Playing with factors and multiples
Why playing with factors and multiples matters
In 2009, a major supermarket chain found itself in court after disputing a payment totalling almost a million pounds for six mountain bikes. How did it find itself in such a position? The court case, which revealed that a simple mistake had resulted in the supermarket erroneously paying a retailer a thousand times more than usual for the bikes, highlighted the importance of nurturing and valuing number sense in both the classroom and our everyday lives.
Developing an intuitive 'feel' for numbers enables young learners to begin to make sensible estimates and identify obvious errors (such as an unexpectedly high payment for bicycles!), and flexibly apply their number skills in different situations. After all, 'without number sense, arithmetic is a bewildering territory in which any deviation from the known path may rapidly lead to being totally lost' (Dowker, 1992). At NRICH we believe that encouraging young learners to play with numbers throughout their primary schooling allows them to develop their number sense in a supportive, engaging environment.
It has been suggested that number sense 'develops continually as the range of known facts and the relationships among them are extended' (Anghileri, 2000). Hence supporting learners with their number sense goes far beyond enabling them to learn and apply their number bonds, it also requires them to understand the properties of numbers and the relationships between them, such as the relationship between factors and multiples.
Story telling and role play
Role play and story telling offer two playful approaches towards exploring number relationships with very young children. We have created a few Early Years activities that are based on books, each of which contains 'Using Books' in their title. Using Books: The Doorbell Rang features the well-loved tale by Pat Hutchins. At the start of this story a plate of freshly-baked cookies is given to two children, to be shared equally between them. As the story progresses, more and more children arrive to share the cookies. Can it be done? The activity offers a lovely context for very young learners to explore different ways to share a number of items. Their reasoning can be further explored through prompts such as 'What will happen if another person turns up now?', 'Can you explain why you think that? and 'Can you show me?'. There are more questions, additional teaching ideas, and links to further resources in the activity.
Once your class has enjoyed investigating ways to share cookies, another useful resource for exploring the concept of sharing equally is Maths Story Time which challenges young learners to help Pirate Panda share his 20 golden coins with his friends:
Pirate Panda has taken all the treasure, 20 golden coins.
Cat, Dog and Rabbit jump about excitedly, “Can we have some too?”
“No! No! No!” says Pirate Panda.
Can the children suggest what Panda ought to do?
Detailed notes are included in the activity suggesting ways to encourage very young learners to reason mathematically, and ways to further develop or vary the context to extend its potential in the classroom.
Making new discoveries
This problem, which requires learners to make sense of information and work in a systematic way, has attracted the interest of both learners and their teachers. 'I love trying to get children engaged in tackling such problems' wrote one teacher. His class adopted a pattern-spotting approach towards solving the problem which you can explore in the Solutions tab of the task, alongside solutions submitted by other schools.
Exploring interactivities
A good knowledge of number properties can enable learners to work in different ways and check their answers, improving their accuracy and making connections in their learning. A very good example is using and applying divisibility tests. Dozens offers a twist on the usual way of assessing learners' knowledge of divisibility tests. Rather than asking learners to check whether a number is divisible by 2, 3, 4, 5... learners have to puzzle over the choices available as they are challenged to find the largest number that meets the necessary criteria.
"To find out if your number is a multiple of 3, all you need to do is add the digits. If the digits add up to a multiple of 3, the original number is also a multiple of 3.
For example: 351"
Taking things further, he reflected on the benefits of drawing on his growing number sense in the classroom:
"To find out if this is a multiple of 3 using normal division would be quite slow, but if you just add the digits it takes only a few seconds:
3 + 5 + 1 = 9
9 is a multiple of 3 so 351 is a multiple of 3."
Sometimes learners might find that it is impossible to meet the criteria. Reailsing that not every problem has a solution, and explaining why that might be the case, is another great opportunity for developing their number sense. Do look at the teachers' resources for this activity for ideas about how to introduce it in the classroom.
Being inventive
Developing a higher level of number sense enables learners to work flexibly, a skill which comes in very useful when exploring activities such as Dicey Array. This low threshold high ceiling game offers a meaningful and motivating context in which learners can deepen their understanding of factors and multiples, and develop their fluency with finding factor pairs. To get started, we suggest watching the video accompanying the activity which models how to play the game.
Playfulness and number facts
In conclusion
Playful activities can support learners develop their sense of factors and multiples in many different ways. They can explore interactivities, enjoy paired games and act out stories. Hopefully, their improving number sense will help prevent them and their families from making costly mistakes when they buy bicycles too!
References
Anghileri, J. (2000). Teaching number sense. London: A&C Black.
Dowker, A. (1992). Computational strategies of professional mathematicians. Journal for Research in Mathematics Education, 23(1), 45-55.
Polya, G. (2004). How to solve it: A new aspect of mathematical method (Vol. 85). Princeton University Press.