IGCSE Q: Sets and Venn Diagrams

Here is a question from an IGCSE past paper (Non-Calc):

This topic is called Sets and Venn diagrams. It’s based on organising numbers into categories.

If you need a review of Set Notation, click here for the Myimaths lesson. But for a quick summary:

• U is known as the universal set. This set, or category, contains all the numbers relevant to this question.
• A is a smaller group within U. This is called a subset.
• B is another group within U. It is also a subset.

In this case, we are told that the universal set is all Natural numbers between 1 and 16 (inclusive). In other words: U= {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}.

Since A are just the factors of 12 inside U, the answer is

A= {1, 2, 3, 4, 6, 12}

By splitting U into this set A, and also into B and considering which numbers A and B share, we can represent this as a Venn Diagram as follows:

You need to familiarise yourself with set notation, and know the meaning of Intersection, Union and Complement.

In summary, the answer to question b is not just which elements are in that group, but how many are there?

The question can be read as A intersect with not B. Visually it looks like this:

So look at our original Venn diagram, and see how many elements are included in the shaded part!

The answer is 3.

🙂

Enlargements

Sometimes, we like to make an object smaller or larger than its original size.

This is similar to rotation, in that we need to know two pieces of information:

• Centre of Enlargement (a point somewhere away from the image, usually the origin)
• Scale factor (usually called k)

The best way to do perform an enlargement is to count how far away each point on the object is from the centre of enlargement, and then multiply by the scale factor.

For example, look at the triangle ABC below.

We are going to enlarge from centre (0,0). As you can see, A is at the point (1,1). If we use scale factor 2, then A’ will be twice the distance of A from O. Luckily this is just (1,1) so you can easily located A’ at (2,2)! Have a look at points B and C and make sure you understand how that works.

HINT: If the centre of enlargement is the origin, it’s quite easy, just multiply the coordinates by the scale factor. But it won’t always be from the origin!

Note, that if we have a scale factor of less than 1 (i.e. a fraction), then the image will be smaller – in other words, a reduction.

Textbook Ex: (you should try and copy these onto paper for better results!)

ANSWERS BELOW

3a) (4.5, 6)
3b) (0,2)

Rotations

Rotations is exactly what it sounds like. Take an object and then spin it around. But what’s important is where do you spin around?

Consider a merry-go-round. All the horses are rotated around the centre obviously right? In transformations, we give this point a special name: the centre of rotation.

So take the object, identify the centre of rotation, and spin around! In reality, using tracing paper is the most effective way to do it when asked to draw rotations.

The two parameters we need are:

• Centre of rotation – (x and y coordinate)
• Angle of rotation – usually given as θ

Step by step:
1) Identify your object
2) Place corner of tracing paper on the centre of rotation
3) Mark your object point on the tracing paper
4) Rotate the tracing paper θ degrees in the correct direction
5) Mark your image point on the number plane

Note: (Tracing paper is provided during exams. If you don’t have tracing paper, you can use normal paper, but obviously it’s a little more challenging to see)

Once you have enough practice, you may be able to visualise it mentally and plot the point accordingly.

Try this Geogebra tool to rotate the object around 90°, 180° and 270°. Create your own object and try changing the centre of rotation to see how it affects your image.

Myimaths Exercises
– Lesson
– Online Homework

If you have access to a printer, try these worksheets.

Click here to return to the Transformations home page.

Translations

The shape we start with is called the object. Once we have performed any of these transformations on that object, we call it the image.

If an object changes location without change in orientation, it’s called a translation (think of it as a ‘slide’) 

More specifically, it’s not enough to just say it slides across. In maths, we like to be exact! So we’re going to introduce our old friend the Cartesian coordinate system! 

Look at this example where the object is triangle OAB being translated to its image O’A’B’

Question: What are the coordinates of the points of interest?  

O ( 0 , 0 ) O’ ( )

A (2 , 3) A’ ( )

B( -1, 2 ) B’ ( )

Have a look at the x and y coordinates of the object and its image. What do you notice?

This pattern can be represented by something called a translation vector.

Vectors are new for us and we will study them in more detail later. For now, all you need to know is that a vector indicates direction as well as distance.

In this example, x has moved 3 units to the right, and y has moved 2 units up. We write this as: 

Textbook Exercises:

Myimaths Exercises:

• Lesson
• Online Homework

Click here to go back to the Transformations Unit page.

Challenge Yourself:

Year 11: Transformations Unit

In this unit, we will be looking at mathematical techniques of changing shapes and figures. They are as follows:

• Translations
• Rotations
• Reflections
• Enlargements
• Stretches

The first 3 are relatively easy, with enlargements and stretches being slightly more challenging. 

You will need: 

• Drawing skills
• Drawing tools: pencil, ruler paper, tracing paper
• Your brain

Logarithms Base 'a' (Year 11)

Here is the review lesson I did for a student who missed last week’s classes on logarithms. I explain logs as the inverse of exponential functions. Below I have attached her worked answers from the textbook for your viewing pleasure.