Parallel lines inquiry

The prompt

Mathematical inquiry processes: Verify; explore; generate diagrams; reason. Conceptual field of inquiry: Angle properties of straight and parallel lines; independent and dependent angles.

The prompt is one of three versions on this page. It is more closed than the others because it contains cues to direct students' thinking. The statement focuses attention on angles and the diagram shows two pairs of parallel lines. The open and semi-open prompts discussed in the next section invite a wider range of responses.

The closed prompt launches an inquiry in which students learn about the angle properties of parallel lines. The teacher should be prepared for students to request an explanation through the regulatory cards. They might also decide to practise recognising alternate, corresponding, and co-interior angles.

Question, notice, and wonder

In the question, notice, and wonder phase of the inquiry, students have made the following responses to the prompt:

What is an independent angle?
The arrows mean the lines are parallel.
One line is not parallel to the others.
When two lines intersect, there are two sets of vertically opposite angles.
The sum of the three angles in a triangle is 180o.
The opposite angles in the parallelogram are equal.
When a line crosses two parallel lines, the four angles at one intersection are the same as those at the other intersection. I know this because I measured them with a protractor.
How can we find the size of the angles in the diagram?

Verify

The statement in the prompt is true. Two independent angles are required to find all the angles in the diagram. Given one red angle (see the diagram below from the slides), we can use angle properties to find the other red angles. Similarly, given any green angle on the transversal once we have the red angles, we can deduce the other green angles.

Exploration

Once students are convinced that the prompt is true, they can explore different cases by creating their own diagrams or using those in the slides. The teacher might help students draw up a list of different diagrams using the number of pairs of parallel lines and the number of other lines (from zero to three pairs and zero to three others). The illustrations below from the slides show two examples.

Pairs of students choose or are allocated one case each to examine. They colour one angle and explain to each other how to deduce the size of other angles. A new independent angle is represented by a different colour.

When drawing their own examples, students should consider different variations of the same case. What happens to the number of independent angles when, for example, a line intersects at the same point where two or more lines already intersect? What happens when lines do not intersect but they would do if extended 'outside' the diagram?

Reasoning about a relationship

As the class shares results from each case, attention turns to finding a formula for the relationship between the number of lines (l), the number of pairs of parallel lines (p), and the number of independent angles (i). The prompt, which required two independent angles, contains five lines and two pairs of parallel lines.

In developing a formula, the teacher encourages students to reason systematically: each additional line adds one to the number of independent angles unless the line is parallel to one already in the diagram. (N.B. If a new line is parallel to one of the existing pairs of parallel lines, then the three parallel lines form two pairs.) Starting with a simple diagram of two lines, the class co-constructs the formula under the teacher's guidance.

Unlike in a discovery lesson, the aim is not for each each student to 'discover' the formula independently. Rather, an inquiry is a collaborative enterprise in which any student that deduces the formula has the duty to share their knowledge if others so wish.

Revised July 2026

Open and semi-open prompts

Open prompt

Mark Greenaway, an Advanced Skills Teacher in the UK, designed the 'straight lines' prompt above. Students used at least six areas of mathematics to find meaning in the prompt (see the picture below). The topics included:

Measurement;
Parallel and perpendicular lines;
Coordinates on a graph, gradients and equations of straight lines, and equations of circles;
Areas of shapes;
Radius of a circle (requiring the use of Pythagoras' Theorem to answer the student's question); and
Vectors.

In the open inquiry that developed from the prompt, students followed multiple lines of inquiry.

One advantage of such an ambiguous starting point is that it gives students the opportunity to make connections between mathematical concepts. It also allows them to apply previous learning to new contexts in creative ways.

However, teachers who have to comply with a compulsory curriculum are often required to reach pre-determined learning outcomes. This pressure militates against using an open prompt.

Indeed, if the aim of using the prompt was to develop or consolidate knowledge about the angle properties of parallel lines, then the teacher would have to ignore the majority of students' contributions and channel the inquiry down a particular pathway.

There is a danger with this approach: it reinforces the idea that there is one correct answer. In the traditional classroom, in which the idea is commonplace, students are averse to taking risks for fear of being wrong.

This is the opposite to the culture of the inquiry classroom. By valuing all contributions, the teacher aims to encourage students to participate and, ultimately, to share responsibility for directing the inquiry.

The contradiction between treating all students' responses to the prompt seriously and the necessity to cover specific curriculum content means that a semi-open or closed prompt might lead to a more successful inquiry.

Semi-open prompt

The semi-open prompt contains more information than the open one. The red dot marks an angle and the arrows indicate parallel lines. Students' responses to the prompt are more likely to focus on those features of the diagram.

The omission of the statement, however, means the focus of the prompt is not so clear. Its absence leaves space for students to make contributions related to other mathematical concepts. Once again, the teacher has to guide the inquiry if it is to meet a specific curriculum outcome.

An angle 'pursuit'

The concept of an independent angle is unlikely to arise spontaneously. The teacher could orchestrate an angle 'pursuit' in which the class derives the size of angles from the one marked with a red dot. (Students might find it easier to use an estimate of the size of the first angle.) As usual, the focus should be on justifying the size of angles.

During the task students realise that to jump to another intersection they need more information - another independent angle - to complete the 'pursuit'.

Classroom inquiry

The classroom inquiries in the reports below are based on the semi-open prompt.

Independent and self-aware learners

A trainee teacher at London Metropolitan University used the prompt on her second school placement. She concludes her report on the classroom inquiry by saying that the Inquiry Maths model is much more than simply an addition to her teaching repertoire: "It has actually displayed an approach to form independent and self-aware individuals."

Language acquisition as a phase of inquiry

Sonya terBorg blogged about using the prompt with her primary class in Idaho (US). The post describes how the class carried out a preliminary inquiry into concepts and language related to angles. The pupils then conducted an open and collaborative inquiry by applying the language acquired in the earlier phase.

Content coverage through open inquiry

Aoife Lynch (a mathematics teacher in Ireland) used the prompt to cover the angle properties of parallel lines with two classes. Her experiences show that it is sometimes difficult balancing the need to cover content and the desire to run an open inquiry.

When Aoife first tried the inquiry she combined the prompt with a question for investigation: How many independent angles would you need to find all the angles?

Students were very reluctant to pose questions or make comments, perhaps because they were unsure of how to answer the question and did not want to take a risk in front of the class. The inquiry very soon became a teacher-directed investigation, with students following along.

On the second occasion, Aoife used the question again, but did not refer to it. When the focus was mainly on the diagram, the class was far more willing to question and comment. This gave Aoife a much clearer picture of students' understanding of the topic.

The inquiry took on a more co-constructed nature because students set the starting point of the lesson through their responses to the prompt. In this case, running a more open inquiry with less emphasis on the teacher's question encouraged students to engage with the concepts and properties underlying the diagram.

Resources

Prompt sheet

PowerPoint

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