As a maths team, we adore using books to help children contextualise, discover and explore mathematical ideas. It could be learning about why we need standard units of measurement in 'How Big Is a Foot' by Rolf Myller, learning about arrays and remainders in ‘Remainder of One’ by Elinor J Pinczes, or solving problems alongside a book character such as ‘Spaghetti and Meatballs for All’ by Marilyn Burns. Each team member has a favourite. What follows is a medley of those favourites and how we might use them in the classroom.
Infinity and Me
By Kate Hosford.
Illustrated by Gabi Swiatkowska
When I looked up, I shivered. The sky seemed so huge and cold. How many stars were in the sky? A million? A billion? Maybe the number was as big as infinity. I started to feel very, very small. How could I even think about something as big as infinity?
This is a book that can be read for its own sake. The maths is not incidental but it is wonderfully philosophical, due to the main character’s quest to make infinity make sense to her. Uma travels through different ways of thinking about infinity by asking for and considering the views of others. This may allow discussions about how we listen and try to accommodate different ways of seeing and that by considering other viewpoints we might better discover our own. This is exactly what happens to Uma when her grandmother compliments her new red shoes.
Infinity comes to life in a sequence of beautifully conceived illustrations. Which of the illustrations makes most sense to the children? How would they describe infinity to Uma?
How Much is a Million? How Big is a Million?
By David M Schwartz By Anna Milbourne and Serena Rigietti
Children are naturally fascinated by big numbers and these are both lovely books that help children to begin to have an understanding of the vastness of our number system.
‘How Big is a Million?’ tells the story of a little penguin, Pipkin, who wants to know exactly now big a million is and on his travels he gets to see not only a million but also 1, 10, 100 and 1000 illustrated as fish, a penguin huddle and snow flakes. However, it is the separate large poster which shows 1,000,000 stars that enables you to inspire children’s curiosity in number. The National Curriculum program of study states that children should have “an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.” This book can help create that.
‘How Much is a Million?’ asks questions that require the reader to conceptualise what at first seems inconceivable, helping children to begin to imagine the size and effect of very big numbers. It does not just cover a million but a billion and trillion also. The concepts covered include: the time needed to count to a million; the height of a tower of a billion children stood on each other shoulders and size of a goldfish bowl for a trillion goldfish. These concepts involve the children using what they know and applying this to ever larger numbers which are near impossible to conceptualize.
The ability to make calculated approximations to solve problems, like the ones in ‘How Big is a Million?’ are sometimes referred to as Fermi problems; named after the physicist Enrico Fermi. He was known for his ability to make good approximate calculations, typically using justified guesses to estimate answers to highly complex problems. However, these estimations provide invaluable checks of reasonableness, once the exact calculation is complete.
The essential skill of being able to estimate is a very important skill that children need so they to can check for reasonableness in their answers but sadly it is also a skill many children lack. An argument for this is that children lack the understanding of the magnitude of numbers within the number system. Consequently, they don’t automatically notice that when calculating for example; 2894 x 208 that 2894 is very close to 3000 and 208 is close to 200 so the answer is going to be in the region of 600,000, but instead they go straight to the process of completing the calculation. This leads to children not noticing their errors because the reasonableness of the answer is not considered as the process of carrying out the calculation is seen as the most important task.
Using books such as these, provide a spring board for children to consider answers to calculations that are near impossible to know the correct answer for. However, it is the process of the consideration of justified guesses that will both improve children’s number magnitude and their ability to judge reasonableness.
One Is a Snail and Ten is a Crab
By April Pulley Sayre and Jeff Sayre.
Illustrated by Randy Cecil.
If one is a snail and two is a person… we must be counting feet! Join in the mathematical mayhem, and count the feet of snails, crabs, dogs and spiders from one to one hundred!
This book is a great book for rehearsing counting in multiples and also for exploring composition of numbers. The numbers are made by combining the feet of the animals, sometimes this is a multiple of a single animal – ten spiders make 80 and sometimes it is a combination of animals – a dog and snail make 5.
Odd and Even
In the story all the odd numbers to 10 are made by adding a snail to another animal’s legs. Using this idea children could investigate:
- even numbers can be halved – animals having the same number of legs on each side.
