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.

Short Practicum Stories

Story #1:

Story number one begins with food. On our first day at Prince of Wales, the admins for the school went to great effort to organize an orientation for all of the student teachers. This included a catered breakfast and lunch during which time, we all got to meet our sponsor teachers. This sort of a welcoming atmosphere was awesome considering that many of us were completely nervous, and we had no clear idea of what to expect.

Story #2:

After the orientation day, we (the student teachers) were encouraged to go around and start observing classes. The first one I watched was a Math 9 class which was being taught by one of my sponsors. She had a tablet laptop with a customized worksheet uploaded, and she was modifying it in real time as the students took notes on their own personal copies. I thought this was just about the coolest thing possible, short of having my own smart board. The next week, I actually taught the same class using a set of notes that I had prepared. Although I was nervous for the first few minutes, I successfully hid my fear, and It went pretty well overall.

Group Micro-Teaching: Feedback and Reflection

Peer Feedback:

After reviewing the peer feedback forms from our micro-teaching activity, I found the general sentiment to be fairly positive. Most of the marks were in the 3-4 range with some above and some below. What I found most beneficial (and also most frustrating) were the written comments. Many of the comments indicated that the pacing of the lesson was too fast, and that more time needed to be spent on explication, especially with respect to development of the summation formula for the geometric series. After reflecting on the exercise, I do agree with this judgment, but I have some reservations which I will elaborate on in the full reflection to follow.

Reflection:

While I enjoyed working with my partners on this activity, I found some aspects of it to be somewhat frustrating. The most challenging obstacles for me were the division of labour during the delivery of the lesson, and the fifteen minute time constraint. With regard to the partitioning of the lesson material, I feel like too much time was spent on the introductory portion. As a result, I felt pressured to race through the main body of the lesson, and there was not enough time left for the experimental activity. This was fairly aggravating for me, as I had put quite a bit of thought into coming up with an interesting lesson plan which incorporated all of the elements required by the assignment. With better preparation and communication with the other members of my team, I feel that this difficulty could have been avoided, or at least minimized. The second main challenge I encountered with this assignment was trying to find a way to accomplish all of the goals of the lesson within the allotted time frame. In the end, it turned out to be impossible to do, and we had to scrap the independent problem-solving activity. While I found this to be slightly exasperating, I also found it to be a useful experience. I learned that it is necessary to be flexible in my teaching practices, and to ‘roll with the punches.’ Circumstances will rarely behave exactly as they are expected to. Thus, it is vital to be aware of this and to be able to adapt one’s strategies accordingly.

Zero Division Poem: Reflection

I was away from class on the day this exercise was assigned, and so I found out about it from my classmates the following Friday. My gut reaction upon hearing about it was something along the lines of: “Seriously? We have to write poetry? This is going to be lame.” I would describe my attitude as being slightly hostile. However, I soon began tossing around ideas, and I kept coming back to the metaphor of black holes, and tears in the fabric of reality. Also, I had been listening to a lot of music by a Swedish math-metal band called Meshuggah, and much of their lyrics deal with concepts like transcendence, insanity and the altering of consciousness. I then mixed these influences together and came up with the idea of zero division as an operation which is so powerful that it can tear the mind apart. Attempting to incorporate imagery was challenging, so I tried to convey the idea of a vortex by using sentence and line structure which got progressively shorter and more abrupt. After writing a few rough drafts, I realized that I was actually getting into the activity, and I really started to enjoy it. As a teaching exercise, I thought it was quite interesting, as it forced me to get very meta-cognitive. I really had to think about how to describe zero division and its relationship to the concept of ‘infinity’ in a very qualitative way. I think this could be an interesting activity to incorporate into a math class, especially to supplement the study of topics like asymptotic behaviour of functions. In the end, I found the experience to be quite rewarding and useful.

2 Column Problem Solving - Polygonal Numbers

They posted in reverse, so start reading from the bottom :)

Singularity

Wrenched. Hurtling through space and beyond the event horizon.
Titanic forces compress and rend thought.
Nothing splits my mind as I
Spiral ever deeper.
The Terminus approaches.
Faster, now.
Spinning.
The Crushing Vortex
Annihilates reality.
Oblivion gives
birth t
o the
inf
in
it
e
.

Micro Teaching Lesson Plan

Geometric Series: Finite and Infinite cases
Bridge/Pretest: Recall the geometric sequence (consecutive terms have a constant ratio). Recall that a series is the sum of a geometric sequence. Give students the bouncy balls for initial investigation.
Learning Outcomes: Understand connection between sequences and series. Be able to compute finite and infinite sums. Begin to formulate an understanding of series convergence.
Teaching Outcomes: Incorporate multiple teaching modes. Prove the formula for calculating a geometric series.
Participatory: Have students bounce different balls. Investigate how successive bounces decrease in size. Lead into the idea that the heights of consecutive bounces form a constant ration = geometric progression. Discuss how to add the total length of 20 (or so) bounces. Prove formula for sum of geometric series. Raise the question: ‘what if the ball never stops bouncing, but continues forever with each subsequent bounce scaled by the same ratio?’ Discuss how powers of r, for r < 1, will approach zero if the exponent is sufficiently large. Segue into infinite series, and display how the formula for the sum will be modified.
Post-Assessment: Have students break into small groups (3-4) to collaborate to complete a problem set. Make some problems tricky by manipulating indices.
Summary: Recap relationship between sequences and series. Discuss infinite series, and the issue of convergence.