2d shape problem solving ks2

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Welcome to the Math Salamanders 2d Shapes Worksheets. We have a wide selection of worksheets on 2d shapes, including symmetry worksheets, naming 2d shapes, shape riddles and puzzles, and sheets about the properties of 2d shapes. There are a range of worksheets at different levels, suitable for children from Kindergarten and up.

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2d Shape Worksheets

There are a range of geometry worksheets set out in different sections as follows:

Kindergarten Section

• identify 2d shapes from pictures
• tracing and shading 2d shapes

1st Grade Section

• sorting 2d and 3d shapes
• identify different types of 2d shapes.
• drawing shapes using positional language - inside, above, below,...

2nd Grade Section

• identify and name members of the triangle family, e.g. equilateral triangles.
• identifying and sorting other 2d shapes
• identify simple properties of 2d shapes - sides, corners, right angles
• solving riddles involving 2d shapes

3rd Grade Section

• identifying a range of 2d shapes;
• knowing and using properties of 2d shapes - sides, symmetry, right angles
• investigate properties and features of 2d shapes
• symmetry worksheets
• 2d shapes worksheets - area and perimeter

4th Grade Section

• classify and measure a range of angles
• using coordinates in the 1st quadrant

5th Grade Section

• knowing properties of angles of triangles and quadrilaterals
• finding missing angles
• area of triangles and quadrilaterals

Geometry Riddles

• Reasoning and exploring the properties of 2d shapes.

2D Shape Clipart

• A range of printable clipart for 2d shapes.

Parts of a Circle Worksheets & Support

• Worksheets and support for finding and naming the parts of a circle.

Identify 2D Shapes Online Quiz

• an online quiz to help develop naming and identifying a range of 2D shapes.

Kindergarten 2d Shapes Worksheets

• Shade the Circles Sheet 1
• Shade the Rectangles Sheet 1
• Shade the Squares Sheet 1
• Count the Shapes 1
• Match the Shapes Sheet 1
• Trace the Shape Sheet 1
• Trace the Shape Sheet 1a
• Trace the Shape Sheet 2
• Trace the Shape Sheet 2a

More Kindergarten Geometry Resources

• More Kindergarten Shape Worksheets
• More Kindergarten Tracing Worksheets
• Top of Page

1st Grade Shape Worksheets

• Identify Squares and Rectangles 1
• Identify Squares and Rectangles 1 Answers
• Identify Triangles and Circles
• Identify Triangles and Circles Answers
• Identify Simple 2d Shapes 1
• Identify Simple 2d Shapes 1 Answers
• Draw 2d Shapes 1

More 1st Grade Geometry Resources

• More 1st Grade Geometry Worksheets

2nd Grade Shape Worksheets

• Identify Triangles 1
• Identify Triangles 1 Answers
• Identify Triangles 2
• Identify Triangles 2 Answers
• Identify 2d Shapes 1
• Sheet 1 Answers
• Identify 2d Shapes 2
• Sheet 2 Answers
• Block Symmetry Sheet 1
• Name and Trace 1
• Name and Trace 2
• Name and Trace 3
• Name and Trace 3 (UK version)
• Name and Trace 4
• Name and Trace 4 (UK version)

More 2nd Grade Geometry Resources

• More Second Grade Geometry Worksheets
• More Block Symmetry Worksheets

3rd Grade Shape Worksheets

• 2d Shapes Properties 1
• 2d Shapes Properties 2
• 2d Shapes - Can you draw it 1
• Can you draw it 1 Answers
• The Alphabet in Symmetry
• The Alphabet in Symmetry Answers
• Block Symmetry Sheet 5
• Sheet 5 Answers
• Line Symmetry 1
• Area Sheet 1

More 3rd Grade Geometry Resources

• More 3rd Grade Geometry Worksheets
• More Line Symmetry Worksheets

4th Grade Shapes Worksheets

• Angle Classification 1
• Angle Classification 2
• Triangle Classification 1
• Angle Measuring 1
• Angle Measuring 2
• Plot the Coordinates Sheet 1
• Area Sheet 4
• Sheet 4 Answers

