Product of factors inquiry
Mathematical inquiry processes: Verify the particular case; generate more examples; conjecture and generalise. Conceptual field of inquiry: Factors and indices.
Helen Hindle designed the prompt for a year 7 mixed attainment classes (see her report on the inquiry below). It can be shown to be true by pairing up the factors of 24:
(1 x 24) x (2 x 12) x (3 x 8) x (4 x 6) = 24 x 24 x 24 x 24 = 244
Indeed, there is a 'rule' for all numbers: the exponent is half the number of factors. So, for example, the product of the six factors of 12 equals 123. Often students take some convincing that the exponent in the case of prime numbers - that is, one - is legitimate.
Furthermore, students regularly believe that there is one exception to the rule. Square numbers, they claim, do not "follow the rule because one factor is on its own." However, it turns out that square numbers do behave in the same way. For example, the product of the three factors of nine equals 91.5 and the product of the nine factors of 36 equals 364.5.
The realisation that square numbers follow the rule can form the basis for an inquiry into square roots and the representation of a square root as a fractional exponent. The inquiry can be extended further into other fractional and even negative exponents, although at this stage the link with factors is severed.
1. Notice, question and wonder
All inquiries, including those the teacher aims to structure, can start with students' questions and observations about the prompt. Even if the students' responses point towards the main line of a pre-prepared structured inquiry, their questions are occasionally novel and suggest new lines that can be built into the structure later.
This is exactly what happened in a year 7 mixed attainment class (see the picture). The students' comments were pointing towards the teacher's plan to test a generalisation and find a rule for the power (exponent).
However, the last contribution from a pair of students was unforeseen: "We wondered whether the factors of 24 are the factors of 244 when raised to the power four."
After taking a moment to think, the teacher knew they would be. However, other questions came to mind:
How many other factors are there?
Is there a way of predicting how many based on the number of factors of 24?
And how are they spaced in relation to the factors of 24 raised to the power four?
The teacher then used the two tasks she had planned and resolved to return to the students' question later in the inquiry.
2. Structured tasks
The second task was a series of questions that led students towards a generalisation connecting the number of factors to the power.
3. Structuring exploration of the students' question
244 has 65 factors, so exploring the question directly risked involving students in a frustrating search.
Instead, the teacher structured the exploration to focus on the first five prime numbers (see an example below), the three square numbers after one and then other composite numbers (six, eight and ten).
The students were able to generalise a pattern for the prime numbers (expressed algebraically by the teacher), but could not resolve the questions for the other cases.
See the structured inquiry PowerPoint for all the cases.
Generating inquiry from students' questions
Miss L Costa reports on the inquiry with her year 8 class at the Coleshill School (Warwickshire, UK).
Students had been presented with this style of lesson before when they investigated the 24 x 21 = 42 x 12 prompt. I displayed the factors statement on the board and asked the pupils to copy the statement down and then write down any questions they had. Some of the pupils went straight to finding out if the prompt was true or false, while others were stuck as to where to start. To help those who were stuck, as I walked around the classroom, I made a note on the board of some of the questions I was asked and some that I had overheard.
I then gave the class a further two minutes before bringing the students together. We discussed each component of the statement:
What does product mean?
What is a factor? For this, we had a small discussion regarding the difference between prime factors and factors.
What does 244 mean? We quickly refreshed what it means if we had a number raised to the power of two and then to the power of four.
One student asked the question, "Does 244 = 482 = 128?" For this, I praised the student because they had started to think outside the box. I then invited this individual to explore it further and find out if it was true or not.
As a class, we then discussed what equals meant and what the statement meant as a whole and asked the class for a vote on whether they thought it to be true or false. Some individuals prior to this had already begun calculating and interestingly they were split as to whether it was true or false. So together we walked through the steps of how we would find it out, firstly listing the factors of 24, then multiplying them and then calculating 244. Those who initially thought it was false were surprised to find that it is true.
I then asked the students to find another integer to which the product of its factors equalled the integer raised to another power. In pairs they worked together to find the following examples that followed the prompt: 304, 404 and 284. I asked for a strategy. Some went with raising to the power of four all the time and others said it had to be an even number. Then I asked the question, "Why does it need to be raised to a power of four?" and "Why does it need to be an even number?" They then discovered the following examples when the product of the factors is the square of the number: 262, 222 and 332.
After this I asked, "Would it work for square numbers, cube numbers, prime numbers?" I left this as an open homework not telling them the rule as some students were clear they wanted to spend some time on it. Overall, the year 8 class loved the inquiry style of learning.
The inquiry in action
Helen Hindle describes the first lesson of the inquiry that developed from the prompt:
This was the class's second experience of inquiry since starting secondary school. Below are the questions and statements that the prompt generated.
