The purpose of this activity is to engage students in recognising patterns in recurring decimals, and linking the size of those decimals to the division of 100 that produces them.
The background knowledge presumed for this task is outlined in the diagram below:
This activity should be used in a ‘free exploration’ way with an expectation that students will justify the solutions that they find.
The procedural approach (show more)
- The student uses trial and error approaches to find the missing divisor.
Students who are learning to apply the properties of multiplication and division to decimals are likely to try various divisors until they get the pattern they are seeking. Look for them to recognise that increasing the divisor reduces the quotient (answer) and reducing the divisor increases the quotient. The size of the number in the pattern should also influence their choice of potential divisors.Click on the image to enlarge it. Click again to close.
The conceptual approach (show more)
- The student applies the properties of multiplication and division, particularly the inverse property to find the missing divisor and creates other patterns by systematically trialling divisors.
Students who understand the inverse relationship of multiplication and division will apply their knowledge of ‘families of facts’ with whole numbers.Click on the image to enlarge it. Click again to close.
Conceptually focused students are likely to use their understanding of divisibility to choose appropriate divisors in the search for patterns.Click on the image to enlarge it. Click again to close.
If students have access to computer spreadsheets, they can find all the quotients easily and look for recurring patterns. Students might notice that multiples of four and five mostly give terminating decimals while multiples of three, seven and eleven usually give recurring patterns. They may recognise that recurring decimals occur when any prime factors of the divisor are not factors of 100. For example, 15 = 3 x 5 (3 is not a factor of 100) and 14 = 2 x 7 (7 is not a factor of 100).