More 4th Grade Geometry Resources

• More 4th Grade Geometry Worksheets
• More Coordinate Worksheets
• More Area Worksheets

5th Grade Shapes Worksheets

• Angles to 90°
• Angles to 90° Answers
• Angles on a straight line
• Angles on a straight line Answers
• Angles around a point 1
• Angles around a point 1 Answers
• Angles in a triangle 1
• Angles in a triangle 1 Answers
• Find the missing angle 1
• Find the missing angle 1 Answers

More 5th Grade Geometry Resources

• More 5th Grade Geometry Worksheets
• Explore 2d Shapes Worksheets

Looking for some geometry worksheets to get children thinking and reasoning about 2d shapes?

The shapes on this page are all about children really understanding what 2d shapes are all about, and using their reasoning skills to justify their thinking.

Using these sheets will help your child to:

• know the properties of a range of 2d shapes;
• recognise that some shapes can also be described as being other shapes; e.g. a square is also a rhombus;
• recognise and understand right angles, parallel lines, lines of symmetry;
• develop their geometric reasoning skills.

Here you will find our free printable geometry riddles from 1st to 5th grade.

These riddles are all about problem solving with 2d shapes.

Using these riddles will help your child to:

• develop their geometry skills;
• develop their understanding of geometric language;
• apply their geometric knowledge to solve problems.

All the geometry riddles in this section support elementary math benchmarks.

• Geometry Worksheets (Riddles)

Here you will find a selection of 2D shape sheets designed to help your child learn their names and properties of 2d shapes.

• recognise a range of 2D shapes;
• recognise that a square is also a member of the rectangle family;
• recognise shapes in different orientations and sizes.

All the free printable shape sheets in this section support Elementary Math benchmarks.

• Shapes Clipart - Basic 2D Shapes
• Shapes Clip art - Triangle and Quadrilateral Families
• Shapes for Kids - Regular Polygons (individual)
• Regular Shapes (groups)
• Printable Shapes - 2D and 3D Shape Sheets

Parts of a Circle Worksheets and Support

Do you know the special geometric names to describe the different parts of a circle?

Well you can find all the support you need on this page, complete with some practice worksheets and an online quiz!

• Parts of a Circle Worksheets

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This quick quiz tests your knowledge and skill at indentifying 2D shapes including types of triangle..

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Teaching Geometry – Properties Of Shapes KS2: A Guide For Primary School Teachers From Year 3 To Year 6

Neil Almond

In Key Stage 2, properties of 2-d shapes and properties of 3-d shapes are an important section of geometry. In this blog, Neil Almond provides his theories for teaching shape, help with lesson plans and plenty of teaching resources. 

For some pupils, shape can be the topic they struggle with the most in KS2 maths , as it is so different to topics like place value which fall under Number. Even pupils who are fluent in numeracy may have difficulties up until upper KS2. 

By looking back at the theory behind the subject and finding alternate ways to teach your lessons, both pupils who excel at shape and those who find it difficult will be able to find a deeper understanding.

• Theory behind teaching geometry: properties of shapes

Properties of shapes Year 3 lesson ideas

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• Properties of shapes Year 6 word problems

Theory behind teaching geometry: properties of shape s

While there is no specific ‘theory’ as such to the teaching behind the property of shapes, there are some dos and don’ts that I feel are important from the outset, particularly in regard to resourcing for this subject. In too many classrooms, there will be a drawer labelled 2-d shapes that we believe to contain 2-d shapes like those below.

The issue here is that they are not 2-d. In the physical plane, where items can be manipulated, there exists exactly 3 dimensions. A piece of paper for example, despite its appearance, has a minuscule measurement of depth, and is therefore 3-d. 

Every shape in that drawer is in fact a 3-d shape. Their purpose however is to represent 2-d shapes in an interactive way. Therefore, the teacher is left with two choices. Keep the label ‘2-d shapes’ but take out all the shapes and replace them with drawn shapes on a piece of paper or change the label to ‘3-d shapes used to represent 2-d shapes’. 

The fact of the matter is that holding up one of these shapes and asking pupils to identify the 2-d shape you are holding is categorically wrong. Asking them what 2-d shape is represented by this 3-d shape is, however, perfectly permissible. It is for this reason that I recommend that 3-d shapes are taught before 2-d shapes.

