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 2 5 6 2 + 2 2 6
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
Indices investigation 1 You have already met powers. For 2 is an of a power. You have also been introduced to fractions, decimals and integers, and have been introduced to algebra with letters being used to stand for numbers. In this investigation you will be exploring what happens when these sorts of numbers (or pieces of algebra) are used in powers.. Task 1 Investigate powers with different types of numbers 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 When you learnt about powers you 'invented' them as a short way of writing repeated multiplications. instead of writing 2 2 2 2 you can write 2 However, if you want to work out the value of 2 unless you memorise the answer, you still have to work it out the long way (by doing the multiplication) 2 2 2 2 2 2 2 8 2 16 as there is no short cut to get straight from the 2 and the in the power 2 to the answer 16 Now it is fairly obvious that if you can write 2 2 2 2 as 2, you should also be able to write and as include brackets to show that -2-2 -2-2 as (-2) likewise for this is the base The task you have been set is to investigate such powers to see if there are any short cuts which will help you work out their values without having to multiply everything out the long way. Here is an to show you how to investigate different sorts of base doing the long way
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
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 -1 2 2-1 2 2-2 2 If you want to include the negative sign as part of the base you must use brackets therefore (-2) 2-2 -2-2 2-1 2 2 -
Indices investigation 2 Task It is possible to think of a power as a sort of number. If we do this we can explore what happens as we use them in problems. In this investigation, you are to investigate what happens when we try to add, subtract, multiply or divide powers. 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 When you operate on numbers you change them in some way. The four operations you know best are addition, subtraction, multiplication and division. If we look at how these operations change the numbers 2 and we find 2+5 2- -1 2 6 2 2 We see that each operation produces a different answer (and uses a different mathematical rule to get that answer.) In this investigation you are to investigate how powers work when they are added, subtracted, multiplied and divided. Once again, use the format from page 2 of the first investigation and look for any short cuts that will help you find the answers more quickly. Also use the same sort of strategy you used to investigate the different sort of bases. Remember to use the investigation flow diagram in writing up your work. Don't forget that finding there is no short cut for a certain type of problem is also an important discovery that you need to record.
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 2 2 2 2 2 2 16 2 2 2 2 8 2 2 2 2 1 2 2 2 0 2? 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 05. 05. 05. 05. 05. 05. 05. 05. 05. i e 1, 2,,, 5, 6, 7, 8, 9, etc keep the base the same but change the index 01. 02. 0. 0. 05. 06. 07. 08. i e,,,,,,,, etc
Both approaches use a system and form a pattern - and produce interesting results when graphed (try it.) You may even be able to explain some weird calculator results by using some of the graphs you draw. By the way, one thing that can happen in this investigation is that you don't always find short cuts which you could use to work out answers much more quickly and easily... Instead you might discover some interesting relationships between powers and other sorts of numbers. (If you want a clue, or to find out more about these other sorts of number, try to find out about the topics of logarithms and/or surds.) Also, sort through your sets of results carefully, as only one or two of the indices you are likely to investigate will give you the clues you need to find the links. As with the first of the indices investigations, to successfully complete this task you will need to plan your approach carefully. Here are some suggestions that might help you construct your strategy. (1) Write down all the different types of index that you should investigate (2) Use sets of problems and create patterns and sequences in them () Use a calculator () Start with the easiest types of problem (5) Think of any special numbers that may produce special results when they are used as indices. Extension Investigate powers with different types of indices and bases. Are there any special rules for evaluating something like 2 1 or 5 2 etc (The problems in this part shouldn't be too difficult, but should be enough to show what happens when the different sorts of indices and bases are combined.)