The prompt was devised by Mark Greenaway (an Advanced Skills Teacher in Suffolk, UK). In his original version, the prompt included values for a and b (2 and 3 respectively). In the prompt’s more open form (as shown above), students might suggest their own values to substitute into the inequality in order to develop conjectures about the relationship between a and b.
To date, students’ questions and observations about the prompt have included:
- What are they trying to find out?
- What values of a and b satisfy the inequality?
- What does the sign mean?
- a could be 2 and b could be 4.
- This can only work when a ≥ 2 and b ≥ 3 with b > a.
- What if a and b are negative numbers? Would the inequalities be reversed?
- What if a and b are fractions?
- Is it possible to change the order of the terms and expressions and find the same values of a and b that satisfy all permutations?
- What if we changed the operations?
- Could we use square roots?
- This one must work all the time: a/b < a - b < a + b < ab.
- Is it possible for a2, a + b, ab, b2 to be equal?
Inquiries have often led students into changing the order of the terms in the inequality. Other pathways include substituting decimal and negative numbers into the inequality, and using more complex expressions. Five examples of the many conjectures about the relationship between a and b that have arisen in classroom inquiry are:
- "If a and b are consecutive whole numbers, then the inequality will never work." By making a = n, a year 9 student showed how the four parts of the inequality become n2 < 2n + 1 < n2 + n < n2 + 2n + 1. The conjecture turns out to be correct, except when n = 2.
- "The values of a and b must be in the same times table to make this correct." A counter-example (like those in the table below) show this conjecture to be false.
- "As long as a is smaller than b, the prompt will work." This can be shown to be false with a counter-example, such as a = 3, b = 5.
- "If a = 1, then the inequality will never work." This is true because when a = 1, ab < a + b.
- "The inequality will not work if both numbers are negative." This is also true because if a and b are negative, a2 > a + b.
One notable result came from a group of students who decided to find the lowest value of b when a is set as a particular whole number.
| Value of a ||Lowest value b can take given a||First difference|
|3|| 7 ||4|
|5|| 21 ||8|
|6|| 31 ||10|
| 7 ||43|| 12 |
|n||(n - 1)2 + (n - 1) + 1 = n2 - n + 1|| |
The students explained their results by focusing on a2 and a + b. In order for a + b > a2, b = a2 - a + 1.
Mark Greenaway runs his own website, which is highly recommended for its comprehensive coverage of ideas for the mathematics classroom. Mark can be followed on twitter @suffolkmaths.