When it comes to the lesson delivery of property of shape, leaning into variation theory can be a useful tool for educators to deploy. The application of variation theory is far more complicated and nuanced than even the best crafted worksheet could ever get across to pupils. 

Rather, one of its possible uses involves the intellectual pursuit of skilful examples, where great care has been considered not only in the examples selected but the order in which they will be revealed.. 

It is very easy to get wrong but the rewards of getting variation theory correct are bountiful and shape, I believe, provides teachers who are interested in its application an easier route into how it works.

Those subscribed to Third Space Learning’s premium content can access a staff CPD powerpoint on variation theory and its application.  

2D Shapes Worksheet Year 3

Download this FREE recognising 2D shapes Geometry worksheet for Year 3 pupils, from our Independent Recap collection.

Geometry – properties of shapes KS2: Year 3

In the national curriculum for maths in England, for each area of maths outlined, there is both a statutory requirement and a non-statutory requirement. The statutory requirement is as follows:

• Draw 2-d shapes and make 3-d shapes using modelling materials; recognise 3-d shapes in different orientations and describe them
• Recognise angles as a property of shape or a description of a turn
• Identify right angles, recognise that 2 right angles make a half-turn, 3 make three-quarters of a turn and 4 a complete turn; identify whether angles are greater than or less than a right angle
• Identify horizontal and vertical lines and pairs of perpendicular and parallel lines

The non-statutory notes and guidance suggests: 

Pupils’ knowledge of the properties of shapes is extended at this stage to symmetrical and non-symmetrical polygons and polyhedra. Pupils extend their use of the properties of shapes. They should be able to describe the properties of 2-d and 3-d shapes using accurate language, including lengths of lines and acute and obtuse for angles greater or lesser than a right angle. Pupils connect decimals and rounding to drawing and measuring straight lines in centimetres, in a variety of contexts.

It is here we can begin our first foray into what variation theory may look like in the classroom. To do this, we will examine the objective ‘ identify horizontal and vertical lines and pairs of perpendicular and parallel lines.’  

As has been a theme throughout this series, the objectives from the national curriculum do not equate to a lesson. It is quite right that objectives taken from the national curriculum are dissected into smaller, more manageable parts. Therefore, I will only be discussing the idea of parallel lines. 

When thinking about the examples we will carefully select, it is best to consider the boundaries of the concept itself. By that, I mean when parallel lines are no longer parallel lines. In order to understand what parallel lines are, they must also know what they are not. 

Secondly, we must also consider whether examples that do demonstrate the concept are shown and whether these are typical or not. For example, many pupils will have no issues in identifying the shape below as a square.

But will struggle to do the same for the following:

Often, this will be called a diamond despite it having all the properties of a square (the rotation of a shape does not alter the name of the shape). The reason for this struggle is that when ‘squares’ are shown to pupils they are often just shown the first example and not the second. 

It is from these cues that we can build up standard and non-standard examples as well as non-examples to build up the concept of parallel lines. How best to introduce these examples is up to you and below are just some examples.

Standard example:

Non-example:

Non-standard example:

Already we can see here that the use of example and non-example demonstrates that parallel lines require more than one line.

When introducing each example, I would do so one at a time telling pupils whether each example does show parallel lines or not. 

Once you have demonstrated a set that draws out differences, probe the pupils’ understanding of the examples with the questions ‘What is the same and what is different?’ 

The quality of conversation and the conjecturing that the pupils will make will be worth the time and effort put into constructing your sequence of examples and non-examples.

There would not be any standard ‘word problem’  that would look at the mathematics behind parallel lines that was so convoluted that it would hold any mathematical relevance.

The most effective use of reasoning and problem solving when working with a concept and drawing on ideas of variation are sentence stems. For example, the following sentence stem could be used when looking at non-standard examples:

Despite the fact that______________, these are parallel lines because___________.

Despite the fact that these lines are not exactly vertical or horizontal, these are parallel lines because they are the same distance apart from each other and will never meet if extended.

