Math 11: Functions Unit Plan

MAED 314A Unit Plan Template
Name: Sam Douglas
Title of unit and grade/ course: Functions/Math 11


1) Rationale and connections:
a) Why do we consider it important for students to learn this topic? Why is it included in the IRPs? (< 150 words)
It is massively important to learn about functions. The idea of a mapping from one set to another can be used to describe a wide assortment of natural phenomena. Moreover, the inclusion of graphical elements allows for a conceptual transition from discrete computational ideas to a more continuous understanding of what a number can be. Additionally, even if a student decides not to take a path in life that explicitly requires a deep understanding of mathematics, a basic knowledge of functions will still be essential. Functions can be used to convey an understanding of causal relationships between data. This is essential for activities such as understanding the media’s use of statistics, or looking at a graph of stock market activity.
b) What are the historical origins and connections for this topic? (<100 words)
The notion of allowing an algebraic expression to vary over a set of values dates back to 12th century Persian mathematicians and their endeavours to solve cubic equations. Our modern understanding of a function as a curve, and the use of f(x) notation are due to Leibniz and Euler.
c) How does this topic connect with life outside mathematics? (<100 words)
Functions are utilized in a vast number of applications. Notable examples include engineering, biology, chemistry, physics, finance and sociology. The main reason for this is the importance of mathematically based models. Functions are used heavily in the physical and social sciences because they allow for easy extrapolation and interpolation when given a discrete set of data.

2) Balanced teaching, assessment and evaluation plan
a) Describe your balanced assessment and evaluation plan. Consider: •teacher, peer and self-assessment; •assessment of student learning, of teaching, and of the unit as a whole•the weighting of marks to take account of summative and formative assessment, instrumental and relational learning
Not having a great deal of experience with the different forms of assessment, I would be interested to try as many as possible to see what works best. In terms of formative assessment, I will mark homework strictly based on completion, and I will allow time in and outside of class for questions. I will also give 2 quizzes. One after the first 4 lessons, and the second after the 8th lesson. The quizzes will be peer-marked in class. Questions will be reviewed as needed. Together, I will count homework and the quizzes for 15% of the total grade. The modeling project will count for 20%, and it will be completed in groups of 4. The final unit test will count for 65%. If need be, I can tweak the homework/quiz mark, or the project mark if I feel that a student’s grade is not accurately reflected by the overall evaluation.

b) Project title and 50-word description

Mathematical Modeling.

To be done in groups of four. Students will be given a set of semi-realistic data. For this data, they will do three functional approximations: i) linear, ii) quadratic, iii) cubic. They will then do a short write-up on each approximation, describing their methods used and which one they felt was most accurate.




c) List of 10 lessons with brief topic outline and teaching strategies to be used.

Lesson topic
Teaching strategies/ approaches used
1)Operations With Functions



· Warm up
· Lecture/ hand out
· Group work / think pair / share
2) Composition of Functions



· Warm up/ activity
· Lecture/ discovery/ hand out
· Group work

3) Inverse Functions



· Introduction and examples
· Inverse pair matching game
· Recap.
4) Polynomial Functions and Inequalities



· Warm up/ hand out to discover
· Lecture/ Graphing
· Group activity
5) Absolute Value Functions 1



· Discovery by hand out
· Paired graphing activity
· Competition review
6) Equations and inequalities with absolute value functions.


· Hand out/ lecture
· Discovery/ examples / Graphing
· Group activity
7) Rational Functions 1




· Warm up / discovery
· Emphasize idea of division by zero and asymptotic behaviour.
8) Equations and inequalities with rational functions.



Graphical interpretation of solutions w/ aid of CAS.
Reason for stating restrictions.
Q&A session and collaborative problem solving.
9) Radical functions, equations and inequalities.



Relate to prior instances of functions.
Extraneous solutions.
Inverse of quadratics with restricted domain.
10) Review class
Focus on concepts relating to asymptotes and extraneous solutions.
Domain and range.


