Inequalities inquiry

The prompt

Mathematical inquiry processes: Verify; generate more examples; identify patterns and reason. Conceptual field of inquiry: Addition and inequalities.

This prompt was devised by Paul Foss, a mathematics teacher in Varndean School,  Brighton (UK), as the start of a unit on place value with year 7 mixed attainment 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. Changing the operation to multiplication, for example, develops a new line of inquiry. 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. Mixed attainment classes have taken this to a higher level by finding the sum of pairs of algebraic expressions. Students 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)] +

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.

Higher attaining students take great delight in finding the product of these expressions by expanding the brackets.

Questions and observations

These are the questions and observations of year 7 and 8 mixed attainment classes.

Notes and resources