Assignment 3
Types of Numbers – Operations with Rational numbers
Fractions and Decimals
Work individually or in pairs. You will produce a paper of two to three pages outlining your ideas (hypotheses) and justifying them in the light of your investigations.
Preamble:
Now that we understand a little about types of numbers and their classification, we need to investigate how they relate to each other. Beginning with rational numbers, I’d like to pose some focus questions about the relationship between decimals and fractions.
We understand that fractions, decimals, percentages and ratio are all ways of expressing rational numbers. A rational number can be expressed as a fraction a/b where a and b are integers. Recognise the terms?
Focus question:
Do all fractions convert to terminating or repeating decimals? Are there any that don’t terminate or repeat?
Is there a relationship between the denominator of the fraction and the nature of its equivalent decimal? ie Can you predict from the denominator, whether a fraction will repeat or terminate as a decimal?
Investigation:
Set up a spreadsheet that converts fractions to decimals. Starting with ½, convert all the proper fractions with denominators from 2 to 14. How many of these are there? How do you change a fraction to a decimal? Use columns labelled numerator, denominator and decimal in your spreadsheet. See below
Divide the decimals and their equivalent fractions into terminating, recurring (or repeating) and cyclic repeaters (what are these?).
Consider the terminating decimals. Is there a common relationship between their denominators? Hint: Consider their prime factors. Write down your ideas.
Now look at the repeating decimals. What patterns can you see? Spend a bit of time on this. Is there a relationship between the denominator and the pattern of the decimal? Record your ideas.
Do the denominators of fractions that produce repeating decimals differ from those that produce terminating decimals? How do they differ? Hint: Prime factors again. Write down your ideas/
What you are doing is called hypothesising. Your ideas are hypotheses. Can you use these ideas to predict the type of decimal that a fraction will convert to? Do your ideas work for fractions with denominators > 15 eg what will something over 20 look like as a decimal? What about a denominator of 30?
Investigation 2:
Check your hypotheses by constructing a second spreadsheet of fractions to decimals where the numerator is always 1 and the denominator varies from 2 upwards. Go as far as you like. Here is a sample:
Test your ideas against your second spreadsheet and refine them as needed.
Write up all your ideas with examples, or counter-examples, if they don’t work.
Post it on the wiki for others to read and comment.
Conclusion:
The process that you have undertaken is closely allied to the ‘scientific method’:
Hypothesise, Investigate, Refine your hypotheses, test them, publish findings.
We will do a number of investigations using the scientific method in this topic and future ones.