# The Indices Investigations Teacher s Notes

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1 The Indices Investigations Teacher s Notes These activities are for students to use independently of the teacher to practise and develop number and algebra properties.. Number Framework domain and stage: Two activities are designed for capable students who are beyond stage 8 Activity 2 is appropriate for students who are at stage 8, advanced proportional Curriculum reference: Number, levels 5 and 6 Numeracy Project book reference: This investigation is not related to any activity in the Numeracy Project resource books Prior knowledge. Students should be able to: Use power notation and the language of powers Work with fractions, decimals and fractions Understand algebraic notation During these activities, students will meet: Powers involving fractions, decimals, integers and algebraic expressions All sorts of things that come from understanding powers Background Each activity can be set as a project to last students a week. Answers are not provided, as students can come up with all sorts of interesting discoveries, and can even follow a single discovery into a branch of mathematics that is new for them. Periodic meetings with students may be needed to check that they have developed a plan of attack and are on track. Rather than marking these investigations, students have been asked to prepare a presentation to the class on what they have done, and what they have learned. This can produce some fascinating sessions. Students even providing notes for the other students on their discoveries and how they work with worksheet and s to practise should this be desired. Be aware however, that even very capable students can find these investigations daunting, as they are unstructured, and they need to develop their own approach. If you feel that they need some initial support, an introductory session could be run. Activity 2 is probably the simplest of the investigations merely involving students looking for patterns and short cuts when adding, subtracting, multiplying and dividing powers. If this is restricted to just looking at what happens with whole numbers (as indices and bases) it is very manageable. The activity is an important one as too many students make erroneous assumptions about when the Great Index Law can be used, as they have never explored its limits. For

2 However even activity 2 may need to be scaffolded if students are unfamiliar with investigations or discovery in mathematics. This can be done by discussing with students what they think they should be looking at (what sorts of numbers etc), working on several sets of s of problems, looking for patterns in what is found, and by explaining how to write up the discoveries. A sheet providing guidelines for writing up an investigation is available for students to follow. (ref investigation sheet here) Investigations: PDF or Word

4 27 6 summarising in power form often identifies the short cut In this there appears to be a short cut to do with using the index outside the bracket for both the numerator and denominator of the fraction. short cut 27 6 Most of this investigation can be done by trying different sorts of problems then looking for patterns and short cuts. Once you have identified a 'short cut', you should write a rule which (a) says how the 'short cut ' works (b) says when you are allowed to use it (c) gives an of its use This rule then needs to be tested on a series of questions which are answered using both the 'long' way and the 'short' way. (If both routes get the same answer, your rule is likely to be a good one.) Note - this task you have been given is too big to be solved without planning. The first thing you should do is decide how you are going to tackle the problem. Here are some guidelines that might help you. (1) Write down all the different types of base that you have to investigate. (Hint - there are lots of algebraic ones) (2) It is always easier to start with the simple stuff and gradually increase the level of difficulty. (Hint - what numbers are easy to deal with? What numbers might give you special results?) () Your overall approach needs to be systematic, carefully structured and wellexplained when dealing with fractional bases you should use different sorts of fractions like one third, three sevenths, nine fourths,... easiest 1 7 special as it has a one on top fraction with top number smaller than the bottom one

5 getting 9 fraction with top number bigger than the bottom one harder 5 6 negative fraction etc () Be careful in the way you set out your problems. (Hint - follow the format from the on page 2.) Finally, there is one thing you need to be aware of. In multiplying variables -1 x 2-1x 2 -x 2 So with powers If you want to include the negative sign as part of the base you must use brackets therefore (-2)

7 Indices investigation Task Investigate powers with different types of number as their base. Write up your investigation in your book, showing each step of the investigation. Be prepared to explain what you have done, and present some of your findings to the class. AC EA AA AM AP+ Explanation In this investigation you are to look at what happens to powers and their values when the indices are things other than natural numbers. (For, if the index is an integer or fraction). However, in investigating various types of index, it will not be possible (initially) to do the problem 'the long way'. 1 2 just what does mean? Consequently you must use other tactics to find the values of such powers. One technique is to use a calculator, the other is to look at patterns. A third could involve the use of a computer and a spreadsheet. Extending this pattern allows you to develop values and expanded forms for zero and negative indices. Powers of ? Furthermore, if your development is systematic, graphing your information will often provide you with useful results. When investigating decimal indices there are 2 basic approaches you can adopt. (a) (b) change the base but keep the index the same i e 1, 2,,, 5, 6, 7, 8, 9, etc keep the base the same but change the index i e,,,,,,,, etc

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