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Maths in Town Squared: shapes and area in Sprite flights

by Playlunch Games

About these maths activities

In this activity, children help cleaned-up sprites find their way home by cutting rifts in the atmosphere. They do this by tracing 2D shapes in the sky using a pre-marked grid.

When they enter this puzzle space, children see a grid and a sign indicating the number and type of shapes they need to create on this grid. The grid is marked with dots, and a mentor explains how to link the dots together to form the required shape, and the other puzzle ‘rules’ (eg: no overlapping lines or shared dots).

A Spright flights game in progress. Eight dots need to be joined together to make 2 rectangles. The [?] button provides guidance on the attributes of a rectangle.

The child’s task is to understand what kind of shapes are required in each grid, then draw from one dot to another until they have made that shape. A successful shape is signalled by a happy sprite flying through the shape and into the space beyond. Children can make different shapes and remove them, experimenting until they work out the solution.

There are two activities in Sprite flights, each with a different level of challenge.

Shape gates

The Shape gates activity asks children to locate and trace the identified shapes within the grid. This could be rectangles, triangles, parallelograms, squares, or a combination of each. Early levels introduce a single shape in different quantities, sizes and rotations. Later levels mix up the shapes in a way that requires reading, planning and testing.

Each puzzle level includes hint pop ups that describe the required shapes in more detail.

The Shape gates grid showing 8 dots ready to be joined up. The sign on the left shows the task: make 2 shapes of two different types.

The same Shape gates grid with 1 of the shapes completed. The child can try joining dots to make lots of different shapes, but will not progress until they have worked out how to make the specific shapes needed for this grid.

Area gates

The Area gates activity invites children to locate and trace triangles with an area of 6 grid squares. Levels start with one or two triangles to find, and progressively require more planning and thought to find triangles of the correct area.

If a child traces a triangle that doesn’t have an area of 6 squares, they receive feedback and hints on how to work out the shape they need. If they have 3 unsuccessful attempts within a game, children are invited to play a simpler puzzle.

Each puzzle level includes hint pop ups that describe how to work out the area of a triangle.

The Area gates grid showing 6 dots ready to be joined up. The sign on the left shows the task: make 2 triangles with an area of 6 squares.

The same Area gates grid with 1 of the shapes completed. The child progresses when they have traced the required shapes for this grid.

Information for teachers and educators

How does this puzzle relate to the Cambridge Big Ideas in Mathematics?

Cambridge University developed the Cambridge Big Ideas in Mathematics, based on concepts proposed by Randall I. Charles1. The Big Ideas described below help represent and teach maths as a connected and coherent area of learning, and they underpin the design of puzzles in Town Squared.

Big Idea 14: Shapes and Solids

Two- and three-dimensional objects with or without curved surfaces can be described, classified, and analysed by their attributes.

In the Shape gates, children can:

Identify and create the correct number of rectangles/squares, parallelograms or triangles, starting with single shapes of each type then creating multiples of the same shape
Make increasingly difficult combinations of points form the required shapes
Explore all the different ways each shape can look or change (position, size, aspect ratio, rotation, reflection) and still be classified as a rectangle, parallelogram or triangle
Access an optional help area at any time, to get a detailed explanation of the required shapes
At the highest levels, make combinations of multiple shapes with less detailed support.

Big Idea 17: Measurement

Some attributes of objects are measurable and can be quantified using unit amounts.

In the Area gates, children can:

Join up points on a grid to form different triangles that have an area of 6 grid squares.
Over time and with in-game support, develop a formula for working out the area of any triangle.
Apply the same formula to increasingly difficult combinations of points, to achieve the target number of triangles so that all triangles have an area of 6 squares.

How do the puzzles scale in complexity?

Both activities in Sprite flights explore the attributes of two-dimensional (2D) shapes.

Shape Gates activity

The Shape gates puzzles start by asking children to make rectangles and squares that are aligned to the grid. Over time, the grid you play on becomes larger to fit more shapes.

