Intersecting lines inquiry
Mathematical inquiry processes: Find relationships; change the prompt; conjecture, generalise and prove. Conceptual field of inquiry: The coordinate plane; gradient of straight lines; simultaneous equations.
Daniel Walker devised the prompt. In his original design, the prompt was y = ax + b and y = bx + a. As Daniel explains, this would have involved students working at an abstract level immediately: "If the students solve these simultaneously either using graphs or algebra, they'll find the solution is (1, a + b)." However, Daniel had a change of mind: "It occurred to me that if I use numbers and maybe throw in an added twist like a + b = 1, then I give more scope for the students to generalise for themselves. Starting with, for example, y = 3x - 2 and y = -2x + 3 will give a solution (1, 1), but will allow them to investigate the effect of varying the relationship between a and b." The final version of the prompt encourages students to explore by changing the gradient and constant. They can then make and test conjectures and generalisations before ending the inquiry with an algebraic proof.
Graphical representation: The equations can be represented graphically, showing the point of intersection is (1, 1).
The solution (1, a + b) for the general case can be arrived at in various ways. On this sheet there are three methods. Asking students to discuss the methods can lead to claims about which one is best and what 'best' means in this context.
At the time of devising the prompt, Daniel Walker was a teacher of mathematics at North Bridge House Canonbury (London, UK).
Lines of inquiry
The questions and observations from year 10 students (pictured) suggest the following lines of inquiry:
Plotting straight line graphs
Solving for x and y by finding the point of intersection
Generalising for a + b = 1 where y = ax – b and y = -bx + a
Solving equations of the type 3x – 2 = -2x + 3
Finding the solution for y = 4x – 2 and y = -2x + 4
Generalising for a + b = 2.
Exploring the case of a + b = 0
Generalising for a + b = n
Rearranging terms in the equations and exploring the graphs.
The teacher designed a structured inquiry sheet (see the resources section) and students selected their own line of inquiry.
Daniel Walker describes how the inquiry developed in his year 9 classroom:
I used the new prompt today and it went really well with a class that have done y = mx + c but not simultaneous equations.It proved to be a really organic way of introducing the topic. Once the two equations were on the board, I gave pupils a minute or two to discuss what they might do with them. Some pupils used this a chance to describe gradient and intercept (and observe that gradient and intercept had been swapped) whilst others immediately realised that two lines would lead to an intersection, which all pupils set about finding on grids.
Once the result (1, 1) had been verified, pupils picked their own values of a and b to investigate. Some stuck to the same format of a + b = 1 (although this wasn't discussed, I like to think they made a conscious decision to do this!), others used positive values for both a and b.
All pupils realised that the x-coordinate is always 1 and a few spotted the fact that the y-coordinate is always a + b. A few pupils made mistakes that meant not all of their graphs agreed with this, but this gave them the motivation to re-check their work.
With 15 minutes to go, I got pupils to feed back, taking a selection of their choices of equations and using Autograph to quickly show and verify their results on the projector. I put a table of their equations and coordinates of intersection on the board (so that all pupils could understand the findings of those who identified patterns). I spent the last 5 minutes showing them an algebraic proof of the result. The investigative element really motivated pupils and those who spotted all the patterns were excited by their discoveries. I'll definitely use this lesson to introduce this topic in the future. Probably the best Friday afternoon lesson I can remember!
Questioning and noticing
These are the questions and observations of a year 11 class at Haverstock School (Camden, UK). During a class discussion, students solved the equation (bottom right) to show that x = 1. The teacher structured the inquiry by directing students to draw graphs of equations in which the sum of m and c is 1. Students then changed the condition (by varying the sum) to pursue their own line of inquiry. After drawing conclusions about the point of intersection of the straight lines, students formed and solved their own equation using one of their examples.
Click here to see a demonstration of the inquiry posted on twitter. It was made at dudamath.com, which is an integrated framework for mathematical explorations and problem solving.