Unit Plan Overview
Algebra, Functions, and Equations
This post has been edited between December 6 and December 20 to meet the course requirements and to respond to feedback.
Teacher Candidate: Jeff Hamilton
School: Semiahmoo Secondary
Class: IB Math 11 (Standard Level)
Students: 15–25 (TBD)
Duration: Five 135-minute classes/week for first 4 weeks of fourth quarter
Preplanning Questions
This will be a large unit for grade 11 students, but I stand
by the recommendations of the textbook because I believe in emphasizing the links between these five chapters as they are taught over the four weeks before moving onto the next topic. It will benefit students to be assessed on these topics together as there are many links, which in turn strengthen the material from each section. The textbook encourages these five chapters to be considered part of the first theme of IB SL math in the two-year course students take during their diploma program. In a way this is like a midterm university experience and I hope to incorporate many non-traditional teaching strategies to provide students with a unique learning journey as well as to provide the experience for what university studies will feel like. Like much of my work in the BEd program thus far, I strive to take the best teaching strategies from secondary and university education and put it into practice in the secondary classroom.
Why do we teach this unit?
This unit is an important development and extension from skills in algebra and linear equations of earlier grades in high school. This course bridges into calculus and integration, and this unit is an important transition to get to such ubiquitous applications of mathematics. In this unit, we discover the relationships of equations beyond lines, namely in parabolas, with their extensions to exponential growth and decay. Topics in the unit will better connect students to the worlds around them with intentionality by the teacher to demonstrate the beauty in the topics. Functions at a glance seems like a dull new thing to learn, however the very nature of representation an equation by the relationships of its variables was once a groundbreaking concept. And this idea can still be groundbreaking to our students, who were prealgebraic just a few years earlier, through demonstrating the applications and art that functions present.
What mathematics projects connect to this unit?
I have two projects I would like to include in this unit.
- Math + Art Assignment: Building off the previous question in why we teach this unit, a way to demonstrate the beauty of the equations we study is to put on our artist hats express these equations in a way to show our artistic expression. Desmos has a global math art contest, which I would adapt to the first project of this unit. In an earlier class, students will have some time to research artwork they like and make sketches they feel like drawing, before we introduce that we will then draw this artwork using equations throughout the unit. What equations fit this artwork? Students may of course chose to create something else on Demos. There are three parts to this project:
- Proposal of piece: This is the context and motivation behind the artwork they wish to produce with equations we learn in the unit. (Week 1)
- The piece: This is the two-dimensional final product on Desmos, which students will submit along with the list of equations (and the bounds) they used to produce it. (Week 2)
- Reflection: Students will have the opportunity to share their work with their peers and they will write a brief reflection of what they learned about equations and their artwork. (Week 3)
- Group Tutorial: This unit covers five chapters from the textbook, which students will have access to. Each chapter will be covered in 2–4 lessons. To promote leadership and communication skills, students will lead a 30-minute tutorial in groups of 3–5 (depending on the class size) for one of the five chapters. This will happen on the final day of the chapter at the beginning of class. Groups will be assigned in the first week. The rationale behind this project is promote communication skills while showing students to take ownership over study habits because they will feel accountable because peers are
counting on them. Our classroom will frequently use vertical erasable surfaces, and the groups are encourages to lead their tutorial in this manner. They may chose which problems to cover. There are three pieces to this project:
- Tutorial overview: The group will check in with me the day before their tutorial with their lesson plan.
- Delivery: On the day of their tutorial, they will have 30 minutes to review the chapter. They must include a link to society and a historical fact for 'this day in math history'. The historical fact in math may be related to society or science as long as it relates to math. Examples include birthdays of mathematicians and an overview of their work or the date of a publication of a paper or book and why it is significant. My first three classes, the reviews of grades 8–10 math, will be structured in this manner so the students have an idea what they can do with their time.
- Reflection: Students will perform self-evaluations and peer-evaluations.
How will learning be assessed?
There will be plenty of opportunities for students to demonstrate their learning in situations low-stake settings where mistakes are encouraged. Students will be able to build off of these opportunities so they may demonstrate their learning in later assessments. The following provide an overview of the formative and summative assessments of the unit.
Formative Assessments:
- Review activities and reflections
- Non-graded quizzes: There will be one on each of the first three days of review and at least one per chapter.
