Progression in primary geometry: thoughts on using tangrams as a concrete resource

    Published: 09 November 2021

    Back in Ancient Greek times, geometry constituted the whole of mathematics. Euclid's famous series of 13 books dealing with geometry, called 'The Elements', are said to be the most studied books in history, apart from the bible. But now geometry receives relatively little attention in the National Curriculum, and it is clear that arithmetic is king. So much so that 'geometry – position and direction' is entirely absent from the Year 3 programme of study.  

    However, there is strong evidence of a correlation between spatial ability and lasting success in science and mathematics. Having the ability to mentally manipulate objects within space supports learning in science, technology, engineering and mathematics (Cheng and Mix, 2014; Lowrie, Logan and Ramful, 2017). Spatial awareness is also crucial in helping children to understand the relationships between numbers, such as magnitude, subitising, proportional reasoning and algebraic thinking.

    Within this blog, I will take a brief look at geometry progression within the National Curriculum, the primary years, using a really nice piece of concrete equipment – the tangram – which is perfect for conceptually developing children's spatial awareness and reasoning.

    Big picture progression of geometry

    Within the National Curriculum, geometry is divided into two domains:

    • geometry – properties of shape
    • geometry – position and direction

    'Properties of shape' focuses on specific geometric knowledge, including names of 2-D and 3-D shapes and the ability to accurately use shape property vocabulary to draw, compare and classify shapes.

    'Position and direction' focuses on spatial sense (spatial awareness) development and reasoning about shape and space relationships, including patterns, sequences and movement descriptions (rotation, translation etc.).

    A quick glance at the National Curriculum may lead people to believe that the emphasis is on rote knowledge. But geometry is much more than vocabulary and naming 2-D and 3-D shapes.

    It is geometric experiences that we need to provide for children. 

    The following tables show the impact of rearranging the curriculum into different subgroups (suggested by Van de Walle) which helps to see the progression. There is a downloadable PDF version of the tables at the end of the blog.

     

    Shapes and properties:

    how to describe and classify a shape using properties and key vocabulary

     

    Table with text

     

    Transformations:

    how to describe a change or movement of a shape; translations, reflection, symmetry, scaling

     

    Table with text

     

    Location:

    how to describe the location an object in space; including co-ordinate plane

     

    Table with text

     

    Visualisation:

    recognising shapes in the environment, ability to visualise and draw objects from different view points

     

    Table with text

     

    Tangrams

    Tangrams are an ancient Chinese geometric puzzle consisting of seven pieces. They could be mistaken for a jigsaw but they are so much more. To make it easier to describe the activities and the possible answers, the different tangram pieces have been assigned numbers.

     

    Tangram

     

    The traditional model comprises five triangles of three different sizes (1, 2, 3, 5, 7), a square (4) and finally, a parallelogram (6).

    It is important to note that tangrams are 3-D shapes as they have depth and can be picked up. I don't view this as a negative; it just needs to be explained to the children. Sometimes we use them to represent 2-D shapes but remind the children that they aren't and discuss why. As the children progress through the primary phase, we can start to consider them correctly as 3-D shapes; they are all very thin prisms.

    Tangrams allow children to develop spatial awareness, experience and enact geometric vocabulary and develop conceptual understanding that shapes can be composed, and similarly decomposed, into other shapes. They are also fun to use and inspire a natural curiosity in children.

    Before embarking on any of the activities below, similar to any other maths manipulative, it is essential that the children are provided with free time to play and explore the resource.

    Through this play, children will be comparing, rotating, translating and flipping the pieces. Geometry is a doing / active language, and tangrams allow children to enact the vocabulary. 

    Key idea: names of shapes and positional language

    This initial exploration provides the opportunity to explore different shape names. They can compare the different shapes and find what the same is and what is different; this will generate a natural need for precise geometric language about faces, edges, vertices, sides, etc.

    Using simple overlays at this stage helps maintain the discussion. Children have to recognise and match shapes from the printed overlay to their tangram shapes in order to complete their pictures. Use overlays that have the tangram shapes in different orientations so that the children can develop an understanding that the same shape can look different depending on rotation.

    Encourage children to slide (translate) their tangrams around on the overlays, rather than taking them off and replacing them. Both of these will strengthen their ability to visualise and start to rotate and translate shapes mentally, avoiding the misconception that shapes have a 'correct' orientation. This often manifests itself in children only identifying one of these shapes as a square:

     

    Shapes

     

    Children can then progress to building the patterns from the overlays on separate paper rather than overlaying directly.

    Grandfather Tang's storybook provides a beautiful opportunity to link geometry with stories. This splendid storybook allows children to illustrate the story through their tangrams. They can use simple shape names and positional language to describe how they built the characters (there is more than a single solution).

    Key idea: sorting shapes

    Throughout the whole of the primary phase, the tangram pieces can be used to allow children to sort based on their own or specified criteria.

