# Prime factors inquiry

# The prompt

Mathematical inquiry processes: Interpret; Analyse and extend structure. Conceptual field of inquiry: Prime and composite numbers: prime factors.

Irfan Rashdi used this prompt during his school-based practice while training to become a secondary school mathematics teacher. The prompt is adapted from the Prime Climb hundred chart. The inquiry has the potential to develop reasoning about the following topics: prime and composite numbers, prime factors, square and cube numbers, structure and pattern, and indices.

## Extending the inquiry

The prompt can be extended by using (or revealing) more of the original chart (below). Irfan suggests the possibility of students from years 6 to 8 filling in their own chart once they have understood the original prompt. This might be done in conjunction with prime factor trees or another method of deriving the prime factors of positive integers. As Irfan says, this approach "promotes learning by doing maths" and encourages students who become "more switched on when interactive activities are introduced."

Acknowledgement

The prime factors inquiry developed out of an Inquiry Maths workshop at London Metropolitan University in January 2017. Alan Benson, the mathematics PGCE tutor, encouraged trainee teachers to teach through inquiry.

# Noticing and wondering

Rebekah Clark, a teacher at the Living Waters Lutheran School (Alice Springs, Australia), reports on how she used the prompt with her year 6 class:

"The prompt was the beginning of our unit on prime and composite numbers. I set the questions ‘What do you notice? What do you wonder?’ and let the students work on the prompt for about 20 minutes. They shared and recorded their thinking in pairs. It brought out some great observations about factors, multiples and primes. The messy writing on the board (picture above) is me trying to quickly get all their ideas on the board! I love the Inquiry Maths prompts and this one was a great introduction to our unit."

## 100 ideas from the prompt

Amie Albrecht writes here about using the Prime Club hundred chart with a group of 50 pre-service teachers. She asked them to consider the chart individually and tell her five things they noticed. Amie then compiled a list of a hundred different ideas, organising them under headings such as structure, the number one and prime factors.

# Making connections through inquiry

Amanda Kirby (a teacher of mathematics at St Clement Danes School, Hertfordshire, UK ) used the prime factors prompt with a small class of ten year 8 students. Even though they had low prior attainment, the students were successful in making connections between composite and prime numbers. Amanda reports on the inquiry:

This is one of my favourite prompts in action. The students managed to extract lots of information from the diagram. One boy's annotations were just simply 8 ➜ 2 x 2 x 2 and 12 ➜ 2 x 2 x 3. He'd never been shown prime factor decomposition but just used the colours and the sections to get straight to the heart of prime factor decomposition. Further work in class involved understanding what was happening with factor trees. After teaching the class about factors and multiples, we'll try to make the links to what they discovered in the first lesson.

# Developing mathematical reasoning

Irfan Rashdi reports on the prime factors inquiry he carried out with his year 9 class during his teaching practice:

During the inquiry, I planned to observe the students' development of mathematical reasoning, individually and in pairs. At the outset, I posed questions: What do you notice? What do you wonder? The purpose of this approach was to promote out-of-the-box thinking in contrast to traditional maths classrooms.

The inquiry promoted a variety of responses as opposed to just one solution; it allowed students to learn by following different pathways (both successful and unsuccessful) as opposed to reaching a single right or wrong answer; and it encouraged students to become more flexible because there were no teacher-controlled parameters.

The pictures of the students' questions and observations (below) show how ideas developed at the start of the lesson. Even though the thoughts and findings might seem basic or obvious, the students were in the process of engaging themselves more in deeper thinking.

In the next part of the inquiry, I led a class discussion about types of numbers, including factors and multiples. Without me making the link to the prompt explicit, the students' were able to gain a greater insight into its structure.

This process extends the idea of Vygotsky’s Zone of Proximal Development - that is, the difference between what a learner can do without help and what he or she can do with help. The teacher stresses the importance of the latter part of the definition, focusing the discussion on what the learner can do with help.

I then required students to inquire into the colour, pattern and structure of the prompt in order to notice and record ideas about prime numbers, composite numbers, square numbers, prime factors, multiples, and divisors.

I observed that students initially studied individually and, generally, they did not establish a clear idea about the prompt. It was when I introduced structure into the inquiry by focusing the students' attention on key mathematical features of the chart that students became engaged in the creative and out-of-the-box thinking that I had hoped to develop.

They became more confident in expressing their ideas as they made connections between their existing (intuitive) knowledge and and the object of the inquiry. In this way, the class constructed new knowledge.

# An alternative prompt

Liz Hill (a trainee secondary teacher studying at London Metropolitan University) devised the prompt about prime factors after attending an Inquiry Maths workshop in January 2018. She used the prompt with a year 8 mixed attainment class while on her teaching practice.

Some students chose to practise finding prime factors after Liz had explained the concept; others explored the statement by testing specific cases. The class concluded that the statement is generally true, although prime numbers themselves are a special case.

One pair of students proposed an alternative statement that they explained to the class: "Odd larger numbers have less prime factors than even large numbers." They defined larger numbers as those above 100.

Other students challenged the concept of an arbitrary point at which the contention becomes true. They argued that numbers in the thirties (excluding prime numbers) largely follow the pattern: 32 (5 prime factors), 33 (2), 34 (2), 35 (2), 36 (4), 38 (2), 39 (2).