Spatializing Area: 3-Part Lesson

Early Years/Primary/Junior (Age 3 – 12)

Curriculum Goal

Kindergarten: Demonstrating Literacy and Mathematics Behaviour

  • Measure, using non-standard units of the same size, and compare objects, materials, and spaces in terms of their length, mass, capacity, area, and temperature, and explore ways of measuring the passage of time, through inquiry and play-based learning (#19).

Primary: Geometry and Spatial Sense/Measurement

  • Estimate, measure, and record length, perimeter, area, mass, capacity, time, and temperature, using standard units.
  • Compare the areas of two-dimensional shapes by matching, covering, or decomposing and recomposing the shapes, and demonstrate that different shapes can have the same area.
  • Use appropriate non-standard units to measure area, and explain the effect that gaps and overlaps have on accuracy.

Junior: Geometry and Spatial Sense/Measurement

  • Show that two-dimensional shapes with the same area can have different perimeters, and solve related problems.

Context

  • Students are sitting in a circle on the carpet.
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Appendix A
Appendix B
Appendix C

Materials

  • Pre-made shapes (Appendix A, B, and C)
  • 2” x 2” square tiles – enough for each student to have at least 12

Lesson

  • Present students with two different sized rectangular shapes (see Appendix A).
  • Ask students: Looking carefully at these two shapes, which one do you think takes up more space? Which shape has more of the shaded area?
  • Students should see that the two shapes are different in size and area.
  • Prompt students to use comparative language: How do you know that shape takes up more space? What shape is longer?
  • Introduce a single square unit and hold up one of the blue square units for all students to see.
  • Tell students: Just to be sure that these two shapes really do have a different area, we are going to measure and compare them. Now, just using your eyes and your imagination, how many of these squares will it take to completely cover the shaded area of this shape?
  • Start with the smaller of the two rectangles and place a square in the top left corner.
  • Tell students: Once you think you know how many it might take, just keep it a secret and put your hands behind your back. Now, keeping your hands behind your back, use your fingers to show how many squares will be needed to cover the entire shape. Ok, everyone, please show me what number you’ve made with your fingers.
  • Once students have their estimates, have a few students share with the group. 
  • Next (or if need be), take two other blue tiles and line them up beside the single blue square to complete the row.
  • Tell students: Now that we have a row of 3 squares all lined up beside each other, does this help us figure out how many it will take to cover the whole space? Get ready to show me your fingers. Ok, let’s see!
  • Complete the next row and ask students to confirm their response.
  • Complete the same process for the square template. 
  • Ask students: How many squares did it take to cover up the rectangle? And how many squares did it take to cover up this bigger square?
  • Ask students: What shape has the larger area? What shape has the smaller area?
  • Show students two different sized shapes with the same area (see Appendix B).
  • Tell students to look carefully the two shapes and think about how much space the shaded part is taking up. 
  • Ask students: Do you think one of these shapes is bigger than the other? Could it also be possible that they take up the same space? What do you think?
  • Go around in the circle and have each student respond. It is expected that most students will believe the ‘b’ shape to occupy a larger area.
  • Present the square unit again and have students mentally iterate how many of the single blue tiles it might take to cover each shape in turn. 
  • Ask students: How many of these blue squares will it take to cover each shape? 
  • Begin with the cross shape. Ask children to see if they can figure out how many squares it will take to cover it. Ask children to use their fingers to show everyone how many they think it might take.
  • Ask students to share their response. Next, move onto measuring the cross shape.
  • Some students might see that the two shapes share the same area. Ask students how this could be possible.
  • Fill each shape with the square units and show students that the two shapes share the same area.
  • Also ask students if there is anything they can do to the cross shape to make it look like the ‘b’ shape.
  • Hand out pre-made shapes to students (Appendix C).
  • Each student will have a different sized shape, but the same area as one other peer. For example, two students will have a shape of 5 square units. Shapes vary in their level of difficulty to adapt to students’ learning abilities.
  • Ask students to look around at all the different shapes.
  • Ask students to see whether they can find the shape with the largest area. What about the shape with the smallest area?
  • After students have responded, hold up the square unit tile and ask them to estimate how many squares it will take to cover their shape.
  • Then, hand out a one square unit tile to each student and see whether having the tile in hand helps or changes their estimates.
  • Ask students how many more tiles they will need to cover their entire shape and hand them that number of tiles.
  • Students will come to recognize whether their estimates were accurate or not and may need to request the addition or removal of tiles.
  • Once students’ shapes are all covered (no gaps, no overlaps), tell students: Some of your shapes need the same number of tiles to cover them. Who thinks they have the smallest shape – the shape that needs the fewest number of squares?
  • Gather the two shapes composed of 5 units (‘L’ and ‘V’ shapes) and place them directly beside each other in centre of the circle for comparison.
  • Ask students: Is there any way to turn this shape (point to ‘L’) into this shape (pointing to ‘V’). Can somebody do it by only moving one square tile?
  • Repeat the above process for the remaining pairs, starting with 7-tile shapes and ending with the 12 tile shapes.

Look Fors

  • Can students accurately predict how many squares will fill the larger shapes?
  • Can students accurately explain and justify their predictions?
  • Can the students accurately predict which shapes have the smallest/largest area?
  • How easily can children move one piece from one shape to make it look like another shape?

Extension

  • Explore further how and why two different shapes can contain the same area in The Garden Patio.

Related Lessons

An activity that develops an understanding of area and size in a “real-life” scenario.

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Appendix A
Appendix B
Appendix C

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