    Year 2 may view them as 2-D shape representations and sort the tangrams themselves, discovering that they can be grouped based on the number of sides or simple lines of symmetry etc.

    Year 4 may also view them as 2-D shapes representations and sort themselves based on number of sides but using the language of quadrilaterals, triangles and polygons; or they may decide to sort based on internal angles.

    Key idea: regrouping shapes into other shapes

    Shapes can be found in other shapes, and larger shapes can be made from smaller ones (composite shapes) in a similar way to numbers.

    Year 2: How many shapes can you make?

    Using the tangram pieces, ask the children to see how many different shapes they can make by combining the tangram pieces.

     

    How many ways can you make a square?

     

    Shapes
    (1 and 2; 3 and 5; 3, 5 and 7)

     

     

    Key idea: exploring more advanced shape vocabulary – giving words a physical understanding

    Year 3: With the children, explore the difference between similar and congruent shapes.

    Can you find shapes that are exactly the same, including size?

     

    Shapes
    (1 and 2, 3 and 5)

     

    What do we call these type of shapes?

    (congruent)

    Can you put two or more tangram shapes together to make shapes that are congruent?

     

    Shapes
    (example shown is 3, 5 and 6 make a shape which is congruent to 1
    and there are lots more)

     

    Can you find shapes that are exactly the same in every other property but are different sizes?

     

    Shapes
    (1, 3 and 7)

     

    Which mathematical word is used to describe the relationship between these shapes? 

    (these shapes are similar)

    Can you put two or more tangram shapes together to make shapes that are similar?

     

    Shapes
    (example shown is 6, 3, 5 and 1 to make a shape which is congruent to 4
    and there are lots more)

     

    Key idea: understanding angles

    Year 3: How many of the shapes have internal right angles? How do you know?

    (1, 2, 3, 4, 5, 7)

    What about the other internal angles? Are they greater or lesser than a right angle?

    Prove it.

    Year 4:

    Which angles are acute and which are obtuse? How do you know?

    (Encourage children to compare unknown angles by stacking on top of a right angle.)

     

    Given the right angle on one of the larger triangles (1 or 2); what are the other angles in the triangles?

     

    Shapes
    (45º)

     

    For example, by laying pieces 3 and 5 in this arrangement on top of the known right angle of 1, we can see that they dissect the 90º exactly in half, therefore each of these smaller internal angles is 45º.

     

    Year 5:

    Shapes

    What is a straight line?

    How many degrees are in a straight line?

    How many different ways can you prove it?

     

     

     

     

     

     

    Shapes

    How many degrees in a whole turn?

    How many different ways can you prove it?

     

     

     

     

     

     

    Year 6:

    What is the size of each internal angle in the parallelogram (piece 6)?

     

    Shapes

     

    By laying pieces 3 and 5 in this arrangement, we can make a shape which is congruent to 6. We already know that the internal angles of 3 and 5 are 45º, 45º and 90º.

    Applying this, we can see that the parallelogram has two internal angles of 45º and two internal angles of 135º

    Can you make a composite shape using tangram pieces to create a shape which has an internal reflex angle? What are the sizes of the other internal angles?

     

    Shapes

     

    Through using one resource, it is easier to see the progression in experience, understanding and language as geometry progresses through the primary years.

    The possible uses of tangrams for geometry has only just been touched upon in this blog. Inspiration for further ideas can be taken from past SAT papers, for example:

    Key idea: transformations

    Tangrams can also be used to explore transformative actions such as reflection, scaling, rotation, flipping, translating.

    Think about these KS2 SATs questions and how tangrams could be used to support understanding.

     

    Shapes
    2019 key stage 2 mathematics paper 3: reasoning

     

    Shapes
    2018 key stage 2 mathematics paper 2: reasoning

     

    Shapes
    2018 key stage 2 mathematics paper 2: reasoning

     

    Shapes
    2017 key stage 2 mathematics paper 3: reasoning

     

    References

    Cheng, Y.-L., & Mix, K. S. (2014). Spatial training improves children's mathematics ability. Journal of Cognition and Development, 15(1), 2–11

    Department for Education (2014) The national curriculum in England: mathematics programmes of study: key stages 1 and 2. Available at: 

    Gov.UK: National curriculum in England: mathematics programmes of study - key stages 1 and 2 (Accessed: 13 October 2021)

    Lowrie, T., Logan, T., & Ramful, A. (2017). Visuospatial training improves elementary students’ mathematics performance. British Journal of Educational Psychology, 87(2), 170–18

    Van de Walle, John A. (2013). Elementary and middle school mathematics: teaching developmentally. Boston: Pearson

    2019 Key Stage 2 mathematics paper 3: reasoning

    2018 Key Stage 2 mathematics paper 2: reasoning

    2017 Key Stage 2 mathematics paper 3: reasoning

    Contains material developed by the Standards and Testing Agency for 2017, 2018 & 2019 national curriculum assessments and licenced under Open Government Licence v3.0

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