume (Mathematics)
Abstract:
Gives a lesson plan for a mathematics activity using balloons which shows connections
between the algebraic concepts of slope, linear functions, and power functions and the
geometric concepts of circle, sphere, volume, and pi. Includes reproducible student
worksheets. (MKR)
Rates and Taxes. (EJ449200)
Author(s):Esty, Warren W.
Source: Mathematics Teacher, v85 n5
p376-79 May 1992
1992-00-00
Journal Articles; Guides Classroom - Teacher
Peer-Reviewed: N/A
Pub Date:
Pub Type(s):
Descriptors:
Algebra; Functions (Mathematics); High Schools; Integrated Activities; Learning
Activities; Mathematical Applications; Mathematical Formulas; Mathematics
Education; Mathematics Instruction; Percentage; Problem Solving; Tax Rates; Taxes
Abstract:
Proposes lessons for algebra students using the context of tax calculations to learn about
the concepts of slope, split functions, averages, rates, marginal rates, and percents.
Students explore ramifications of possible tax revisions. (MDH)
Utilizing the Spreadsheet and Charting Capabilities of Microsoft Works in the
Mathematics Classroom. (EJ415499)
1990-00-00
Author(s):Wood, Judith Body
Pub Date:
Source: Journal of Computers in
Pub Type(s): Journal Articles; Reports Mathematics and Science
Research
Teaching, v9 n3 p65-71 Spr 1990 Peer-Reviewed: N/A
Descriptors:
Algebra; Computation; Computer Graphics; Computer Uses in Education; Experiential
Learning; Graphs; Mathematical Applications; Mathematics Education; Secondary
Education; Secondary School Mathematics; Spreadsheets
Abstract:
Presented are ideas for the utilization of Microsoft Works in the secondary mathematics
classroom. The spreadsheet and graphing capabilities of this software package are
demonstrated for several topics including the concept of slope for first year algebra, and
creating and interpreting graphs for a general mathematics class. (A Note:The following
two links are not-applicable for text-based browsers or screen-reading software. Show
Full Abstract
Activities: Using Linear Functions. (EJ378123)
Author(s):Wallace, Edward C.
Pub Date:
Source: Mathematics Teacher, v81 n7
Pub Type(s):
p560-66 Oct 1988
1988-00-00
Journal Articles; Guides Classroom - Learner;
Guides - Classroom Teacher
Peer-Reviewed: N/A
Descriptors:
Algebra; Equations (Mathematics); Functions (Mathematics); Graphs; Instructional
Materials; Learning Activities; Mathematics Instruction; Secondary
Education; Secondary School Mathematics; Worksheets
Abstract:
This activity exposes beginning algebra students to the concepts of slope and intercept.
Four worksheets pertain to the "electrician problem." (MNS)