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Course Profile
Foundations of Mathematics, Grade 9 applied, Public
Course Overview
Course
Profiles are professional development materials designed to help teachers
implement the new Grade 9 secondary school curriculum. These materials were
created by writing partnerships of school boards and subject associations. The
development of these resources was funded by the Ontario Ministry of Education
and Training. This document reflects the views of the developers and not
necessarily those of the Ministry. Permission is given to reproduce these materials
for any purpose except profit. Teachers are encouraged to amend, revise, edit,
cut, paste, and otherwise adapt this material for educational purposes.
Any
references in this document to particular commercial resources, learning
materials, equipment, or technology reflect only the opinions of the writers of
this sample Course Profile, and do not reflect any official endorsement by the
Ministry of Education and Training or by the Partnership of school Boards that
supported the production of the document.
© Queens
Printer for Ontario
Acknowledgments
Public
District School Board Writing Teams - Mathematics
Course
Profile Writing Team
Myrna Ingalls, Lead Writer, York
Region District School Board
Shirley Dalrymple, York Region
District School Board
Carolyn Gallagher, Kawartha Pine
Ridge District School Board
Mary Howe, Ontario Association
for Mathematics Education
Irene McEvoy, Peel District
School Board
Lionel LaCroix, Peel District
School Board
Christine Surtamm, Peel District
School Board
Reviewers
Bill Clarke, Mark Pankratz, Kelly Searle, Ottawa Carleton DSB: Angela
Con, Kawartha Pine Ridge DSB; Donna Del Re, Peel DSB; Sandra Emms Jones,
Waterloo Region DSB; Gary Flewelling, Ontario Mathematics Co-ordinators Association; Ron Lewis,
Rainbow DSB; Bob McRoberts, York Region DSB
Lead Board
Peel District School Board
Allan Smith, Project Manager
Partner
Boards
Kawartha Pine Ridge District School Board, Ottawa Carleton District
School Board, Rainbow District School Board, Waterloo Region District School
Board, York Region District School Board
Associations
Ontario Association for Mathematics Education (OAME)
Ontario Mathematics
Co-ordinators Association (OMCA)
Applied Course
Profile Overview
Foundations of Mathematics, Grade 9
Identifying Information
School: Course
Developer:
Department:
District:
Course Title: Foundations
of Mathematics
Grade: 9 Development
Date:
Course Type: Academic Course
Revisor(s):
Ministry Course Code: MFM1P
Credit Value: 1
Credit Revision
Date:
Description/Rationale
This course enables students to develop mathematical ideas and methods
through the exploration of applications, the effective use of technology, and
extended experiences with hands-on activities. Students will investigate
relationships of straight lines in analytic geometry, solve problems involving
the measurement of three-dimensional objects and two-dimensional, and apply key
numeric and algebraic skills in problem solving. Students will also have
opportunities to consolidate core skills and deepen their understanding of key
mathematical concepts. p.18 The Ontario Curriculum, Grades 9 and 10,
Mathematics
This grade nine mathematics course is designed to
develop students proficiency as learners and thinkers in mathematics, and
their ability to apply mathematics in their daily lives. Students will
interpret, analyze, and create mathematical models to solve realistic problems,
within and beyond the mathematics classroom.
Unit Titles and Time
|
Unit 1: |
Constructing Graphical Models through
Investigation |
35 hours |
|
Unit 2: |
Algebraic Models and Rates of Change |
30 hours |
|
Unit 3: |
Dynamic Geometry and Measurement |
30 hours |
|
Unit 4: |
Summative Assessment Activities |
15 hours |
Unit Descriptions
Unit 1: Constructing Graphical Models through
Investigation
Time: 35 hours
Description
This Unit will introduce
abstract concepts through activities which will engage grade 9 students, and
embed the teaching of skills within contexts.
