Solving equations inquiry
Mathematical inquiry processes: Identify structure; create example sets and generalise. Conceptual field of inquiry: Equations with the unknown on both sides; rearrangement of algebraic terms.
The prompt acts as a template for students to explore linear equations. It has often led to an inquiry that combines inductive and deductive reasoning in a mutually supportive process.
The inductive side can be developed from a consideration of whether equations of this form always have a solution. As students solve particular cases, the conjecture soon arises that they do (unless the coefficients of x are the same). Deductive reasoning is a feature of the solving process. It can also be evident in students' attempts to solve the general form ax + b = cx + d, giving x in terms of a, b, c and d.
In a structured inquiry, the teacher could start the inquiry by requesting integers to place in the boxes. Even with this closed start, the prompt has the potential to generate various suggestions for further inquiry. Some examples from classes in lower secondary school are:
Change the operation(s) to subtraction.
Use fractions instead of integers.
Re-design the equation with the unknown on one side only.
Include a second variable on both sides of the equation.
Re-design the equation to include an algebraic fraction.
Richard Goodman (Principal Lecturer at the University of Brighton, UK) contacted Inquiry Maths to make the following comment about the prompt: "I like the solving equations inquiry, but I am surprised you describe it as closed at the start. My reaction was to close it down a bit further - asking what happens if we know the solution? Or what about using the same two numbers in different order on either side (2x + 5 = 5x + 2)?"
Exploring a line of inquiry
The exploration of equation sets is a line of inquiry that has developed from the prompt, particularly when students change the operations. The sets are created by using the same numbers for a, b, c and d and varying the operations in a systematic way (++, +-, -+, --). In a more structured inquiry, the teacher might give the class the first set. The set might contain other properties, such as using consecutive even numbers:
2x + 4 = 6x + 8
2x + 4 = 6x - 8
2x - 4 = 6x + 8
2x - 4 = 6x - 8
The equations give a curious set of solutions. In the case of those above, the solutions are, respectively, x = -1, 3, -3 and 1.
The pattern of two pairs (one positive and the other negative in each pair) occurs for all sets. Considering a set using consecutive odd numbers in which a = 3, b = 5, c = 7, and d = 9, the solutions are x = -1, 3.5. -3.5, and 1. If the numbers go from highest to lowest (a = 9, b = 7, c = 5, and d = 3), the solutions are x = -1, -2.5. 2.5, and 1.
The order of the negative and positive numbers in the solution set depends on the relationship between a and c and b and d. For example, the randomly generated set with whole numbers less than 20, in which a = 2, b = 17, c = 12, and d = 3, gives the solutions x = 1.4, 2, -2, and -1.4. The two negative numbers are the third and fourth solutions this time because b > d.
The line of inquiry might end with students trying to explain the pattern in the solution set or even with a proof of the general case.
The proof below comes from a year 11 student in a UK secondary school. His teacher wrote to Inquiry Maths:
"My class was exploring the solving equations prompt and noticed that there was a pattern in the solutions when they created groups of equations. One student who had just moved into my class from the set above wanted to explain the pattern. He did the first proof in class and I asked him to finish them off at home. When he'd completed them I put them on the board and asked him to explain to the class. The inquiry really re-engaged him with maths and he has been working hard ever since. That shows how the creativity of inquiry can stretch students and increase their levels of confidence."
Multiple inquiry pathways
The picture shows the questions and observations of students at The Bishop Wand Church of England Secondary School and Sixth Form (Sunbury, UK). There are a number of interesting lines of inquiry that could develop:
Using sequences of numbers (2 4 6 8);
Using fractions or decimals;
Finding numbers that give different types of answers, such as negative numbers;
Deciding if there is always a solution; and
Establishing a rule for each side of the equation, such as using different numbers on either side (4 4 5 5), reversing the same numbers (6 2 2 6) or using numbers with the same relationship (3 3 2 4).
The final idea could lead into a proof that for equations of the type nx + n = (n - 1)x + (n + 1), the solution is always x = 1. The teacher reports that the lesson involved "impressive and thoughtful inquiries," which included lots more pathways than the few shown in the picture.
Finding a general solution
The inquiry ended with the demonstration below of a general solution for solving an equation in the form ax + b = cx + d . As Caitriona comments, it is a "great example of pupils stretching just beyond their current knowledge", which is precisely the aim of the inquiry prompts.
At the time of the inquiry, Caitriona Martin was a maths coordinator at St. Andrew's School, Leatherhead (UK).
Questioning and noticing
Amy Flood recorded these questions and observations from her year 8 class at Haverstock School (Camden, UK). The students are attempting to understand the prompt by applying their existing knowledge about equations. Amy was able to assess the students' existing level of knowledge in order to construct a guided inquiry.
Amy was second-in-charge in the mathematics department at the time of the inquiry. She is now a head of department.
Classroom inquiry in year 7
These are the questions and comments of a year 7 (grade 6) mixed attainment class in a UK secondary school. As the students were relatively experienced in mathematical inquiry, the teacher decided to run an open inquiry. The questions and comments led to three hours of classroom inquiry.
Alison Browning, a secondary school teacher of mathemtics in the UK, used the solving equations prompt with her year 7 class. She was particularly interested in developing resilience and asked the students to reflect on how they felt during the different phases of the inquiry. The students drew diagrams to show how they had moved from confusion to understanding and solving their own equations. Alison sent Inquiry Maths a sample of the diagrams.