- The two projects each have three parts, as outlined above, and parts 1 and 3 will be formative. The first guides them towards their second, which is summative, while the third guides them towards their next project or assessment.
Summative Assessments:
- Friday quizzes (weeks 2 and 3)
- Delivery of group tutorial (week varies by group)
- The submission of their piece for the math + art assignment (week 3)
- Final test (week 4)
Elements of Unit Plan
The course in the quarter system lasts about 9 weeks. The first 4 weeks will be taught by me and I will be responsible for 50% of their course grade. I will provide my 4-week overview to the students on the first day, so they know what/when they are learning and being assessed.
Tentative Breakdown:
Note: These are just the summative assessments. Formative assessments will be nearly every day, with larger ones outlined in the projects overview above.
- 5% – Quiz 1 (Friday of Week 2)
- 10% – Art Piece (Wednesday of Week 3)
- 10% – Group Tutorial (depends on group)
- 5% – Quiz 2 (Friday of Week 3)
- 20% – Unit Test (Friday of Week 4)
This course is similar to pre-calculus 11 in the BC curriculum.
Big Ideas
- Algebra allows us to generalize relationships through abstract thinking
- The meanings of, and connections between, operations extend to powers, radicals, and polynomials
- Quadratic relationships are prevalent in the world around us
Curricular Competencies
Key one for each of the four is included
- Reasoning and modeling: think creatively and with curiosity and wonder when exploring problems
- Understanding and solving: develop, demonstrate, and apply conceptual understanding of mathematical ideas through play, story, inquiry, and problem solving
- Communicating and representing: take risks when offering ideas in classroom discourse
- Connecting and reflecting: connect mathematical concepts with each other, with other areas, and with personal interests
Content Objectives
- Real number system
- Powers with rational exponents
- Radical operations and equations
- Polynomial factoring
- Rational expressions and equations
- Quadratic functions and equations
- Linear and quadratic inequalities
Adaptations for English language learners
- Vocabulary for each chapter will be hung up around the room
- Students will have opportunities to teach other formally and informally, which gives them exposure to using the mathematical language prior to larger assessments
- Word problems and activities will have intentional language such that students do not get stuck on a non-math word they do not know (additionally, people represented in word problems and activities will represent the many minds of math around the world)
Materials
- Whiteboards and markers
- Document camera
- Projector
- Textbook
- Ch 1: Quadratic Functions
- Ch 2: Exponents and Logarithms
- Ch 3: Algebraic Structures
- Ch 4: The Theory of Functions
- Ch 5: Transformations of Graphs
- Course website (to accommodate for students learning from home if applicable)
- Smartphones + WiFi (students will be permitted to use their own for specific activities)
- Cribbage boards
- Children's basketball net + ball
The unit contains 20 lessons, which commence at the start of the
quarter. This is why the first week is largely dedicated to review of
math the students have learned in high school thus far. Taking the time
to work through these quizzes will show myself and the students where
their weaknesses lie. The following weeks list the general topics for
each lesson and some the key assessments. Furthermore, there will be a "puzzle of the week" introduced each Monday after the halfway break and reviewed on Friday at the end of class.
