# A Model World

Purpose

The purpose of this unit is for students apply and refine their measurement skills and to develop an understanding of enlargement and reduction, through making scale models to be used in a short animated film.

Achievement Objectives
GM4-1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
GM4-2: Convert between metric units, using whole numbers and commonly used decimals.
GM4-8: Use the invariant properties of figures and objects under transformations (reflection, rotation, translation, or enlargement).
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
NA4-2: Understand addition and subtraction of fractions, decimals, and integers.
NA4-3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
NA4-5: Know the equivalent decimal and percentage forms for everyday fractions.
Specific Learning Outcomes
• Understand and describe proportion using the language of mathematics.
• Make accurate metric length measurements.
• Understand the principles of scale.
• Understand that the value of the scale factor of a reduction is less than 1 and for an enlargement is greater than 1.
• Understand that the scale factor of a reduction can be found by multiplying by a fraction or by dividing by its whole number reciprocal.
• Make accurate angle and circuference measurements.
• Calculate relative proportional relationships between two body parts.
• Make a model using scale measurements.
Description of Mathematics

There has been a burgeoning of special effects within the modern film industry. One skill required to create historical, real, fantastical or futuristic worlds, is being able to produce accurate scale models of people and places. Companies such as WETA Workshop (NZ), have teams of people who are skilled in creating ‘miniature’ worlds, which then come to life in larger-than-life size upon the big screen. They also create collectible items that are small models of large screen characters and places. Much mathematics is involved in this process of creating scale models.

Enlargement (or reduction) is a transformation that changes the size of an object represented in two dimensions or three dimensions, into an image of the object in which the component parts are all in the same proportion or ratio to the original. The ratio of any two corresponding measurements in the object and its image is called the scale factor.

The understandings that need to be developed with the students are that the two figures (the original object and its image) have the same shape, and that the centre point and angle sizes do not change (are invariant). An enlargement, or a reduction, involves a direct transformation, in which length measurements are multiplied by the same multiplier, or divided by the same divisor, resulting in a proportionally adjusted length, area or volume measurement.

The relationship between the scale factor, going from a large figure to a small figure, and that of going from the (same) small figure to the (same) large figure, is a reciprocal one. For example, if the original object is made 3 times bigger, it is increased by a scale factor of 3. If the original object is made 3 times smaller, the scale factor is one third. Students learn that the scale factor is the number used as the multiplier in the scaling process. An important understanding to be developed, is that in calculating scale measurements of reductions, they can multiply by the fraction or divide by its whole number reciprocal: the result will be the same.

It is important that students learn to work between different ways of expressing and recording scale. One expression is as a ratio, 1:3, another is as a description in words, such as one centimetre to three centimetres, and another is as a graphic. Being able to interpret and record scale in each of these ways is important.

At this level students are becoming comfortable and efficient in measuring and estimating, using appropriate standard units of length. They are learning to apply measurement skills and formulae to practical tasks, using a range of additive and simple multiplicative strategies with whole numbers, fractions, and decimals. They are learning to understand and work with proportions and ratios. The human body is a fascinating subject/context for exploring proportions and the relative size of component parts. By building scale models and by selecting appropriate scales, using ratio to translate the dimensions of the original object into its scale model, students come to understand the consistent proportional nature of this type of transformation.

As students create their own proportional sculpture, they learn about the conventions of model making, and apply their new knowledge of the practical design elements, materials and processes particular to this art form. They learn from exploring the animation work of other artists, and they recognise the power of this art form in communicating messages and in providing entertainment.

Note: the scale factor of 1/10 is used in these sessions. This means that the models are small in size. Should the availability and cost of construction materials not be a limiting factor, a 1/6 scale can be used. This is the preferred scale of many model makers of figure replicas. Calculators can be used to support these calculations.

Associated Achievement Objectives

The Arts
Visual Arts

• Investigate the purpose of objects and images from past and present cultures and identify the context in which they were made, viewed and valued.
• Explore and use art-making conventions, applying knowledge of elements and selected principles through the use of materials and processes.
• Develop and revisit visual ideas, in response to a variety of motivations, observation and imagination, supported by the study of an artist’s works.
• Explore and describe ways in which meanings can be communicated and interpreted in their own and others’ work.

English
Speaking, Writing and Presenting
Purposes and audiences

• Show an increasing understanding of how to shape texts for different purposes and audiences.
INDICATORS:
- construct texts that show an awareness of purpose and audience through deliberate choice of content, language and text form;
- convey and sustain personal voice where appropriate.

