This prompt was developed in collaboration with Paul Foss, a maths teacher in Brighton (UK), as the start of a unit on place value with year 7 mixed ability classes. Students notice that the digits are in the same order in each inequality (and in the prompt the digits are consecutive, descending on the left and ascending on the right) Experienced inquirers have often made the following conjecture:
"If the add sign is in the same place on both sides, then the inequality sign is greater than. If the add sign is in a different place, the inequality sign is less than."
This proves to be false, but in the process of finding that out classes have produced a full list of all the possible inequalities using descending and ascending consecutive digits.
One advantage of the prompt is the obvious changes that can be made to it. A favourite, in my experience, is to change the operation to multiplication. Students who find 43 + 21 < 123 + 4, but 43 x 21 > 123 x 4 are often intrigued to explain why the inequality sign is reversed. All-ability classes have taken this to a higher level by finding the sum of pairs of algebraic expressions. For example, we can generalise for an inequality of the form 43 + 21 < 123 + 4 in the following way:
[10n + (n - 1)] + [10(n - 2) + (n - 3)] <
[100(n - 3) + 10(n - 2) + (n - 1)] + n
where n is an integer, 4 ≤ n < 10.
Summing both sides of the inequality gives:
[10n + (n - 1)] + [10(n - 2) + (n - 3)] = 22n - 6 [100(n - 3) + 10(n - 2) + (n - 1)] + n = 112n - 6
Thus, the right hand side of the inequality is 90n greater than the left-hand side.
Talented students take great delight in finding the product of these expressions by expanding the brackets.
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