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Number line inquiry

The number line prompt is suitable for most secondary school classes, although it has been developed mainly with those in years 7 and 8 moving from arithmetic to algebraic reasoning. It invites students to generalise about the difference between two products. In the initial phase of inquiry, students (under the teacher's guidance if necessary) explain the procedure shown in the prompt
. They often ask if the difference is always two, but cannot believe the result will be the same with larger numbers .
All-attainment classes have readily set about exploring more examples before taking great pleasure in proving that, with 
four consecutive numbers, the difference between the products will always be two. If n is the first number on the number line, the products are, for the 'outside' two numbers, n(n + 3) = n2 + 3n and, for the 'inside' two numbers, (n + 2)(n + 3) = n2 + 3n + 2. 
Lines of inquiry
Students have gone on to create their own inquiries by changing the prompt. The following changes have led to algebraic proof:
(1) Use different intervals between the numbers
Systematically change the intervals by going up in 2s, 3s and so on. List the differences between the products and identify a pattern.
Combine different pairs of numbers on the number line
Link the numbers differently (for example, the first with the third and the second with the fourth). Is the difference between the products always the same? How could you work out the difference from the first number?
Extend the number line to six consecutive numbers
Link the numbers systematically and work out the differences between the products.
Inquire into different sequences
Use terms from quadratic and cubic sequences or the Fibonacci sequence.

See the mathematical notes 
for more details on the lines of inquiry.
Proof after changing the prompt
A year 7 student proves that the difference between the products of the top two and bottom two numbers is 4n + 6 where n is the lowest number of the four consecutive numbers.
A year 8 student constructs a proof (in terms of n) for each difference between the products in the first eight cases.


Alternative prompt 
7 + 6 + 5 + 4 = 7 x 6 - 5 x 4
This prompt was posted on the internet here with the instruction to ‘explore’. In a classroom, students will not be able to explore without at first noticing and discussing the structure of the equation. An initial phase of questioning and observing would allow students to identify that structure and create more equations of the same type. It is soon evident that any four consecutive natural numbers will follow the same pattern: 4 + 3 + 2 + 1 = 4 x 3 – 2 x 1. What happens if the difference between the natural numbers is greater than 1?
Both sides of the final equation can be shown to equal 4n – 6a
The alternative prompt is similar to the situation when students change the original prompt to find the difference between the products of the first and second numbers and the third and fourth. It could be used as an assessment prompt by inviting students to inquire without guidance after they have been supported through the number line inquiry.

This prompt has been developed in sessions with Richard Goodman (Principal Lecturer, University of Brighton) and cohorts of teachers training to teach maths on the Mathematics Development Programme (2009-2012) and the Developing Mathematical Practice course (2013-14). In 2014, Liam Richman (Oakwood School, Horley, UK) produced this paper on the prompt to fulfil part of the course requirements. 

Teachers' reflections on using the prompt
A maths department's reflections
An evaluation of the inquiry by members of a mathematics department who collaboratively planned and then taught the number line inquiry.

Caitriona Martin's reflections
Caitriona (who was a teacher of mathematics at St. Andrew's School, Leatherhead, UK at the time of the inquiry) has been using inquiries since she became a qualified teacher in 2011. She reflects on how exciting inquiry can be for students and teachers alike.
Questioning and noticing
These are the questions and observations of a year 8 mixed attainment class at Haverstock School (Camden, UK). Students described the procedure, suggested changing the types of numbers and wondered if the difference between the products is always two. They have also proposed joining the numbers in a different way and started to explore the use of algebra by suggesting expressions for the consecutive numbers. As the inquiry proceeded, students verified the difference is two when the numbers are consecutive, eight when the numbers increase by two, 18 with gaps of three and so on. The class then went on to use the algebraic expressions n, n + 1, n + 2, n + 3 to prove the results. The student who suggested combining consecutive numbers in a different way (see the diagram in the bottom right-hand corner) finished her inquiry by proving the difference between products is 4n + 6.
Regulating inquiry
, a year 7 student at Haverstock School (Camden, UK) used the regulatory cards to suggest a new direction for the inquiry. In 'changing the diagram', he wanted to join the numbers in a different way. Would the difference between the products be the same each time? Jabril soon came to the conclusion that the differences are not the same (being, respectively, 17, 9 and 25 on the card above). However, he went on to show that the difference is 2n + 3 (where n is the lowest number of four consecutive numbers).
Inquiry in mixed attainment classes
These are the questions and conjectures of a year 8 class at the start of the inquiry. The students suggest other consecutive numbers give the same result, although one pair of students would like to try three-digit numbers. Others conjecture that the result will always be two. One pair notes that the sums of the 'internal' and 'external' numbers are equal. The inquiry teacher could use this observation to encourage further changes to the prompt.

