Spatializing Area

St. Andrew Catholic School Public Lesson Team: Nancy Valentini, Francesca Lisi, Julie Fiorucci, Debby Culotta & Monica Rohel and Joan Moss, Bev Caswell & Zack Hawes

To print detailed lesson plans and materials, please click here

Curriculum – Measurement

  • Visual comparisons
  • Mental and physical iterations
  • tiling
  • Conservation of area
  • Composing/decomposing shapes/area
  • Transitivity

Summary

In this lesson students will be asked to directly compare and measure shapes of various sizes and dimensions. The lesson begins by having students compare and measure two shapes that differ in both appearance and area. Next, students are asked to compare and measure two shapes that differ in appearance but share the same area. Finally, students are asked to compare and measure an assortment of different sized shapes, some of which share the same area and some that do not. Throughout the lesson, students will approach measurement in through visual comparisons, mental iterations (imagining how many of a given unit – a 2” x 2” square tile – it will take to cover a given shape), physical iterations (using the given unit to physically determine the area), and tiling (using the given unit to completely cover the shapes with no gaps or overlaps). For shapes that are different in appearance but identical in area, students will be given opportunities to demonstrate how one shape can be transformed and made into the other shape.

Materials

  • Download: Pre-made shapes
  • blue square foam tiles (2″ x 2″) – enough for each student to have at least 12 of their own if need be 

Lesson

Part 1: Introduction – Different shapes, different areas

  • Present students with two different sized rectangular shapes (Appendix A). “Looking carefully at these two shapes, which one do you think takes up more space? Which shape has more of the shaded area?”
  • Students are expected to 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?”
  • Next, introduce a single square unit. “Just to be sure that these two shapes really do have a different area, we’re going to measure and compare them.” Slowly hold up one of the blue square units for all students to see. “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. while saying this, deliberately place the square in the top left corner.
  • “Once you think you know how many it might take, just keep it a secret and put your hands behind your back. Now, still 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 it is clear that 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. “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 other rectangle (i.e., start with one square in the corner and have students estimate how many, and if need be, start adding rows). 
  • “So just remind me. How many squares did it take to cover up this shape (point to smaller rectangle)? And how many squares did it take to cover up this shape (point to the larger rectangle)?”
  • Confirms students answers. “That’s exactly right. So what shape has the larger area? What shape has the smaller area?” 

Part 2: Different sized shapes, same area

  • Next, present students with two different sized shapes but with the same area (Appendix B). “Look carefully at these two shapes and think about how much space the shaded part is taking up. 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. “Again, just like before, how many of these blue squares will it take to cover each shape? Now, don’t say anything just yet. Let’s start with this shape (point to the cross shape). Look carefully and see if you can figure out how many squares it will take to cover it. You can use your fingers to show everyone how many you think it might take.”
  • Take a look at the students’ responses and ask each student to share her/his response. Next, move onto measuring the next shape.
  • At this point, some students might come to see that the two shapes share the same area. Ask around to see whether other students might also see how this could be possible.
  • Next, tile each shape and show students that in fact the two shapes do share the same area. If it hasn’t come up yet, ask students: ” Here’s a challenge for you. Is there anything you can do to this shape (pointing to the cross shape) to make it look like this one (pointing to the ‘b’ shape). Is there any way to turn one of these shapes into the other one just by moving one square tile?”

Part 3: Composing shapes of more, less, and the same area

  • Hand out pre-made shapes to students (Appendix C). Each student will be given a different sized shape, but will have 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 and for this reason can be deliberately handed out to scaffold students’ learning according to their comfort and proficiency with the task. “Look around at all the different shapes. just by looking, see whether you can find the shape with the largest area. What about the shape with the smallest area?”
  • After students have responded, hold up the square tile and once again ask them to estimate how many squares it will take to cover their shape.
  • Hand out a single bleu square tile and see whether having the tile in hand helps or changes their estimates. Instead of asking students to share their estimates with others, place the role of tile distributer and ask students how many more tiles they will need to cover their entire shape: “Let’s pretend that i’m the tile store shopkeeper, and you’re going to order tiles from me and my helper. You will need to let me know exactly how many more tiles you will need to cover your entire shape. Remember that you already have one tile. So, how many more tiles would you like to order to cover the entire shape?”
  • 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), engage students in a number of questions that deal with ordering and comparing their shape to others: “Hmm. This is interesting, Some of your shapes need the same number of tiles to cover them. Let’s start wit the smaller shapes. who thinks they have the smallest shape – the shape that needs the fewest number of squares?”
  • At this point, 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. “Is there any way to turn this shape (point to ‘L’) into this shape (pointing to ‘V’). Now, look very carefully. 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.

Part 4: Composing shapes with 6 square tiles

  • For this last part of the lesson, students will be presented wit ha game-like challenge. It is important that before students are provided with the opportunity yt build their shapes that they are shown the rules of the game (entire edges of the squares must touch). Model both correct and incorrect alignment of tiles.