Foundations for College Mathematics
This course enables students to broaden their understanding of real-world applications of mathematics. Students will analyse data using statistical methods; solve problems involving applications of geometry and trigonometry; solve financial problems connected with annuities, budgets, and renting or owning accommodation; simplify expressions; and solve equations. Students will reason mathematically and communicate their thinking as they solve multi-step problems. This course prepares students for college programs in areas such as business, health sciences, and human services, and for certain skilled trades.
Register now- Department: Math
- Course Developer: The Educators Academy
- Development Date:
- Revision Date: 2021
- Course Title: Foundations for College Mathematics
- Course Reviser: Meenakhshi Shah
- Grade: Grade 12
- Course Type: College Preparation
- Ministry Course Code: MAP4C
- Credit Value: 01
- Prerequisite: Foundations for College Mathematics, Grade 11, College Preparation or Functions and Applications, Grade 11, University/ College Preparation
- Ministry Curriculum Policy Document: The Ontario Curriculum, grades 11 and 12, 2007 (Revised)
Overall Curriculum Expectations
Mathematical Models
-
i. develop, select, apply, and compare a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding;
ii. develop and apply reasoning skills (e.g., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make mathematical conjectures,
iii. assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;
iv. demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by assessing the effectiveness of strategies and processes used, by proposing alternative approaches, by judging the reasonableness of results, by verifying solutions);
v. select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems;
vi. make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);
vii. create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems;
viii. communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
Personal Finance
-
i. evaluate powers with rational exponents, simplify algebraic expressions involving exponents, and solve problems involving exponential equations graphically and using common bases;
ii. describe trends based on the interpretation of graphs, compare graphs using initial conditions and rates of change, and solve problems by modelling relationships graphically and algebraically;
iii. make connections between formulas and linear, quadratic, and exponential relations, solve problems using formulas arising from real-world applications, and describe applications of mathematical modelling in various occupations.
Geometry and Trigonometry
-
i. demonstrate an understanding of annuities, including mortgages, and solve related problems using technology;
ii. gather, interpret, and compare information about owning or renting accommodation, and solve problems involving the associated costs;
iii. design, justify, and adjust budgets for individuals and families described in case studies, and describe applications of the mathematics of personal finance.
Data Management
-
i. solve problems involving measurement and geometry and arising from real-world applications;
ii. explain the significance of optimal dimensions in real-world applications, and determine optimal dimensions of two-dimensional shapes and three-dimensional figures;
iii. solve problems using primary trigonometric ratios of acute and obtuse angles, the sine law, and the cosine law, including problems arising from real-world applications, and describe applications of trigonometry in various occupations
Unit Outline
# | Unit | Approx. Time |
---|---|---|
1 | Mathematical Models | 27 Hours |
2 | Personal Finance | 27 Hours |
3 | Geometry and Trigonometry | 27 Hours |
4 | Data Management | 27 Hours |
5 | Final Examination | 02 Hours |
Total | 110 Hours |
Unit Description
Mathematical Models
In this unit, we will look at various means of graphically representing relationships. We will consider different situations, and determine the form of graphical representation that would best illustrate the relationship. We will develop the skills to both produce and analyse such graphs. We will also look at ways of representing relationships using algebra. We will again look at a number of different relationships, and determine how they might best be described using the language of mathematics. We will use these representations to further study and develop the relationships.
Personal Finance
Mathematics becomes a critical life-skill when we apply it to our finances. We will look at various financial applications including annuities and mortgages, and gain the skills that will allow us to make educated and rational choices when faced with some of the biggest decisions we will ever make. We will also bring those skills into the personal budget, and look at the cost of living. We will consider savings plans, renting versus owning a home, and maintaining a budget. We will develop skills that everyone, regardless of their path thorough life, will need and use.
Geometry and Trigonometry
In this unit, we will look at the basics of Trigonometry, the study of triangles. We will consider different methods of determining information about triangles, the lengths of sides and their angles, and develop the skills that will enable us to choose an appropriate strategy based on the information we have. We will also consider the measurement of length area and volume. We will consider the different ways of measuring, and develop an understanding of their origins. We will tackle one of Mathematics' most useful tools- optimization- the process of maximizing one quantity given certain restraints in another.
Data Management
In this unit, we will gain some vital skills in data management. Perhaps the most important of those is the ability to analyse information, to spot patterns, and to be able to make predictions based on them. We will use tools that will help us process information and develop the skills that will enable us to analyse our findings.