#### Number Framework Stage

Numeracy stages 7-8

Required Resource Materials

Pencils, erasers and rulers

A4 paper

Wool or string

A set of calculators

Tape measures

Wire clippers

Model making wire (armature wire)

Plasticine

A set of protractors

Camera (cellphone)

Video camera

Activity

Previs is the abbreviated term for ‘previsualisation’, which includes the conceptual design and storyboard for a movie.
The conceptual design of the story for the short animated film (about a child and a pet) is not specifically developed within this mathematics unit of work. It is assumed that the development of each student’s film storyboard will be undertaken within the literacy programme, and will parallel and complement the mathematics and art within this series of lessons.

Learning activities
Whilst this unit is presented as sequence of five sessions, more sessions than this may be required. It is also expected that any session may extend beyond one teaching period.

Session 1

This session is about understanding the animation context for making scale models and for learning key ideas about human body proportions.

SLOs:

• Understand and describe proportion using the language of mathematics.
• Understand the principles of scale.
• Make accurate body length measurements.

Activity 1

Write the words film animation on the class chart/computer screen.
Brainstorm and record the students’ related knowledge and ideas.
Explain that film animation will be the focus of this and subsequent lessons.

Activity 2

1. Explain that the students will watch two short movie animations after which they will work in pairs to consider and record the similarities and differences between each of the animation clips. Show:
Wallace and Gromit (1:46)
The Adventures of Tintin (first 2 minutes of 42:20)

2. Make chart paper available.
Have students work in pairs to create a venn diagram, in which they consider and record the similarities and differences between each of the animation clips.
Have them pair share their ideas.

Activity 3

Highlight the physical process of simple animation in 2 dimensions.
Illustrate the 2D animation process by showing flip cards.
Make available to students sets of small pieces of paper and have them draw and create their own flip animation sequence. (For example, of a person performing a simple sporting action.) Have them swap and compare their 2D animations.

Activity 4

1. Highlight the physical process of animation in 3 dimensions. (Wallace and Gromit)
Make available to pairs of students small moveable plastic figurines.
Have student talk about how a simple animation feature can be created.
Agree that both processes involve a series of tiny movement changes, which, when sped up create the animation effect. This is called stop motion animation.
2. Have students research information about stop motion animation, or make available Attachment 1. Discuss.

Activity 5

1. Ask what the designers may have had to consider in their creation of a 3D model for animation. List ideas and lead discussion to proportion and scale.
Write agreed definitions of proportion and scale, using student knowledge and recognised sources.
(For example:
Proportion: the correct relationship between the size, shape and position of the different parts of something.
Scale: an indication of the relationship between the distances on a map and the corresponding actual distances, and/or the relationship between two sets of dimensions or measurements of an object and a different size copy of the same object.)

2. Distribute Attachment 2.
Discuss and have students complete this.
Explain they will be making scale models of a friend.
Have them predict the head to body ratio they expect to find when they measure their friend.

Activity 6

Conclude with a summary of learning about stop motion animation and the mathematics they anticipate using in creating a scale model of a friend.

Session 2

This session is about creating an enlargement of a 2D image (drawing), and exploring the concept of scale.

SLOs:

• Understand and apply the process of making a scale drawing.
• Understand that the value of the scale factor of a reduction is less than 1 and for an enlargement is greater than 1.
• Understand the reciprocal relationship between the scale factors of a transformation.
• Understand that the scale factor of a reduction can be found by multiplying by a fraction or by dividing by its whole number reciprocal.

Activity 1

Begin by reviewing together notes made about model making and the definitions recorded in Session 1, Activity 7.

Activity 2

Explain that the ideas will be explored in the following task.
Have students undertake a simple enlargement task.
Make available A4 blank paper, pencils, rulers, string and the following set of instructions.
On a landscape A4 page:

1. Draw a rectangle 8cm x 10cm.
• Inside the rectangle draw a simple picture or shape.
• Over the top draw a grid of squares 2cm x 2cm.
• Draw a second rectangle 16cm x 20cm.
• Draw a grid of squares 4cm x 4cm.

2. Enlarge the image drawn in Step 2.

Activity 3

1. Distribute a copy of Attachment 3 to each student and have students record their answers to the following questions:
What is the size relationship between the Step 3 shape (Figure 3) and the Step 6 shape (Figure 6)? (Double in size)
Explain how you know, using measurements. (16 cm are double 8 cm, 20 cm are double 10 cm, 4 cm x 4 cm squares are double 2 cm x 2 cm squares)
How do you write this as a ratio? (2:1)
What is the Scale Factor? (2)
Explain how you know. (All measurements in Figure 3 are multiplied by 2 to create Figure 6.)
What is the size relationship between Figure 6 and Figure 3? (Half in size)
Explain how you know, using measurements. (8 cm is half 16 cm, 10 cm is half 20 cm, 2 cm x 2 cm squares are half 4 cm x 4 cm squares)
How do you write this as a ratio? (1:2)
What is the Scale Factor? (1/2)
Explain how you know. (All measurements in Figure 6 are multiplied by 1/2 to create Figure 3.)
What do you notice about the relationship between the enlargement scale factor and the reduction scale factor? (They are reciprocals: 2/1 and 1/2)
The scale factor of an enlargement is always_______________ (greater than 1).
The scale factor of a reduction is always_______________ (less than 1 but greater than 0).
If the scale factor connecting the object with its image is usually shown by the letter k, use these symbols to write a statement about:
An image that is larger than the original object: enlargement. (k > 1)
An image that is same size as the original object. (k = 1)
An image that is smaller than the original object: reduction. (0 < k < 1)

2. Discuss student responses to the questions.

Activity 4

Make available lengths of wool or string.
Have each student investigate the accuracy of their enlargement by carefully placing the thread over a particular part of their original picture of image (Figure 3), measuring and doubling the thread and laying it on the same part of the Figure 6, image to check that it is exactly double (x2).
Conclude by emphasising that when enlargement or reduction is undertaken, every length measurement of the original (object) is multiplied by the scale factor to create the image.

Session 3

This session is about students applying their knowledge of scale to making a small 3D model of a friend.

SLOs:

• Make accurate metric length measurements.
• Accurately multiply length measurements by the scale factor of 1/10.
• Check the accuracy of length measurements and calculations made by another.

Activity 1

1. Explain that in this session they will prepare one character for their short animation film of their friend and their pet. As they watch the Wallace and Gromit video clip once again, they are to consider the practical process of making a scale model of their friend.

2. Review the Wallace and Gromit video clip.

Activity 2

1. Place in front of the students a set of tape measures, lengths of wire, wire clippers, and plasticine, length of coloured thread.
Tell the students they will need their pencils, paper and a ruler. Make calculators available as appropriate.
Have students discuss in pairs and record, a suggested logical sequence of steps for making an operational 1/10 scale model of their partner.

2. Share these as a class, discuss, and agree on a sequence of steps such as:
1. Draw a stick figure picture of their partner with at least one arm and one leg bent in some kind of action. (This does not have to be exactly in proportion as it is for recording purposes only.)
2. Use the tape measure to accurately measure from joints, the key body length measurements (lower leg, upper leg, forearm, upper arm, body length to hips, neck, head, hand foot). Record these measurements on the matching part of the stick figure drawing.
3. Draw a second identical stick figure diagram of their friend. Calculate, using the scale factor of 1/10, the length measurements for the scale model they will be constructing, and record these on the second figure. Check calculations with a friend.
4. Using lengths of wire and wire clippers, make a wire armature scale model of their friend, using the measurements recorded on their second diagram. Practical considerations:
• Highlight the need for accurate length measurements, with an additional small allowance for joins.
• Two wire lengths can be cut and twisted together for strength. (This will depend on the wire. It should have both firmness and flexibility.)
• Legs can be made in one piece, but at the knee joint (where the upper and lower leg measurements meet) tie a coloured thread to the wire. (Once plasticine is layered onto the frame, these will mark the correct ‘bend points’.) Apply this procedure to the arms as well.
• Allow extra wire bent on each foot to insert into a small ball of plasticine so the model can be secured in a standing position.
5. When each partner’s model is complete, check their model measurements with their second diagram, to ensure that the model is to scale.

6. Mold the ‘flesh’ onto the wire armature by wrapping the plasticine around the wire and leaving exposed, the thread that is tied to mark the joint.

Activity 3

1. Before the students begin their model making process, discuss and have a student model several measurements on a friend. Discuss and agree on start and end points for each measurement. For example, begin the upper leg measurement at the outer hipbone and stop beside and just above the knee joint.

2. Use each of these measurements to model and record calculations, applying the scale factor and its reciprocal. Ensure that examples include multiplying an actual limb measurement by 1/10 or 0.1 and dividing the same limb measurement by 10. Recognise and agree that these multiplication and division operations achieve the same result.