Geometry – properties of shapes KS2: Year 4

• Compare and classify geometric shapes, including quadrilaterals and triangles, based on their properties and sizes
• Identify acute and obtuse angles and compare and order angles up to 2 right angles by size
• Identify lines of symmetry in 2-d shapes presented in different orientations
• Complete a simple symmetric figure with respect to a specific line of symmetry

Pupils continue to classify shapes using geometrical properties, extending to classifying different triangles (for example, isosceles triangle, equilateral triangle, scalene triangle) and quadrilaterals (for example, parallelogram, rhombus, trapezium). Pupils compare and order angles in preparation for using a protractor and compare lengths and angles to decide if a polygon is regular or irregular. Pupils draw symmetric patterns using a variety of media to become familiar with different orientations of lines of symmetry; and recognise line symmetry in a variety of diagrams, including where the line of symmetry does not dissect the original shape.

The objective this will look at will be ‘Identify lines of symmetry in 2-d shapes presented in different orientations.’ In this objective, ‘orientation’ is the key word and here we can draw on variation theory again. The objective makes no reference if the shapes should be regular or irregular, so it is best to spread this over two days and tackle both. Ideally, mirrors will be provided for each pupil. 

When using mirrors it is best to provide pupils with two types of questions. One where they are required to find a line of symmetry or denote whether an already given line represents a line of symmetry or not. 

Once pupils are comfortable with this, they should then use their skill with a mirror to draw the other half of a shape.  At the beginning stages, this should be done on isometric paper so that the drawings themselves can be as accurate as possible. 

I would start with simple shapes and gradually, over a sequence of a couple of lessons, make the shapes harder by including irregular shapes. I would not, at this stage, expect pupils to complete a ‘picture’ only shape. 

There would not be any standard ‘word problem’ that would look at the mathematics behind symmetry that was so convoluted that it would hold any mathematical relevance.

A great investigation for symmetry would be to dislodge the misconception that a circle only has one side. 

To do this you can look at lines of symmetry of regular shapes. When pupils are aware that regular shapes have all equal sides and equal angles, they can then investigate what this means for lines of symmetry. 

As long as you provide regular polygons, they will quickly see a pattern where the number of lines of symmetry are associated with the number of sides. A square has 4 lines of symmetry, a regular pentagon has 5 lines of symmetry etc. 

Once pupils notice the pattern you can get them to conjecture about all manner of regular shapes. 

At this point, I would ask about a circle. Pupils are often taught that a circle has one continuous side and so therefore, if our pattern holds true, should only have one line of symmetry. 

A quick attempt of getting 30 pupils to find just one line of symmetry will, in more cases than not, provide at least two cases where pupils have drawn the one line in different places of the circle. 

Here we can draw the distinction between a circle and a polygon (must have straight sides) and that there are actually an infinite number of lines of symmetry and this leads some mathematicians to conjecture that a circle also has an infinite number of sides. 

Geometry – properties of shapes KS2: Year 5  

• Identify 3-D shapes, including cubes and other cuboids, from 2-D representations
• Know angles are measured in degrees: estimate and compare acute, obtuse and reflex angles
• Draw given angles, and measure them in degrees (°)
• Angles at a point and 1 whole turn (total 360°)
• Angles at a point on a straight line and half a turn (total 180°)
• Other multiples of 90°
• Use the properties of rectangles to deduce related facts and find missing lengths and angles
• Distinguish between regular and irregular polygons based on reasoning about equal sides and angles

Pupils become accurate in drawing lines with a ruler to the nearest millimetre, and measuring with a protractor. They use conventional markings for parallel lines and right angles. Pupils use the term diagonal and make conjectures about the angles formed between sides, and between diagonals and parallel sides, and other properties of quadrilaterals, for example using dynamic geometry ICT tools. Pupils use angle sum facts and other properties to make deductions about missing angles and relate these to missing number problems.

In this lesson, it will take its objective from being able to identify angles in a straight line and, as the non-statutory guidance says, ‘ Pupils use angle sum facts and other properties to make deductions about missing angles and relate these to missing number problems’.

Prior to teaching this objective, it is important that the following have been learn by the pupils:

Pupils should draw on their knowledge of right angles and know that 2 right angles next to each other will constitute a half turn and that a half turn is equal to 180 degrees. 