Students will gather, analyze,
manipulate, and display data from primary and secondary sources to model and
communicate results about both linear and non‑linear situations. Many
contextual problems will be studied to ensure that students gain depth of
understanding through meeting the same specific expectations in different
contexts. Students will conduct investigations to verify or refute their own
conjecture, using lines or curves of best fit, tables and pattern descriptions.
They will communicate their findings and describe trends. A rich contextual
foundation for subsequent algebraic studies will be built in this unit. Several
different types of technologies will be introduced for gathering, analyzing and
displaying data.
Overall Expectations: all those from the Relationships Strand
Specific Expectations: all those from the Relationships Strand and as identified in the activities from each of the other Strands
Unit 2: Algebraic
Models and Rates of Change
Time: 30 hours
Description
This unit is designed to:
weave the expectations of the Analytic Geometry strand together with
expectations from each of the other strands in the policy document.
introduce the abstraction of xs, ys, and vocabulary like
slope and intercepts.
allow the teacher and students to tie these abstractions back to
contexts from the first unit.
highlight the power of these abstract symbols and concepts as a means to
summarize similar models (e.g., y = 2x summarizes H = 2A, total
points are double the number of baskets, etc.) and to communicate with technology.
help students consolidate and extend their algebraic and numeration
skills.
Students will use the ideas and contexts of the
first unit to develop algebraic models of linear relations. Students will
explore and determine the characteristics of lines and their corresponding
equations through the use of spreadsheets, graphing technology, and paper and
pencil. To solve problems, students will recognize and model realistic situations
that involve constant rates of change. The need for algebraic techniques,
numeric skills and the laws of exponents will emerge from problems in context.
Overall Expectations: all those from the Analytic Geometry Strand
Specific Expectations: all those from the Analytic Geometry Strand and some from each of the other Strands
Unit 3: Dynamic Geometry and
Measurement
Time: 30 hours
Description
This unit is designed to:
weave
the expectations of the Measurement and Geometry strand together with
expectations from each of the other strands in the policy document.
help
students to extend their skills in exploring geometric relationships, forming
and testing reasonable conjectures, using dynamic geometry software and other
means to manipulate and transform, communicating their findings and applying
geometric relationships to solve problems.
help
students to consolidate and extend their algebraic and numeration skills
through work with formulas and multi step problems.
Students will use concrete materials, diagrams, drawings and dynamic geometric software to investigate the properties of three dimensional objects, optimal measurements and geometric relationships of two dimensional figures. Students will confirm and extend their intuitive understanding of geometric properties through inquiry. They will pose questions, make observation with the help of technology, judge the reasonableness of answers and solve multi step problems.
Overall Expectations: all those from the Measurement and Geometry Strand
Specific Expectations: all those from the Measurement and Geometry Strand and some from other Strands
Time: 15
hours
Description
This unit will be used to
model a final assessment in grade 9 mathematics. Individual and group
performance skills will be assessed using traditional and performance based
tasks, over a period of several days. Thirty percent of the final evaluation
for the course will be based on this summative assessment unit and it is
recommended that at least 2/3 be based on performance tasks, and at most 1/3 be
based on pencil and paper tests. It is suggested that the form and substance of
this summative assessment unit be shared with students and their parents near
the beginning of the course, so that their energies can be directed towards
acquisition of the required skills and knowledge.
In this summative assessment
unit, students will demonstrate their achievement of the expectations of the
course. They will do this by solving problems which require them to:
form
and test conjectures.
model
situations.
gather,
organize, and display data for a purpose.
identify
necessary and/or sufficient conditions in a problem
decide,
with awareness, what is important and what can be ignored in a problem.
communicate
reasoning and results.
demonstrate
their skills using technology for a purpose.
carry
out pencil and paper routines.
Course Notes
Key
messages:
Mathematical
modeling is a primary focus.
The high school mathematics curriculum is
designed to ensure that students understand the power of mathematics in
modeling authentic problems and situations, and acquire the various skills
needed to create, interpret, and analyze such models.
The
problem comes first.
