My year 9 bottom set have been doing probability recently (in fact they finished the topic today with some healthy topic assessment results - quite a few Level 6s in the old NC parlance). In order to help them prepare for the new GCSE, we have looked at some problems that link probability with proportion, beyond the idea of just fractional representation. I thought I would share a few of my favourites here.

1) "In a bag there are only red, white and yellow counters. I am going to take a counter out of the bag at random. The probability that it will be red is 1/4. It is twice as likely to be white as red. Give an example of how many counters of each colour there could be."

This was a great starter problem to link these ideas together. Many of the pupils did see the relatively obvious 1, 2, 1 solution; linking to 1 out of 4. But this problem allows us to explore other solutions, with the proportions linking the idea of solutions in the ratio 1:2:1. We can then talk about different probabilities for red (i like 1/6 with white being 3 times as likely so you get the ratio 1:3:2).

2) "A bag contains red, white and blue counters. The probability of taking a red counter is 1/6.

(a) John says that there are 40 counters in the bag. Explain why John cannot be right.

(b) There are 12 red counters in the bag. Work out the total number of counters in the bag."

This one is a great problem for exploring the links between probability and fraction calculations, in particular fraction of an amount and equivalent fractions. We can talk about mixed numbers, and that they are not appropriate for this context, What is nice then is to discuss changing the numbers in the question, and which numbers make it easier, harder, or the same? This allows the exploration of the idea that multiples of 6 in part (a) means that John can be right, that reducing the number of red counters to 1 makes it easier to answer the question, but that it doesn't necessarily need to be a multiple of 6, and that 12 is a 'coincidence' in that case - however change the fraction to a non-unit fraction and suddenly multiples of the numerator do become important.

3) "A die has three red faces, two blue faces, and one green face. Sam rolls the die 300 times and gets these results (in a table) Red = 153, Blue = 98, Green = 49. Is the die biased? Explain your answer."

I love this problem. Once you get past the idea of biased because it is more likely to land on red, you can talk about the relative proportions of the number of sides compared to the relative proportions of the results, and these being the same. Again it can be ratio, fractions of 6 compared to fractions of 300, or simply something like each one is nearly 50 times bigger (and in reality of course it would be nice if it were all of these).

There are others and again sharing is welcome through the comments or otherwise, but given the fact that proportional reasoning is such a big part of our new GCSE, in my opinion these sorts of problems are the sort that need exploring with pupils of all levels.

1) "In a bag there are only red, white and yellow counters. I am going to take a counter out of the bag at random. The probability that it will be red is 1/4. It is twice as likely to be white as red. Give an example of how many counters of each colour there could be."

This was a great starter problem to link these ideas together. Many of the pupils did see the relatively obvious 1, 2, 1 solution; linking to 1 out of 4. But this problem allows us to explore other solutions, with the proportions linking the idea of solutions in the ratio 1:2:1. We can then talk about different probabilities for red (i like 1/6 with white being 3 times as likely so you get the ratio 1:3:2).

2) "A bag contains red, white and blue counters. The probability of taking a red counter is 1/6.

(a) John says that there are 40 counters in the bag. Explain why John cannot be right.

(b) There are 12 red counters in the bag. Work out the total number of counters in the bag."

This one is a great problem for exploring the links between probability and fraction calculations, in particular fraction of an amount and equivalent fractions. We can talk about mixed numbers, and that they are not appropriate for this context, What is nice then is to discuss changing the numbers in the question, and which numbers make it easier, harder, or the same? This allows the exploration of the idea that multiples of 6 in part (a) means that John can be right, that reducing the number of red counters to 1 makes it easier to answer the question, but that it doesn't necessarily need to be a multiple of 6, and that 12 is a 'coincidence' in that case - however change the fraction to a non-unit fraction and suddenly multiples of the numerator do become important.

3) "A die has three red faces, two blue faces, and one green face. Sam rolls the die 300 times and gets these results (in a table) Red = 153, Blue = 98, Green = 49. Is the die biased? Explain your answer."

I love this problem. Once you get past the idea of biased because it is more likely to land on red, you can talk about the relative proportions of the number of sides compared to the relative proportions of the results, and these being the same. Again it can be ratio, fractions of 6 compared to fractions of 300, or simply something like each one is nearly 50 times bigger (and in reality of course it would be nice if it were all of these).

There are others and again sharing is welcome through the comments or otherwise, but given the fact that proportional reasoning is such a big part of our new GCSE, in my opinion these sorts of problems are the sort that need exploring with pupils of all levels.

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