# Number line inquiry

# The prompt

**Mathematical inquiry processes: **Interpret; test particular cases; conjecture, generalise and prove; extend to other cases. **Conceptual field of inquiry: **Multiplication; algebraic notation, terms and expressions.

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) = *n*^{2} + 3*n* and, for the 'inside' two numbers, (*n* + 2)(*n* + 3) = *n*^{2} + 3*n* + 2.

## Lines of inquiry

Students have gone on to create their own inquiries by changing the prompt (see the **mathematical notes** for more details). 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.

### (2) 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?

### (3) Extend the number line to six consecutive numbers

Link the numbers systematically and work out the differences between the products.

### (4) Inquire into different sequences

Use terms from quadratic and cubic sequences or the Fibonacci sequence.

### (5) Proof

A year 7 student proves that the difference between the products of the top two and bottom two numbers is 4*n *+ 6 where *n* is the lowest number of the four consecutive numbers.

A year 7 student proves that the difference between the products of the top two and bottom two numbers is 4*n *+ 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.

# Online inquiry

Over the course of the coronavirus pandemic, the Department for Education has closed schools in England on two occasions. During the second period, starting in January 2021, the minister ordered state schools to teach remotely. At Inquiry Maths, we have argued** ****here** that the online environment can be detrimental to inquiry processes in mathematics. Nevertheless, it is important in the current climate that students' education continues to include elements of inquiry. **Andrew Blair **reports on an online inquiry he ran with his year 7 class:

Our school is situated in an area of high social and economic deprivation in London (UK). Over 70% of students are eligible for free school meals and at the start of the lockdown many did not have devices to engage effectively with online learning. Over half of students on roll have now received Chromebooks from the school. The majority have been donated by charities as the government supply continues to be unreliable.

### Online lessons

My year 7 mixed attainment class has 24 students. Online attendance to lessons is high with just one or two absences each lesson. During the first term in school (September to December 2020), we regularly used inquiries, directing the process through the **regulatory cards**** **and following lines of inquiry that originated in students' questions and observations. Lessons were full of curiosity and conjectures. Moreover, the class invariably developed a deep understanding of the conceptual field on inquiry.

For remote learning, we work in Google Classroom and each lesson I upload differentiated slides that students can edit. After an introduction from me in a Google Meet, students ask questions by 'unmuting' or through the chat box. They stay in the lesson as they work through the slides. I monitor progress and ask individuals to explain their solutions by 'presenting a window' for the rest of the class to see.

After a couple of weeks of establishing this way of working, I wanted to get back to inquiry. Students were comfortable with the platform and had got used to their new Chromebooks. I presented the number line prompt on a Jamboard and invited students to post notes. Before doing so I considered going over the social and mathematical norms of an inquiry classroom, but I felt we had established such a strong culture of inquiry that I trusted them to use the freedom to post responsibly.

The results are pictured above. I arranged the observations into sections that gave structure not only to their contributions, but also to the inquiry that developed out of them. The sections are:

What do we notice about the prompt?

The difference between the products

Testing with other numbers

Changing to the prompt

A new line of inquiry.

Once I had established the structure, it proved difficult, if not impossible, to deviate from it. I had to create and upload the **slides** before the lessons so the direction of the online inquiry was less responsive to students' interests than in the classroom.

By the end of the first lesson, we had explored cases when the number line goes up in intervals greater than one. By the end of the second, we had completed a table of our results and discussed the patterns we noticed. During that lesson I invited students to listen to an explanation of how we could use an algebraic representation to prove the results. Four students, including Mohammed who had posted a comment about algebra on the Jamboard, joined me.

However, it proved much more difficult to promote the sharing of ideas in the virtual environment. Even though Alim presented and explained his proof that the difference between the products is always eight when the interval is two (above) at the start of the third lesson, fewer students advanced onto an algebraic approach than I expected.

In the classroom, I would have tried to use the interest and excitement generated by the first group to involve more students. Perhaps I would have asked another one of them to explain to the class or for them each to work with individuals. Unfortunately, in lacking that immediate peer support, many students found constructing the proof a step too far.

The inquiry finished with students working on the difference between the products when the numbers are connected in different ways. Although the majority position was that there is no common result for consecutive numbers with the first 'linked' to the second (using Halima's term) and the third to the fourth, Alim and Mohammed both proved the difference is 4*n* + 6.

This was a strong ending to our inquiry, but I felt we had not tapped the potential of the prompt - or, perhaps it is better to say, not as many students did so as would have done in normal times.

February 2021

# 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 4*n* + 6.

# Lines of inquiry

**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.

**Jhahida Miah **gave the presentation to trainee teachers at London Metropolitan 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

## An algebraic approach

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.

## Arithmetic to algebra

**Emma Rouse**, a Lead Practitioner at Britton's Academy (Rainham, UK), used the prompt to develop an inquiry with her year 8 mixed attainment class. The initial questions and observations (above) 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 makes predictions about what will happen when the numbers on the line increase by two and then three (see picture below).

He goes on to attempt to prove that the difference between the products in the prompt will always be two (see picture below). In giving three examples connected to consecutive numbers, he shows *transitional *thinking between arithmetic and algebra. Any one of the examples would constitute a general proof, although the use of *n*, *n *+ 1, *n* + 2 and *n* + 3 is more formally correct. Emma was excited by the inquiry process in her classroom and praised the students for their reasoning and progress.

# Initiating inquiry

**Samuel Down**, deputy leader of the mathematics department at Durrington High School (Worthing, UK), 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.

# Regulating inquiry

## Using 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."

## Re-directing the inquiry

**Jabril**, 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 2*n* + 3 (where *n* is the lowest number of four consecutive numbers).

# Inquiry in mixed attainment classes

This year 7 mixed attainment class came up with a rich and varied set of questions and comments. One pair of students 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.

These are the questions and conjectures of a year 8 mixed attainment 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.

# An alternative prompt

The prompt was posted on the internet **here** with the instruction to ‘explore’. In a classroom, students will not be able to explore without first *noticing *and d*iscussing* 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 - for example,

4 + 3 + 2 + 1 = 4 x 3 – 2 x 1

5 + 4 + 3 + 2 = 5 x 4 - 3 x 2 and, in general,

*n* + (*n* - 1) + (*n* - 2) + (*n* - 3) = *n*(*n* - 1) - (*n* - 2)(*n* - 3)

What happens if the difference between the natural numbers is greater than 1?

Both sides of the final equation can be shown to equal **4****n**** – 6*** a*.

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.

# Resources

An evaluation of the inquiry by members of a mathematics department who collaboratively planned and then taught the number line inquiry.

### Acknowledgement

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.