3. Have individual students explore both methods using a calculator, and check their results with a partner.

Activity 4

Have students complete the process up to the end of Step 4, stopping at regular intervals to check progress with reference to the agreed steps, and to answer questions.

Activity 5

Conclude the session by sharing models and reflecting on the process of creating a 3D scale wire model. Reflect on Session 1, Activity 5, Step 2.

Session 4

This session is about having students recognise that angles stay the same when a scale model is made, and about students making, scaling and applying circumference measurements.

SLOs:

• Understand and demonstrate that when the length of sides changes by the same scale factor, angles remain the same (are equal).
• Make accurate angle measurements.
• Make accurate circumference measurements.
• Accurately multiply circumference measurements by the scale factor of 1/10.

Activity 1

Begin this session by reviewing the process of making a model 1/10 the size of the original object. Review the reciprocal relationship between the model size and the size of the friend.

Activity 2

1. Point out that measurements of length have been made to make the models.
Using one wire model figure, bend it into an action pose with a leg and arm bent.
Ask: What other aspects of this figure can be measured?
Guide the conversation to discussing angle measurements. Agree that we measure angles.
Write this (incorrect) statement on the class chart:
When scale models are made, the scale factor applies to all length and angle measurements.
Have students discuss in pairs whether they agree or disagree with this statement, have them state their position to the class and explain how they know that they are correct.

2. Have students share their ideas.
If necessary, suggest and follow this process.
Make available a camera or cellphone.
Have one student model the action pose of the wire figure.
Take a profile photo of the person and of the action figure.
Print the photo and use a protractor to measure the identified angle in both pictures. For example: 3. If cameras and photo printing facilities are available, have student pairs repeat this process.

4. On the class chart write a concluding statement:
When lengths change by the same scale factor, angles remain constant (equal).
Draw a diagram like this: Together record the enlargements and reductions (reciprocal scale factors).
Together measure the angles using the protractor.
Repeat this with other examples.

Activity 3

Explain that students will now complete their scale models for animation, by adding ‘the flesh and clothes’ to the wire skeleton. They will do this by wrapping plasticine around the wire armature, molding it into a realistic body shape, and by adding texture and detailed features as appropriate.
Have students recognise the need to continue to keep the ‘rounded’ body measurements in proportion, and not have models that are too fat or thin.
This should include making and recording at least 4 circumference true measurements (head, arms legs and waist) of their partner, and multiplying each of these by 1/10 or (dividing by the reciprocal) to calculate the 3D scale model values.

Activity 4

Have partners check each other’s calculations and allow time in the remainder of the session for the models to be completed, ready for animation.

Activity 5

Display and share models. Have students give constructive peer feedback on the proportions and level of realism and detail in each other’s models.

Activity 6

Explain that, if possible, for the next session, each student is to bring a photograph of their pet standing or walking, and at least one true length measurement of one part of the pet: tail, back leg, or body.

Session 5

This session is about making a scale model from a picture when all true measurements are not available.

SLOs:

• Accurately measure a photograph and record measurements.
• Calculate relative proportional relationships between two body parts.
• Calculate scale dimensions.
• Make a model using scale measurements.

Activity 1

Have students share and discuss their models and identify the most challenging part of the process so far.

Activity 2

1. Remind students that, just like Tintin and Snowy and Wallace and Gromit, their animation should include a child and their pet.
Have students share pet photos.

2. Show Attachment 4 (A4 size) and discuss the profile photo features. Have students decide if their own photo is suitable for measuring.
Distribute A4 copies of Attachment 4 and make available lengths of wool or string.
Explain to student they will need pencils, rulers and erasers and have them complete the measurements only for Table 1. Compare these with a partner and agree on a value for each.

Activity 3

Together discuss how to complete Table 1. Model and discuss how to approach Table 2.

Activity 4

Have students repeat 2 and 3 above for Tables 3 and 4, concluding by comparing these with a partner and agreeing on a value.

Activity 5

Make available plasticine and tape measures.
Explain that the pets will not have wire armatures, but will be made from plasticine only.
Have students use the scale measurements to create their pet models, adding texture and details as appropriate

Activity 6

Share plasticine pets and reflect on the mathematical process of making a scale model from a photograph.

Activity 7

Refer to Attachment 1 (Stop motion animation) and/or research information that students have already contributed.

Activity 8

Make available video camera/s, cameras.
Have students display the storyboards creating during the literacy programme.
Have students work in pairs and prepare to begin filming, using their storyboards.
Allow sufficient time for students to make a stop motion animation movie of a few seconds duration, evaluate their product and process. Continue filming.

Attachments