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y - x = 4

The prompt is inspired by Mike Ollerton's booklet Learning and Teaching Mathematics Without a Textbook. His starting point is x + y = 7, which is then followed by a list of questions to investigate. In contrast, inquiry maths takes a step back in the learning process by encouraging students to come up with the questions. Moreover, the prompt y - x = 4 challenges students, in finding pairs of numbers that satisfy the equation, to face the concept of subtracting negative numbers. In my experience, students in year 7 quickly link x and y to the axes of a graph, proceed to find coordinate pairs, and plot them on a graph.
  
This inquiry builds students' confidence to suggest changes to the prompt. They readily come up with alternatives to the operation and y-intercept. If you want the class to focus more on the gradient, you could change the prompt to y - 2x = 2 or a similar equation.
This inquiry has led to the introduction of rearranging equations when, for example, a student notices that the lines produced by y - x = 4 and y = x + 4 are the same. At GCSE level, students have looked at the angle of elevation using trigonometry and developed questions about similar triangles (see mathematical notes).

A common misconception that arises during this inquiry is that "y cannot be less than 4." The student reasons that if the value of y is below 4, then subtracting another amount can only make the answer smaller and, hence, y - x
can never equal 4 when y < 4. The idea of subtracting negative numbers can be introduced by using a table of 
x and y values that satisfy the equation (as illustrated on the left).
 
Another issue that can arise when students use the prompt relates to plotting coordinates from the equation. The prompt is presented with y and then x, yet students write a list of coordinate pairs 
with the value of x first and plot the x-coordinate first. This is a deliberate obstacle to slow students down and make them think about the relationship between x and y. However, when students are not used to inquiry, the teacher might want to facilitate the change in representation (from equation to graph) by using the prompt x - y = 4.

Moving between forms of representation
   
Research carried out in a Norwegian secondary school with year 11 classes* used Mike Ollerton's original starting point - that is, x + y = 7. For the three teachers involved, it was a first taste of inquiry. Understandably, they were cautious and designed four differentiated cards of questions to act as prompts to guide students. One question (or rather instruction) on the first card - "Draw these number pairs [values for x and y] on millimetre paper" - caused the teachers most problems. In not wanting "to give the game away by telling too much", the teachers found it difficult to decide how to respond when students could not interpret the instruction to "draw" number pairs. Not wishing to mention Cartesian graphs directly, one teacher suggested x- and y-axes to a group. The teachers later concluded that the stumbling block caused by the word "draw" could have been avoided if they had used words like "mark" or "plot". Throughout this episode, the teachers were reluctant to lead students into graphing functions in favour of letting them discover the alternative form of representation for themselves. This process, however, took a longer time than they thought appropriate considering the curriculum constraints they faced.
    
The academics carrying out the study (who were also advising the teachers) over-estimated students' ability to construct their own legitimate mathematical representations. In my experience, moving from one form of representation to another is rarely a step secondary school students take spontaneously. The step needs to be co-constructed with the inquiry teacher offering as much guidance as necessary - even "telling" if this allows students to inquire into their questions further. This should not be a cause of guilt or disappointment to the teacher who might feel she has not held out long enough. Inquiry is not about discovering a pre-determined outcome; rather, it is a joint mathematical exploration initiated by the student and supported by knowledgeable others, be they peers or adults.

* Goodchild, S., Fuglestad, A., Jaworski, B. (2013). Critical alignment in inquiry-based practice in developing mathematics teaching. Educational Studies in Mathematics, 84, 393-412.

   

Notes
Lesson notes
  
Resources
Prompt sheet
Note sheet (first quadrant)
Note sheet (four quadrants)
Promethean flipchart   download
Smartboard notebook  download

Reasoning through inquiry
Yolanda Lewis used the prompt with her year 8 class as part of a unit about graphs. She reports that the students engaged with the prompt straight away, contributing a variety of ideas and questions.
Yolanda continues, "The questions that caused the most discussion were 'Is it true?' and 'Are there an infinite number of solutions?'. As all the students wanted to give an opinion on these I allowed them some time to discuss with the people around them before bringing it back together as a class. The debate got heated and Issey and Elena vehemently disagreed with the rest of the group, but were struggling to communicate their ideas effectively. I decided to structure the inquiry by giving the class two options from which to choose, plotting the line by either using algebraic manipulation and their knowledge of y=mx+c or finding values that would work. Meanwhile, the girls continued to discuss their ideas and put them down on paper (see below)."
   

