It would be a fair and honest reflection to say that lots of resources I used during my first few years of teaching were quite repetitive. While sets of repetitive questions can have their purpose, they’re not always appropriate. Since then, I’ve been making a more conscious effort to find or write exercises that at least mix up the ways that questions are presented and include more twists and challenges.
I’ve found this to be easier for some topics than others. For example, even a sequence of fairly similar equations can be spiced up a little by simply switching round the orders of the terms.
For practice exercises on calculating areas of triangles, resource writers often mix things up by changing the orientations of the shapes or providing increasing amounts of information in each question.
However, one topic that I’ve struggled to do this for is areas of circles.
Let’s say that my class have got to the point where they have learned the formula for the area of a circle and now I want them to practise using it to calculate the areas of lots of whole circles. How can I provide a variety of questions without veering off just yet into areas of semicircles, sectors or compound shapes?
I’ve been searching through lots of textbooks and online resources, looking for ideas for ways to do this. I’d like to use the rest of this blog post to collate and share some of these ideas for questions on calculating areas of whole circles. No doubt there are plenty more ideas to find, but here is a selection…
1. Varying whether students are given the length of the radius or diameter:
This seems the most obvious and common way to mix up circle area questions. The main thing students need to ask themselves while working through these sorts of questions is, “Do I need to halve the number before substituting it into the formula or not?” I quite like exercises that include even numbers for radii and odd numbers for diameters, like in the third and fourth examples. If students aren’t thinking carefully about what they are doing, they might be tempted to halve the 6 cm because it’s even or doubt whether they should halve the 9 cm because it would give a decimal.
2. Varying the ways that the lengths are labelled:
I only came across examples like the third and fourth ones a couple of years ago. Since using them, I’ve noticed that students seem to find these a little trickier than the first example because the length of each radius is not quite as obvious. However, labelling lengths in such a way in the early stages could make it easier for students later on, when they begin calculating areas of shaded regions.
3. Questions that don’t provide diagrams:
The bottom two questions require students to think a little more carefully about the meaning of ‘radius’ and ‘diameter’ before deciding what numbers to substitute into the formula. There were plenty of alternative worded descriptions for this kind of thing.
5. Questions that provide more than one length for students to choose from:
These questions can catch out students if they have got into the habit of just substituting what ever number they see (or half of it) into the formula. They have to think a little more carefully about the fact that they need the length from the centre to the edge of the circle.
5. Questions where students need to work out the length of the radius themselves before they calculate the area:
I found a bunch of nice questions like this. Before students can calculate the area, they have to consider what information they need and how they can get it.
6. Questions where students need to measure the radius or diameter with a ruler before they calculate the area:
I’ve noticed that students tend to find the empty circle the most difficult because they instinctively try to measure the radius. This is usually by attempting to guess where the centre is, but it can be very hard to do this accurately. However, if they understand that the diameter is the furthest distance from one side of the circle to the other, then they could try to measure that instead. For example, they could slide their ruler up the circle, watch the measurements initially increase and then decrease again as they pass the half way point, slide back to the greatest measurement and take that as the length of the diameter.
7. Questions where students need to measure the diameter with a ruler, but part of the circle has been covered:
These questions seem to back up point from the third example in the previous section. In the second question, students can be sure that they are able to measure the furthest distance across the circle because it starts to curve back inwards again just before it’s covered. However in the third example, the circle is still curving outwards at the point where it starts being covered.
8. Questions where students need to calculate the length of the radius before they calculate the area of the circle:
If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: Thinking About Calculating Areas of Circles
If you found this post interesting, then I would also recommend the following:
- These websites by ‘Boss Maths’ (@Boss_Maths) and Craig Barton & friends (@mrbartonmaths) provide lots of free resources that are usually good for incorporating variation into their practice exercises: BossMaths.com and VariationTheory.com
- For more about circles, here is an interesting video of a lesson that introduces students to the properties of circles through exploring different ways of cutting circles out of square pieces of paper. 4th Grade Shanghai Math Lesson – Introduction to Circles
Previous blog posts:
This series, ‘Planning Thought’:
‘What I Learned From Shanghai’ Series:
- Not As Complicated As It Looks
- Do Sweat The Small Stuff
- A Shanghai Lesson on Adding Negative Numbers
‘Making Connections’ Series
- One Method: Many Topics
- Crafty Starter Activities
- Many Methods: One Topic (Part 1)
- Many Methods: One Topic (Part 2)