Traditionally in mathematics, common
practice would be to first teach skills in isolation or limited contexts, and
then use the skills to solve problems, contextual or not. In this sample course
profile, the problem to come first, although it is still the desired
mathematical skills and knowledge that drive the choice of problems and
activities.
Teach
skills as they are needed.
It is intended that the need for the
skills (including basic manipulative skill) of the course emerge from rich,
contextual settings rather than in isolation. Teachers may well have to take
time to develop specific skills, once the need has been identified. An inquiry
approach to teaching and learning demands that teachers use flexibility to
capture teachable moments. An analogy might help:
The goal of this course is similar to creating a beautiful piece of furniture. It might be necessary, at times, to take time to practise the use of a lathe or sanding on scraps of wood, separate from the finished product. However, all work is directed towards making our fine piece of furniture.
New
classroom routines are needed.
A range of teaching strategies is expected. To implement the inquiry-related expectations of the curriculum, teachers will likely find it necessary to broaden their repertoire of strategies to facilitate investigations, explorations, and communication of findings. Because there will not be a lot of class time available to engage in large amounts of review of skills, it is recommended that diagnostic testing be used to determine the extent of focus required on numerical skills. Practice and other remediation activities can be assigned as homework. As implementation of the grade 1-8 mathematics program proceeds, teachers of Grade 9 mathematics may find that the need for diagnosis and remediation of student background knowledge and skills will become less time-consuming and leave more time for students to work on enhancements and extensions. Research and preparation for the next days investigation might also be included as homework assignments, along with consolidation and extension of classroom activities
Other points to
consider:
Much
of the material in the profiles is common to both the Grade 9 Applied and Grade
9 Academic courses. The content, procedures and processes are important for all
students to learn in both programs. The two programs allow for different
approaches and emphases as related to time allocation, instructional
approaches, and needed supports and extensions.
To
ensure that all students performing at or above the provincial standard have
the necessary knowledge and skills to succeed in either the Grade 10 Academic
or Grade 10 Applied Mathematics course, summative assessment activities should
be similar or identical for the two grade 9 courses, except where additional
expectations are involved in the Grade 9 Academic course.
Different
developments for some activities within a unit are provided for the applied and
academic streams, but students might benefit from the teachers borrowing
from the other stream.
Unit 1
is very similar in the Academic and Applied profiles since it is important for
all students to develop conceptual understandings of linear and non-linear
relationships from a concrete point of view. Unit 4 is also very similar in the
Academic and Applied versions since students who are working at level 3 or 4 in
either course should be well prepared for either the Applied or Academic grade
10 course.
It is
envisioned that Units 2 and 3 will show more differences between the Academic
and Applied courses. Unit 2 introduces the abstract concepts of lines. The
Applied version will return more frequently to concrete examples and
applications in different contexts while the academic version will extend
farther and more frequently into abstract applications. In Unit 3, geometric
properties will be explored more concretely in the Applied version and more
abstractly in the Academic. Problems will be posed in a more open-ended fashion
in the Academic course and contain more scaffolding (guiding, supporting)
questions in the Applied course.
When
doing investigations, teachers are advised to carry out the experiments
themselves, beforehand. Considerations should include timing, questions to
pose, means of drawing closure, prerequisite skills, appropriate connections
and links to other disciplines, and the choosing of investigations having
regional or current interest. Availability of computer resources may determine
the sequencing for Unit 3, since computer labs may not be available for all of
the grade 9 classes at the same time.
Technology
can be useful in learning, doing, and assessing achievement in mathematics is
desirable and essential. When using technology for an activity, teachers are
advised to practise its use beforehand. This profile will identify which
graphic calculator and dynamic geometry software skills are needed for each
activity. Occasionally, teachers may wish to demonstrate the use of technology
as a tool for gathering, organizing and displaying data, and at other times
students must learn to use the technology themselves.
