Decreasing differences

Prior knowledge.

• Skip count forwards and backwards in numbers up to 10

Background

These activities can help reinforce and develop students' skip counting ability, as well as develop pattern generalisations. When students are writing their rule in words, insist on them using a sentence like "take each number and subtract five to get the next number". Rather than "minus 2" which is the desire of most students to write. The full sentence better highlights the repeated process of moving from one to the next, rather than the gap between the first 2 numbers. It also allows a further stage of development - symbolising - where after students have written patterns in word sentences for a while, this can be abbreviated. "Subtract five" easily becomes "-5", while "take a number" becomes "n".

Another useful development is to highlight two similar sequences with the same difference. For example, 26, 22, 18, 14,... and 20, 16, 12, 8,... This develops the idea that when describing sequences in this way we need both the first number and a rule to get from one number to the next.

In the 'extra for experts' part, students are subtracting decimals and fractions to make their sequences. Students really need to be at stage 7 in addition subtraction to deal with problems. Likewise, the integer introduced in the investigation can be found at stage 7 of that domain.

Comments on the Exercises

Exercise 1
Asks students to complete number sequences involving whole numbers.  For example, 59, 53, 47, 41. It also asks students to fill in the missing number in a sequence for example 65, __, 51, 44, 37

Exercise 2
Asks students to complete number sequences involving fractions and decimals for example 6, 5 1/2, 5, 4 1/2, ...