Powers and Standard Form: Book 12

INforM – Interactive Notebooks for Mathematics Book 12 - Powers and Standard Form

Powers and Standard Form

The activities are related to work in Ma2 Number and algebra. Students should be taught to use index notation [for example, 8 2, 8 to the power minus two thirds] and index laws for multiplication and division of integer powers; use standard index form, expressed in conventional notation and on a calculator display

A variety of real life situations are used to place the topic in context.

Organisation of the materials The Smart Notebook file is saved as ‘Book 12 Powers and standard form.notebook’ It consists of 7 pages of which the first is the title page, shown above. There are 4 pages to support the activity. Page 6 is a blank page. Page 7 contains teacher notes which are amplified here.

The first activity On page 2 of the Notebook is the first activity ‘Work out the size of Jupiter’ and serves as an introduction to the notation.

The information about the earth is presented in both standard form and long integer formats to encourage discussion about how standard form ‘works’. Pupils can come to the board to measure both the Earth and Jupiter. The rulers and pictures have been scaled so that the earth measures ‘1cm’. This will allow pupils to carry out a simple multiplication to find the diameter of Jupiter. This could be done as a whole class and then pupils can work individually on mini-white boards to write down how they think the number would be written in Standard Form. Discussions that arise could cover the fact that the first part of the notation has to be <10, also the number of decimal places used, rounding and the subsequent effect on accuracy.

The main activity Page 3 moves on to consider the relative sizes of some of the other planets.

One approach is to rewrite the planet sizes as ordinary numbers and discuss the relative sizes. Pupils could do this individually on a mini-whiteboard or with a partner to ensure that they have understood the basic notation.

The planets have been selected such that Mars is about half the size of the Earth, Pluto is approximately a third the size of Mars, so a sixth of the size of the Earth and Saturn is approximately 10 times the size of the Earth. Therefore, a good size to make the Earth on the board is 12mm.

Having completed the task, a discussion point is what happens to a number written in Standard form if it is multiplied by 10 or 100? What if a number is halved? Examples will need to be chosen carefully to cover different outcomes eg 8 x 105 halved is 4 x 10 5, whereas 1.2 x 105 halved is 6 x 104.

of the Notebook introduces powers of 2 in a real life context and can then be used to reinforce standard form whilst also giving pupils the opportunity to apply their problem solving skills.

It is recommended that the video is simply played initially so that pupils gain some idea of exponential growth.

The 2 questions on the left hand side give pupils a problem solving opportunity.

One suggestion is to calculate how much video footage represents one hours worth in real time (approximately 8 seconds). Select the ‘count up’ option on the stopwatch. Start the video and then, because there is some lead time on the video, start the stopwatch as soon as the footage begins. Once the stopwatch reaches 8 seconds, stop the video. This will require some co-ordination and it is suggested that the teacher tests it out beforehand to become familiar with it, but perhaps allows pupils to carry out the tasks in the lesson. After 8 seconds have elapsed and the video has been stopped, it is possible to count how many cells there are on the screen and consequently work out how many divisions there have been. This does provide pupils with opportunity for debate since a number of pupils will not initially realise that all cells divide. For example, if there are 16 cells on screen, implying 4 divisions, some pupils will think there have been 8 divisions since 2 x 8 = 16.

The last 2 questions give an opportunity to explore both powers of 2 and Standard form. Possible routes of questioning are if 210 is roughly one thousand, then which power of 2 would we need to have roughly one million? What is the output on a calculator when we put in 2100?

The extension task The final page for pupils is designed to allow exploration of numbers in standard form using negative indices. The ruler is locked in place so objects will need to be moved to it. Shorter pupils may need the objects moved towards the bottom of the board, but the ruler is placed low enough to allow access to all.

SMART specific issues

The Smart Notebook Gallery has an education area within which there is a Mathematics area. This contains some useful Notebook resources, including Flash resources such as the calculator and the stopwatch which is used in this activity. Additional resources can be downloaded from the EdCompass website. (http://edcompass.smarttech.com). Follow the links to Notebook collections and Mathematics. If you are in an open Notebook page, Select Import and Import Online Content and, providing you have an internet connection, the relevant web page will open automatically.

The science area of Smart Notebook Gallery has a number of resources that provide real life applications for Mathematics.

Flash resources return to their original position, regardless of their state when the Notebook is saved. This means that the stopwatch has to be reset to ‘count up’ each time it is used.