Not
all Specific Expectations are of equal value. Those which are critical to the
development of mathematical literacy are emphasized in the learning activities,
and are often revisited. These are expectations which are taught, assessed,
evaluated and, where necessary, revisited using alternate instructional
strategies in a cyclic process that stops only when students have achieved the
Expectations to their optimal Levels.
Use of
the Achievement Levels Chart of Mathematics is the basis of assessment of all
aspects of the course.
The
implementation of Grade 9 Mathematics is a process, not an event. It is
necessary that those at all levels of the system make continuous and measurable
progress towards full implementation.
The
expectations of the Number Sense and Algebra strand will be woven into the
first three units rather than separated out.
Teaching/Learning
Strategies
Only through the use of a wide variety of teaching, learning, and assessment strategies and tools can the wide range of expectations in this course be addressed.
Teachers will:
include
a balance of whole class, small group and individual instruction.
include
a balance of student-centred and teacher-directed activities.
provide
students with materials, technological tools and software for use in
experiments, demonstrations, and investigations.
address
a variety of learning styles in each unit.
plan
so that sufficient class time is spent engaging students in the solution of
rich contextual problems.
be
accountable to addressing the overall and specific expectations in their planning,
and accountable to tracking student progress in the overall expectations,
including the most important specific expectations.
assume
a variety of roles in the classroom, including guide, facilitator, and director
of learning..
provide
many opportunities for students to demonstrate their ability to meet course
expectations.
ensure
that the culmination of an activity helps the students to build a solid
understanding of the mathematical concepts arising from that activity and sets
the stage for future learning.
prompt
at the beginning of an activity, provide suggestions in the middle, and support
and challenge at the end, as needed by individual students, and by the class as
a whole.
provide
verbal instruction to accompany written procedures to avoid the frustration and
uncertainty that so often undermine the learning opportunities afforded by a
complex task.
use
learning/performance tasks that are designed to link several expectations and
give the students occasion to demonstrate their optimal levels of achievement
through the communication of results, the ability to pose extending questions
following an inquiry, and to provide the solution to unfamiliar problems.
provide
remediation or extension opportunities.
provide
opportunities for students to practise or extend their skills and knowledge,
outside of the classroom.
provide
regular, informal assessment which provides the feedback that students need in
order to improve their achievement.
modify
instructional and assessment strategies for special needs students.
Students will:
develop
increasing responsibility for their own learning.
explore,
hypothesize or formulate, manipulate or transform, infer or conclude, and
communicate during an inquiry.
engage
in explorations involving the use of technology (e.g., graphing software,
dynamic geometric software, data bases, the Internet, statistical programs,
spreadsheets and multimedia resources) and the collection of data.
follow
examples and Socratic developments of concepts and take notes provided by the
teacher.
apply
individual and group learning skills.
pose
and answer questions in a context.
describe
the patterns that emerge verbally, algebraically and visually (using tables,
graphs and posters).
demonstrate
an understanding of concepts, and ability to select and perform algorithms
accurately in order to solve problems.
practise
prerequisite skills.
Assessment/Evaluation
Assessment is a systematic process of collecting information or evidence about student learning; evaluation is the judgement we make about the assessments of student learning based on established criteria. This profile will focus on providing specific examples of assessment strategies and tools, and general statements about how these assessments might be used in evaluation. Evaluation requires that the teacher not simply average marks. In forming an evaluative judgement, the students highest, most consistent level of achievement should be used.
The focus of this course is on inquiry, problem solving, communication, and acquisition of high levels of knowledge and skills and application of mathematics. Knowledge and understanding continue to be important. Assessment looks at students meeting course expectations at a variety of levels, with an emphasis on growth over time. Assessment should be used to gather information for diagnostic, formative and summative purposes. It is important to note that assessment and evaluation will be criterion referenced, comparing student performance to the Ministry standard, not to other students. Level 3 is defined as the provincial standard. A student achieving at this level is well-prepared for work in the Grade 10 Academic or Applied course. Level 4 performance requires a consistent demonstration of well-communicated higher level thinking and not simply technically correct solutions.