Following the discussion, students understood the meaning of the statement and decided, initially, to work as a class to decide if the statement was true. Students used calculators to find 244, which led to a discussion of place value when some students had difficulty saying the resulting number. The class then listed the factors of 24 by generating factor pairs (as suggested in the statement above). They did not initially recognise that 1 and 24 are factors of 24 until I hinted that they were missing a factor pair. This then led to a discussion about how you could make sure that you had not missed out any factor pairs.
Students wanted to continue to work as a whole class and investigate another number. Forty-five was suggested and the product of its factors was found to be 91 125. We changed the prompt to The product of the factors of 45 equals 454. On realising that this is not true, the students did not know where to go next with the inquiry. I suggested that we should change the prompt to The product of the factors of 45 equals 45?. It quickly became clear that 91 125 is 453.
At this point pupils decided that they wanted to work either individually or in pairs or groups to investigate other numbers. The following observations or issues arose:
Some of the pupils investigated the number 60, which lead to a discussion about standard form when the calculator could not display the answer in normal notation. One student asked, “How do I know how many zeros to add to the end of the number if I want to write it as a normal number?” This question generated a discussion about how to multiply by powers of ten.
Another pupil investigating 60 discovered that 603 x 603 gave her the same answer as the other pupils on her table who had worked out the value of 606.
A different pupil claimed that 17 "did not work" because "the factors of 17 are 1 and 17, and 17 can’t be the answer to 17 to the power of a number.” This led to a discussion about what 171 meant.
The students who made these observations agreed to discuss them further in the second lesson of the inquiry. By the end of the first lesson, students were yet to notice that the power will always be half the total number of factors. As the inquiry progresses, we will reflect on why this is the case. When asked what they had learnt in the lesson pupils made the comments shown below.
Year 7 students in a mixed attainment class at Haverstock School, Camden (London, UK) asked the questions and made the observations. They decided to list factors of numbers and determine whether "cubing worked" in all cases as it had in the case of 18. Some started to explain that the exponent is half the number of factors. The inquiry ended with pairs of students presenting their findings, including two pairs explaining that, as square numbers have an odd number of factors, the exponent is not a whole number. Using the example of nine (1 x 3 x 9 = 1 x 9 x 3 = 9 x 3 = 91.5), it was decided that the 0.5 in the exponent represents the square root of the number.
Questions and conjectures
These are the questions and observations of Phoebe Stow's year 7 class at Seahaven Academy in East Sussex (UK). For their first inquiry, Phoebe reports that the students were very excited. The comment that "you need to do it to the power of half its factors" shows one student has already generalised to other cases.
These conjectures have arisen in year 7 or 8 lessons:
Is there a number n for which the sum of the factors of n, apart from n and 1, sum to n?
The number would be similar to a perfect number without using 1 when summing the factors. The student who asked this question called the number, if one existed, a 'Sant Number' (after his name). In a discussion about the question on twitter, @mathstermaths referred to a paper titled "Almost Perfect Numbers" in Mathematics Magazine - journal of the Mathematical Association of America - from March 1973. The authors did not know if a Sant Number, as we shall call it, existed, but they postulated some conditions that it would have to fulfil if it did.
Bigger numbers have more factors than smaller ones;
Odd numbers have fewer factors than even numbers; and
If you multiply a number by 10, the new number has 6 factors more - for example, 3 and 30, 7 and 70, 8 and 80.
Adapting the prompt
The prompt can be adapted for the class you are teaching. Using 18 instead of 24 can encourage students to look at smaller numbers in their initial exploration. In one year 7 class, the prompt gave rise to these questions and comments:
What does the 3 mean?
Product means 'times'.
Is it true that 18 is a multiple of 3?
We wonder if 18 equals 183.
183 = 5832
What does "equals 183" mean?
We wonder if you have to multiply all the factors together to get the product.
The factors of 18 are 1, 2, 3, 6, 9 and 18.
After clarifying the definition of 'factors' and 'product', the teacher established that the prompt was true by multiplying the factors of 18 to give 5832. He also initiated a whole-class discussion to ensure the students understood the meaning of the prompt in its entirety.
As the class was new to inquiry, the teacher restricted the students' choices when deciding what to do next. Rather than require them to select from the full range of regulatory cards, students chose from three options:
Ask a student to explain.
Make up more examples of your own.
Use a worksheet to practise finding factors.
The students that explored soon realised that the exponent would not always be three. They began to make conjectures, such as "the more the factors a number has, the bigger the power will be." By the end of the first lesson, the students who had chosen the worksheet option were confident to make up and present their own examples to the class. In the next lesson, some students went on to generalise from their results; others who had identified the connection between the number of pairs of factors and the exponent tried to explain the 'exception' of square numbers. They used a calculator to answer their own question about the meaning of 0.5 as an exponent.