Week 1: Review + Introduction
Lessons 1–5
Monday: Grade 8 Review (quiz not recorded)
- First hour of class is orientation to our learning environment with emphasis on how we will participate in a safe space and use collaboration with the whiteboards to make our mistakes before we are assessed
- Each class will have a 15 minute break at the midway point where students may do what they wish (I will encourage them to stand up and walk around to get blood flowing to their legs)
- Introduce puzzle of the week
- The final hour of this class will begin with an online 20-question quiz on grade 8 math
- The remainder of the class will be working through further problems from grade 8 math
Tuesday: Grade 9 Review (quiz not recorded)
- This class will begin with an online 20-question quiz on grade 9 math
- The remainder of the class will be working through further problems from grade 9 math
- During independent work, grade 8 math from Monday may be revisited
Wednesday: Grade 10 Review (quiz not recorded)
- This class will begin with an online 20-question quiz on grade 10 math
- The remainder of the class will be working through further problems from grade10
- During independent work, grade 8 and 9 math from Monday and Tuesday may be revisited
Thursday: Quadratic Functions 1
- Recap of grades 8–10 math through a Kahoot
- Sections 1 and 2 of chapter
Friday: Quadratic Functions 2
- Sections 3 and 4 of chapter
- Solution to puzzle of the week
- Art activity (see detailed lesson plan)
Week 2: Quadratic Functions + Transformations of Graphs
Lessons 6–10
Monday: Quadratic Functions 3
- Sections 5 and 6 of chapter
- Introduce puzzle of the week
- Sections 7 and 8 of chapter
- Group 1 checks in with me regarding their facilitation
Tuesday: Quadratic Functions 4 (group 1 presents)
- Group 1 leads tutorial to begin class
- Teacher-lead review for second half including 5-question non-recorded quiz which students will peer-evaluate (0 or 1 for each question)
- Inquiry activity with quadratic functions in society
Wednesday: Transformation of Graphs 1
- Sections 1 and 2 of chapter
- Sections 3 and 4 of chapter
- Group 2 checks in with me regarding their facilitation
Thursday: Transformation of Graphs 2 (group 2 presents)
- Group 2 leads tutorial to begin class
- 5-question non-recorded quiz which students will peer-evaluate (0 or 1 for each question)
- Outdoor learning activity for second half (see detailed lesson plan)
Friday: Quiz (recorded)
- Quiz on "quadratic functions" and "transformations of graphs" in 30 minutes
- Students who finish early may work on their solution for the puzzle of the week
- Solution to puzzle of the week
- Second half of class will be game time – today students will learn how to play cribbage (students who already know how to play will have the opportunity to teach their peers) and we will do this on Fridays in weeks 3 and 4 as well
- The math involved with playing cribbage does not directly relate to the course content but it develops mathematical thinking. Games on Fridays is an incentive for students to get through their work during the week.
Week 3: Exponents and Logarithms + Algebraic Structures
Lessons 11–15
Monday: Exponents and Logarithms 1
- Sections 1 and 2 of chapter
- Sections 3 and 4 of chapter
- Introduce puzzle of the week
- Case study on virus transmission and exponential growth
- Group 3 checks in with me regarding their facilitation
Tuesday: Exponents and Logarithms 2 (group 3 presents)
- Group 2 leads tutorial to begin class
- 5-question non-recorded quiz which students will peer-evaluate (0 or 1 for each question)
Wednesday: Algebraic Structures 1
- Sections 1 and 2 of chapter
- Sections 3 and 4 of chapter
- Inquiry activity to establish which concepts are arbitrary and which are necessary
- Group 4 checks in with me regarding their facilitation
Thursday: Algebraic Structures 2 (group 4 presents)
- Group 2 leads tutorial to begin class
- 5-question non-recorded quiz which students will peer-evaluate (0 or 1 for each question)
- Formal review for quiz on content covered this week
Friday: Quiz (recorded)
- Quiz on "exponents and logarithms" and "algebraic structures" during first 30 minutes
- Solution to puzzle of the week
- Cribbage during second half of class
Week 4: The theory of Functions + Recap + Assessment
Lessons 16–20
Monday: The Theory of Functions 1
- Sections 1 and 2 of chapter
- Introduce puzzle of the week
- Sections 3 and 4 of chapter
- Group 5 checks in with me regarding their facilitation
Tuesday: The Theory of Functions 2 (group 5 presents)
- Group 2 leads tutorial to begin class
- 5-question non-recorded quiz which students will peer-evaluate (0 or 1 for each question)
- Debrief on facilitation
Wednesday: Review 1
- Formal review session
- Students may request questions and topics to be covered
- Handout for independent or group review
- Debrief on math and art
Thursday: Review 2
- Learning games for review (see detailed lesson plan)
Friday: Unit Test (recorded)
- Unit test for first half of class
- Solution to puzzle of the week
- Final week of cribbage
- Debrief on strategy and mathematical thinking in the game
Week 5 will begin with sequences and series or binomial expansion. Furthermore, the content covered this unit will reappear in May of 2022 when students completing the IB diploma program write their final exam. I hope to leave students with useful resources to hold onto when they do their final revisions later on.
Resources:
Mathematics Standard Level for the IB Diploma. Cambridge. 2012.
Menu Math
Desmos
Ideas to consider following discussion with peers:
- Math assignment with board game or card game (Ex. cribbage and function for points)
- Peer-to-peer teaching (to improve communication skills and understanding)
- Outdoor
education (link content to theory of knowledge and place-based
learning, outdoor teaching with small whiteboards, and discussing the
outdoors as they relate to mathematical thinking)
- Math art (STEAM)
- Math movie (like Imitation Game, but one that applies to this unit)
Feedback from peers:
- Build your own stock portfolio (limitations on deterministic model)
- Virus transmission and exponential growth (modeling)
- Kinematics with quadratics outside (videotracking software available at https://physlets.org/tracker/)
- Literature to support "why are we learning this?" and "why are we learning it this way?"