Assessment strategies and tools must address the variety of teaching and learning styles as well as the variety of expectations. High quality assessment can measure individual and group performance, and individual performance within a group. A balanced assessment program will include methods:
to
assess Understanding of Conceptual and Procedural Knowledge/Understanding:
tests, quizzes, and observation of performance tasks.
to
assess Thinking/Inquiry/Problem Solving, and Application in unfamiliar
settings: performance assessment, observation, and conferencing.
to
assess Communication: journals, portfolios, performance assessments,
observations and presentations.
to
assess Application in familiar settings: tests, quizzes, performance
assessments.
to
assess Learning Skills and to set goals: journals, portfolios, observations and
conferencing.
Assessment tools to be used throughout the course include:
the
four level Achievement Chart.
rubrics
(both teacher‑created and student‑generated).
checklists.
rating
scales.
anecdotal
comments.
analytic
marking schemes.
A selection of these tools has been designed to accompany specific assessment activities. Teachers are encouraged to use them, then develop similar tools for other assessment activities. Some suggestions for increasing scoring consistency include:
involve
other teachers in the department in the creation of rubrics for assessment.
involve
students in the setting of criteria, and the use of self and peer assessments.
gather
exemplars of student work at the four levels, so that teachers and students can
get a better image of what achievement at these levels looks like.
Assessment of the expectations using the 4 levels of the Achievement
Chart will be ongoing throughout the Unit. There will be a summative
performance activity and a summative pencil and paper test for each Unit.
Assessment tools will be designed to allow students to demonstrate performance
at the full range of their learning (levels 1 to 4). Some smaller pencil and
paper assessment tools may not allow for demonstration of Level 4 performance.
Accommodations
The following accommodations for students with special needs should be made throughout the course:
Accommodations for ESL/ESD students:
The
unit provides many activities to learn that are not language driven, such as
data gathering and recording. When students make submissions, they should have
opportunities to use point form rather than prose, and graphical or pictorial
representations. Alternatively, a member of their group could help them record
their observations and conclusions. A glossary of terms may assist with
explanations and definitions.
Accommodations for
students with learning disabilities:
Students
may have directions read to them by a member of their group or the teacher.
Some students may benefit from being given more detailed directions identifying
key components of problems. Students with communications difficulties may be
given the option to present their solutions orally to the teacher, practising
with a group member first. Information can be taped or the student provided
scribed notes. Accommodation for extra time should be made. Time has been
allotted for consolidation of numeric skills from earlier grades. Teachers
could modify the amount of work assigned. Calculators can be used throughout
the course as needed.
Resources
Practical
Flewelling, Gary
and Lemenchick, Chuck, Mathematics
Assessment Grade 9. Gage
NCTM Activities For Active Learning and
Teaching
EQAO Grade 9 Sample Assessment Document. 1998
OMCA, OAME Linking Assessment and Instruction in Mathematics: Connecting to the Ontario Provincial Standards. 1995
ProQuest, http://www.umi.com/proquest, This website provides access to more than 3000 journals, magazines, dissertations, newspapers, and other publications, for a fee. This is a good source of secondary data. There are several similar services available
Professional Reading
NCTM, Mathematics Assessment: Myths, Models, Good Questions and Practical Suggestions
Stenmark, Jean Kerr, Assessment Alternatives in Mathematics. Assessment Committee of the California Mathematics Council
Bennett, Barrie et. al., Cooperative Learning: Where Hearts Meet Minds. 1991 (available from Educational Connections, P.O. Box 249, Station P, 704 Spadina Avenue, Toronto, Ontario)
Clarke, Judy et. al., Together We Learn: Co-operative Small Group Learning, Prentice Hall, 1990
Course
Evaluation
Course improvement should be viewed as an ongoing and collaborative process among mathematics teachers. As new resources, new technology, and new insights on the programs develop, teachers will adapt their programs to better serve the needs of their students.
