Unit Plan

Your name: Helin Wang 

School, grade & course: Eric Hamber, Math 8

Topic of unit (NOTE: This should be a unit you will actually be teaching on practicum!):

Understanding Percent, Ratios, Rates, and Proportional Reasoning

 

(1) Why do we teach this unit to secondary school students? Research and talk about the following: Why is this topic included in the curriculum? Why is it important that students learn it? What learning do you hope they will take with them from this? What is intrinsically interesting, useful, beautiful about this topic? (150 words)

 

This unit is foundational because percent, ratios, rates, and proportional reasoning appear everywhere in students' daily lives-prices, discounts, taxes, recipes, maps, data, and comparisons. These ideas form core mathematical tools for financial literacy, scientific thinking, and data interpretation. The BC Curriculum includes these topics because they help students understand multiplicative relationships, develop number sense, and recognize patterns across different representations. Strong proportional reasoning predicts success in later mathematics, including algebra, linear functions, and statistics.

 

 

This content is also intrinsically interesting and beautiful: percent offers a consistent "out of 100" lens for comparison; ratios reveal hidden relationships in nature, art, and culture; proportions show how scale governs the structure of buildings, canoes, and snowshoes across Indigenous nations. By the end of the unit, students should feel confident switching between representations, interpreting real-world situations, and using proportional thinking flexibly and confidently.

(2) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form part of this unit. Describe the topic, aims, process and timing, and what the students will be asked to produce, and how you will assess the project. (250 words)

 

 

Project Title: "Understanding Our World Through Percents and Proportions"

 

Aim:

Students investigate a context of their choice where percent, ratios, rates, or proportional reasoning play a meaningful role (e.g., sports stats, clothing discounts, climate data, music tempo ratios, recipe scaling, Indigenous canoe proportions, or art scaling). The goal is to integrate mathematics with personal interest, community relevance, and real-world interpretation.

 

Process & Timing:

The project unfolds over four class days plus at-home work.

- Day 1: Students brainstorm contexts and choose a topic. Teacher provides exemplars.

- Day 2: Students gather real data, organize it into tables, and decide which mathematical tools apply (percent change, ratios, rate comparisons, proportional scaling, etc.).

- Day 3: Students create visual representations such as percent bar models, ratio tables, double-number lines, scaled diagrams, or percent growth charts.

- Day 4: Students produce a final product and practice explaining their reasoning. Students then submit a polished two-page report, poster, or digital slideshow.

 

Final Product Requirements:

1. At least two mathematical representations (e.g., percent change + ratio table, or scale diagram + proportional equation).

2. A written explanation that justifies why the mathematics fits the situation.

3. A reflection on how the math helped them understand the context better.

 

Assessment:

Rubric includes accuracy, clarity of reasoning, use of multiple representations, communication, and reflection. Emphasis is on understanding, not perfection.

(3) Assessment and evaluation: How will you build a fair and well-rounded assessment and evaluation plan for this unit? Include formative and summative, informal/ observational and more formal assessment modes. (100 words)

Assessment will be balanced across formative, summative, observational, and performance-based modes. Formative assessments include warm-ups, whiteboard checks, exit slips, and mid-lesson conferences to target misconceptions. Observational assessment happens during partner and group work, especially in open-ended problem-solving tasks. Summative assessments include a chapter test, the culminating project, and selected practice problems that model exam-style questions. Students also complete self-assessment checklists to reflect on progress with key competencies. Throughout the unit, assessment prioritizes reasoning, flexible strategy use, and understanding rather than procedural memorization

Elements of your unit plan:

a)  Give a numbered list of the topics of the 10-12 lessons in this unit in the order you would teach them. 