- Students will love playing cribbage
Detailed Unit Plans
I hope to implement the thinking classroom as the normal for the
teaching of this unit. I used it for my two lessons I taught during the
short practicum, and before February I would like to read Professor
Liljedahl's book to better prepare my lesson plans. I am fortunate that
the math classroom I will be teaching in is equipped with three walls of
vertical erasable surfaces.
The following three unit plans are not in the traditional lecture/exercise/homework format.
Lesson 5: Quadratic functions 2
Note: The first 60 minutes of class will be conducted in the Liljedahl thinking classroom environment. I will continue working through the material in Chapter 1: Quadratic Functions. This will be our fifth day in the quarter system together, which will be plenty of time for the students to become oriented with our classroom dynamics, to dig into the review of grades 8–10 math, and to see what the first few sections of the first chapter are like. They will be ready to get their first exposure to our first project, in STEAM, which will take place in a non-traditional format, in the final hour of lesson 5.
Topic: An Introduction to the Art of Math
Pedagogical Goals:
- Get students out of the classroom
- Put emphasis on connecting artistic expression and mathematical thinking
- Have students thinking about why they like math and how they think about math
Preparation:
- At the end of lesson 3, students will post a 250-word response to the online forum as to what the think a function is and to write what functions they know
- At the end of lesson 4, students will be asked to write about their favorite visual artwork (this could be album artwork if they are interested in music, architecture, or sculptures – it does not just apply to paintings, drawings, etc.), which will be posted in another 250-word response
- Connect with an art teacher for a senior art class to link our hours of teaching together for this day as the final hour of the first Friday of quarter 4 will take place in the same space
Assess Background Knowledge:
- The two discussion posts from lessons 3 and 4 assess background knowledge in math and art
- Review of grades 8–10 inherently features lots of work with functions and this is another way to get a feel for where students are at with knowledge of functions
Materials:
- Artwork (from the art class, either made by the students or presented by the students in that class)
- Means to take notes (paper or digital)
- Means to make sketches (paper or digital)
- I encourage a physical notebook for math, especially for material connecting to art
Timing: 60 minutes
- Art students open class with a welcome message (5 minutes)
- They will say some rules of the classroom and encourage participation from the guests
- The art teacher and myself will write an overview of the whole activity
- Groups from tutorial facilitation will work with small groups of art students (15 minutes)
- There will be five groups with 3–5 art students and 3–5 math students in each group
- Students will be asked to think about the how the artwork can be represented with functions
- Students will be asked to prepare questions to ask their peers in the art class
- Math students will bring our puzzle of the week for the art students to think about in groups
- Our puzzle of the week will be accessible to non-math students
- We will then go to the full group (15 minutes)
- One math and one art student from each group comes to the front of the class to debrief their small group discussion
- Teachers may ask further questions to guide discussion to the connections between mathematical and artistic thinking
- Math puzzle debrief (5 minutes)
- Final debrief on mathematical and artistic thinking (10 minutes)
- Students will share what they learned about the other subject
- Reflection (10 minutes)
- Students will have the time to write out their reflection (proposal draft) for their math + art project
- During this time, the two teachers will share ideas about how the class went (they will make a collaborative document to express what worked well, what didn't, and what they could do better next time)
- Students will participate in a closing activity where each student summarizes the activity with one word
Learning assessment:
- Students will hand in their reflection/proposal draft with links to functions thus far
- This will be the first step towards the first part of their math + art project (outlined above)
Feedback:
- Students will be asked to fill out an anonymous Google form with the following queries:
- What did you like about today's activity?
- What didn't you like about today's activity?
- What could we do better next time?
Lesson 9: Transformation of graphs 2
Note: The first 60 minutes of class will be conducted in the Liljedahl thinking classroom environment. Group 2 will have lead their tutorial and we will have completed our five-question non-recorded quiz for students to see how they are doing with this chapter of the textbook. The next 75 minutes will be conducted outside and follow the subsequent details.