Comments from this year 7 class focus on clarifying the procedures shown in the prompt. Unsurprisingly, many in the class chose to 'make up more examples' when it came to choosing a regulatory card. One pair, however, suggest changing one number in the sequence. Another poses a 'what-if-not' question - what if they are not consecutive numbers? Lastly, two pairs attempt to move directly to using algebra by labelling the consecutive numbers a, a + 1, a + 2, and a + 3. 
The teacher moved the four students who had initiated the algebra pathway to the same table and instructed them in the multiplication of algebraic expressions. They went on to present their proof that the result must always be two to the rest of the class. The picture (above) shows one of the students in the course of proving that if the sequence goes up in steps of two, then the difference between the products will always be eight. The four were able to extend their approach to take account of any changes their peers suggested to the prompt.

This year 7 class came up with a rich and varied set of questions and comments. One pair asks if the difference will always be two, while another has already started to change the prompt by linking the top two and the bottom two numbers. Other students suggest using consecutive multiples of ten, which leads to the suggestion that the result will be twenty - that is, ten multiplied by the difference when the numbers are consecutive. Already in the initial phase of the inquiry, the class has generated multiple pathways.
Inquiry pathways
Jhahida Miah, a teacher of mathematics, recorded these questions and observations from her year 8 class at Haverstock School (Camden, UK). She suggested that the students could develop the inquiry along a number of pathways based on their questions and observations: 
  • Check the generalisation that the answer will always be 2;
  • Use numbers that are not consecutive – i.e. gaps of 2, 3, etc.;
  • Use more than four numbers;
  • Change the operation;
  • Explain why you always multiply an odd by an even number (and why that gives an even answer);
  • Find a connection between the difference in ‘outside’ and ‘inside’ numbers and the difference of their products.
Below is a presentation that Jhahida Miah gave to trainee teachers at London Met University in January 2018. She designed the second slide to support her students' questioning and noticing. They go on to form groups of three with each student taking on one of the following roles: scribe, speaker and manager.
Exploration and proof
Emma Rouse, a Lead Practitioner, used the prompt to develop an inquiry with her year 8 mixed attainment class. The initial questions and observations (top picture) show that students have already formed a conjecture about the difference between the products. They have also begun to suggest changes to the prompt. One student (middle picture) makes predictions about what will happen when the numbers on the line increase by two and then three. He goes on (bottom picture) to attempt to prove that the difference between the products in the prompt will always be two. In giving three examples connected to consecutive numbers, he shows transitional thinking. Any one of the examples would constitute a general proof (although the use of n, n + 1, n + 2 and n + 3 might be more formally correct). The picture shows the student making the transition from empirical to deductive and abstract reasoning, which is a key development in secondary maths classrooms. Emma was excited by the inquiry process in her classroom and praised the students for their reasoning and progress.

Emma Rouse is Lead Practitioner in the maths department at Britton's Academy (Essex, UK).
Initiating inquiry  
Samuel Down used the number line inquiry with his year 9 class. The picture shows the students' initial questions. For a first experience with inquiry, it is notable how creative the students have been in proposing changes to the prompt. The question "What if we add the pairs then divide the answers?" always gives an answer of one. The proof that [n + (n + 3)] ÷ [(n + 1) + (n + 3)] equals one could lead the class into more complex proofs.

Samuel Down is deputy leader of the maths department at Durrington High School (Worthing, UK). You can follow Samuel on twitter @SDown4.
Directing inquiry through the regulatory cards
Jay Pringle (a secondary school maths teacher in Plymouth, UK) tried out the number line prompt with classes in several year groups and with different prior attainment. He commented that it was "a really enjoyable experience with all the classes." Prior to the inquiry, Jay had contacted Inquiry Maths about how to use the regulatory cards in lessons. He reported back that: "The regulatory cards worked really well in getting students to think about what they needed to do next. By getting all the opinions up on the board (by ticking the selected cards) it was easy for students to see how different actions were interlinked and 'how working with another student to find more examples in order to prove that it is always true' flowed together."

Overall, Jay said, students "seemed genuinely interested in the exploration and were very excited to find interesting examples - for example, negative numbers or starting with a decimal and still going up in ones - and then to show the class their examples. The lesson flowed very well into a second lesson on algebra for the higher sets and it was easy to engage students in algebra worksheets when their goal was to learn how to expand n(n + 3) and (n + 1)(n + 2)! They were interesting lessons for me and for the students. I'm looking forward to doing some more again soon."