- odd numbers have an ‘extra 1’ – Numicon and tens frames could help the children show this.
- when odd and even numbers occur in nature.
Making numbers to 10
A snail, person, dog, insect, spider and crab are used in the story to count to ten. The children could:
- use the animals to make all the number bonds to ten
- use different combinations to make all the numbers to 10 and this could be extended to 20. How many different combinations could you find?
The multiples of 10 in the story are made by groups of an animal or a group of an animal and a crab and for each multiple of 10 there are two ways of making the number. The children could:
- explore the commutative law. 4 lots of crabs = 10 lots of dogs - 4x10 = 10x4. The children could make arrays to support the reasoning of this.
- explore the distributive law - 7 lots of crabs = 10 lots of insects plus one crab – 7x10 = (10x6) + (1x10) The children could investigate other ways of making other numbers using combinations of animals.
The Greedy Triangle
By Marilyn Burns
Marilyn Burns is one of our maths heroes. Many of us on the Primary Maths Team have several of her titles on our own bookshelves. Here we have the captivating story of a young triangle who is discontented. By visiting his local ‘Shape Shifter’, he is gifted another side and another angle. Lo and behold the triangle is transformed into a quadrilateral! But he becomes bored and unhappy, and even greedy, making several more visits to the Shape Shifter who keeps giving him one more angle and one more side. So begins a visual journey through polygons.
This book naturally links to practical exploration of polygons, their properties and generates lots of maths talk. Given large elastic loops (about 1 metre of broad elastic) or geoboards and elastic bands, the children could form and explore the polygons that are described. After each magical encounter with the Shape shifter, children can be encouraged to explore the resultant possible shapes. Be sure to stop and give the children time to observe and comment about ‘What is the same? What is different?’ in the elastic shapes. For example, the triangle that they start with could be equilateral, right angled, isosceles etc. Many different forms of quadrilateral could be produced rectangles, squares, trapezoids. Does adding one more side always result in one more angle? Encourage the children to explore the possibilities and ‘Go extreme!’
The Tangram Cat
By Maranke Rinck and Martijn van der Linden
Part of the charm of this book are the tangram illustrations, rendered as animal companions created for the Tangram Cat. High jinks occur when the tangram animals are not what are expected by their creator. At the back of the book is a set of tangram pieces for children to use so that they can re-create the animals made by the boy. Tangrams are a seven-piece Chinese puzzle that allow learners to explore the properties of shape and also consider how larger shapes can be created from smaller shapes. Composition and decomposition allow them to become more geometrically fluent. Show the children and watch them borrow the book and play with the puzzle. Can they make another animal? Can they use the pieces to make a boat for the cat to escape from the crocodile in?
We think you'll agree that maths in stories can be a novel experience for children, firing up their imagination and allowing them to consider bigger maths questions.
Here are some further books we think you and your children will also enjoy.
- The Doorbell Rang – Pat Hutchins (division by sharing)
- Remainder of One – Elinor J Pinczes (division with remainders)
- Counting on Frank – Rob Clement (estimation and comparison)
- Spaghetti and Meatballs for All – Marilyn Burns (a division and perimeter problem)
- A Place for Zero – Angeline Sparagna LoPresti (place value)
- Sir Cumference and the First Round Table – Cindy Neuschwander (Circles)
- Cut Down to Size at High Noon – Scott Sunby (Scaling)
- The Librarian who Measured the Earth – Kathryn Lasky (Working mathematically)
- Alice in Wonderland – Lewis Carroll (Scaling – think potion and cake)
Rinck, M and Van der Linden, M. (2017). The Tangram Cat. Lemniscaat Limited.
Burns, M. (2008). The Greedy Triangle. Scholastic.
Schwartz, D. M and Steven Kellogg, S. (1993). How Much is a Million? Eos; 1st Mulberry Ed edition
Millbourne, A. (2007). How Big is a Million? Usborne Publishing Ltd
Hosford, K. (2012). Infinity and Me. Carolrhoda Books.
Pulley Sayre, A and Sayre, J. (2004) One is a Snail, Ten is a Crab. Walker Books; New Ed edition
Pitt, C. and Smith, J. (2012). Pro PHP MVC. Berkeley, CA: Apress