Last year we set up a branch of the Association of Teachers of Mathematics (ATM) in Singapore. The ATM is a UK professional organisation and I used to go to their meetings in the UK and in Hong Kong.
Last year, one of my colleagues in the Infant school decided to set up an ATM branch in Singapore. (Setting up a branch is free and astoundingly easy.) So far the ATM branch has met four times with about 50 teachers each time.
We had a meeting this week and it was brilliant to meet both primary and secondary colleagues from around Singapore to talk about maths teaching ideas. And eat cakes and fruit drink wine!
The theme for our meeting was shape and geometry teaching ideas. One good idea that was shared in the Secondary teachers chat was a gift-wrapped sphere with a note that said, “This Secret Santa present is yours if you can estimate its volume to within 50 cm2.” I made an estimate (without a calculator) but apparently I was 52 cm2 off the correct answer! Doh. I never did find out what was in that package.
I shared an idea for teaching about the surface area of a sphere using an orange that you unpeel. It’s described on this wonderful blog by William Emeny. (His blog’s name is eerily similar to my own!)
If you work in Singapore, please come along to our next meeting! We meet once a term (three times a year) and email invites are sent out.
If you don’t live in Singapore, consider joining a professional group near you. I highly recommend it for meeting new friends, hearing about other schools, and sharing ideas.
This question is quite vague – a useful feature in my opinion! Describing the area of the triangle as “slightly smaller” means that there are many possible correct areas. Students are not intimidated and willing to try to make a triangle.
The task is also able to be differentiated to many levels. At the most basic, students can draw accurate triangles and count squares and parts of squares. Students who know how to find areas of right-angled triangles using the formula (1/2)bh can draw these. To find the final side length requires Pythagoras, though the teacher can decide not to request this. More advanced students can draw triangles and find their areas using the formula (1/2)ab sinC. Finding the third side length of these triangles requires the cosine rule.
This is great starter or ender to lessons. It can be used to consolidate or review area concepts.
Do you like vague questions?
An open question like this one:
- is easy to pose,
- is easy to understand,
- can lead to lots of practice (if that’s your goal for the students),
- can be kept simple or extended to suit lots of students,
- can lead to an interesting discussion,
- can lead to a generalisation.
I used this with a year 11 class that are revising for their Foundation IGCSE exam. I asked students to come up to the board and sketch some triangles that fit the criteria. Some triangles had 10 cm as the length of one of the shorter sides. Others had 10 cm as the length of the hypotenuse. Some students had chosen to draw isosceles triangles; other students drew scalene triangles.
This task can be made more simple by asking students to draw accurate triangles and measure the sides. Then they can extend this to checking using Pythagoras’ theorem.
On the other hand, this task can be extended by asking students to generalise what they have found. If 10 cm is length of the hypotenuse, what can be said about the other two sides? If the triangle is isosceles, there is only one answer. But if it is scalene, perhaps students will call the length of the second side x. Then they can come up with a formula for the third side in terms of x. And how does this change if 10 cm is not the length of the hypotenuse but of one of the other sides?
I have been reading More Good Questions by Marian Small. One differentiation strategy she recommends is to give students 2 options. The questions are related and the follow-up discussion engages students who did either option. For the question above, we talked together after a few minutes.
- What is the area of each of the full circles in your picture?
- How can you test that the answer is reasonable?
I set these cheese questions for some year 11 students who are revising for their IGCSE Foundation exam. After a few minutes we talked about the two answers using these questions.
- Which measurements did you use?
- Did you look up any formula?
- What units did you use for your final answer?
These two questions are available here.
Here is a classroom activity I used with my IB Maths HL students to practice the double and half angle formulae. It is a set of dominoes that the student has to connect from the Start card to the Finish card by solving problems with the double and half angle formulae. The question on the right (above) needs to be connected to its answer on a another card. The answer on the left fits with a different question. I know that the students have completed the task correctly if all the cards are used in a chain from Start to Finish.
I made this activity with Formulator Tarsia, brilliant free software for teachers. I have used it to make matching cards, dominoes, and hexagon puzzles. It’s designed for maths and it can handle maths notation including fraction, exponents, roots, integration, matrices, and so on. I converted the Tarsia file to a pdf using Cutepdf.