Lesson	Topic
1	Chapter 8 Get Ready – Understanding Percent
2	Chapter 8.1 Understanding Large and Small Percents
3	Chapter 8.2 Fractions, Decimals, and Percents
4	Chapter 8.3 Percent of a Number
5	Chapter 8.4 Combined Percents & Percent of a Percent
6	Chapter 8.5 Percent & Financial Literacy
7	Chapter 8 Unit Review
8	Chapter 7 Get Ready – Ratios, Rates, and Proportional Thinking
9	Chapter 7.1 Ratios
10	Chapter 7.2 Rates
(11)	Chapter 7.3 Using Proportional Reasoning
(12)	Chapter 7 Unit Review

 

b) Write a detailed lesson plan for three of the lessons which will not be in a traditional lecture/ exercise/ homework format.  These three lessons should include at least three of the following six elements related to your mathematical topic. (And of course, you could include more than three!) 

Lesson 1 — Understanding Percent (Get Ready)

Integrated Theme: Indigenous perspectives on sharing, community ratios; visual arts through percent grids

 

Pedagogical Goals:

- Build conceptual meaning of percent as "out of 100."

- Connect percent to visual models and community-based proportional relationships.

- Introduce Indigenous perspectives on representing parts of a whole (e.g., resource sharing, community distribution).

 

Materials:

Hundred grids, colored pencils, chart paper, Indigenous-designed visual patterns (non-sacred), sticky notes.

 

Timing & Sequence:

1. Welcome + Background Knowledge Check (5 min)

Quick whiteboard prompt: "What does 50% mean? How do you know?"

 

2. Indigenous Context Mini-Story (10 min)

Teacher shares a non-sacred example:

- In many Indigenous communities, resources are shared proportionally (food, fish allocation, community duties).

- Percent grids help represent "equal sharing" visually.

 

Students discuss: How does representing parts of a whole help communities make fair decisions?

 

3. Visual Arts Activity - Designing Percent Tiles (20 min)

Students shade 100-square grids to create pattern tiles (inspired by beadwork, basket patterns, or weaving).

Each tile must represent:

- one "simple" percent (e.g., 25%),

- one "fractional" percent (e.g., 7%),

- one "mixed" percent (e.g., 3.5%).

 

4. Thinking Classroom Activity (15 min)

Groups at vertical surfaces classify pre-made grids into categories:

- "greater than 50\%,"

- "between 1% and 10 %"

- "fractional,"

- or "uncertain."

 

Students justify visually.

 

5. Consolidation (5 min)

Students write one sentence: "Percent helps us compare because..."

 

Assessment:

Observations during group work; completed percent tiles; exit slip on percent meaning.

 

Lesson 5 - Combined Percents & Percent of a Percent

 

Integrated Theme: History of percent in trade; arts through multiplicative patterns

 

Pedagogical Goals:

- Understand percent of a percent.

- Recognize historical contexts of interest rates and trade percentages.

- Visualize multiplicative relationships through geometric patterning.

 

Materials:

Area models, double grids, Renaissance trade examples, pattern blocks.

 

Timing & Sequence:

 

1. Historical Hook (10 min)

Short story:

- Merchants in 15th-century Italy used "per cento" to calculate goods, tax, and interest.

- Percent of a percent appeared in compound fees.

 

Students ask: "Why would merchants need percent of a percent?"

 

2. Percent-of-a-Percent Modelling (15 min)

Use area models to show:

- 40 % of 50 %

- 30% of 65%

 

Students create area-model art tiles showing multiplicative layering.

 

3. Thinking Classroom Challenge (20 min)

Vertical groups solve three tasks:

1. A store takes 70 % of the price, and a designer gets 30 % of that.

2. A population decreases by 20 % then increases by 20 %.

3. GST + PST combined.

 

Groups choose strategies: decimals, grids, or ratio reasoning.

 

4. Whole-Class Discussion (10 min)

Emphasize: percent changes are multiplicative.

 

Assessment:

Exit slip: "Why doesn't a 20% increase cancel a 20% decrease?"

Teacher notes from group reasoning.

Lesson 11 - Using Proportional Reasoning

 

Integrated Theme: Indigenous technologies (canoe design, snowshoes), art scaling

 

Pedagogical Goals:

- Deepen understanding of proportional relationships.

- Explore how proportional reasoning shapes Indigenous technologies.

- Use open-ended problem-solving.

 

Materials:

Images of Indigenous canoes/snowshoes (non-sacred), grid transparencies, ratio tables, whiteboards.