Prior to this lesson, I would spend considerable time getting pupils to complete missing number type questions with two and three integers where the answer is equal to 180. 

This would be done under the guise of regular fluency practice and I would not explain its true purpose at the time. 

To begin with, I would ensure that all missing angles can be solved by measuring it with a protractor and expect the questions to be solved this way. Furthermore, the orientation of the questions would greatly vary so pupils get practice in using a protractor in a manner of different situations.  

Once pupils have begun to grow comfortable with measuring, only then would I introduce questions not drawn to scale and where pupils will have to rely on their application of number to solve the missing angles. Once again, this would take place where there are 2 or 3 given angles that pupils would have to solve. It would be at this point that all the prior training of finding missing values that equal 180 would begin to make sense and ensure that this objective is taught as efficiently as possible.

Angles in a straight line allow for the retrieval of other number facts that have been taught previously. A common reasoning activity in my classroom involved pupils recollecting squared and cubed numbers to solve missing angles in a straight line. This worked in multiple ways. 

First, questions such as the one below where angle b was a square number between 20-50.

Pupils would then have to use their knowledge of square numbers and angles in a straight line to find all the possibilities. 

B = 25 degrees A = 155 degrees

B = 36 degrees A = 144 degrees

B = 49 degrees A = 131 degrees

The other way it would be used would be with cubed numbers where A was an obtuse angle and also a cubed number. What are angles A and B?

Again, here pupils would have to recall their knowledge of cubed numbers and obtuse angles and know that there is only one possible outcome here. A must be 125 degrees as it is the result of 5 cubed and the only cubed number that is an obtuse angle. Therefore, angle b must be 155 degrees.

Geometry – properties of shapes KS2: Year 6

• Draw 2-D shapes using given dimensions and angles
• Recognise, describe and build simple 3-D shapes, including making nets
• Compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
• Illustrate and name parts of circles, including radius, diameter and circumference and know that the diameter is twice the radius
• Recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles
• Pupils draw shapes and nets accurately, using measuring tools and conventional markings and labels for lines and angles.
• Pupils describe the properties of shapes and explain how unknown angles and lengths can be derived from known measurements.
• These relationships might be expressed algebraically for example, d = 2 × r; a = 180 − (b + c).

Somewhat surprisingly, it is not until Y6 that we see the use of the net explicitly mentioned. A primary favourite is the deconstruction of 3d shapes into their component pieces to see how they are created. 

It is quite common for this to be done earlier in school and so when Y6 teachers wish to look at this objective, it is important to extend pupils’ thinking.

The joy of the above lesson would be to allow pupils to physically cut the shapes and attempt to match them up – though it would be possible to look at pupils’ special thinking by only using the 2D representations. 

When modelling the activity, I would be sure to look at K and E as this is the most obvious pairing that pupils may go for. By using this as the model, we are getting pupils to extend their thinking as we are eliminating the easier option from the list. 

The answers and the name of the shapes can be found below.

Properties of s hapes Year 6 word problems

A simple true or false question can provide valuable insight into pupils’ understanding of shape. For example, 

True or false?

In order to be regarded as a quadrilateral, a shape must have at least one right angle. 

While pupils may know the answer is false, you can push this thinking on by getting pupils to draw some quadrilaterals that do not contain a right angle. These would include a parallelogram and a trapezium.

Looking for some more ideas of how to do this? You can find plenty of free resources and shapes worksheets on the Third Space Learning maths hub . You may also find our blog, What Are Vertices, Faces and Edges? , useful to pair alongside this topic!

For guidance on other KS2 subjects, check out the rest of the series:

• Teaching Decimals KS2
• Teaching Place Value KS2
• Teaching Fractions KS2
• Teaching Percentages KS2
• Teaching Statistics KS2
• Teaching Ratio and Proportion KS2
• Teaching Multiplication KS2
• Teaching Division KS2
• Teaching Addition and Subtraction KS2
• Teaching Geometry – Position, Direction and Coordinates KS2

Looking to get ahead on other KS2 maths topics? We have the lowdown from expert primary teachers on all the trickiest KS2 maths concepts to teach, including teaching times tables , telling the time , as well as the long division method and the long multiplication method .