Issey and Elena argue that the solutions for the prompt could not be infinite as they did not include all pairs of numbers, such as 2.5 and 2.25. Although it is true that not all pairs satisfy the equation, it remains the case there is an infinite set of pairs that do. Once y (the independent variable) is defined, then x (the dependent variable) is fixed. Nevertheless, the students' reasoning can be linked to Cantor's argument that some infinities are bigger than others. He showed how, for example, the set of real numbers constitutes a larger infinity than the infinite set of natural numbers. In the girls' case, the set of pairs of numbers that satisfy the equation is infinite, yet the infinite set of all pairs of numbers is larger.

   
Yolanda was pleased at how the students developed their mathematical inquiry skills: "Students were very positive about the lesson and I felt it was our most successful inquiry yet. As well as giving them the opportunity to work collaboratively and in a more open manner, it also exposed misconceptions around plotting graphs, such as x and y coordinates being plotted on separate graphs. Moreover, the inquiry encouraged students to think about the infinite nature of straight line graphs."

Yolanda is Maths Learning Leader at Steyning Grammar School, West Sussex (UK). She is planning to include inquiries in the Key Stage 3 (ages 11-14) schemes of learning. You can follow Yolanda on twitter at @MathsPanda.
  
First-time inquiry
Amanda Kirby
used the inquiry with her year 8 class. The questions and comments show the students’ attempts to make sense of the prompt: 
  • The question about the use of even numbers (perhaps originating in the four on the right-hand side of the equation) is quickly answered with values of seven and three that satisfy the equation. 
  • The attempts at rearranging the equation are impressive, particularly considering that the year 8 class was a ‘bottom’ set. Indeed, the two attempts could lead to a rich discussion about why one is correct.
  • The change made to the prompt (four is replaced by five) displays the confidence to initiate a new pathway for inquiry. 
Amanda was pleased with the students’ enthusiasm, commenting that she was “very proud of their first attempt." Indeed, the class embraced inquiry more than Amanda’s top set in year 10. Her experience is not uncommon. When planning levels of support during inquiry, teachers should bear in mind that measures of ability or prior attainment are not necessarily good indicators of students’ independence and creativity.

Amanda Kirby teaches maths and is the Numeracy Across the Curriculum Coordinator at St Clement Danes School, Hertfordshire (UK). Inquiry maths prompts, Amanda comments, have proved successful in increasing personalisation, differentiation and students' independence. You can follow Amanda on twitter @mathsteach2000.
  
The inquiry in years 7 and 8
These are the comments and questions from a year 7 class (middle ability). They provided an excellent starting point for rich inquiry. The girl who claimed "y has to be greater than 4" taught herself about subtracting negative numbers and went on to explain to the class that y could indeed be less than 4. This strand of the inquiry developed out of graphing the equation when students asked what happened to the straight line when it got to (0, 4). The comment about x and y taking any value led to a discussion about dependent and independent variables, while the one on the use of x and y developed into considering conventions in maths. However, the main part of the inquiry saw the students change the prompt to determine how the graph changed.

Below are the questions and comments of Ann Macdonald's year 8 class at Longhill High School, Brighton (UK). Ann describes her approach: "I had not
told the class that they were about to begin a unit of work on co-ordinates and lines and was interested to see whether they would make statements or ask questions relating to the unit. When they didn't, I initiated a discussion based around their responses which led to some thinking about dependency of variables and rearranging equations which I was able to revisit and link to in a later lesson. I then directed the class to consider the Cartesian grid. Even when students were very unsure where the inquiry might lead, they had confidence in the inquiry process itself. After using the cards, they were more engaged when being instructed on (or reminded or enabled to figure out) how to draw lines given equations."