Mathematical inquiry processes: Make connections; generate examples; express in general form. Conceptual field of inquiry: Direct and inverse proportion; multiplicative relationships; reciprocal; equation.
The prompt could be used with students after the ratio prompt. It consists of two unlabelled tables.
The pairs of numbers in the left-hand table are directly proportional. Considering the horizontal pairs, the multiplier is 12/5. The multiplier for the vertical pairs is 4/5.
In the right-hand table of the prompt, the situation is different. The vertical pairs are inversely proportional, with the multipliers (4/5 and 5/4) being the reciprocals of each other. The relationship between the horizontal pairs (where the first number is a and the second b) is 60/a = b.
Students invariably notice that three numbers in each table are the same and will try to fit the fourth number into a rule. If they are familiar with multipliers from the ratio inquiry, then they quickly find the horizontal and vertical multipliers for the left-hand table.
The right-hand table will provide more intrigue. Looked at horizontally, one number increases and the other decreases. Can we still use a multiplier? Or must we divide to make a number smaller?
The same is the case when we consider the vertical pairs - one number decreases and the other increases. What are the vertical multipliers? Is there a connection between them?
As the inquiry develops through different lines of inquiry (see suggestions below), students can develop an understanding of the formal algebraic notation associated with direct and inverse proportion.
Lines of inquiry
Students should be encouraged to think about the multipliers, their expression as fractions, and the connections between them. The concept of the reciprocal helps students to reason about the vertical multipliers for the inversely proportional numbers.
2. Generating examples
Once the class has compared the tables and identified the similarities and differences between them, students could make up their own examples to understand the mathematical structure of the prompt more deeply.
3. The unitary method
The teacher might take the opportunity to introduce the unitary method by adding a middle row to the tables.
4. General form
The teacher might guide students to express the relationships between the numbers algebraically.
5. Forming equations
By labelling the two columns x and y, students can write an equation of y in terms of x. The teacher might show the class or individuals how to do this formally by substituting in one pair of values and using the second pair to check the equation.
6. Reasoning backwards from two equations
Once students are familiar with the form of the equations for direct and inverse proportion, they can reason backwards. They create their own equations in order to form two tables - one showing direct proportion and the other showing inverse proportion - in which three of the numbers are the same. At this stage, the teacher could encourage students to experiment with the square, cube, square root and cube root of x.