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Ideal for pupils who struggle to tie together the multiple concepts required to effectively tell the time.

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Exploring 2D Shapes

Children often  enjoy playing with long strings or ribbons and making and describing shapes. Adults could  build on this by providing long loops of string or elastic to stimulate conversations about 2D shapes.

Encouraging mathematical thinking and reasoning:

The Mathematical Journey

Same and different:

• developing the use of language to compare, contrast and describe. For example: straight, wiggly, curved, rounded, pointy, bigger, smaller, longer, shorter, corner, like, different, side, curve 

Recognising and labelling common shapes:

• making connections with shapes in their own environment and beginning to use names such as triangle, rectangle, square, corner, side...
• counting the sides and corners of the shapes they make

Describing position:

• using words such as on, inside, outside, under, left, right, above and below to describe position

Comparing sizes:

• beginning to use language such as bigger, smaller, longer and shorter to introduce ideas about measuring

• Shape searches in a series of photos or in books and drawings
• I-spy shapes games
• Using educational games that involve shape sorting or matching
• Building pictures from precut shapes or cutting shapes from coloured paper or magazines
• Painting on the ground or wall with big wallpaper brushes and water
• Using natural materials such as leaves, twigs and daisy chains to make shapes
• Printing with potato or sponge shapes
• Drawing shapes in the sand tray, with paints, through cornflour gloop or with a finger
• Riding wheeled toys through puddles and looking at the tracks

Download a  PDF  of this resource.

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2d shape problem solving activity

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

18 November 2011

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Good stuff! A useful resource. Many thanks.

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catherinemitri

A great resource to reinforce the concepts linked to diagonals on 2-D shapes. Particularly liked the algebra extension activity. Will be using this resource with my Y5 class. Thank you.

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2-dimensional shapes inquiry  

Mathematical inquiry processes: Explore and test cases; find rules; generalise. Conceptual field of inquiry: Measurement of angles; interior angles of polygons; i nscribed shapes inside circles.

The prompt is adapted from one designed by Theo Giann (a PGCE student at the University of Brighton, UK, during 2011-12). In his original version, the fourth circle contained a question mark, rather than a hexagon. However, it is possible for students to interpret the question mark as a teacher's closed question and, hence, limit the inquiry to finding the next polygon in the sequence. In its current form, the prompt might lead to more open lines of inquiry, such as the ones below that have been generated by students' questions and comments:

Names of two-dimensional shapes;

Perimeters of polygons;

Construction;

Methods to inscribe shapes in a circle;

The circumference of a circle;

Regular polygons (including interior and exterior angles); and

Areas of polygons.

Theo tried his prompt with a year 8 class as part of a Masters-level research project on encouraging autonomy. As this was Theo's first inquiry with an unfamiliar class, he decided not to use the regulatory cards and instead gave the students a limited choice of working in their books, making a poster or completing a worksheet. 

New line of inquiry

A new line of inquiry is suggested by the following questions from a year 7 student in Ollie Rutherford 's class at Haverstock School (Camden, UK):

What is the ratio between the number of sides of a polygon and the area of the segments left over? 

Is there a sequence to the ratios?

The answers connect multiple topics in the curriculum: ratio, sequences and expressions for the n th term, area of polygons and a circle, Pythagoras' Theorem, trigonometry (including the cosine rule) and properties of individual shapes (such as the centroid of a triangle) amongst others.

Making thinking visible through inquiry

Amelia O'Brien , a PYP teacher at the Luanda International School (Angola), posted the picture (above) of her year 5 pupils' responses to the prompt on twitter . She reports that the pupils made interesting observations connected to pattern. There are a number of lines of inquiry suggested by the observations. In one line, for example, pupils tested the conjecture* that "the more sides that the inside shape has, the less space there is left over inside the circle" by working with the areas. 

The pupils used their visible thinking cycle ( claim-support-question ) to develop the inquiry, which included an impromptu word inquiry into the suffix quad- . Overall, Amelia says, students were engaged and there was "so much excitement" in the classroom.

* The conjecture could lead into this inquiry on the 'fit' of one shape inside another.

