Angle Estimation

Task 214 ... Years 4 - 8


Estimation is a technique sometimes hook students into learning. If you have made a guess at something you have a reason to see how close you were. This is the basis of Angle Estimation and the power of the activity comes from being able to record the estimate with a Rotagram, so it can be checked against the accuracy of a protractor. Another feature of the Rotagram is that it always shows both the angle estimated and its complement to 360.

This cameo has a From The Classroom section with a use for Rotagrams described by an adult student artist.



  • 2 Rotagrams
  • 1 protractor card


  • average
  • concept of proof
  • estimating angles
  • estimating fractions
  • fractions, whole & parts
  • measurement, angle
  • mental arithmetic
Angle Estimation


Rotagrams were designed by Geoff Giles (1929 - 2005), renown Scottish educator and founding director of DIME Projects (Developments in Mathematics Education). He was responsible for an immense number of innovations in mathematics education. Several of our tasks were gifted by Geoff and use materials that were his creations. We recorded our respect for Geoff in our August 2005 eNews. When Rotagrams went out of production in UK, Geoff's wife, Bet Sampson, gave us permission to source production in Australia. In 2019 Bet gave permission for Geoff's Rotagram Booklets to be freely distributed through Mathematics Centre to support learning about angles, their relationships and applications using Rotagrams.


A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The card begins by building the concept of an angle as the amount of turn between two rays. The rest of the card is about estimating and checking with one of the two protractors. One is on the card itself and is marked in 5 intervals. The other is 1 intervals and on a separate card. But this is certainly only the tip of the iceberg.

Ask the students to draw any triangle in their journal.

It's possible to use the Rotagram to find the total of the three angles. I want you to find out how.
Of course the Rotagram won't give a result as a number of degrees, but it will give a very clear visual result. The clue is to set the rotagram to one of the angles, then move it to another vertex. Now the issue is how to place it so the next angle is added to what is already recorded.

Ask the students what it means to do a 180 on a skateboard.

So how many degrees of turn do you think there are in a straight angle?
Once this is discovered ask students to record the fact in their journal. Then it's natural to ask:
What would happen if you explored the angles of a quadrilateral?
...and connect it with doing a 360 on a skateboard.
Okay, so you know about 180s and 360s. Can you show me what it means to do a 90?
...which can be given the name right angle, which leads on to the number of degrees in a half right angle, a third right angle, two thirds right angle ... and exploring other polygons or angles within other shapes. For example:
Using only the Rotagram what relationships can be discovered about the angles in each of these rectangles. The size of one angle is marked?

You can create more Investigation Guides like this yourself, or one of the extra activities for this task could be for each pair who uses it to design a new IG to add to the class collection, or you could use these sheets already prepared by Heather Scott, an experienced UK Mathematics Education consultant.

Heather's details are in the Rotagrams section of Mathematics Centre Resources & Ordering, which also supplies class sets of Rotagrams and DIME Rotagram Booklets by Geoff Giles.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

To run a whole class lesson you will need a class set of Rotagrams. However consider introducing the lesson outside on a larger scale. Students can be asked to estimate angles made by tying a fixed cord to a pole in the quadrangle and rotating another cord around the pole in relation to this fixed cord. A large 'blackboard protractor' can be used to check estimates. Sometimes using a piece of paper can check an estimated angle. For example if the required angle is 30 fold a piece of paper like a dart to fit between the two cords. If this is correct, then three of them should make a right angle. A right angle is very easy to create from any scrap of paper by folding it anywhere and then folding the crease onto itself.

Back inside continue the investigation with the Rotagrams along the lines suggested above. Following a lesson or two like this, text book exercises usually look easy.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 133, Angle Estimation, which also includes support software to extend the estimating angles challenges.

The Rotagram can also be fruitfully used in conjunction with the option Fraction Pie in the software for Lesson 33, Fraction Estimation. Before using the software to show the requested fraction, students can be asked to estimate the fraction with the Rotagram. When the software shows the correct result, the Rotagram estimate can be easily checked by placing it against the screen. That is:

  • Step 1: Guess the required fraction of a turn with the Rotagram.
  • Step 2: Guess it with the software using the mouse.
  • Step 3: Ask the software for the answer.
  • Step 4: Check the Rotagram estimate against the answer.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Angle Estimation task is an integral part of:

  • MWA Chance & Measurement Years 5 & 6
  • MWA Chance & Measurement Years 7 & 8

The Angle Estimation lesson is an integral part of:

  • MWA Chance & Measurement Years 7 & 8

From The Classroom

Dave Miller-Stinchcombe

Artist in Training
This letter was first published in the November 2013 Mathematics Centre eNews.

Perhaps it will give you a reason, if you have Rotagrams, to talk with the art teacher(s) in your school.

Dear Ina,
I'm a maths teacher based in the UK. I have recently taken up drawing as a hobby, and have started using a Rotagram as a way of checking angles, particularly for perspective drawing and for portraiture. This works much better than a protractor, as I'm not cluttered up with numbers and can just use my eyes, plus the Rotagram has square sides so it can rest on a ledge, or be fixed to a clear surface and still used.

I showed the art teacher at school what I was doing and she had never seen a Rotagram before, but loved the idea of using them to help students to draw in perspective. The Rotagram is superior to (...alternatives...) for the same purpose (I feel), as well as having further applications in art. So a quick bit of googling, and I find you, who appear to be the only makers and retailers of Rotagrams I can find.

(I have included) a couple of pictures showing how the Rotagram can be used to help sight angles when making a drawing. Please forgive the drawing, I've only been learning for a few months. I hope these are useful to you.


Sighting an angle.

In composition scale.

The angle on paper.

One of the very good reasons why Geoff Giles designed this tool to assist with learning about angles is, exactly as Dave has realised, because it isn't cluttered with numbers.

Green Line
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