 

Timing & Sequence:

1. Indigenous Context Story (10 min)

 

Share an example:

- Many First Peoples designed canoes using proportional systems (length-to-width ratios, balance, glide efficiency).

- Snowshoes use proportional scaling to distribute weight on snow.

 

Ask: Why is proportion essential for stability and movement?

2. Art + Scaling Activity (15 min)

Students scale a canoe/snowshoe outline by a factor (e.g., 1:4) on grid paper.

Discuss: What happens to area when dimensions scale?

 

3. Thinking Classroom Problem Set (25 min)

Groups solve open-ended tasks at vertical boards:

- "Design a canoe $20 \%$ longer but with the same proportions."

- "A snowshoe is scaled down for a youth-what must stay proportional?"

- "Create two proportional and one non-proportional table-explain differences."

 

4. Reflection (5 min)

Students write: "Proportional reasoning helps design..."

 

Assessment:

Observational notes; accuracy of scaled drawings; group explanations.

Response to Promoting Flow

Watching this talk made me realize that I’ve actually felt “flow” many times, especially when I’m working on math or planning lessons. It usually happens when the challenge matches what I can handle, neither too easy nor too overwhelming, and I just naturally sink into work without noticing time. 

I do think students can feel this in secondary math classes, but it depends on the environment. When the task is reachable, interesting, and they’re given enough time to think, students really can get absorbed. We can’t force flow, but we can make it easier for them to get there by giving space, choosing the right tasks, and letting them feel confident in their thinking. 

This talk made me think about how I design lessons, and how I might create moments where students can genuinely enjoy the thinking progress

Response to The Giant Soup Can of Hornby Island

I tried to solve this puzzle by using only what I could see in the photo. The bike became my measuring tool since a medium hybrid bike usually has a wheel diameter of about 0.70 m and a handlebar height around 1.05 m. When I compared the bike to the tank, the diameter looked like it covered about 5 bike wheels, so I estimated the diameter to be around 3.5 m . Campbell's cans usually have a height that is about one and a half times their diameter, so I estimated the tank's height to be close to 5.25 m.

Once I treated the tank as a cylinder, the volume followed naturally. I used V= 𝝅 r^2 h 

with r ≈ 1.75 m and h ≈ 5.25 m. This gave me V ≈ 𝝅 × (1.75)^2 × 5.25 ≈ 5.2  × 10^1 m^3. That is roughly 52000 litres, or about 13700 gallons. This amount of water would still help with the early stages of a house fire.

What I liked about this puzzle was how much I could figure out from a single picture. It reminded me that a simple detail like a bicycle can give you enough structure to build a reasonable estimate without relying on outside numbers. If I extended this for students, I would ask them to compare two different reference points from the bike to see how their estimates change, or to calculate how much paint is needed to cover the red and white sections of the tank. 

Arbitrary and Necessary

Hewitt's idea of arbitrary and necessary made me rethink how I usually plan math lessons and even whole units. I realized I often try to "explain everything well," but I do not always separate what students actually need to remember from what they should be able to figure out on their own. This distinction changes the way I look at my planning. If something is arbitrary, like notation or vocabulary, then I can simply introduce it clearly and move on. But if something is necessary, then I need to create moments where students can actually see why it must be true. That means I cannot jump straight to formulas or polished steps just because it feels efficient. I need to slow down at the right places and design experiences that let the idea appear naturally.

This also reshapes how I plan a unit. Instead of listing pages to cover or examples to finish, I start thinking about the "awareness moments" I want students to have. For example, in geometry, I want students to notice angle relationships by cutting and rearranging shapes. In algebra, I want them to understand why certain operations preserve equality instead of memorizing slogans like "two negatives make a positive." During my short practicum, I often heard my SA say that he really did not like students using the phrase cross multiplication. What exactly is cross multiplication? We are not doing it because we need to memorize a rule. We do it because we are multiplying or dividing both sides of the equation by the same number. These moments require time and space, so my pacing needs to be flexible rather than packed with every exercise. I also find myself paying more attention to which parts of a lesson might overload students with things they can't reason out yet. If I ask them to memorize too many arbitrary details before giving them something meaningful to reason through, they lose motivation.

