Below, I’ve included a few lessons I’ve taught and a short description of why I liked them. Feel free to use/adapt them – if you have constructive critiques, I’d be happy to hear them in the comments!

**Table of Contents**

Least Common Multiples (Grade 10)

Pattern Recognition (Grade 6-12)

Injective/Surjective/Bijective Functions (University)

## Calculating Eclipses using Least Common Multiples (Grade 10ish)

Ancient Languages and Eclipses

This “lesson plan” is just a brief powerpoint, so I’ll give a few notes about implementation:

- We had an eclipse the day before, so I thought it would be neat to see how long it would be before we had an eclipse on the exact same day.
- The earth’s rotational period is ~365 days. The moon’s is ~28 days. The LCM of 28 and 365 is 10 220, so the earth, moon, and sun will return to their exact location every 10 220 days (or,
**drumroll**, every 28 years). That means there will be an eclipse on the same day every 28 years. - This theory doesn’t work. It’s off by about a week (28 years from that date, the eclipse fell a week early). It’s a beautiful chance to talk about approximations (of the earth and moon’s rotational periods) and why simple mathematical models can only give approximate answers.

I taught this class as a guided discovery. I asked students to make predictions about when the next eclipse would fall on this date and how they might calculate it. I gave them some information about what an eclipse was, and how the moon’s shadow works. I had huge buy-in and engagement from them, which was just excellent. Being able to look up our answer on Time and Date added some realism to the math, as did the fact about half of the student had watched the eclipse the night before.

After we had our answer, I threw in some history on the etymology of the words “moon” and “month” and how they come from a common root word. This was part of my cross-curricular approach to math. I intentionally gave a real world scientific connection, a historical connection, a predictive connection, a connection to their experiences, and a connection to language. My hope is that at least one element would stick out as interesting and catch each individual student and give them a path to access their knowledge of Least Common Multiples in the future.

After that, I gave three more standard word problems. I kept them light and a little bit flippant. The students very easily solved them, across the range of the class. Were there other ways to do it? Of course. Probably more time efficient ways, with more serious applications. But I don’t believe that efficiencies and seriousness are the best approaches to learning. Much too adult.

## Introduction to Symbology (Grade 10ish)

I saw this assignment as a “priming of the pump.” I wasn’t concerned with students learning all of these symbols – I also didn’t care if they made mistakes. We were talking about relationships, which leads into functions, and there are a lot of symbols there. We start talking about sets (Integers, Reals, Complex numbers), we change our Y into f(x), and often, there’s very little understanding of purpose. This assignment was more of a puzzle, a game. Students had to recognize patterns and make guesses about meaning.

I was very impressed with two things:

- How well the students did; and
- The mistakes they did make.

The students, only in Grade 10, lapped this up without much trouble, and I gave them zero explicit instruction. I just gave them the assignment and asked them to try. I’ve seen lots of math majors in university courses who mess these symbols up. On the other hand, the mistakes they did make we mostly my fault. My description of “is an element of” caused particular trouble, because of the note I gave afterwards, saying that elements are sometimes called members. If competency and fluency with these symbols had been a goal of mine, a little bit of explicit instruction and my students would have been completely comfortable. As it was, they had an intro and I think it lowered their apprehension around symbology, which was the main intent.

## I’ve Got Your Number (Grade 6-12)

I’ve used this across several age groups – Grade 7 and 8 students at a math camp; a pre-AP grade 9 class; a Workplace and Apprenticeship grade 11 class. I think it’s applicable across a huge swath of age groups and competencies. Here are the essential steps:

- Memorize all the numbers and which box they’re in;
- Give out copies to students;
- Have them quiz you:
- They give you a number, and you tell them which boxes it’s in;
- They tell you all the boxes a number is in, and you tell them which number it is.

- Have them analyze looking for patterns and strategies to copy you; and
- Have students who think they’ve solved it attempt to answer challenges from their peers.

There is, of course, a simpler way than memorization to learn all the numbers and boxes. I’m not going to write what it is here. If you want to use the activity and really can’t figure it out, send me an email and we can talk it through.

## Bias In Graphing (Grade 11ish)

This is a simple collection of different graphs I found online that are biased or misleading in some way. We went through, and I had students analyze and make suggestions for what they saw as flaws. We also discussed interest groups and who would want to present data in a biased way for each situation.

Overall, students appreciated the examples, and, by the end were able to give quite detailed analyses. I would say the powerpoint dragged on a bit too long – if you’re going to use it, I’d recommend cutting a few of the graphs, just for the sake of freshness. However, the repetition did give every student a chance to analyze a graph, and that was invaluable as diagnostic evidence. I knew precisely which students were struggling.

## Injective/Surjective/Bijective Functions (Grade 10- Second Year Undergraduate)

This isn’t a lesson plan – it’s a handout that I made for a university student I was tutoring. I think it nicely showcases two aspects of my instructional practice:

- This structure follows an instructional technique known as “Concept Attainment.” This technique focuses on describing and learning large concepts (often abstract, but not necessarily so). It does so by providing critical attributes of concepts – those things that are inherent features to all objects within a concept. It then focuses on providing an exemplar object and then both examples and non-examples.
- When teaching a class through concept attainment, I tend to be fairly explicit and direct with my instruction. Specifically, I use a deductive process: I give students the critical attributes and an exemplar, then I have them analyse examples and non-examples. However, it can be run indirectly and inductively by allowing students to induce the critical attributes from examples and non-examples.

- This handout makes very purposeful use of colour and white space. There’s also a very particular pattern and structure followed throughout. The idea is to allow the structure and colour and visual cues to assist the learning and comprehension of the reader. Everything is written in blue (just the colour of ink I chose). When the text changes from blue, there’s a purpose. Throughout, examples are associated with green checkmarks and non-examples are associated with red Xs. In the relationship map, I chose black as a darker, firmer colour to constrain thinking – students, at this point, don’t need to think about what set contains relationships. Injective and Surjective are, respectfully, red and blue. Where their Venn Diagram overlaps (Bijective) I’ve used purple (the combination of red and blue). Terms, circles, and shading are all coherently coloured to indicate groupings of ideas.
- Learning Styles don’t exist. Multiple Intelligences are questionably scientific. However, making appropriate use of colours and visual cues to prompt and support understanding is still valid. It adds a cohesion to presented material.