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The prompt
The prompt
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, a teacher of mathematics, 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.
Structured lines of inquiry
Structured lines of inquiry
Andrew Blair used the prompt with his year 10 class at Haverstock School (Camden, UK). In the question, notice, and wonder phase of the inquiry, students responded to the prompt in the ways shown in the picture.
Andrew grouped the responses into four categories, with each category suggesting a different line of inquiry:
Category 1 (blue) The category contains questions about the values of x and y. When x (the independent variable) is given a value, students can calculate y (the dependent variable).
The question about whether x and y are the same in both equations leads to a line of inquiry in which the equations are solved simultaneously by, for example, uisng 3x - 2 = -2x + 3.
Category 2 (orange) A pair of students has interpreted the values of x and y as coordinates, which connects them to a Cartesian graph. This observation opens a line of inquiry in which students plot the coordinates and represent the equations as straight lines. This leads to the realisation that the lines intersect.
Category 3 (red) Students have noticed that the coefficient of x (m) and the constant (c) have been 'flipped around'. The observation opens another line of inquiry in which students change the difference between m and c.
The class has already generated two examples (4 + (-2) and 5 + (-5)) where the sums of m and c are two and zero respectively. Do the points of intersection change as we vary the sum?
Category 4 (green) Students have suggested rearranging the terms in the equations or using different operations. Such rearrangements, if carried out within a logical system (and not randomly), leads to the exploration of new graphs.
An example of a logical system can be developed from one of the students' suggestions: 3x = y - 2 and -2x = y + 3, 4x = y - 3 and -3x = y + 4, and so on.
By the end of the first lesson of the inquiry, all the students had plotted the equations in the prompt on a graph and found the point of intersection. Andrew designed a structured inquiry sheet that led students through the rest of the lines of inquiry.
October 2018
Student-led inquiry
Student-led inquiry
At the time of devising the prompt, Daniel Walker was a teacher of mathematics at North Bridge House Canonbury (London, UK). He describes how the inquiry developed with his year 9 class:
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.
Show and verify
Show and verify
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!
Resources
Resources
Dynamic demonstration
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.
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