Overall, Hewitt's framework pushes me to design lessons where understanding is not handed to students but discovered step by step. It reminds me that math is not just a collection of rules but something that makes sense when students have the chance to look for structure. As I keep planning future lessons, I want to build a classroom where students remember the arbitrary because they have to, but they understand the necessary because they saw it happen.

Microteaching Lesson Plan

                                                              

Response to Battleground Schools

I found it interesting that debates about how math should be taught in North America have been repeating for over a century. The constant shift between focusing on basic skills and emphasizing understanding made me think that education reform is often more about ideology than actual learning. It feels like schools keep changing direction without really solving the core problem of how to make math meaningful for students.

I was also struck by how the New Math movement, which aimed to modernize math education, ended up failing because teachers and parents weren't ready for it. It reminded me that even a well-intentioned reform can't work if people don't fully understand or believe in it. Real change in education probably depends more on teacher support and communication than on new theories or curricula.

Lastly, I noticed how politics and media shaped public opinion during the Math Wars. What surprised me was that something as neutral as math became tied to ideas about authority, competition, and national identity. It made me realize that discussions about curriculum are never just about knowledge, they also reflect the values and power structures of the time.

Post Microteaching Reflection

To be honest, I started facing challenges with this microteaching from the very beginning when I was choosing the topic. I am actually more comfortable with things like table tennis or photography, but since those are not really classroom activities, I decided to choose magic instead. That choice brought two main difficulties. One was figuring out what kind of activities to do after teaching the trick. The other was how to tell if everyone had really learned it, because once the secret is revealed, any kind of test no longer makes sense.

Another question I kept thinking about was what it actually means to teach magic. Is it just about showing techniques, or is there something more meaningful behind it? I once read a book that explained the difference between teaching magic and simply exposing secrets. What makes a magic performance memorable is the mystery of not knowing how it works, and the most enjoyable part is the process of trying to figure it out. When someone tells you the answer right away, the magic suddenly loses its charm. That is the hardest part of teaching magic for me. I did not want to focus on tricks that rely on hand skills, but rather on the methods of deception in magic, such as how magicians lead people to make certain choices or shift their attention. However, I think I did not manage the timing and emphasis very well. Some people mentioned in their feedback that it was hard to follow the goal of the lesson, and I understand why. I wanted everyone to experience the idea first instead of being told directly at the beginning, but I probably did not highlight those key points clearly enough during the lesson.

Aside from that, most parts of my lesson went as I expected. I think this experience was very valuable because it helped me clearly see the gap between lesson planning and actual teaching. I really enjoyed the whole process, and I am very thankful to my group members. They gave me a lot of support and feedback while I was teaching, which made me feel encouraged and confident.

Response to Lockhat's Lament

After reading Lockhart's Lament, I really agree with his point that math should be about curiosity and creativity instead of just following rules. I like how he describes mathematics as a kind of art, and that part really stood out to me. I also don't agree with the idea that students should only memorize formulas or follow the exact steps shown in practice exams. When students only plug numbers into formulas, they lose the chance to actually think and see patterns. I think giving students space to explore and even make mistakes is what makes math meaningful.

But I don't fully agree with how strongly Lockhart dismisses the practical side of math. I understand his frustration with "usefulness" being overemphasized, but I still think connecting math to real-life examples or applications is beneficial for students. In my own experience, my professors and classmates often say that math is the foundation of all sciences. I actually developed my mathematical thinking by exploring how math connects to the fields I am interested in. I think that a physics student, a statistics student, and a computer science student would each look at the same math problem differently, and that to me is also part of the beauty of mathematics.

This also reminded me of Skemp's idea of relational and instrumental understanding. Lockhart clearly values relational understanding, knowing why something works instead of just how to do it. Both authors care about genuine understanding, but Skemp seems more balanced. He admits that instrumental understanding can sometimes help students build confidence or make progress, while Lockhart mostly rejects it. From my perspective, I find myself agreeing a bit more with Skemp's view.