To meet these goals, teachers should evaluate the effectiveness of their courses using a variety of information sources. While students performance on summative evaluations such as class tests, the final assessment, and the grade 9 EQAO mathematics assessments are obvious indicators of a courses success, many other sources of information are available to teachers as well. These include students reflections on their learning in their mathematics journals, parental feedback, and student performance in subsequent mathematics courses as well as other subject disciplines which build on grade nine mathematics.
Anecdotal evidence can be gathered from observing the following indicators:
the
care taken by the students in their work
students
efforts to complete their work and seek help as needed
students
pursuit of extension activities
students
growth in independence and persistence when completing tasks
Coded Expectations: Foundations of Mathematics, Grade 9, Applied
Number
Sense and Algebra
Overall
Expectations
NAV.01
consolidate numerical skills by using them in a variety of contexts throughout the course;
NAV.02
demonstrate understanding of the three basic exponent rules and apply them to simplify expressions;
NAV.03
manipulate first‑degree polynomial expressions to solve first‑degree equations;
NAV.04
solve problems, using the strategy of algebraic modelling.
Specific Expectations
Consolidating Numerical Skills
NA1.01
determine strategies for mental mathematics and estimation, and apply these strategies throughout the course;
NA1.02
demonstrate facility in operations with integers, as necessary to support other topics of the course (e.g., polynomials, equations, analytic geometry);
NA1.03
demonstrate facility in operations with percent, ratio and rate, and rational numbers, as necessary to support other topics of the course (e.g., analytic geometry, measurement);
NA1.04
use a scientific calculator effectively for applications that arise throughout the course;
NA1.05
judge the reasonableness of answers to problems by considering likely results within the situation described in the problem;
NA1.06
judge the reasonableness of answers produced by a calculator, a computer, or pencil and paper, using mental mathematics and estimation.
Operating
with Exponents
NA2.01
evaluate numerical expressions involving natural‑number exponents with rational‑number bases;
NA
2.02
substitute into and evaluate algebraic
expressions involving exponents, to support other topics of the course (e.g., measurement, analytic geometry);
NA2.03
determine
the meaning of negative exponents and of zero as an exponent from activities
involving graphing, using technology, and from activities involving patterning;
NA2.04
represent
very large and very small numbers, using scientific notation;
NA2.05
enter
and interpret exponential notation on a scientific calculator, as necessary in
calculations involving very large and very small numbers;
NA2.06
determine,
from the examination of patterns, the exponent rules for multiplying and
dividing monomials and the exponent rule for the power of a power, and apply
these rules in expressions involving one variable.
Manipulating Polynomial Expressions
and Solving Equations
NA3.01
add
and subtract polynomials, and multiply a polynomial by a monomial;
NA3.02
expand
and simplify polynomial expressions involving one variable;
NA3.03
solve
first‑degree equations, excluding equations with fractional coefficients,
using an algebraic method;
NA3.04
calculate
sides in right triangles, using the Pythagorean theorem, as required in topics
throughout the course (e.g., measurement);
NA3.05
substitute
into measurement formulas and solve for one variable, with and without the help
of technology.
Using Algebraic Modelling to Solve
Problems
NA4.01
use
algebraic modelling as one of several problem‑solving strategies in
various topics of the course (e.g., relations, measurement, direct and partial
variation, the Pythagorean theorem, percent);
NA4.02
compare
algebraic modelling with other strategies used for solving the same problem;
NA4.03
communicate
solutions to problems in appropriate mathematical forms (e.g., written
explanations, formulas, charts, tables, graphs) and justify the reasoning used
in solving the problems.
Relationships
Overall
Expectations
REV.01
determine
relationships between two variables by collecting and analysing data;
REV.02
compare
the graphs of linear and non‑linear relations;
REV.03
describe
the connections between various representations of relations.