Structured and open inquiry

These are the questions and observations of Sally Pearson 's year 7 mixed attainment class. During the inquiry, students were involved in structured or open inquiries depending on their levels of initiative and independence. Sally reports on the development of the inquiry:

It was the class's second inquiry so far. I was impressed with the variety of the responses and the way that students were eager not only to provide a new question or comment but also how they were listening so carefully to each other and building on or amending what others were saying.

Lots of answers to questions were provided by other students and they were all so desperate to speak we ended up spending about half of the lesson on debates about whether a square is a rectangle, how many lines of symmetry a rectangle has compared to a square, whether there are 360 degrees in a circle and whether a right angle is acute, obtuse or neither.

It was a great start to a unit on angles with a mixed attainment class as some naturally gravitated to exploring angles in triangles, some changed the prompt to see if the sum of the interior angles of any quadrilateral is 360 degrees  and some were intrigued by the interior angles in the pentagon. For those that needed it, I closed the inquiry down fairly quickly as they wanted support investigating angles in triangles or polygons with direction from me, but I was interested in the number (about eight out of 30) that were happy to structure their own further inquiry based on changing the prompt with quadrilaterals or extending the pattern they saw to hexagons and beyond.

Sally Pearson is a Lead Practitioner in the mathematics department at Brittons Academy (east London, UK).

Claim-support-question inquiry cycle

Amelia O'Brien 's grade 6 PYP class at the Luanda International School (Angola) asked questions and made observations about the prompt to initiate an inquiry. The students used a cycle of claim-support-question to frame their thinking. Amelia reports that "the students were very engaged as they chose their own 'claim'." The cycle led to an advanced mathematical inquiry that included claims such as "irregular polygons cannot fit into circles" and angles in a heptagon sum to 820 o (see pictures).

Claim-evidence-reasoning inquiry cycle

Courtney Paull (a grade 7 math guide at the International School Manila) posted this picture on twitter . Students created their own claim-evidence-reasoning cycles from the prompt. The picture shows one claim: "Shapes with sides that are parallel to each other have an even number of sides." Courtney said that she was not sure where the inquiry was going, but the children made "some really awesome claims."

After the claim that shapes with sides that are parallel to each other have an even number of sides was posted on twitter, one mathematics educator shared the picture below as a counter-example to the claim. To maintain the validity of the claim, the wording could be changed to specify 'regular shapes'.

Co-regulating inquiry

Co-regulation refers to a process in which the teacher and students direct an inquiry together. In the account that follows, the teacher offered a year 7 mixed attainment class the opportunity to decide what to do within a restricted set of options. At the start of the inquiry, the class posed questions and made observations about the prompt (see below). The students focused on the names of the shapes and the interior angles of the polygons, and on the fact that the polygons are inscribed inside circles. As the students read out their questions and observations, the teacher took the opportunity to introduce formal vocabulary, such as ‘segment’, ‘inscribed’ and ‘interior' angle.

The teacher then used a protractor on the interactive whiteboard to measure the angles in the triangle and quadrilateral, confirming that they were all, respectively, 60 o and 90 o . 

The teacher introduced the six regulatory cards he had chosen for the class. In previous inquiries, students had started to choose particular cards automatically instead of thinking afresh about how to regulate their activity. Those cards described how to inquire (for example, Work with another student ), so now all the cards were about what to do in the inquiry.

However, before asking students to select a card, the class discussed what each card might mean in the context of the current inquiry. Under the guidance of the teacher, the students proposed the following:

All the students went on to select “Use a worksheet” or “Look for a pattern”. The teacher showed those who chose the latter how to split a polygon into triangles so they were able to calculate the sum of interior angles in multiples of 180 o before working out the value of one angle in a regular polygon.

Thus, co-regulation in this case involved the teacher in designing the regulatory cards and deciding when instruction was appropriate, while students developed the meaning of the cards in the specific context and selected a card that enabled them to learn.   

Inquiry in the primary classroom

Adapting the prompt.

Amanda Klahn adapted the prompt for an inquiry with her grade 4  IB Primary Years Programme class. The reduction in the number of inscribed shapes to three makes the inquiry potentially more open. Now the pupils can suggest moving to the next case with more confidence because the hexagon might be more familiar than the heptagon (the next case in the original prompt). As Amanda says, the pupils' questions and observations provide enough material to keep the class inquiring for weeks - from area and perimeter to constructions using a pair of compasses.

