# My Take:

I recently had the privilege of designing a circle Geometry Workshop for the UCT 100Up Programme. All in all I really enjoyed the experience. Below are the resources I created for the Project.

Learning without Limits

These are posts that relate to Mathematics.

I recently had the privilege of designing a circle Geometry Workshop for the UCT 100Up Programme. All in all I really enjoyed the experience. Below are the resources I created for the Project.

Recently I had the opportunity to attend a Maths workshop presented by Mark Philips and the Mind Action Series Textbook publishers. I signed up on a whim because of the topic, which was Euclidean Geometry. In my experience it’s one of the most poorly understand topics by learners. I hoped to learn some new approaches that would make the time I spend on this section more productive and ultimately improve the learning outcomes.

- Focus on the Process
- In Geometry, do exercises first & then theorems. (Build’s awareness before complexity)
- Allow learners to do well in early tests to build confidence
- Build confidence through slow ascent in difficulty
- Don’t be so preoccupied with Time
- Spend 80% of the Time on Basics and 20% on the Harder Questions for most classes
- Encourage learners to move around and do physical stuff – Eg Circle Dance
- Use narratives and stories in your teaching – especially those involving relationships
- Make Learners Feel about the Topic, see above
- Humour & Novelty are important – keep learners on their toes
- Integration between Mathematical Topics important
- Get Learners to make their own questions – higher cognitive levels of thinking
- Understanding Mathematical vocabulary is important – specific practice on this is warranted

So the biggest surprise for me was how for the most part, our attention was held throughout the 3hr workshop. It’s no easy feet to keep a room of 40 + teachers on a Saturday morning engaged, talking about content that they cover every year. The presenter was really skilled at keeping people on their toes and you never quite knew what was coming next. The amount of movement in the session also stood out for me as well as the clarity and size of the visual aids. The presenter also showed a depth of understanding on the topic and vast experience teaching it, which meant his comments were specific and helpful.

I was pleasantly surprised by how much I learn’t in the session. It is not always the case when going to Maths Development workshops. I took detailed notes and they are reflected in the key ideas section above. I also managed to grab a clip of the Circle Dance (below), which was a fantastically creative way to remember the circle geometry theorems. I am struck by how useful it is to have a development session with a teacher who is a real master of their craft. More of these types of workshops would be brilliant for the development of best practice among Maths teachers.

With the amount of Mathematics content online now a days, I often find the biggest challenge is sorting through what to share with my students in class. Often resources appear usable at first, but then as you dig down deeper there are issues with alignment to the curriculum, notational differences or just confusing accents. However every now and again, you find a stand alone unit of work that has been handled particularly well and is ideally suited to being used in your class. In this case I try to do a blog post about it, so I don’t forget to use it in Future! The content below is taken from the Khan Academy program on Algebra 1 and is a fantastic introduction to the concept of a Function in Mathematics. This is a foundational concept but can be tricky to explain. This unit allows students to proceed at their own pace and build up a solid understanding of the concept.

Link to the Unit on Functions from Khan Academy

I also added some additional resources at the end that I sourced from the Web to test my students understanding, since I largely run this unit as a self study module

- Kahoot Linear Functions Basics
- Understanding Functions
- Functions with different Visual Representations
- Relation & Functions Determination

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A while ago I found the most delightful animated series on YouTube that explained the foundations of Euclidean Geometry delightfully. I decided it was so good that I needed to start off my GR9 unit, showing these videos to my learners. Below are the videos together with some other resources I have collected, related to this unit of work Euclidean Geometry.

Solving for Angles with Parallel Lines

Solving Basic Calculations using Euclidean Geometry Facts

Angle Relationships in Parallel Lines and Triangles

Every now and again I come across Fun Social Media Posts about Mathematics, normally in the context of solving some sort of interesting problem. I thought it would be cool to capture them for use in class and so I started this gallery.

Recently I had a very interesting discussion with one of my grade 9 students around the idea of why co-ordinates switch positions when reflecting over the line y = x . This section is covered in GR9 Maths under Transformations. However a lot of the time students tend to memorise the formula with very little curiosity as to why it actually works. What follows in the rest of this blog post is bits from our discussion, together with artifacts of the learning process.

My drawing to understand why the rule actually works (Geogebra)

Consider object A, when it gets reflected over the line y = x , it follows a path perpendicular to that line and reappears on the opposite side as A’ the image. It is a reflection therefore AE = A’E.

Now consider triangles AEB & A’EB, they must be congruent due to (SAS)

1) BE is common

2) Angle BEA = Angle BEA’ = 90 degrees

3) AE = A’E

Now consider Triangle ABC & A’BD, they must also be congruent due to (AACS)

1) Length g = Length h – congruency above

2) Angle ABC = Angle A’BD – sum of angles

3) Angle ACB = Angle A’BD = 90 degrees – construction

So why do co-ordinates switch when going from getting the reflection over the line y = x

eg A(5;1) to A’ (1;5)

Take ABC and imagine trying to place it on top of congruent triangle A’BD. Can you see BC now lines up with BD & AC with A’D. When that happens, BC which used to be on the x axis goes to the y axis and AC which used to be parallel to the y axis is now parallel to the x axis.

The effect of this is that x becomes the y and the y becomes the x, hence the rule is:

**(x;y) -> (y;x)**

I will have a crack at a formula for the more general concept of a formula for reflecting over any arbitary line in a later post , but I have a feeling it may have some fairly interesting mathematics in it … (:

If you enjoyed the post or just want to clarify anything you read, please feel to leave a comment.

A while ago I started compiling a collection of fun resources related to the basics of Trigonometry. They will be most useful when acting as a supplementary resource to classroom instruction.

**Kahoots**

**Google Quizzes**

**Understanding the Unit Circle Definition of Trig**

I recently put together a collection of Calculus Resources for GR12 Maths in the SA Curriculum, using the tool EDpuzzle and learning content available by Mindset Learn as the base. Hopefully they can prove useful to other Maths Teachers out there.

Calculus Intro + Functions

Average Gradient

Understanding Limits

Definition of Derivative

Example of Derivative first principles y = 1/x

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