Fractions inequality inquiry
Mathematical inquiry processes: Identify structure, generate examples; generalise and prove. Conceptual field of inquiry: Subtraction of fractions; inequalities.
Students verify the truth of the inequality, using a number line or by converting to equivalent fractions with common denominators. If the prompt is the first time the students have come across finding the difference of two fractions, they are likely to select the regulatory card Ask the teacher to explain. As the class explores other examples in which the numerators and denominators on the left-hand side have differences of two, the students become fluent in the procedure of subtracting one fraction from another.
The fact that the fraction on the right side of the inequality is not half way between the other two, despite both numerator and denominator being exactly in the middle, can form the basis of an interesting discussion about the concept of a fraction.
The inquiry has the potential to move into different pathways, including:
Exploring what happens when the difference between the two numerators and denominators on the left-hand side of the inequality are greater than 2;
Seeking an inequality that reverses the symbol while maintaining the condition that the right-hand fraction should be in the 'middle' of the other two;
Explaining when the properties of the prompt 'work' and when they do not;
Making a generalisation from the prompt, such as
Finding the values of n that satisfy the generalisation.
The general case simplifies to the inequality 0 < n3 + n2 - 6n - 4, which can be shown on a graph with n on the x-axis. Although n = -1 and n = -2 satisfy the rearranged inequality, the original inequality is not defined for those values of n.
Creating examples and non-examples through inquiry
Saleshni Cook, a PYP educator in Istanbul (Turkey), posted the picture on twitter. She reports that her grade 6 class used the question stems before coming up with examples and non-examples. The class verified that the inequality in the prompt was true and also in other cases. One student ended the inquiry by expressing the relationship between the fractions algebraically.
The middle term of the inequality is made up by adding the numerators and denominators of the other two fractions. Students notice how the inequality is constructed and verify its truth, perhaps by using equivalent fractions. They can then conjecture that the relationship holds for any pair of fractions. As they find more examples, they become firmer in their conviction and can start to think about the general case expressed algebraically. The general case is always true (see a proof here).