Topic: Place-Based Learning in Math
Pedagogical Goals:
- Get students out of the school to give them an exposure to outdoor learning
- Place emphasis on environmental appreciation and provide links to social justice which support the environment
- Touching on Indigenous perspectives through place-based learning
Preparation:
- I would like to read "Braiding Sweetgrass" by Robin Wall Kimmerer before February to develop my oration around Indigenous perspectives and place-based learning
- At the beginning of class, students will share examples of quadratic equations in any form on their portable whiteboards
- I will prepare the problems we will do in the activities
Assess Background Knowledge
- Group 2 will have completed their tutorial at the beginning of class and we will have done a five-question quiz
- I and the students will have an idea of how comfortable they are with the content
- Students will demonstrate understanding in material through completing the games
Materials:
- Class set of portable whiteboards with dry erase markers
- Excerpt from "Braiding Sweetgrass", of which I will read an excerpt that relates to mathematical thinking in the outdoors in relation to Indigenous perspectives
Timing: 75 minutes
- Class break (15 minutes)
- As usual, students will have their 15-minute break, however this time they will be independently walking around the nature behind the school
- They will be asked to take one picture that connects their mathematical thinking to nature and they may approach this prompt however they wish
- Debrief of finding place activity (15 minute)
- Students will share how they felt in walking around
- Students will share why they chose the photo they captured
- Relay game (30 minutes)
- In teams, we will play Telustrations using equations and graphs of quadratic equations as the prompts
- I will tell the first team member what to draw (or I will provide a sketch of which they will deduce the equation)
- Other team members will be located around the school ground (but not too far away)
- Debrief (15 minutes)
- Students will discuss the activity and learning in the outdoors
- I will read a story of place, the chosen excerpt from "Braiding Sweetgrass"
Learning Assessment:
- I will see how students do in going between graphs and equations
- Students will reflect on their learning through debrief activities
Feedback:
- Students will be asked to fill out an anonymous Google form with the following queries:
- Would you like to have more math classes taught outside? Yes/No
- Why or why not?
Lesson 19: Review
Note: Two games will be played through which students will review the unit content.
Topic: Unit Review through Mathketball and Jeffpardy
Pedagogical Goals:
- Engage the full class in implementing vertical erasable surfaces in our thinking classroom
- Students will work through problems in groups and then on their own
- Utilize a gamified approach to keep up participation
Preparation:
- I will read "Building Thinking in Mathematics" by Peter Liljedahl before February so that I have further ideas for using the whiteboards
- Students will complete my formal review activities from lesson 18 and these will guide the questions I use in the games as I prepare them for the final test
- The top team in Mathketball and the top three scorers in Jeffpardy will receive journals or math puzzles (their choice)
Assess Background Knowledge
- The nature of this class is to assess content from throughout the unit over the four weeks
- Challenging questions will connect topics together and students will see which chapters the questions came from should they wish to further revise that specific area
Materials:
- Whiteboards (for groups to work on)
- Projector (for myself to display the problems)
- Children's basketball net and ball
Timing: 135 minutes
- Part 1 – Mathketball (45 minutes)
- Teams (from tutorial groups) will work through 15 longer problems on the board
- Each team will have a section of the board to work on
- Problems will be given out randomly (so that groups cannot see the work of the same problem on another board)
- Once a problem is completed they will ask me to inspect it
- If it is correct on the first try, they receive the next problem
- They receive hints until they get it right
- Correct on the first try means they get one shot on the net at the end of the activity and if they require hints, they do not get a shot (additionally, if they pass on the question without having time to come back to it – they lose a shot)
- After 30 minutes, I will review the solutions to the problems
- Groups will then get to take their turns taking their shots (up to 15 per group)
- The winning time is the team that makes the most shots
- Break (10 minutes)
- Part 2 – Jeffpardy (45 minutes)
- This is just like Jeopardy, however each of the categories corresponds to one of the five chapters
- There will be five questions increasing in difficulty for each chapter
- They will enter their answer their score in an online format like Kahoot so they have the incentive to work quickly
- The winners are the top three scores on the Kahoot
- Break (10 minutes)
- Part 3 – Independent work (25 minutes)
- Students will work independently on that which they find the most challening
- The classroom will be organized by groups from the tutorials, so they may serve as experts to be asked about questions from different chapters by their peers (mini office hours)
- I will walk around the room and answer questions the groups cannot answer to their peers
