Crossing the Bridge

1. Crossing the Bridge Four friends must all cross a bridge in 17 minutes. They start on the same side of the bridge. A maximum of two people can cross at any time. It is night and they have just one torch. People that cross the bridge must carry the torch. Each person walks at a different speed. A pair must walk together at the rate of the slowest: Rachel: - takes 1 minute to cross Adam: - takes 2 minutes to cross Jenny: - takes 5 minutes to cross Harry: - takes 10 minutes to cross How can they all cross in 17 minutes?

2. Crossing Bridges Jennifer Piggott 2007 www.nrich.maths.org

3. Outline • NRICH: • Philosophy • Content • Some maths • Reflections on crossing bridges

4. The Yonghy Bonghy-Bò On the Coast of CoromandelWhere the early pumpkins blow,In the middle of the woodsLived the Yonghy-Bonghy-Bò.Two old chairs, and half a candle,--One old jug without a handle,--These were all his worldly goods:In the middle of the woods,These were all the worldly goods,Of the Yonghy-Bonghy-Bò,Of the Yonghy-Bonghy-Bò. … … By Edward Lear

6. Enrichment • Content: Posing and solving problems and tackling rich tasks. • Teaching: Creating an atmosphere of sharing, critical evaluation and difference. Modelling what it is to be mathematical. • Pupils who: Are creative and imaginative, are comfortable with feeling uncomfortable.. Work within a community. • Longer term effects: Confident independent learners who can apply knowledge beyond the classroom.

7. Make 37 Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any four numbers from the bags so that their total is 16. Pickany ten numbers from the bags so that their total is 37.

9. Good Problems • Are very much about and for individuals • Initial impact • Have particular content outcomes

10. Crossing Bridges NRICH Teacher • A model • Rationale • Assessing need • Testing • Optimising input • Building • Supporting not leading • Crossing • Keeping a watchful eye Learner NRICH Teacher Learner Teacher NRICH Learner Learner NRICH Teacher

11. www.nrich.math.org • Four main aspects to the site: • Monthly magazine • Archive • Maths finder • Mapping documents • Articles and games • Packages • Ask NRICH • Thesaurus • Plus the Newsletter and other publications.

13. Rich Tasks • accessible, • intriguing, • challenging, • low threshold - high ceiling, • contexts for problem posing, • offer potential for elegant or efficient solutions, • broaden mathematical content knowledge and skills, • encourage creativity and imaginative application of knowledge, • reveal patterns or lead to generalisations or unexpected results, • reveal underlying principles, • enable learners to make connections, • encourage collaboration and discussion, • develop confident and independent critical thinkers.

14. Rich Contexts • have time to explore starting points and alternative routes, • encourage dialogue and interactions, • Include modelling and metacognition, • Make use of props and cues, not hints and clues, • involve a community or practice, • develop and use appropriate language, • allow sharing, • encourage creative, independent thinkers, • value different approaches, • use critical evaluation of effective and efficient methods, • develop learners’ confidence in being mathematical, • support the application of knowledge beyond the classroom, • are appropriate to everyone.

15. What bridges to build? • For pupils • Safety  Challenge • Dependence  Independence • Wary of the new  Exploration • Solitude  Communication • For teachers • Keeping control  Letting go • Cautious  Trusting • Talking  Listening • Leading  Supporting

16. Basket Case A woman goes into a supermarket and buys four items. Using a calculator she multiplies the cost (in pounds) instead of adding them. At the checkout she says, "So that's £7.11" and the checkout man, correctly adding the items, agrees. What were the prices of the four items? www.nrich.maths.org – May 2005

17. Finding out more Go to: www.nrich.maths.org Search for:- Crossing Bridges Email:- Jennifer Piggott – jsp38@cam.ac.uk

18. Keep Your Distance There are four points on a flat surface: How many ways can you arrange those four points so that the distance between any two of then can be only one of two lengths? Example: www.nrich.maths.org – Sep 2005

19. Concrete wheel 100 mph 100 miles nrich.maths.org – May 2006

20. Smith and Jones Mr Smith and Ms Jones are two maths teachers, who meet up one day. Mr Smith lives in a house with a number between 13 and 1300. He informs Ms Jones of this fact, and challenges Ms Jones to work out the number by asking closed questions. Ms Jones asks if the number is bigger than 500. Mr Smith answers, but he lies. Ms Jones asks if the number is a perfect square. Mr Smith answers, but he lies. Ms Jones asks if the number is a perfect cube. Mr Smith answers and (feeling a little guilty) tells the truth for once. Ms Jones says she knows that the number is one of two possibilities, and if Mr Smith just tells her whether the second digit is 1, then she'll know the answer. Mr Smith tells her and Ms Jones says what she thinks the number is. She is, of course, wrong. What is the number of Mr Smith's house? http://nrich.maths.org/public/viewer.php?obj_id=2194