Straight line graphs inquiry
Mathematical inquiry processes: Explore; make connections; change the prompt; reason and analyse structure. Conceptual field of inquiry: Substitution; the coordinate plane; gradient of straight lines; intercepts of the y- and x-axes.
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 and observations from which the class launches its inquiry. In helping to set the context and direction, students find the inquiry has more relevance and meaning and are more motivated to pursue different pathways.
Furthermore, changing the prompt to y - x = 4 challenges students, in finding pairs of numbers that satisfy the equation, to face the concept of subtracting negative numbers. Students in year 7 can quickly link x and y to the axes of a graph, proceed to find coordinate pairs and plot them on a graph.
The inquiry builds students' confidence to suggest changes to the prompt. They readily come up with alternatives to the operation, y-intercept and coefficients of the variables from which they can compare changes to the graph. The teacher might structure this part of the inquiry by requiring students to change only one feature of the equation at a time. If you want the class to focus more on the gradient, you could change the prompt to y - 2x = 2 or a similar equation.
Reasoning through inquiry
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 picture)."
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.
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.
Amanda Kirby, a teacher of mathematics and the Numeracy Across the Curriculum Coordinator at St Clement Danes School, Hertfordshire (UK), 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
Making sense of the prompt
These are the questions and observations of a year 8 class at Haverstock School (Camden, UK). There are a number of ways students have attempted to make sense of the prompt:
Meaning and representation The most basic questions relate to the meaning of x and y. Another question about what the variables represent could lead to a discussion about the difference between meaning and representation.
Substitution Students have substituted values into the equation to make it correct for an individual case.
Generalisation A general approach to substitution is given in the contention that both x and y have to be even or odd for the equation to be correct.
Manipulation Further comments relate to ‘swapping around’ the variables with students going on to rearrange the equation.
Changing the prompt Suggested changes to the prompt include using different letters or operations. The question about using x and y twice could lead the inquiry into changing the equation with coefficients greater than one, such as 2y – x = 4, y – 2x = 4 or 2y – 2x = 4.
Amy Flood, second-in-charge of the mathematics department at Haverstock School, conducted the inquiry.
The picture was posted on twitter by the Mathematics Department of Wellfield High School (Leyland, UK). It shows the questions and observations of a year 8 class. The students showed that the equation could have different solutions by plotting the graph, using decimals, fractions and negative numbers for the values of x and y. The students also changed the operation from addition to subtraction and went on to compare the gradients and y-intercepts of their graphs. The department reported that the result was “a fabulous maths inquiry that developed in lots of ways.”
These 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 that 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."
These are the comments and questions from a year 7 class (middle set). They provided an excellent starting point for rich inquiry. The girl who claimed that 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.
Lines of inquiry
(1) Rearranging equations
This line of inquiry emerged when a year 8 student noticed that the graph of y - x = 4 is the same as the graph of y = x + 4. What other equations could we rearrange? Why are the graphs the same?
(2) Angles of elevation and similarity
Year 10 students wanted to use trigonometry to see if there was a connection between the angle of elevation of a line and its gradient. Is there a connection between the two? (See mathematical notes).
(3) Using a third variable
Introducing another variable introduces three-dimensional coordinates. How would you graph y - x - z = 4?
(4) Tackling misconceptions
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.
Another issue that can arise when students use the prompt relates to plotting coordinates from the equation. The equation in the prompt presents y first followed by x, yet the mathematical convention when writing coordinate pairs is to write the x-value 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 coordinate axes) by changing the prompt to x - y = 4.
Akram and Ilyas created the regulatory cards when their year 7 class considered how to change the original prompt. They have suggested other lines of inquiry, including a change to the operation and the use of indices. The question marks on the second card indicate that the equation could equal different values, which the class went on to explore systematically.
Moving between representations
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.