After a few days working on surface area, I wanted to challenge my students and address the standard listed below by with a task that applied area in a different context. I debated about using this task because I couldn’t envision how I would engage students and encourage them to persist in proving that a square of certain areas on a coordinate grid is not possible.

**HSG-GPE.B.7: **Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Here is the task from Illustrative math (click on the image below to go to their site)

I started class by providing each student with a grid whiteboard and marker and I asked them if they could draw a square whose area is 1 square unit. They did this with no problem and sat looking at me like I was crazy.

Next, I asked if it was possible to draw a square whose area was 2 square units, with the corners on the grid. Common student responses below:

- “It’s impossible!”
- student draws a 1 x 2 rectangle
- student grabs calculator and computes the square root of 2, then attempts to draw a square with sides measuring exactly 1.41421356 units long.
- student divides 2 by 4 draws a square with sides 1/2 a square long, then realized its area is 1/4.
- student divides 2 by 2 and draws a 1 x 1 square, realizes this is their square with an area of 1.

So THIS is what Robert Kaplinsky means when he talks about Depth of Knowledge! It was very insightful to show student understanding of squares, length, area, and what a square unit means.

Eventually a few students said they figured it out, and were so excited to explain to me how they knew it was a square and how they know its area was 2 square units.

I asked students to try drawing a square with integer each area from 1-10 & students were engaged all period. We kept track of the areas that students thought were not possible.

The next class period I gave students whiteboards and a sheet of graph paper. I asked them to draw the squares that they could, and also determine the length of each side, and then double check the area using the formula for the area of a square to ensure that their graphic solution matched their calculated area.

After about 20 minutes I asked the class if they noticed anything about the relationship between the side lengths and the areas of the squares. Many said it appears that if the area is x, then the side length is the square root of x. AHA!

With additional discussion students were able to explain that since there are no pair of integers squared that sum to 3, 6, or 7 it is impossible to draw a square with those areas whose corners were on the grid. I also asked students what would be areas of the next 3 largest squares that would be possible to draw.

They were able to tell me and explain how they knew!

Through this task students learned more than I possibly could have taught them through 2 days of direct instruction and drill like practice. Students developed a more intuitive understanding of:

- perpendicular lines having opposite reciprocal slopes
- the meaning of slope
- the Pythagorean Theorem
- area
- properties of squares
- squaring & square roots
- proof.

Annie was right in saying that teachers who argue that there is no time to implement these tasks with such a jam packed curriculum are missing the point. I have to constantly remind myself, that the hurrier I go the behinder I get!