Micro-teaching Lesson Plan

       

The locker problem

First, I tried smaller numbers (like 10 or 20 lockers) and looked for patterns in which lockers stayed open. I noticed that only lockers with numbers 1,4,9,16..... were open. Then I thought about why, I found that each locker is switched once for every student whose number divides that locker. That means the total number of switches equals the number of divisors of that locker. Normally divisors come in pairs, so lockers get switched an even number of times and end up closed. But perfect squares have one unpaired divisor (like 4 with itself in 16), so they are switched an odd number of times and end up open.

From this reasoning, the general conclusion is: after 1,000 students finish, the open lockers are the perfect squares up to 1000, what means there are 31 of 1000 lockers are still open.

                                          

Math Art Project - Individual Response

In this group project, I really learned a lot. It showed me how many math concepts can actually be found inside these artworks. Even though our group only picked one project that we were most interested in, and we didn't get the chance to try others, I already saw how much potential there is in bringing these kinds of hands-on activities into a math class. This feels very different from just teaching math concepts directly or giving students word problems that don't connect to their lives. When students actually try to make something, the clear steps help them see those math concepts in a more direct way. They also get space to be creative and discover math through their own ideas.

But this requires teachers to try these activities first and think from the student's point of view. The projects should keep students curious but not be so hard that they lose confidence or spend too much time stuck on it. Just like in our group project, it's only after we try different art projects, understand the ideas behind them, and connect them to the BC curriculum that we can design activities that work for students and get them involved in learning.

I really enjoyed this activity. My favourite part was when everyone finished their own small piece and signed their names on it. At the end, when all of our pieces and names were connected together to form a larger artwork, I really felt a strong sense of connection with the whole class.

Math Art Project - Group Write-up

 Group members: Damanjit, Elvie, Helin, Yuki

Original Artwork and Artist: Flowering Grid by Eric Gjerde by Tejom Patel

After our group collectively chose this artwork as our project topic, we faced a big issue of not having resources that lets us perfectly remake the original artwork. The artist, Tejom Patel, had creatively expanded their flower tessellation (based on Eric Gjerde’s work), by adding different folds to produce a new, unique piece. We started off attempting to perfectly mimic Patel’s art. However, due to our lack of knowledge on how the folds and designs work, we had to start off by choosing a design that had guides and tutorials. This was the Spread Hex Tessellation.

Similar to the original artwork, we kept the idea of hexagons and reflectional symmetry, but folded a tessellation where the hexagons overlap and pile up.

 

Here is the link to the video tutorial of the spread hex tessellation: 

https://youtu.be/3BTu2Hih39A?si=jRJpq3Fw6cG6oauz 

Through our research phase, we came across Eric Gjerde’s book Origami Tessellations: Awe-Inspiring Geometric Designs. This resource provided detailed folding tutorials for many origami tessellations built from triangles, squares, and hexagons, and also explained key techniques such as Pleat Intersections, Triangle Twist, Square Twist, and Hexagon Twist. With this reference, we gained a deeper understanding when looking back at our own work, and it also gave us the idea to design an activity more suitable for a short classroom session.

Our interactive activity with the class was a hands-on origami activity where each student folds their own piece of flower that will then combine to create a big multi-piece flower tessellation. Since abstract origami tessellations take a long time to fold, we designed it such that all prep is done (fold lines created beforehand) and students are to follow instructions while helping each other to collectively create one piece of art with the class. 

Here is the link to the origami flower we made in class:

https://youtube.com/shorts/tQMteMhp1Dk?si=Hxtkz7Yhg_-RoIt7 

In addition to experimenting with hexagon-based tessellations, our group created a variation called the Layered Compass, which is folded from square paper rather than a hexagonal grid. This shift gave us a chance to explore the mathematical flexibility of tessellation design. Whereas hexagons naturally lend themselves to 120° rotational symmetries and interlocking flower-like patterns, the square base highlights 90° rotations, reflections, and layered symmetry. By adapting the same folding principles—pleats, twists, and repeating units—to a different polygonal foundation, we were able to compare how tiling properties change with shape and how symmetry groups are expressed through origami art. Using square paper also made the process more accessible, since it is a common format and easier for classroom folding activities. Through this variation, we not only made the project our own but also deepened our appreciation of tessellations as a versatile mathematical art form that can be reinvented through creative folding choices.

