NCETM_Mathematics_Department_Workshops_Index_Numbers(1)

Mathematics Department Workshops Topic: Operations with Fractions Overview Objectives related to this topic are: • use index notation for small positive integer powers (Year 8) • use index notation for integer powers; know and use the index laws for multiplication and division of positive integer powers (Year 9) • use index notation with negative and fractional powers, recognising that the index laws can be applied to these as well (Year 10) • know that n1 = n and n1 = 3n for any positive number n (Year 11) 2 3 Materials required • Activities HT2.IDX.1a,b,c – A selection of introductory activities • HT2.IDX.2 – This requires Autograph or similar graphing software • HT2.IDX.3 – A Powerpoint resource • HT2.IDX.4 – A number investigation requiring scientific calculators • HT2.IDX.5a,b – Jigsaw files (one set printed in two parts) and .pdf versions • Resource sheet HT2.IDX.6 • Data projector and internet access Suggested activities Activity 1: Getting Started Consider as a team the following introductory activities and assess their suitability for the learners in your school • HT2.IDX.1a – “Invitations” • HT2.IDX.1b – “Penny Millionaire” • HT2.IDX.1c – “Chess Rice” Activity 2: Sample Learning Activities As a team, use the following resource sheets to explore the suggested learning activities. Resource Sheet HT2.IDX.2 - Graphical software The use of dynamic graphic software (e.g. Autograph) will greatly assist the learners in appreciating both the growth and decay of the exponential function, especially after seeing the rapid growth aspect in the first activity on doubling. As a department discuss whether it might now be appropriate to introduce the laws of indices for multiplying, dividing, using negative indices and finding powers of powers. Use of calculators in class will help learners to reinforce these rules for themselves. Slideshow HT2.IDX.3 – Fractional Powers As a team discuss how you would introduce the fact that fractional powers are equivalent to corresponding roots. The slideshow offers one approach to helping learners appreciate the equivalence of roots and fractional powers. The approach used is one based on squares and cubes. www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

Resource Sheet HT2.IDX.4 Friedman Numbers In this number investigation, learners are invited to tackle some number puzzles based on a curious property of certain integers. Consider who you would use this with. Are some of the ideas more appropriate for some learners than others? Resource Sheet HT2.IDX.5a and HT2.IDX.5b - Jigsaws The two files together form a single set of cards. The learners are encouraged to reinforce their knowledge of index numbers by combining cards with the same value in a group. • How many different groups are there? • Do they all contain the same number of cards? Activity 3: Reflection - so what’s good about these activities? Read ‘Eight Principles for Effective Teaching’ on resource sheet HT1.IDX.6 (taken from Malcolm Swan in the Standards Unit ‘Improving Learning in Mathematics’ resource) In pairs or small groups discuss the ways in which the sample activities fit and don’t fit with these principles. Collect other related activities which fit with some of the Eight Principles. Sort these activities into a progression which can be developed into a unit of work to be included into your team’s scheme of work. Embedding in practice Hooks for Learning • Powers of 10 Movie - http://www.youtube.com/watch?v=WYg_oSEV-XM (With Monty Python’s Galaxy Song in the background!) [2 mins 47 s] Teachers are advised to view this movie before presenting it to a class in order to assess its suitability for the age group of the learners. - http://www.youtube.com/watch?v=BBsOeLcUARw (With original commentary in the background) [9 mins 03 s] This version is longer but is more universally suitable for younger audiences. • Towers of Hanoi Puzzle – How many moves are required to move the discs? Legend has it that the world will end when the original set from the Temple of Hanoi has been moved: all 100 discs of it! Is this likely? Action points At the end of the session, spend time recording some actions. What do you need to do: • Next day? • Next week? • Next year? Further reading • Mathematics: From the Birth of Numbers Jan Gullberg • The Universal History of Numbers Georges Ifrah • Images of Infinity Ray Hemmings and Dick Tahta (Leapfrogs INSIGHT series) www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching