4. Interior Designs

• Stick figure handout (two per page): docx PDF
• 1cm grid paper: docx PDF
• Computer, laptop or tablet
• Pattern blocks

This stage builds on Exterior Designs by exploring the correlation between symmetry and the geometric transformation of rotation and reflection. Students will have the opportunity to explore the effects of transformation on shapes and use this knowledge to create designs for the interior decor of their house.

Introduce the vocabulary to describe geometrical transformations by engaging students in a game of Line Tag, where players move along lines on the floor to tag others. In this game, the lines on the floor of the gym or an outdoor court can be used as a designated playing area. One or two players can be selected as the ‘taggers’. Players are only allowed to travel along the lines and if they are tagged, they sit down on the line as ‘roadblocks’. If players come to a roadblock, they must turn around and go in another direction.

After playing the game, ask students to describe the movement of the ‘tagger’. Students may describe that they moved forward along the line or left and right along the line. Explain to the students that if the player is a shape and the playing area is a flat plane, this movement of a shape on a plane is called translation.

Continue to explain the mathematical terminology to describe the movements. When students describe that they turned to move another direction, explain that mathematicians call this rotation. Explain that rotation is measured in angles and introduce related vocabulary, including right angles, acute angles, obtuse angles, and reflex angles. Once students are familiar with the game, you can introduce this vocabulary to build student understanding and knowledge of the types of angles, for example by introducing rules such as 'all turns to the right must be right angles, and all turns to the left must be obtuse angles'.

Encourage students to describe what they had to do when they came to a roadblock. Students may explain that they turned around. Model to students that when they turn around, they are facing the opposite direction and in geometry, mathematicians call this movement reflection.

Ask students to observe what is happening to the size and shape of the players as they are moving. Explain to students that translation, rotation and reflection are rigid motions because the movements do not change the size or shape of the object moved. Explain that two shapes are considered congruent if you can apply rigid transformations from one shape to the other.

Explain to students that there is also another form of transformation that has not been modelled, which involves non-rigid motions. That means that the movements will change the size or shape of the object. Support students to discuss what that transformation may be and what will happen to the object when that geometrical transformation is applied.

Introduce students to the term dilation, used to describe transformations where the size of the object is altered, such as enlargement. Dilation is described in terms of a scale factor. Draw links to the process of dilation when students were reading and drawing scaled maps.

Explore transformations in the real world by observing patterns and designs on objects or buildings around the school. For example, transformations can be applied to shapes to form designs on doors, windows, furniture, upholstery, and walls. 

Enable students by teaching them the vocabulary of shape and transformations language at the commencement of the learning sequence. Students can create shape and transformation picture dictionaries or make entries on the word wall using drawings or photos to help build their familiarity with the vocabulary. Support students to make connections between formal transformation language and 'flip, slide and turn'.

Extend students by encouraging them to describe each movement with greater precision and detail. For example, if students describe a movement as a translation, prompt them to explain the distance moved. A rotation can be expressed in terms of its angle and a reflection can be described as a vertical or horizontal process.

The following sequence of activities offers students the opportunity to deepen their understanding of geometric transformation and to study how transformation can be applied to shapes when creating patterns and designs. Students are encouraged to relate these ideas in their own experience and to apply them to create an interior design for a room in their house using Minecraft: Education Edition.

1. Understanding the vocabulary

This activity has been inspired by the ‘Motion Man’ problem published by Van De Walle, J. et al (2019). It is designed to draw students' attention to the formal vocabulary used to describe transformations and identifying the effects of these transformations.

Make connections between the Line Tag game and transformation of shapes by recreating the game onto dotted grid paper and representing the tagger as a stick-figure token, such as:

It would be useful to make copies of the stick figure and attach the mirror image on the back to make a double-sided stick-figure. This allows the effect of reflection to be observed.

Explain to students that they are going to make and play a board game version of Line Tag. Give students a double-sided stick-figure token and dotted grid paper and explain that they will use the dotted grid paper for the playing field, and their partner will place blockers in their way. Player 1's aim is to make it to "home", while Player 2 aims to block them from reaching the goal by making Player 1 undertake a transformation after each turn. 

To begin, Player 2 places a counter on a location on the grid paper. This marks 'home' for Player 1 to reach. Player 1 starts at one of the four corners of the grid, and rolls a 6-sided die, moving that many places in a straight line in any direction on the grid paper. After they move, they describe their movement, for example '5 steps up' or "3 steps at an acute angle to the right".  Player 2 then places a counter as a blocker on the grid next to Player 1, to block their movement. Player 1 rolls the dice again, makes a transformation and moves that many places, describing the transformation. For example, "I make a right angle turn to the left, and move left 3 places". Player 2 then adds another counter to block Player 1's movement. Player 1's aim to make it to the goal, while Player 2 aims to block them from reaching the goal by making Player 1 undertake a transformation after each turn. 

Encourage students to be specific about their description of their movements. For example, when they rotate their token, they describe the angle as a right, acute, obtuse, or reflex angle. If they reverse direction, they describe the transformation as a reflection. Provide students with enough time so that each player takes a turn as Player 1 and Player 2.