 

Favourite and least favourite math teachers

My favourite math teacher was my high school calculus teacher. He also graduated from the education program at UBC, and he was one of the reasons I decided to come here. 

I liked his class simply because I always wanted to challenge him. Even when my ideas were not full developed, I would raise my hand, walk up to the board, and think through the next step while writing out the solution.  He often gave us very difficult calculus problems with multiple layers of integrals that included functions like sine cosine, exponentials, and inverse trigonometric functions. These problems made me want to try and he made me feel like I could beat him, like I could solve these problems faster than he could.  At the time I really believed this, until graduation day when he showed us his UBC transcript filled with A's, can't find a single B on that paper. Now, as I study Education myself, I realize how hard and special it is when a teacher can make students feel smarter than him.

In class, he often shared stories from his university days. One line that stayed with me was: "When you are in university, don't always eat lunch with the same group of people." He explained that the campus was full of people from different backgrounds, and by hearing their stories, you could open up your perspective.

To be honest, I never really had a math teacher I disliked, because I truly love math. But some teachers were more inflexible. They didn't care if you had a clever idea or a different way of solving a problem. They just gave you one set method, the quickest way to solve the answer. Day after day we repeated the same kinds of problems, and it felt like there was no interest or motivation.

Responses to Second Reading: What is meant by 'curriculum'?

As I was reading Eisner's article, I was struck by his discussion of the explicit, implicit, and null curriculum. The first thing that caught my attention was how described the explicit curriculum as an "educational menu." On the surface, schools seem to let students choose the subjects they like. But when he raised the question of the null curriculum, I started to think more carefully. I recognized that I had never asked myself during my own schooling why my school didn't offer courses like law or economics. Which now seem to me some of the most practical skills an adult need to survive in society.

The second point that stood out to me was his explanation of the implicit curriculum. He gave the example of students reading books to reach a smiling face and then getting a goldenrod ticket. Three tickets allowed them to leave for lunch five minutes early. This example really struck me because I also had similar experiences in school, and I can still see these practices in BC schools today. I used to think this was a good way to make students interested. But now I realize there is a big difference between being interested in learning itself and being interested only in the reward. This is a reminder for me as a future teacher. I want to help students build a true interest in math, not just work for small rewards or grades.

Finally, this article gave me new perspectives on things I used to see as normal and ordinary. Eisner also pointed out that schools often miss not only subject content like law or economics but also ways of thinking. This makes me ask myself how I can plan classroom activities, especially tests, to guide students to build creative thinking and to truly enjoy math.

Responses to First Reading: Relational Understanding and Instrumental Understanding

As I was reading Skemp's article, I felt genuinely inspired. When he contrasted instrumental and relational understanding, it was like he was describing my own experience as a student. In class, professors often just introduced a formula with a typical example, which made it seem perfectly fitted. But when I later worked on assignments and quizzes, I often ran into "surprises" where the formula did not quite apply. At that point I had to practice many similar but slightly different problems, and by comparing them I would slowly discover new patterns. That is why Skemp's " 20 cms by 15 yards" example struck me. He mentioned about instrumental understanding involving "a multiplicity of rules rather than fewer principles of more general application" also helped me realize why math education often relies so much on endless practice, because no single example or formula can really cover it all.

What also stood out to me was Skemp's analogies. The football story is interesting, but the music one made the most sense to me. Thinking about students just memorizing notes without ever hearing the sound reminded me of my own math learning. It was lots of symbols but not much meaning. I can still remember teachers saying, "Even if you don't get it now, just memorize it, you'll understand later." That way of learning shaped a lot of my early math and it is exactly what Skemp is pointing out as a problem.

Finally, I agree with Skemp's call for relational understanding. For me, math makes more sense when I can connect rules with reasons, see how topics fit together, and try methods on new problems. His article made me think about my own experiences and also about my future as a teacher. If I only pass on rules without reasons, I might help students get through exams, but I will not give them the confidence, flexibility, or joy that comes from really understanding math.