Finding surface area

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Purpose

The purpose of this activity is to support students in calculating the surface areas of cuboids in an efficient manner, using multiplication and addition.

Achievement Objectives
GM4-3: Use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
Required Resource Materials
  • Cardboard boxes with dimensions that are whole numbers of centimetres. You need at least 1 for every pair of students in your class
  • Rulers
  • Calculators
  • Cardboard or paper, glue sticks, scissors
Activity
  1. Presents students with one of the cardboard boxes.
    Suppose I want to paint the outside of this cuboid shaped box. What amount of area surface would I cover with paint?
    If needed, provide students with a definition of horahanga mata - surface area (e.g. it is the area on the outside of a shape).
    Let students discuss how they would calculate the surface area. Look for them to:
    • Recognise that the cuboid has six faces, arranged in three pairs of identical parallel faces.
    • Use multiplication to calculate the area of each face.
       
  2. Model the process of finding surface area using a diagram like this, with the dimensions of the box you have:
    Ask students to calculate the area of each rectangle and to explain how they calculated it.
    Reflect on the calculated areas. Did we need to calculate all of these amounts of area? Is there a way we could have been more efficient?
    Look for students to recognise that the amounts of area are the same for the parallel faces - the top and bottom, the left and right faces, and the front and back faces.

    Diagram showing the different spaces of surface area on a cuboid.
     
  3. Work together to find the total surface area. Students might explore adding all of the surface area measurements together, or might explore multiplying the parallel measurements by two and then adding these measurements together. Links could be made here to the order of operations. 
     
  4. Open the box and lay it flat to form a net. Cut off the tabs.
    Does our calculation of surface area look right?
    Write dimensions on the surfaces of the net to confirm the calculation.

    Cuboid net.
     
  5. Organise students into pairs that will encourage peer scaffolding and extension, and productive learning conversations. Provide each pair with a box and ask them to calculate and record the surface area of the box, then draw a diagram to prove their calculations. Roam and look for students find the area of all six faces, efficiently and accurately combine the areas to form a total amount of surface area, and record the surface area using the correct unit of measurement, square centimetres (cm2). Calculators should be freely available. Support students by helping them draw a diagram or write measurements on the box, as required. 

Next steps 

  1. Give students open-ended challenges like this:
    • Design five cuboids that have a surface area of 600cm2.
      Can you make one that is a cube?
      Note that a 10cm x 10cm x 10cm cube has a surface area of 600cm2.
      What is the volume of that cuboid? Why is it significant? (1 litre)
    • Design different nets for a cuboid that is 8cm x 7cm x 6cm.
      Which net is the most compact and will fit on the smallest sheet of cardboard?
      Note that packaging is a major industry and small savings in area of one box can produce huge savings if large numbers of boxes are produced.
       
  2. Make links to meaningful, real-world situations in which calculating the total surface is useful.
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Level Four