HCF and LCM inquiry 1
Mathematical inquiry processes: Test the generalisation with particular cases. Conceptual field of inquiry: Highest common factor and lowest common multiple.
This prompt was created collaboratively by the Mathematics team at Brittons Academy (Rainham, east London, UK). It arose from a comment made by a year 9 student during a lesson in which the class was using prime factors to find the lowest common multiple (LCM) and highest common factor (HCF) of pairs of numbers. By studying Venn diagrams, the year 9 student noticed that the LCM of a pair of numbers could always be found by dividing the product of the two numbers by their HCF. He claimed: "The product of two numbers divided by their highest common factor is their lowest common multiple."
The student gave the following explanation to his teacher.
24 as a product of its prime factors is: 2 x 2 x 2 x 3 = 24
60 as a product of its prime factors is: 2 x 2 x 3 x 5 = 60
The product of 24 and 60 must therefore be 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 = 1440.
The HCF is given by the product of the common factors (shown in the intersection of the Venn diagram).
If you divide the product of the two numbers by their HCF, you are left with all of the factors in the Venn Diagram, which is the LCM: 24 x 60 ÷ (2 x 2 x 3) = 2 x 2 x 2 x 3 x 5 = 120.
Written more formally, LCM(a,b) = a x b ÷ HCF(a,b) where a and b are natural numbers.
The teacher was intrigued and shared the student's reasoning with the department. Teachers had just begun using Inquiry Maths prompts with their classes and were excited about turning the observation into a prompt to use when introducing HCF and LCM to mixed attainment classes in year 7.
An alternative prompt
The product of two numbers is the product of their HCF and LCM.
Rachel Mahoney, a mathematics teacher at Carre's Grammar School in Sleaford (Lincolnshire, UK), used the alternative prompt with her year 9 class. She blogs about the inquiry that developed here, concluding that Inquiry Maths classrooms are "an excellent learning environment" in which "all thoughts and ideas are welcome and one in which it is OK to be wrong."
Chris McGrane, Principal Teacher of Mathematics at Holyrood Secondary School (Glasgow, Scotland), posted these pictures on twitter. They show the inquiry of his S1 (grade 6, year 7) class. Chris reported that almost everything on the board came from the students. The inquiry brought out some brilliant work from the class, he added. It was "one of those lessons that makes you love being a teacher."
At the end of the inquiry, Chris encouraged the students to reflect on the inquiry process by thinking about what they had done. The students' list included questioning, recall of relevant knowledge, discussion, exploration, testing, further questioning, and wondering. The students concluded that, "We were being mathematicians."
In the next lesson, the class learnt about prime factors and representing them in Venn diagrams to develop more lines of inquiry.
Chris McGrane is author of Mathematical Tasks: The Bridge Between Teaching and Learning. He discusses the Inquiry Maths model between pages 202 and 206.
Inquiry in a mixed attainment class
The questions and observations of a year 7 class at Brittons Academy (see the illustration) reveal the range of prior knowledge within the class. Some students asked questions about the meaning of each component of the statement whilst other students, with a good understanding of product, lowest common multiple and highest common factor, began to explore the prompt with different numbers to see if it was true. In the class discussion generated from the initial responses, students shared their understanding of the key words with their peers, building upon each other’s prior knowledge.
When asked to consider the next steps in the inquiry, students devised four options based on individuals' prior knowledge of the topic:
Practice finding factors of numbers.
Practice finding the highest common factor and lowest common multiple of two numbers.
Try to find some examples of when the prompt is true.
See if the prompt works when you have more that two numbers.
Early in the inquiry students began to ask if there was another way to find the lowest common multiple, rather than listing multiples of each number.
At this point the teacher introduced the concept of prime factors and showed how they could be used in Venn diagrams to find both the highest common factor and lowest common multiple of pairs of numbers.
Overall, the teacher reports, the inquiry allowed pupils to explore factors, multiples, HCF, LCM and prime factors, providing high levels of challenge for all pupils within the mixed attainment class. It could be extended further by exploring what happens with 3 or more numbers.
Conjecture for three numbers
One student who began to explore the prompt with three numbers (see picture above) found the HCF and product of 15, 20 and 25. In this case, the statement in the prompt does not lead to the LCM: 7500 ÷ 5 ≠ LCM(15,20,25). In collaboration with another student who introduced Venn diagrams to the exploration (see picture below), the students found the LCM is 300, which results from 7500 ÷ 25. This observation led to a conjecture that for 3 numbers: The product of the three numbers divided by the (HCF)2 = LCM.
Yassine and Tabassum, year 7 students at Haverstock School (Camden, UK), verify the prompt is true in two cases.