Sum of consecutive numbers inquiry
Mathematical inquiry processes: Explore other cases; identify patterns; conjecture and generalise. Conceptual field of inquiry: Sum of integers; prime factorisation; factors.
The prompt comes from Samia Henaine's website Maths Teachers as Bridge Builders. We are grateful to Samia for giving us permission to feature the prompt on the website.
The prompt originates in the classic mathematical investigation, which normally starts with a teacher's problem and follows a pre-determined course (see this activity as a good example). Students consider the first few cases of the sums of consecutive positive integers and attempt to spot patterns. The sums of two consecutive numbers, for example, give all the odd numbers greater than one: 1 + 2 = 3, 2 + 3 = 5, 3 + 4 = 7, and, in general, n + (n + 1) = 2n + 1. The sums of three consecutive numbers give the multiples of three greater than three itself (3n + 3); the sums of four consecutive numbers are in the form 4n + 6; and the sum of k consecutive numbers is kn + [k(k - 1) / 2].
In addition to pattern-spotting, the inquiry prompt has the advantage of focusing students' attention on mathematical structure. By starting with a number that can be expressed as a sum of consecutive numbers in multiple ways, the prompt leads students into deeper lines of inquiry. Initial questions might include:
Are there other ways to express 45 as the sum of consecutive numbers?
Why cannot 45 be expressed as the sum of four consecutive numbers?
Is 45 the first number that can be expressed as the sum of consecutive numbers in five ways?
If 45 is the first, why is it?
What is the first number that can expressed as the sum of consecutive numbers in two, three or four ways?
Can numbers greater than 45 be expressed as the sum of consecutive numbers in more than five ways?
As students explore, they come to a surprising result. There are no numbers lower than 45 that can be expressed as the sum of consecutive numbers in four ways. The first time this occurs is for the number 81.
This is an intriguing result that leads students into questioning and speculating. What is the property of 45 that means it can be expressed in so many ways? What is the property of 81 that means it is the lowest number that can be expressed in four ways? (See the 'General rule and theorem' section below.)
The differences between investigations and inquiries
Math teachers as bridge builders
Samia Henaine is a math teacher, and PYP and ICT Coordinator at Houssam Eddine Hariri High School in Saida (Lebanon). Her mission is to support teachers to build bridges by connecting concepts and forms of thinking within the mathematics classroom. Moreover, Samia encourages teachers to develop a growth mindset in relation to their students' mathematical potential.
Samia's website contains other inquiry prompts.
Samia's consecutive numbers prompt
In her original prompt, Samia included a second case alongside four sets of consecutive numbers that sum to 45:
Samia explains the origins of the prompt: "'Writing given numbers as a sum of two consecutive numbers' is a question for students at primary level. "When I gave it to my 10-year-old child, I wondered if this question would trigger his mind and develop his reasoning skills. So, I invented the prompt and trialled it by asking him to write what he sees and wonders. The results were outstanding, and he came up with different generalisations about the numbers that could be written as a sum of two, three, four, five, six, seven, eight, nine, or ten consecutive numbers.
"In brief, any direct question can be turned into an inquiry prompt that requires students to cultivate their thinking and reasoning skills and deepen their understanding of concepts. Moreover, inquiry prompts are 'Age-Boundless'; any child aged from eight to 16 can investigate and think about the prompt and get the answer whether by observing patterns and using basic operations or by using the properties of number sequences and writing expert solutions."
The design of the inquiry prompt in mathematics classrooms is dependent on two main factors: the experience of the students and the aims of the teacher. If students have carried out inquiries before, then the more open prompt featured on this page (with the cases for 45) might be more appropriate. However, if the teacher wants to structure the start of the inquiry, then Samia's prompt might be better. In containing two cases, it suggests a line of inquiry in which students compare numbers that can be expressed as the sum of consecutive numbers in multiple ways.
General rule and theorem
This activity from the Implementing the Mathematical Practice Standards Project requires students to look for evidence that might help them answer the question: Is there a general rule for how many different ways a number can be written as a sum of consecutive positive integers?
It turns out that there is a rule, which can be presented as a theorem: The number of ways a positive integer can be written as the sum of consecutive numbers is one fewer than the number of its odd factors.
The number of odd factors can be worked out using prime factorisation. For example, the prime factorisation of 18 = 2 x 32. Ignoring two, which will give even factors, add one to the exponent of 3 (that is, 2) and there are three (2 + 1) odd factors. Therefore, 18 can be written as the sum of consecutive numbers in two (3 - 1) ways (5 + 6 + 7 and 3 + 4 + 5 + 6).
In general, the number of odd factors can be calculated from the prime factorisation 2p x aq x br (a and b being odd) by finding the product of (q + 1)(r + 1). It follows that the number of ways of expressing a number as the sum of consecutive positive integers is (q + 1)(r + 1) - 1.
45 as the product of prime factors is 32 x 5. The number of the odd factors of 45 is (2 + 1)(1 + 1) = 6. Therefore, 45 can be expressed as the sum of consecutive positive integers in five (6 - 1) ways.
81 as the product of prime factors is 34. The number of the odd factors of 81 is (4 + 1) = 5. Therefore, 81 can be expressed as the sum of consecutive positive integers in four (5 -1) ways.