Over multiple levels, the number of shapes and mix of sizes increases and the shapes are rotated. Instructions and feedback reinforce the idea that these shapes are all being defined by their attributes, not by the way they are rotated on the grid.

Non-aligned, non-‘traditional’ representations of a shape (eg: a rectangle rotated on an angle, or a triangle with a corner pointing down) help to develop children’s understanding that the attributes of a shape are consistent, even when you rotate or reflect that shape.

There are a total of 15 levels of difficulty in this activity, which introduce different shapes in groups as shown below.

Difficulty level	This group of puzzles addresses these concepts
Level 1	Introduces the core concept of forming a shape on the grid. Uses the smallest grid (a 5x5 grid).
Level 2	Make 2 rectangles/squares that are aligned to the grid. Begin to explore how to make shapes of the required type using the available options. Recognise that a square is a type of rectangle.
Level 3	Make 3 rectangles/squares that are aligned to the grid. It is gradually harder to make the specified number and type of shapes on the grid, requiring more attention to the attributes.
Level 4	Make 2 rotated rectangles/squares on the grid. Explore how a shape has the same attributes even when rotated.
Level 5	Make 3 rotated rectangles/squares on the grid. t is gradually harder to make the specified number and type of shapes on the grid, requiring more attention to the attributes.

The next group of puzzles focus on parallelograms without right angles. The shapes start out with gentle angles, so they all look like gently sloping rectangles. Gradually more angled shapes are introduced, where the top and bottom of the shape fall outside their starting points. Help and feedback popups reinforce the message that these are all the same type of shape because of their basic attributes.

Difficulty level	This group of puzzles addresses these concepts
Level 6	Make 1 parallelogram. Introduces a new shape with different attributes to the previous shapes.
Level 7	Make 2 parallelograms. Begin to explore how to make shapes of the required type and in different rotations or reflections.
Level 8	Make 1 sharply-angled parallelogram. Further exploration of the ways a shape can be represented and still have the same attributes.
Level 9	Make 2 sharply-angled parallelograms, exploring how this more angled shape has the same attributes even when rotated or reflected.

The next levels begin mixing the rectangle/square and parallelogram shapes shown before. At first the dots are arranged to make the different shapes quite obvious; over time, there are many dots and multiple ways to make the mix of shapes. This means the child has to really think about the type of shapes needed on each grid.

Difficulty level	This group of puzzles addresses these concepts
Level 10	Make 2 shapes: rectangles/squares and parallelograms. Explores a mix of shapes to encourage thinking about the different attributes of each shape. Some puzzles have more than one possible solution.
Level 11	Make 3 shapes: squares and parallelograms. It is harder to make the required number and type of shapes on the grid, requiring more attention to the attributes. Some puzzles have more than one possible solution.
Level 12	Make 3 shapes: squares and parallelograms, including rotated rectangles; encourages comparisons between rotated rectangles and parallelograms.

The next levels add triangles to the mix. Grids start out with odd numbers of dots, and the first few grids show a clear separation between the four-sided and three-sided shapes. Over time the dots are arranged to make it harder to separate the shapes, requiring more thought about the definition of the shapes required.

Difficulty level	This group of puzzles addresses these concepts
Level 13	Make 2 shapes: introduces triangles, a shape with different attributes to the previous shapes, and mixes these with the previous shapes. Explore how to make shapes with the required attributes.
Level 14	Make 3 shapes, using a mix of shapes to encourage thinking about the different attributes of each shape. Many puzzles have more than one possible solution.
Level 15	Make 4 shapes, using any shapes that have been used before. The larger grid and higher quantity of shapes makes for more complex puzzles. Many puzzles have more than one possible solution. Uses the largest grid (an 8x8 grid).

Total number of puzzles: 150

By Level 15 children have completed many unique puzzles with clear guidance about the types of shapes required on each grid and the attributes of those shapes. From this point on, children are given random puzzles from the upper levels.

