Teach Secondary Issue 14.7

Students could make a table with some examples, like this: speed time reduction 1 0 1.5 ? 2 1 2 3 2 3 4 3 4 5 4 5 Note that if the podcast runs 4 times as fast, it takes 1 4 of the time. That’s a reduction in time spent of 3 4 , not 1 4 . Clearly the ? in the table needs to be a fraction between 0 and 1 2 . This is a good way to think about reciprocals. The time it takes to listen to a � -speed podcast is � 1 of the normal speed. So, the time it takes to listen to an �� speed podcast is going to be �� of the normal speed. This tells us that for a 1.5 = 3 2 speed podcast, the listening time will be 2 3 of the normal speed, so the reduction will be 1 3 . Don’t rush through this – let students create lots of examples to see why it works. With a 1 3 reduction corresponding to 15 minutes, in the original question, the original podcast time, at normal speed, must have been three times as much, so it was a 45-minute podcast. That’s the thing we can deduce. Checking for understanding Can you invent a similar problem to this one that has a nice neat answer, like this one did? Students will need to think of a speed that has a ‘nice’ reciprocal in order to make this work. [ M A T H S P R O B L E M ] PROPORTIONALITY Proportionality is a notoriously difficult idea for students, observes Colin Foster... Colin Foster (@colinfoster77) is a Professor in Mathematics Education in the Department of Mathematics Education at Loughborough University. He has written many books and articles for mathematics teachers. foster77.co.uk , blog.foster77.co.uk THE DIFFICULTY Yesterday, I listened to a podcast at 1 1 2 speed, rather than at normal speed. It saved me 15 minutes. What can you work out from this? This is a hard question that students will likely struggle to answer. Part of the difficulty is that the question doesn’t even tell you what it wants you to work out – only that something can be deduced! If your students are completely stuck, then it’s fine to leave this question hanging, because by the end of the lesson they will be able to answer it. THE SOLUTION What does ‘ 1 1 2 speed’ actually mean? If this is too hard for students to express, try this instead: What does ‘double speed’ mean? This is easier. It means the podcast will take half as long to play as it would at normal speed. Give me some more examples like this. Students will offer things like ‘4 times the speed means it takes a quarter of the time’ – although at 4 times the speed, it might be impossible to follow what anyone on the podcast is actually saying! In this lesson, students see how the speed at which they listen to a podcast affects how much time they save. 69 teachwire.net/secondary