At the time of the inquiry, Amanda was an IB PYP teacher at the Western Academy of Beijing (China).

Kindergarten inquiry

Ju Bonilla Garcia , a teacher at the United World College Thailand in Phuket, used the 2-dimensional shapes prompt with her kindergarten class. She tweeted during the inquiry that, "Tons of knowledge [is] being revealed through discussions!" ( 1 ) and students are asking "interesting questions which will move our inquiry forward" ( 2 ). Ju reports on the development of the inquiry:

A s part of our transdisciplinary inquiry into shape and space, kindergarten students showed an interest and curiosity in a series of structures that were built by Grade 4 in the playground at UWC Thailand. We looked for shapes, lines and angles within the structures, which led us to the Inquiry Maths prompt. We started by viewing the prompt as a whole class and student observations were sorted through the concepts of form, function and connection. This initial exploration of the prompt did not result in any student questions.

However, students demonstrated great enthusiasm when asked, 'What would happen if we continued to play with the lines?' Each student was then provided with a shape from the inquiry prompt to play with individually. This encouraged experimentation with lines within the shapes which, in turn, led to student questions. 

Interestingly, students drew additional lines within the shapes which formed various sized triangles. We then looked at student examples and many learners made conjectures about how many triangles were present within each shape (as drawn by their peers) and the characteristics of triangles.

Through this inquiry, many gained the understanding that shapes can be partitioned to form other shapes, just like numbers. I have used IM prompts with older grade levels (upper primary) and was curious to see how this approach to learning maths would work with kindergarten. The key differences included the extent to which I needed to scribe for students (obviously!), the time frame and how questions were elicited. It was certainly just as valuable, engaging and rich as when facilitating this approach with older learners.

Discussions about the prompt

Inquiry and misconceptions  .

In his Masters-level essay, Theo Giann considered the impact of inquiry on one student and raised an important question about misconceptions during inquiry:

One boy, who usually was unmotivated in my lessons, produced a poster with many examples of different inscribed polygons and clear mathematical labelling of each one. He went on to eloquently explain his work to the rest of the class; however, he made an error in his work – he deduced that because he had drawn an irregular inscribed pentagon, all irregular pentagons could be inscribed. I asked him to investigate this statement but he did not see his error and I ran out of time to explain where he had gone wrong.

On the one hand, the student clearly developed his own line of inquiry, explored relevant aspects of the problem and quite frankly worked harder than I had ever seen or ever saw afterwards. The freedom of being able to decide his own workload, his own methods and his own definition of success inspired him to have a really fantastic lesson. On the other hand, his work contained errors which, due to their complex and unique (to him in that lesson) nature, could not be fully addressed – I cannot guarantee that this has not led to misconceptions being formed or even supported. 

As a mathematics teacher, I was now faced with a seemingly impossible question – to what extent can mathematical accuracy be discarded for the sake of enthusiasm? No doubt, there is ready at hand an army of purists who would yell from the hilltops “NONE! EVER!” – I myself verge very closely towards that camp – but can we really cast out of hand a method that promotes a child’s enjoyment of mathematics simply because it can lead to unresolved misconceptions? Surely no system can be completely free of that unfortunate phenomenon.

Planning and evaluating inquiry

Maria Esteban (a trainee teacher at London Metropolitan University) used the prompt on her second school placement. She discusses how she planned and evaluated the inquiry in her project report .

Functioning with Geometry and Fractions

In this article from the ATM 's Mathematics Teaching 207 (March 2008), Derek Ball and Barbara Ball describe the types of students' thinking that arose when considering the prompt above.

Alternative prompts

This statement seems plausible to pupils. Even when it is clear that the statement is false, it has encouraged classes to seek a pattern in the number of different types of polygons. At its most open, the inquiry has led to students asking why pentagons are not classified in the same detail as triangles and quadrilaterals. One possible line of inquiry might be to attempt to develop a classification for pentagons based on the length of sides and type of interior angle. 

Prompt sheet