Area Gates activity

Shapes and objects can be measured and quantified using unit amounts. Children usually start out measuring areas with regular shapes like squares and rectangles, using standard units (eg: centimetres, inches).

Measuring triangles is a little harder, as triangles don't conform to the straight lines of a grid. The typical approach is to measure the base and height of a triangle and halve it. The Area triangles game uses a similar approach, as players can count grid squares of 12 and halve it to work out how to make a triangle with an area of 6.

This game helps children practise applying a consistent formula to increasingly complex layouts of grids and points, solving a problem that has real-world applications.

Each level adds a little more complexity. While the descriptors here are simple, the complexity is in completing the activity - each additional triangle added is an extra 3 points on the grid to negotiate.

Difficulty level	This group of puzzles addresses these concepts
Level 1	Make 1 triangle of 6 squares. Introduces the core concept of forming triangles of a specified area and the use of the grid to work out the area of a triangle. Uses the smallest grid (a 6x6 grid).
Level 2	Make 2 triangles, both with an area of 6 squares. Begin applying the formula to more than one shape at a time. Explore how all triangles must be of the required area to achieve a successful result.
Level 3	Make 3 triangles that have an area of 6 squares. Shapes of the required area are harder to calculate than previously.
Level 4	Make 2-3 triangles of an area of 6 squares. Layouts make creating shapes of the required area more difficult, so children must take more care in calculating the area.
Level 5	Make 4 triangles of an area of 6 squares. The larger quantity of triangles and complex layouts make creating shapes of the required area more difficult, so children must take more care in calculating the area.
Level 6	Make 4-5 triangles of an area of 6 squares. The larger quantity of triangles and complex layouts make creating shapes of the required area more difficult, so children must take more care in calculating the area.
Level 7	Make 5 triangles of an area of 6 squares. Uses the largest grid and largest quantity of shapes. All puzzle layouts make creating shapes of the required area very difficult and each layout requires hard thinking to solve. Uses the largest grid (a 10x10 grid).

Total number of puzzles: 44

How do these activities improve disposition towards maths and learning?

Describing physical space in two and three dimensions is essential to geometric thinking and the skill of organising visual information.

For young children, being able to recognise shapes is helpful when learning to read letters and numbers. Knowing the names of shapes also helps children recognise and describe other signs or objects around them.

Defining and grouping 2D shapes is also an important part of learning to organise information generally, and think about how we group and categorise things in the world around us.

Paying attention to differences between shapes encourages children to focus on details when they’re comparing items. Sorting shapes into categories also helps children learn to describe the differences between similar items in precise terms. These skills are useful in maths generally, and in other areas such as reading, writing and science.

Feedback in the activity is part instantaneous, part gradual. Successfully tracing a required shape plays an animation of a sprite flying home - an unsuccessful attempt doesn’t. But this doesn’t always mean the puzzle has been completed successfully. The puzzle is only done once all the required shapes (indicated in the instruction panel) have been traced.

In Shape gates, if you make the correct number of shapes, but they’re the wrong type (and/or dots are left over) a pop up screen lets you know. You can go back to the puzzle to change your choices and keep trying until you crack the puzzle.
In Area gates, if you trace a triangle that doesn’t have an area of 6 grid squares, a pop up screen lets you know, and you receive a strike. After 3 strikes you’re invited to try an easier puzzle.

All activities include a Help button. This can be accessed at any time during play and provides instructions the child can use to complete the activity.

Shape gates helps to identify different shapes.
Area gates helps to calculate the area of a triangle.

References

Charles, Randall I., (2005). Big ideas and understandings as the foundation for elementary and middle school mathematics. Journal of Education Leadership, 7(3), 9-24. Available from: https://www.jaymctighe. com/wp-content/uploads/2011/04/MATH-Big-Ideas_ NCSM_Spr05v73p9-24.pdf