Scaling our school

Purpose

In this unit we estimate the length and height of buildings at the school. We compare these estimates with the actual heights or lengths and build scale models. Students select appropriate scales using ratio to translate the real dimensions into the dimensions for the scale model.

Achievement Objectives
GM4-1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
NA4-4: Apply simple linear proportions, including ordering fractions.
Specific Learning Outcomes
  • Calculate measurements for simple scale models.
  • Determine measures of height indirectly.
  • Make reasonable estimates of the height and length of a building.
  • Measure length accurately to produce simple scale models.
Description of Mathematics

Estimation is about getting an approximation of the size of something that is appropriate for a purpose. For example, a painter might estimate the wall area of a school building for the purpose of buying paint or quoting for a job. She might measure the length and width of the building or use stride lengths to approximate the dimensions. A key feature of estimation is the use of trusted benchmarks that are used to estimate with. The painter might know that their stride is about 60cm so 10 strides are equivalent to 6 metres. an important aspect at the beginning of this unit. Students will estimate the height and length of buildings within their school grounds and then use indirect methods of measurement to determine the actual height. Measurements of length can also be determined by accurate use of tape measures.

To build a scale model of the school students need to understand the concept of ratio. A ratio compares two measures. For example, a scale model that uses the ratio 1:10 can mean that 1 metre on the scale model is equivalent to 10 metres in real life. That would mean a building that is 15 metres long will be 1.5 metres long in the scale model. Obviously, a model that is 1.5 metres long is too big so choosing a useful ratio is important. A 1:100 scale would mean that 15 metres becomes 15 centimetres in the scale model. That seems a more useful size. In this unit students gain a sense of the size of measurement units to create accurate estimates and build functional models.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

  • Modelling the use of trusted benchmarks to estimate lengths, such as, using a metre ruler to estimate the length of the room.
  • Providing students with experiences on digital platforms, like Google Maps, such as, asking them to work out the distance from their home to school.
  • Building models with cubes that are in proportion so students can see what changes and what stays the same.
  • Choosing scales that make conversions easier for students, such as 1:100.

The context for this unit is to create a model of the students’ own school. You might choose accessible building complexes that have significance to your students, such as a Marae or community centre. Alternatively, use Google Maps to create scale models of important buildings from around the world.

Required Resource Materials
  • PowerPoint One
  • Measuring equipment: 30 cm rulers, metre rulers, tape measures, trundle wheels.
  • Photos of buildings in the school grounds using Google Maps.
  • Digital cameras.
  • 1cm grid paper.
  • Construction materials: cardboard, scissors, tape, pens, paint.
Activity

Getting Started

  1. Explain to the students that the Board of Trustees/Principal wants a scale model of their school for display in the administration area (or similar).
    Can someone explain what a scale model is?
    How can we create a scale model?
    What will we need to make it?
    Look for ideas about material resources but ensure that the students consider other items like photographs and measurement instruments.
  2. Use Google Maps to get a bird’s eye view of the school layout. Zoom in on the site using the + - symbol and go to photo view. The scale appears in the bottom right of the screen. For example, a 2.5 cm length on screen may represent 20 metres.
  3. Discuss the meaning of the scale and how it might be used to find the real length and width of each building. Use a ratio table to work out the dimensions of one building. For example:

    Map length

    Building length

     

    Map width

    Building width

    2.5 cm

    20 m

     

    2.5cm

    20 m

    1 cm

    20 ÷ 2.5 = 8m

     

    1cm

    20 ÷ 2.5 = 8m

    4.8 cm

    4.8 x 8 = 38.4m

     

    2cm

    2 x 8 = 16m

  4. Provide the students with a photocopy of the Google Maps bird’s eye view. Ask them to work in small groups to work out the length and width of each major building using the scale of 1cm:8m.
  5. After a suitable time bring the class together to discuss strategies and check answers.
  6. Use Google Maps on street view mode to tour the school from a side on view, as if walking around the school.
  7. Put the students into small groups. Assign a building to each group. There may be duplications but that is fine.
    I want you to take photographs of the building you have been assigned. You will need enough photographs to construct several views of the building so you can make a scale model. What might you need to consider before you go out?
    Students might make a list of considerations such as:
    • All four side views are needed. A side might need several shots.
    • Use a measurement benchmark so lengths can be estimated. The benchmark might be an object, such as a metre ruler, or a person from the team whose height is known.
    • Important details might be focused on, such as doorways, artwork, or objects such as alarm boxes or arches.
  8. After the discussion send the groups out with a digital camera to take their photographs. When the groups return save the images to a drive or USB key for safe storage.