Program Considerations
Assessment and Evaluation
- Projects
- Assignments
- Tests
- Classroom Discussions
- Questions and Answers during Investigation
- Presentations
- Final Exam
- Worksheets
- Group Discussions
- Investigations
- Homework
- Practice Worksheets
- Pre-Tests
- Portfolios
- Self Evaluations
- Exit Cards
- Conversations
- Checklists
- Rubrics
- provide a common framework that encompasses the curriculum
expectations for all courses outlined in this document;
- guide the development of quality assessment tasks and tools (including
rubrics);
- help teachers to plan instruction for learning;
- assist teachers in providing meaningful feedback to students;
- Seventy per cent of the grade will be based on
evaluations conducted throughout the course. This portion of the grade
should reflect the student’s most consistent level of achievement
throughout the course, although special consideration should be given to
more recent evidence of achievement.
- Thirty per cent of the grade will be based on a
final evaluation in the form of an examination, performance, essay, and/or
other method of evaluation suitable to the course content and administered
towards the end of the course.
A Summary Description of Achievement
in Each Percentage Grade Range
and Corresponding Level of Achievement |
||
Percentage Grade
Range
|
Achievement Level
|
Summary
Description
|
80-100%
|
Level 4
|
A very high to outstanding level of achievement.
Achievement is above the provincial standard.
|
70-79%
|
Level 3
|
A high level of achievement. Achievement is at the
provincial standard.
|
60-69%
|
Level 2
|
A moderate level of achievement. Achievement
is below, but approaching, the provincial standard.
|
50-59%
|
Level 1
|
A passable level of achievement. Achievement
is below the provincial standard.
|
below 50%
|
Level R
|
Insufficient achievement of curriculum
expectations. A credit will not be granted.
|
Categories
|
50–59%
(Level
1)
|
60–69%
(Level
2)
|
70–79%
(Level
3)
|
80–100%
(Level
4)
|
Knowledge
and Understanding
|
The student:
|
|
|
|
Knowledge of content
(e.g., facts, terms,
procedural skills, use
of tools)
Understanding of
mathematical concepts
|
– demonstrates limited
knowledge of content
– demonstrates limited
understanding of
concepts
|
– demonstrates some
knowledge of content
– demonstrates some
understanding of
concepts
|
– demonstrates
considerable knowledge
of content
– demonstrates
considerable understanding
of concepts
|
– demonstrates
thorough knowledge
of content
– demonstrates
thorough understanding
of concepts
|
Understanding of content (e.g., concepts, ideas, theories, principles,
procedures,
processes)
|
demonstrates
limited
understanding
of content
|
demonstrates
some
understanding
of content
|
demonstrates
considerable
understanding
of content
|
demonstrates
thorough
understanding
of content
|
Categories
|
50–59%
(Level
1)
|
60–69%
(Level
2)
|
70–79%
(Level
3)
|
80–100%
(Level
4)
|
Thinking/
Inquiry
|
The student:
|
|
|
|
Use of planning skills
– understanding the
problem (e.g., formulating
and interpreting
the problem, making
conjectures)
– making a plan for solving
the problem)
|
– uses planning
skills with limited
effectiveness
|
uses planning
skills with some
effectiveness
|
uses planning skills
with considerable
effectiveness
|
– uses planning skills
with a high degree
of effectiveness
|
Use of processing skills
– carrying out a plan (e.g., collecting data,
questioning, testing, revising, modelling,
solving, inferring, forming conclusions)
– looking back at the solution (e.g.,
evaluating
reasonableness,
making convincing
arguments, reasoning,
justifying, proving,reflecting)
|
uses processing
skills and
strategies with
limited
effectiveness
|
uses processing
skills with
some
effectiveness
|
uses processing
skills with
considerable
effectiveness
|
uses processing
skills with a
high degree of
effectiveness
|
Use of critical/creative
thinking processes (e.g.,
problem solving, inquiry)
|
uses critical/
creative thinking
processes with limited
effectiveness
|
uses critical/
creative thinking
processes with some
effectiveness
|
uses critical/
creative thinking
processes, with considerable
effectiveness
|
uses critical/
creative thinking
processes with a high degree of
effectiveness
|
Categories
|
50–59%
(Level
1)
|
60–69%
(Level
2)
|
70–79%
(Level
3)
|
80–100%
(Level
4)
|
Communication
|
The student:
|
|
|
|
Expression and organization of ideas and
mathematical thinking (e.g.,
clarity of expression, logical
organization), using oral, visual, and written forms
(e.g., pictorial, graphic, dynamic, numeric, algebraic
forms; concrete materials)
|
expresses and organizes
mathematical
thinking with limited
effectiveness
|
expresses and organizes
mathematical
thinking with some
effectiveness
|
expresses and organizes
mathematical
thinking with considerable
effectiveness
|
expresses and organizes
mathematical
thinking with a high degree of
effectiveness
|
Communication for different
audiences (e.g., peers, teachers) and purposes
(e.g., to present data, justify a solution, express a mathematical argument)
in oral, visual, and written formsin oral, visual, and/ or written forms
|
communicates for
different audiences and
purposes with
limited effectiveness
|
communicates for
different
audiences and
purposes with
some
effectiveness
|
communicates
for different
