How would you score it?

6957 /www/nrich/html/content/id/6957/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> A game was played at a fair in which children guessed the weight of Mr. Jones, the rugby teacher. Five children made the following guesses: 1. Between 70 kg and 90 kg 2. Between 75 kg and 100 kg 3. Between 88 kg and 140 kg 4. Between 90 kg and 95 kg 5. 85 kg exactly It turns out that Mr Jones weighs 88 kg. Each child argues that they should win the prize. How might each child argue that their guess was best? If you had to award the prize, who would you choose to be the winner, and how would you justify your choice? Devise a scoring system that penalises those who guess very wide ranges, and rewards guesses where the middle of the range is very close to the actual weight. Does your scoring system reward the child you thought should be the winner? </mdoxml> <mdoxml version="1.0"> Victor from St. Pauls School in Brazil, and Alice from the British School in Manila suggested similar arguments for each child: Child number 1 would argue that they have the smallest correct range. Child number 2 would argue that 88 is almost exactly in the middle of their range. Child number 3 would argue that 88 was the first number in their range. Child number 4 might say that the correct weight is just 2 away from the first number in their range. Child number 5 could argue that he was the only one to enter a weight, not just guess a range of weights (so in an unfair way, he had the smallest chances of getting it right). They would choose child 5 as he had by far the smallest chances and even so he almost got it right. Huw from Cowbridge Comprehensive School in Wales devised <a href="/content/id/6957/Huw.pdf">this scoring system</a> that penalised students who had chosen a wide range of values and rewarded students whose median value was close to Mr Jones' actual weight. Ben and Joe, from the same school, and Daniel from King's School in New Zealand, suggested the same scoring system. Here is how Ben and Joe worked out who had done best: <a href="/content/id/6957/Ben.pdf">Ben's solution</a> <a href="/content/id/6957/Joe.pdf">Joe's solution</a> Matthew from Moonee Ponds Central School in Australia suggested this scoring system: Find the difference between the actual weight and both ends of the range. Pick the larger number and subtract it from 100 to get your score. The highest score wins. Thank you to Sophie from The King's Junior School in Chester and to the many students from Mount Pleasant Primary School who also offered their opinions. </mdoxml> <mdoxml version="1.0"> <h3>Why do this problem?</h3> This problem offers a fairly informal introduction to the importance of considering spread as well as average when working with data. Students will intuitively make arguments for particular winners, and these intuitive ideas can be honed into more formal statistical statements about why some guesses should be valued more highly than others. <h3>Possible approach</h3> Display the question with the five guesses and give students a short amount of time to rank the five guesses in order. Collect a few of the students' rankings on the board. Now give students time to look at the results in more detail and to make a case for their preferred ranking. Explain that they will be expected to justify their ranking to the rest of the group, so they will need to think about the arguments others will make, and how to counteract them. After students have had time to consider their arguments, give those with differing views a chance to convince the class of the merits of their ranking. (If there is consensus within the class, challenge them to convince you.) Key ideas that should emerge are: <ul> <li>whether the true value lies within the range suggested</li> <li>how far the true value is from the middle of the range suggested</li> <li>how wide the suggested range is</li> </ul> Finally, challenge students to use these key ideas to produce a fair scoring system that could be published in advance of future "Guess the Weight" competitions. They could test their scoring system to check it agrees with their suggested rankings, modifying it if needs be, or they might reject their initial ranking if they believe their proposed scoring system is fairer. Perhaps students could also use their system to score a real "Guess the Weight" competition. <h3>Possible extension</h3> <a href="/7722">Retiring to Paradise</a> provides a different context for considering the importance of spread as well as average when working with data. <h3>Possible support</h3> Use the analogy of hitting a target in archery to help students to think about how to rank the guesses. "Is it easier to hit a large or a small target?" "Do you score more when you hit the target at the edge, or in the middle?" </mdoxml> <mdoxml version="1.0"> Imagine trying to hit a target in archery. Is it easier to hit a large or a small target? Do you score more for hitting the target at the edge or in the middle? Now imagine that the children's guesses are like shots at a target... </mdoxml> <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/7413">Warm-up</a></li> <li><a href="http://nrich.maths.org/7366">Try this next</a></li> <li><a href="http://nrich.maths.org/6956">Think higher</a></li> <li><a href="http://nrich.maths.org/7352">Read: mathematics</a></li> <li><a href="http://nrich.maths.org/7352">Read: science</a></li> <li><a href="http://nrich.maths.org/7352">Explore further</a></li> </ul> <div> </div> One method would be to mark each guess on two criteria: 1) Is the actual weight within their guess? 2) How large an interval does their guess cover? We could give them a fixed number of points, say 100, for choosing an interval that contains the actual weight, and then divide it by a number proportional to the size of the interval. For example, we might score the 5 guesses: 1) 100/21 = 4.76 2) 100/26 = 3.85 3) 100/53 = 1.89 4) 0 5) 0 However, this seems a bit unfair on those giving precise guesses. Let's write the guesses in the form $a\pm b$. For example, the first child's guess would be: $80 \pm 10$. Suppose the actual weight is m kg. Then we could choose the mark as $\frac{M}{(b+1)(a-m)^2}$. We can choose M to scale the marks to make them not have too many decimal places. Choosing M=10000 gives the marks as, making 5 the winner: 1) 1.291322314 2) 219.478738 3) 0.020292043 4) 40.31242126 5) 1111.111111 According to this scoring system, the winner is somebody who gave an interval that didn't contain the correct answer. Is this what we want? We could choose M=10000 if the interval contains the correct answer, and M=1000 if it doesn't. This gives: 1) 1.291322314 2) 219.478738 3) 0.002029204 4) 4.031242126 5) 111.1111111 which makes 2 the winner. </mdoxml> 2 3 0 0 1 0 0 How would you score it? Invent a scoring system for a 'guess the weight' competition. Handling, Processing and Representing Data Averages Using, Applying and Reasoning about Mathematics Selecting and using information Handling, Processing and Representing Data Processing and representing data Handling, Processing and Representing Data Measures of spread