3) In detail:

a) Lesson plans for three lessons, showing a balanced instructional approach. (Note that you cannot use only lectures, homework, quizzes and tests to pass this assignment – that does not characterize a balanced approach!) Each lesson plan should be one page long.
b) Project plan for the unit project. Include a description, a rationale and a marking scheme (one page total).

Mathematical Modeling Project
Step 1
Students will be split into groups of 4, and they will be given a set of data relating to some real life situation which they are to approximate.
Examples include: sales figures, pollution levels, etc.
Step 2
Students will utilize the data to create linear, quadratic and cubic approximations.
Students will use systems of linear equations to find coefficients which will fit the curves to the data.
Students will use their approximations to solve a problem related to extrapolation or interpolation of the data.
Step 3
Students will prepare a 6-8 page report including an introduction, each of their approximations (with full calculations), a write-up discussing which they felt was most accurate, and their findings related to their given problem.

Marking Scheme
The project will be marked mainly based on the accuracy and effective presentation of the calculations. 40% for the computations, 40% for effective choice of points to be used in the curve fitting, and 20% for overall presentation of the report (includes introduction, comparison, summary of findings and effective use of graphical aids).








Math 11 – Operations with Functions

Bridge:
- Introduction to Functions and some mathematical history. Sharaf al-Din al-Tusi, Euler and Leibniz.

Warm-up and Discovery:
- Ask students to simplify some Algebraic statements.
- Have students complete tables of values for selected f(x), g(x) and operations with each.
- Relate the idea to linear functions (already familiar by this point).

Learning Objective:
- Students will be able to do operations related to functions.
- Students will understand the idea of mappings between sets.
- Students will be able to identify domain and range for a given function.

Teaching Objective:
- To make them familiar with operations involving functions and how to use the graphing calculator.
Pre-test:
- Asked students during warm-up and discovery what they know about functions.
Participation:
- Students will be asked to answer questions all the time during the lecture. Asking students to participate and do group work activities.
Post-test:
- Exercises to be done by students in groups.
Summary:
- Summarizing how to use the graphing calculator and what functions are again.



Math 11 – Inverse Functions
Bridge:
- Introducing what inverse functions are.
Warm-up and Discovery:
- Get students to do some tables of values for some of the inverse functions, provided for them in a work sheet, and let them compare the graphs of these functions with their original functions.
- Idea: encourage intuitive development of what inverse functions are and how they relate to their original function.
Learning Objective:
- Inverse Functions and how to graph them with / without a graphing calculator.
Teaching Objective:
- To make the students familiar with the concept of “inverse functions”
- To teach students how to graph an inverse function.
- To make students realize the relationship between a function and its inverse.
Pre-test:
- Ask students during warm-up and discovery to answer questions related to this part.
Participation:
- Matching activity. Pass out function examples and have students find inverses.
- Continue activity by having students travel around the class to find their ‘inverse matches.’
Post-test:
- Group problem sets. Assess based on effective collaboration and unique solution methods.
- Emphasize the importance of trying multiple solution methods.
Summary
- Summarizing what inverse functions are and how to graph them with/ without a graphing calculator.







Math 11 – Absolute Value Functions 1.

Bridge:
- Introduction to how to graph functions involving an absolute value.

Warm-up and Discovery:
-Ask students to do the exercise provided for them related to absolute values (computational).
- Have students use tables of values to graph functions of form y = ax+b.
Learning Objective:
- Students will be able to graph absolute value functions.
- Students will be able to use graphing calculator and graph absolute value functions.

Teaching Objective:
- To make the students familiar with the absolute value functions.
- To teach students how to graph an absolute value function.
Pre-test:
- Ask students during warm-up and discovery to answer questions related to the above.
Participation:
- Paired graphing activity. Students will work with desk partners and collaborate to solve the graphing exercises.
Post-test:
- Number students into 4 teams and have them compete to complete different problem sets.
Summary:
- Summarizing what these functions look like when we graph them.