Disclaimer: the resource section of this blog risks becoming a Don Steward re-blogging service.
I have to credit my first mentor Dr Nick Sterno for this lesson or series of lessons. It is necessary for pupils to have strong prior knowledge of circles before taking on these tasks.
I haven’t yet tried any of the other 3 act mathematics from Dan Meyer but this one is good. It is important to give pupils the task of estimating the solution first (they always predict they must have the same volume). Also give them the big reveal at the end. It somehow means more to them seeing it in practice rather than just calculating. The suggested extension is good too.
A solid class will progress onto finding a given height/radius and rearranging the formula. Weaker students can be supported by giving them an A4 sheet of paper to experiment with. I quite like comparing the two methods for accuracy.
Presume you are confident their knowledge on cylinders is secure you can now look at cones and spheres. The link is nice without the need for any feeding of the formulae.
Always a go to website for all your resource needs. These visualisations are the king. Get the pupils to explain what the images demonstrate in their own words and test them on it later in the lesson. I normally find they can get most of the lesson through without ever needing to be fed a formula. This in my opinion promotes an understanding of the calculations. Once pupils were successful they can be fed the formula, it is more efficient than going through this rigmarole each time. This idea can be less successful with pupils who have previously been fed formulae but only half remember them.
The limits of pythag, rearranging and fractions will determine exactly how far you can push on from here. No point just repeating it just like this 100 times, once they get a couple right they get it. Just whether they can start finding it from slope length, frustums etc…
Make sure you dig around the various appropriate pages on the blog there are good questions hidden all over the place. A measuring cylinder question, a half full cone, limits of a sphere and such all come to mind. Make sure you are keeping your answer in pi when calculating answers Don has a habit of making the answers very marginal and wrong answers can be produced easily.
Build a cone and use the string demonstration to show how the curved surface area is derived. Helps pupils who struggle to visualise pi x r x l. As you open up the layers of string from around the cone and lay them flat, the base of the resulting triangle is obviously the circumference of the cones base and the height is the slant height not the vertical. Pupils can measure them if they really need to. Not all groups will benefit from this, but if they understand it becomes hard to forget.
Get them to draw the nets accurately. That will force the use of the compass and stress the link between arc length, and base circumference. You could reinforce the area of a circle by getting them to calculate the area of the circle.
What it is missing?
I want a selection of those questions where solids are emptying at a given rate. If I could have one of these (with a worked solution) for each 3D shape then the pupils could get practice at that style of question in 5 different lessons thus reducing the extent to which this is a shocking question. I might make it my business to find them next time I teach this topic. Unless there are any helpful soles out there…
Access maths has some. I love maths twitter.