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Prime factors inquiry

Irfan Rashdi used this prompt during his school-based practice while training at London Metropolitan University (London, UK) 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 numbers
  • prime factors
  • multiples
  • divisors
  • structure and pattern
  • squares and square roots; cubes and cube roots
  • indices.
Notice and Wonder  
Amie Albrecht (senior lecturer and applied mathematician at the University of South Australia, Adelaide) writes in this post 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 that the group came up with, organising them under headings that include structure, the number one, prime numbers, composite numbers, square numbers, prime factors and divisor and factor-oriented.
Extending the inquiry
The prompt can be extended by using (or revealing) more of the original chart. 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."

The prime factors inquiry developed from an Inquiry Maths workshop at London Metropolitan University in January 2017. Alan Benson, the mathematics PGCE tutor, encouraged trainee teachers to try an inquiry lesson during their teaching practice. Irfan Rashdi went on to devise his own prompt and Jhahida Miah added to the Inquiry Maths model by requiring students to take on different roles (see this slide).


Alternative prompt
Liz Hill (a trainee secondary teacher studying at London Metropolitan University) devised this prompt after attending an Inquiry Maths workshop in January 2018:  
Larger numbers have more prime factors than smaller ones.
Liz 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. They concluded that the statement is generally true, although prime numbers themselves are a special case. One pair 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, although other students challenged the concept of an arbitrary point at which the contention becomes true.
Making connections through inquiry
Amanda Kirby (a secondary school teacher) 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."

Amanda Kirby teaches mathematics at St Clement Danes School, Hertfordshire (UK). You can follow Amanda on twitter @mathsteach2000.
Developing mathematical reasoning
Irfan Rashdi reports on an inquiry based on the prime factors prompt he carried out with his year 9 class during his second 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.