Specific
Expectations
Determining
Relationships
RE1.01
pose
problems, identify variables, and formulate hypotheses associated with
relationships (Sample problem: Does the rebound height of a ball depend
on the height from which it was dropped? Make a hypothesis and then design an
experiment to test it);
RE1.02
demonstrate
an understanding of some principles of sampling and surveying (e.g.,
randomization, representivity, the use of multiple trials) and apply the
principles in designing and carrying out experiments to investigate the
relationships between variables (Sample problem: What factors might
affect the outcome of this experiment? How could you design the experiment to
account for them?);
RE1.03
collect
data, using appropriate equipment and/or technology (e.g., measuring tools,
graphing calculators, scientific probes, the Internet) (Sample problem:
Drop a ball from varying heights, measuring the rebound height each time);
RE1.04
organize
and analyse data, using appropriate techniques (e.g., making tables and graphs,
calculating measures of central tendency) and technology (e.g., graphing
calculators, statistical software, spreadsheets) (Sample problem: Enter
the data into a spreadsheet. Decide what analysis would be appropriate to
examine the relationship between the variables a graph, measures of central
tendency, ratios);
RE1.05
describe
trends and relationships observed in data, make inferences from data, compare
the inferences with hypotheses about the data, and explain the differences
between the inferences and the hypotheses (Sample problem: Describe any
trend observed in the data. Does a relationship seem to exist? Of what sort? Is
the outcome consistent with your original hypothesis? Discuss any outlying
pieces of data and provide explanations for them. Suggest a formula relating
the rebound height to the height from which the ball was dropped. How might you
vary this experiment to examine other relationships?);
RE1.06
communicate
the findings of an experiment clearly and concisely, using appropriate
mathematical forms (e.g., written explanations, formulas, charts, tables,
graphs), and justify the conclusions reached;
RE1.07
solve
and/or pose problems related to an experiment, using the findings of the
experiment.
Comparing Linear and Non-linear
Relations
RE2.01
construct
tables of values, graphs, and formulas to represent linear relations derived
from descriptions of realistic situations involving direct and partial
variation (e.g., the cost of holding a banquet in a rented hall is $25 per
person plus $975 for the hall);
RE2.02
construct
tables of values and scatter plots for linearly related data involving direct
variation collected from experiments (e.g., the rebound height of a ball versus
the height from which it was dropped);
RE2.03
determine
the equation of a line of best fit for a scatter plot, using an informal
process (e.g., a process of trial and error on a graphing calculator;
calculation of the equation of the line joining two carefully chosen points on
the scatter plot);
RE2.04
construct
tables of values and graphs to represent non‑linear relations derived
from descriptions of realistic situations (e.g., represent the relationship
between the volume of a cube and its side length, as the side length varies);
RE2.05
demonstrate
an understanding that straight lines represent linear relations and curves
represent non‑linear relations.
Describing Connections Between
Representations of Relations
RE3.01
determine
values of a linear relation by using the formula of the relation and by
interpolating or extrapolating from the graph of the relation (e.g., if a
student earns $5/h caring for children, determine how long he or she must work
to earn $143);
RE3.02
describe,
in written form, a situation that would explain the events illustrated by a
given graph of a relationship between two variables (e.g., write a story that
matches the events shown in the graph);
RE3.03
identify,
by calculating finite differences in its table of values, whether a relation is
linear or non‑linear;
RE3.04
describe
the effect on the graph and the formula of a relation of varying the conditions
of a situation they represent (e.g., if a graph showing partial variation
represents the cost of producing a yearbook, describe how the appearance of the
graph changes if the cost per book is altered; describe how it changes if the
fixed costs are altered).
Analytic Geometry
Overall
Expectations
AGV.01
determine,
through investigation, the relationships between the form of an equation and
the shape of its graph with respect to linearity and non‑linearity;
AGV.02
determine,
through investigation, the properties of the slope and y‑intercept of a
linear relation;
AGV.03
graph
a line and write the equation of a line from given information.
