Powers / Indices / Exponents

INTRODUCE

PRESENTATION: Light the Lights Puzzle

NRICH have a great puzzle which can be used to introduce square numbers and therefore, as a consequence, the idea of powers. The idea is that the students test different values which each light up particular lights – either red, yellow, green or blue. They have to try to determine the rules that either turn a light on or off and therefore the smallest number that would turn all four of the lights on.

Preferably, the students would have access to a computer themselves which would allow them to try their own values. However, this does mean that they would be able to click on Solution to see the answer. An alternative would be to have the students decide on some numbers that they would like to test as a class. As they believe they have found particular rules, you could give them the opportunity to try more values to test them.

I would advise giving each student this HUNDRED GRID, provided by NRICH in the Teachers’ Resources. As numbers are tested I would suggest that they colour in each number accordingly, which will allow them to begin to see patterns. I have provided an example of a fully coloured-in 100 grid for you to use as a reference, along with the rule for each colour, which can also be found on the presentation.

IMG_2637

As an extension, you could ask the students to provide an algebraic expression to describe each of the rules, which are also included on the 100 grid and on the presentation.

 


APPLY

PRESENTATION: Binary

There are 10 types of people in the world…

…those that understand binary and those that do not.

Knowledge of binary is necessary to encode information which allows computers to function. Computers store all information in binary bits or, more often, in octal or hexadecimal. In addition, sound and video is stored digitally via binary on CDs, DVDs and Blu-Ray discs.

All number systems rely on powers to work. The decimal number system that we use in our daily lives relies on powers of 10, binary relies on powers of 2 and hexadecimal on powers of 16. Without a good knowledge of powers we would be unable to write numbers in any number system, even if we do not realise that it directly relies on powers.

The presentation goes through how the decimal number system works and then extends this to explain how the binary number system works. After explaining the mechanics of the binary number system, the presentation then gives some applications of binary to computer programming, for example.

Numberphile produced a good video which explains binary and an interesting application of it to Pacman, the video game, which you may wish to show as an accompaniment or alternative.


AMAZE

VIDEO: Graham’s Number

Graham’s Number is one of the most famous numbers in mathematics because, at the time, it was the largest number ever used in a published mathematical paper; it even made the 1980 edition of the Guinness Book of World Records. Ronald Graham was trying to find the solution to a problem in Ramsey Theory, which involved colouring connections between the vertices of hypercubes, and managed to prove that the solution must lay between 6 and Graham’s Number. In 2008, the lower bound was increased to 13 by Jerome Barkley.

Graham’s Number is so ridiculously large that there are not enough atoms in the universe to write one digit of Graham’s Number on each. Let that settle in for a moment…

The definition of Graham’s Number relies on Knuth’s up-arrow notation, which involves raising numbers to powers upon powers upon powers upon powers, and so on. Numberphile do a fantastic job of explaining what Graham’s Number is in many different videos. The two linked below are very good; the first involves a number of Numberphile regular contributors and the second has Ronald Graham himself. The link above takes you to the Numberphile playlist in which they have a number of videos all focused on discussing Graham’s Number.

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