Exploring

  1. Ask students to estimate the length, width and height of the building they are assigned using the chosen benchmark.
  2. Discuss the students’ estimates of the height and length of their building and how they arrived at these estimates. What is the height of the tallest building? The shortest? What is the length of the longest building? The shortest?
  3. Discuss the effect of perspective on the estimates. Structures in the foreground look larger than those at a distance. The students may need to use other visual clues to counter this effect (e.g., vehicles, trees, people).
  4. Compare the estimates of length and width with the measurement obtained from Google Maps. The images on Google are very accurate.
  5. Discuss how the actual length and height of the buildings can be measured. The length can be measured using a metre ruler or a tape measure. Elicit some ideas from the students as to how they might measure the height indirectly (They are not allowed to climb the buildings or climb a ladder). Some ways they can measure the height include:
    • One student holds up a metre stick beside the building. Other students walk away from the building until a referent such as a paper clip, or their thumb held at arm’s length, appears to be as long as the metre stick. Using the referent students count the number of paper clips, or thumbs high the building is, then convert this to metres.
    • Take a photo of a metre stick (or other known measure) being held upright beside the building. Use the metre stick as a referent to find the height of the building.
    • Measure the shadow cast by a metre stick. Then measure the building’s shadow at the same time of day. Use the measurements to draw a scale model on grid paper. Read the height of the building from the drawing.
    • Use trigonometry although that is not expected at this level.
  6. Get students to choose an appropriate method to measure the actual length, width, and height of their building. Once the measurement is complete put the data in a table.

    Building

    Length

    Width

    Height

    Office Block

     

     

     

    Senior Block

     

     

     

    Library

     

     

     

    Was there a large difference between the estimated dimensions of your building from your photographs and the actual measurements? Why may that have happened?

  7. Explain that to make a scale model of the school, the model of each building will need to be in proportion.
    What does ‘in proportion’ mean? (The corresponding lengths of the real building and the model are in the same ratio)
    Demonstrate with a hypothetical building as an example.
    Show Slide One of PowerPoint One
    What is the length, width, and height of this building?
    Use the squares in your book to draw a scale drawing.
  8. Discuss what is different and what is the same between their scale drawing and the original. Lengths change but angles stay the same. Most importantly the ratios of the lengths within the figure stay the same. Show the students Slides Two and Three to show what happens if the ratios are not kept constant.
  9. Discuss why the scale used by each group needs to be the same or the model of the school will not look proportional.
    Using the table of actual building dimensions, ask:
    Architects usually use ratios of 1:100, 1:150, 1:300, etc.
    Why do you think they do that?
    The scale needs to allow for easy conversion of lengths and the scale model has to be the right size for viewing.
  10. What scale can we use for our models that will work for these dimensions?
  11. Choose a suitable scale.
    If we use a scale of 1cm:1m (1cm:100cm) how long will out model of the Admin Block be?
    1:100 is feasible as a 44m long building will need a model that is 44cm long.
    1cm:2m is also feasible as it means a 44m long building will need a model that is 22cm long.
  12. Record the selected ratio symbolically, e.g., 1:200.
    Create a table of referent measures to support students to make their models. For example:

    Actual dimension

    Model dimension

    2m

    1cm

    10m

    5cm

    20m

    10cm

  13. Ask students to work in their groups to determine the dimensions of their scale model.
  14. Students will need to develop scale drawings (blueprint) of their building including length, width, and height. Discuss features such as doors, windows, and other features of their buildings.
    Students could use Google Sketchup as a tool for constructing their blueprints.
  15. Once the blueprint has been completed get the students to reflect on their drawings. Are the proportions correct? Will your building fit with the other buildings in the school model?
  16. The students, in groups, can construct their school buildings. Some students will need support to construct nets for the models. The Unit called Representing 3D models in 2D drawings provides useful lessons on creating nets for models.
  17. Models can be painted with the colours of the actual buildings.

Sharing

  1. Allow the students to share and reflect on the scale models.
    Convince another group that your scale model is in proportion to their model.
    What things would you change if you were going to create the model over again?
  2. Assemble the scale models of buildings into a complete scale model of the school. Include playing fields, paths etc. This is an exercise about scale as well since the areas and locations of the building must also be in proportion to the scale that is used.
    How can we locate each model in the correct place? (Using a co-ordinate system is useful way to do that.)
    How might we use the Google Map image to guide where we locate each model building?
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Level Four