audiences and
purposes with
considerable
effectiveness
|
communicates
for different
audiences and
purposes with a
high degree of
effectiveness
|
Use of conventions,
vocabulary, and terminology
of the discipline (e.g.,
terms, symbols) in oral,
visual, and written forms
|
uses conventions,
vocabulary, and
terminology of the discipline with limited
effectiveness
|
uses conventions,
vocabulary, and
terminology of
the discipline
with some
effectiveness
|
uses conventions,
vocabulary,and terminology of
the discipline
with considerable
effectiveness
|
uses conventions,
vocabulary, and
terminology of
the discipline
with a high
degree of
effectiveness
|
Categories
|
50–59%
(Level
1)
|
60–69%
(Level
2)
|
70–79%
(Level
3)
|
80–100%
(Level
4)
|
Application
|
The student:
|
|
|
|
Application of knowledge
and skills in familiar
contexts
|
applies
knowledge and
skills in familiar
contexts with
limited
effectiveness
|
applies
knowledge and
skills in familiar
contexts with
some
effectiveness
|
applies
knowledge and
skills in familiar
contexts with
considerable
effectiveness
|
applies
knowledge and
skills in familiar
contexts with a
high degree of
effectiveness
|
Transfer of knowledge and
skills to new contexts
|
transfers
knowledge and
skills to unfamiliar
contexts with
limited
effectiveness
|
transfers
knowledge and
skills to unfamiliar
contexts with
some effectiveness
|
transfers
knowledge and
skills to unfamiliar
contexts with
considerable
effectiveness
|
transfers
knowledge and
skills to unfamiliar
contexts with a
high degree of
effectiveness
|
use of equipment, materials and technology
|
uses equipment, materials and technology safely
and correctly only with supervision
|
uses equipment, materials and technology safely
and correctly with some supervision
|
uses equipment, materials and technology safely
and correctly
|
demonstrates and promotes the safe and correct
uses of
equipment, materials and technology
|
Making connections within
and between various contexts (e.g., connections
between concepts, representations, and forms
within mathematics; connections involving use of
prior knowledge and experience; connections between
mathematics, other disciplines, and the real world)
|
makes connections
within and between
various contexts with
limited effectiveness
|
makes connections
within and between
various contexts with
some effectiveness
|
makes connections
within and between
various contexts
with considerable
effectiveness
|
makes connections
within and between
various contexts
with a high degree
of effectiveness
|
- no accommodations or modifications; or
- accommodations only; or
- modified expectations, with the possibility of
accommodations.
Teaching & Learning Strategies
- Communicating: To improve student success there will be
several opportunities for students to share their understanding both in
oral as well as written form.
- The use of technological
tools and software
(e.g., graphing software, dynamic geometry software, the Internet,
spreadsheets, and multimedia) in activities, demonstrations, and
investigations to facilitate the exploration and understanding of
mathematical concepts;
- Learning
Goals and Success Criteria is explained to the students before starting any unit, task or
activity.
- Problem
solving: Scaffolding
of knowledge, detecting patterns, making and justifying conjectures,
guiding students as they apply their chosen strategy, directing students
to use multiple strategies to solve the same problem, when appropriate,
recognizing, encouraging, and applauding perseverance, discussing the
relative merits of different strategies for specific types of problems.
- Reasoning
and proving: Asking
questions that get students to hypothesize, providing students with one or
more numerical examples that parallel these with the generalization and
describing their thinking in more detail.
- Reflecting: Modeling the reflective process, asking
students how they know.
- Selecting
Tools and Computational Strategies: Modeling the use of tools and having
students use technology to help solve problems.
- Connecting: Activating prior knowledge when introducing
a new concept in order to make a smooth connection between previous
learning and new concepts, and introducing skills in context to make
connections between particular manipulations and problems that require
them.
- Representing: Modeling various ways to demonstrate
understanding, posing questions that require students to use different
representations as they are working at each level of conceptual
development - concrete, visual or symbolic, allowing individual students
the time they need to solidify their understanding at each conceptual
stage.
- Group
Work: Working
cooperatively in
groups reduces isolation and provides students with opportunities to share
ideas and communicate
their thinking in a supportive environment as they work together towards a common goal.
- Comparison and
evaluation of written work is very important in this course. This course
focuses on giving many examples of correct work, and helping students
build the skills needed to peer-correct and self correct.
- Oral presentations are a good tool
for learning. Students can learn from one another, and from their teacher.
Charts and graphs are used to present effective learning opportunities of
concepts and skills to different students.