Extend students by asking them to give a degree measure of rotation when they rotate their token. Add parameters to their play to increase the challenge, such as 'all rotations must be reflex angles', or restrict the number of moves that Player 1 can make to get back to 'home'

At the end of play, students discuss the strategies they used to successfully make it to home, or to block their opponent with the class. 

2. Exploring non-rigid transformations

Review the difference between rigid and non-rigid transformations with students. Explain that while rigid transformations such as rotation, reflection and translation cause the original object to change position, a non-rigid transformation causes the object itself to change. Examples are dilation (including enlargement and shrinking, making an object larger or smaller in all directions) and shearing, where part of the object is pushed to the side (such as 'pushing' the top of a rectangle over while keeping the bottom fixed, to form a rhombus). 

Hold up a copy of the stick-figure and ask students to predict what will happen to the stick-figure if enlargement is applied. Invite students to suggest how we can describe the effects of enlargement accurately. Invite students to draw what the stick-figure looks like if it is enlarged two times its size. Ask student to draw what the stick-figure will look like if it is reduced to half its original size. Describe how students used non-rigid transformation to make scaled drawings of their house plans in previous stages of this sequence. 

More information and activities to support student understanding and skills in non-rigid transformation of 2D shapes can be found on the Mathematics Curriculum Companion. 

3. Applying transformations to create tessellating designs

This activity aims to consolidate students’ understanding of geometric transformation by applying translation, rotation and reflection repeatedly to form tessellating designs.

Establish the concept of tessellation by posing this question:

Can you use a shape to create a repeating pattern with no gaps or overlaps? Can you use 2 different shapes?

Introduce the term tessellation to students and ask them to find the mathematical meaning for the word. Add to the class word wall. Prompt students to notice the correlation between transformation and tessellations with guiding questions, such as: 

• Can you see a relationship between tessellation and rotation?
• Can you make a connection between… and …
• What does this remind you of?

Provide pattern blocks for students to explore which shapes tessellate, and which do not. Ask students what characteristics they notice about the shapes that do tesselate, compared to the shapes that don't.

Encourage students to find combinations of different shapes that will tessellate (such as a square and an equilateral triangle). Once students have created a tessellating pattern using pattern blocks, provide grid or dot paper for them to draw their design. Invite students to share their designs. Encourage them to describe how their designs were created in terms of translation, reflection, or rotation.

Extend students by asking them to explore the similarities between shapes that tessellate. Encourage students to create a table displaying shapes that allows regular tessellation, the size of each angle and the total internal angle of the shape. Observe students as they explore the correlation between tessellation and the angles. Do they notice that if the angle measures add up to 360 degrees, the shapes will fit together with no overlaps? Further prompt students to examine the significance of 360 degrees.

Further ideas about tessellation are available on the Maths Curriculum Companion. 

3. Decorating a room

This activity allows students to apply their understanding of geometric transformation to decorate a room using Minecraft: Education Edition. One of the benefits of creating interior designs on Minecraft: Education Edition is that it allows a three-dimensional view of the design.

Present the context for the activity:

Now that you have built your new home and have designed its floorplan, you are going to become interior designers for one of your rooms, using Minecraft: Education Edition. Your design needs to meet the purpose of the room and incorporate aspects that demonstrate the different geometric transformations - translation, rotation, reflection and enlargement.

Students discuss with a partner or small group how they might incorporate geometric transformations in their interior designs. Invite students to share their ideas and create a brainstorm list. Some ideas that students may use include:

• A floor design that demonstrates tessellation and symmetry.
• Wallpaper designs.
• Various artefacts and decorations in the room.
• Modular shelving units.
• Hanging wall pictures depicting enlargement.
• Border around a picture frame, clock, or furniture.
• Rotation of a shape to create a design for a lamp shade.

Encourage students to be creative when designing their room. Students add labels to features of the room that demonstrate each type of transformation.

Enable students requiring support by encouraging them to select a one or two shapes and asking them to perform a transformation on them. For example, students may choose to perform rotation on a triangle. Prompt students to think about where this design can be used in a room.

Extend students by asking them to include a combination of transformations in their design.

Areas for further exploration

1. How did they make Ms Pac-Man?

This activity from the Mathematics Curriculum Companion allows students to develop understanding of translation, rotation, and reflection. Students explore how Ms Pac-Man moves around the maze and how its motion will differ without these geometrical transformations.

Students create patterns for a wallpaper border, using isometric dot paper or digital technology. Encourage students to use a combination of transformations to design their pattern.

Ask students to describe the transformation that has been applied to the shapes and check their understanding of various geometric transformations.

Enable students requiring further support by guiding them to use pattern blocks to create their designs. Encourage students to use accurate vocabulary to describe the transformation process, referring to the mathematics word wall.

Extend students by asking them to indicate the distance a shape has moved in translation or the angle of rotation. Make a shape on a geometric drawing app and ask students to predict its coordinates when it is translated, rotated, or reflected.

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