Specific
Expectations
Investigating the
Relationship Between the Equation of a Relation and the Shape of Its Graph
AG1.01
determine,
through investigations, the characteristics that distinguish the equation of a
straight line from the equations of non‑linear relations (e.g., use
graphing software to obtain the graphs of a variety of linear and non‑linear
relations from their equations; classify the relations according to the shapes
of their graphs; focus on the characteristics of the equations of linear
relations and how they differ from the characteristics of the equations of non‑linear
relations);
AG1.02
select
the equations of straight lines from a given set of equations of linear and non‑linear
relations;
AG1.03
identify
y=mx + b as a standard form for the equation of a straight line,
including the special cases x=a, y=b.
Investigating the Properties of
Slope
AG2.01
identify
practical situations illustrating slope (e.g., ramps, slides, staircases) and
calculate the slopes of the inclines;
AG2.02
determine
the slope of a line segment, using the formula m = rise/run;
AG2.03
identify
the geometric significance of m and b in the equation y=mx + b through
investigation;
AG2.04
identify
the properties of the slopes of line segments (i.e., direction, positive or
negative rate of change, steepness, parallelism, perpendicularity) through
investigations facilitated by graphing technology, where appropriate.
Graphing and Writing Equations of
Lines
AG3.01
plot
points on the xy‑plane and use the terminology and notation of the
xy‑plane correctly;
AG3.02
graph
lines by hand, using a variety of techniques (e.g., making a table of values,
using intercepts, using the slope and y‑intercept);
AG3.03
graph
lines, using graphing calculators or graphing software;
AG3.04
determine
the equation of a line, given the slope and y‑intercept, the slope
and a point on the line, and two points on the line;
AG3.05
communicate
solutions in established mathematical form, with clear reasons given for the
steps taken.
Measurement and Geometry
Overall
Expectations
MGV.01
determine
the optimal values of various measurements through investigations facilitated
by the use of concrete materials, diagrams, and calculators or computer
software;
MGV.02
solve
problems involving the measurement of two‑dimensional figures and three‑
dimensional objects;
MGV.03
formulate
conjectures and generalizations about geometric relationships involving two‑dimensional
figures, through investigations facilitated by dynamic geometry software, where
appropriate.
Specific
Expectations
Investigating the
Optimal Values of Measurements
MG1.01
construct
a variety of rectangles for a given perimeter and determine the maximum area
for a given perimeter;
MG1.02
construct
a variety of square‑based prisms for a given volume and determine the
minimum surface area for a square‑based prism with a given volume;
MG1.03
construct
a variety of cylinders for a given volume and determine the minimum surface
area for a cylinder with a given volume;
MG1.04
describe
applications in which it would be important to know the maximum area for a
given perimeter or the minimum surface area for a given volume (e.g., building
a fence, designing a container).
Solving Problems Involving
Measurement
MG2.01
solve
problems involving the area of composite plane figures (e.g., combinations of
rectangles, triangles, parallelograms, trapezoids, and circles);
MG2.02
solve
simple problems, using the formulas for the surface area of prisms and
cylinders and for the volume of prisms, cylinders, cones, and spheres;
MG2.03
solve
problems involving perimeter, area, surface area, volume, and capacity in
applications;
MG2.04
judge
the reasonableness of answers to measurement problems by considering likely
results within the situation described in the problem;
MG2.05
judge
the reasonableness of answers produced by a calculator, a computer, or pencil
and paper, using mental mathematics and estimation.
Investigating Geometric
Relationships
MG3.01
illustrate
and explain the properties of the interior and exterior angles of triangles and
quadrilaterals, and of angles related to parallel lines;
MG3.02
determine
the properties of angle bisectors, medians, and altitudes in various types of
triangles through investigation;
MG3.03
determine
some properties of the sides and the diagonals of quadrilaterals (e.g., the
diagonals of a rectangle bisect each other);
MG3.04
communicate
the findings of investigations, using appropriate language and mathematical
forms (e.g., written explanations, diagrams, formulas, tables).
Continue to Unit 1
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