Simply Surds

Play with these curious, irrational numbers and discover the key relationships between them!

The aim of this activity is to explore working with these numbers known as surds (square roots of numbers). Surds are irrational numbers and so they are left in ‘root form’ so that they can be expressed exactly. As decimals, they go on forever, with no pattern. As part of this exercise we will discover some of their properties. What happens when you multiply two of them together? When do two irrationals make a rational? These and other questions will be tackled as you go through the activities below. I do recommend starting with the great poem below from the Kumars!

The root three poem

Although somewhat 'numberist' in nature, you might want to check out the text, which you can at this link

The activities

See the essence of the activities on offer on this gallery.



These are the resources you need to run this activity.

You'll need the Simply surds activity sheet shown below

And then you will need this   Simply Surds practice worksheet

Description/Teacher notes

This activity is all about getting students to play with familiar contexts so that they can discover and explore relationships for themselves.


  • Part 1 - The context of a rectangle and its area is familiar. The notion of square root is familiar. Putting them together requires a bit of thought and care. Students can start in all sorts of places looking for solutions for a, b and c and, in discussion with each other, should home in on an effective strategy (worth teachers playing with this themselves for a moment). Consider what happens when a and b have to be whole as well. The idea is to look for general statements about groups of solutions. Depending on time and students, you might explore the length of the diagonal.
  • Part 2 - Again, this is easily accessible. Students play with their calculators here and may be informed by Part 1 of the activity. Worth pausing on the idea that two irrational numbers can combine to create a rational one. Many of the ideas about simplifying surds are here in this activity and the idea is that students start making claims and conjectures about those rules.
  • Part 3 - Here we are looking more specifically at generalising about rules for simplifying surds. Again, the first two parts of the activity should inform this one.
  • Part 4 - This part is about recognising the different ways in which surds can be simplified. Being able to speculate about this is quite important.

Always, never, sometimes.....

These are important mathematical ideas and I recommend that teachers make sure that students have reflected properly on these ideas. 'Sometimes' is easier to justify because we just need one example. 'Always' and 'Never' need proof to be accepted, although either can be 'disproved' with just one example. Take the opportunity to focus on these important mathematical ideas here.


Then you will want to have a look at the practice activity which does contain some good, challenging practice as well as some good consolidation work. You can see the solutions to the practice worksheet in the hidden box below.....

Further puzzles

You might try these related puzzles on other websites to extend work on this idea. The idea is not to fill up gaps in time, but to employ some of the new skills students have in the context of some next level problem solving skills.

These surds puzzles are of a similar nature from nrich and the CEMP.

Try this Irrational Arithmagon from nrich.

And then there is this surds related puzzle!


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