July Problem to Discuss

Here is July's Problem of the month for us to discuss:

Finding area of a triangle with a shaded area within a square

Questions to reflect on as you answer this question:

  1. What do you notice?
  2. How would your learners solve this problem?  What would be their strategy?  What is your strategy? 
  3. How does the area change if the spaces are 2 units? 1/2 units?
  4. What other questions do you have about this problem?
  5. Can you come up with a story that would represent this problem?

Please share out your experiences and thoughts about this problem.  Let's have fun thinking and being creative!

Brooke Istas
Moderator, Math and Numeracy CoP

Comments

I love this problem and it got me thinking in a number of ways. I am not sure I can answer each of your questions correctly but here goes :)

1  What do you notice? I immediately jumped into triangle formulas trying to find lengths of the big triangle and determined that was not going to work. I noticed that my standard algorithm for finding polygonal areas were not going to easily apply. I also noticed that the only measurement that was given was the scale unit of the grid and I love that. Finding the lengths of the sides I wanted to find looked like too much work. Way too many of our problems today give too much information.

2. Learners might struggle if they are used to simply one or two strategies when presented with an area problem. I suspect they would start by using area formulas of squares and of triangles and they would try to find lengths of sides which would frustrate them. It will be interesting in the fall when I have students to see if they persist beyond that strategy or if they derive other strategies.

I personally jumped to the ratio of the square's area to the area of the small triangle's area. I remembered that two triangles make a rectangle and by eyeball it looks really close that 3 full rectangles made from pairs of the small triangle would fill that 2x2 square area. Since that is 6 triangles to one big square I figured the area of the triangle would be 1/6 the square's area (which is 4square units) making the area of the shaded region 2/3 square units.

My mind then jumped to the triangle area formula of 1/2 b h and I realized that 1/2 of b (which is 2 units long) gives us just one so the area of the triangle has to be just the height of the triangle. Comparing the height of the triangle to the length of the square's side it looked like 3 heights would = the 2 square length of the large square. 3h = 2 which yields a result of h=2/3 square units. This helps me feel like I might be on the right track :) 

3. Since my focus was on ratios, I am not so sure my processing would change for me. Find the area of the square then divide by 6 because there are 6 triangles still that make up the square. So 2 unit square would be 1/6 * 4^2 = 1/6 * 16 = 8/3 or 2 2/3. Likewise 1/2 units would be 1/6 * 1^2 = 1/6. 

4. I wonder what the un-shaded area of the large triangle might be :) I wonder if student were to construct the shapes in question out of paper, done to scale, if they might spatially see how to derive a solution easier? 

5. Ugg the story for this one is tough for me and I really look forward to reading some good stories from others. Everything I come up with seems so artificial and I so dislike fake "real life" stories. One thought was that a person could wish to create a triangular mosaic to fill a room. The person wishes to have the triangles fill small square areas in such a way to be able to randomize the square tiles around the room and yet have the triangular mosaic still make a pleasing arrangement. The person thinks they might want to use 4 colors to make up the small triangles but she wants to know if that is a good choice? How many colors would you suggest the person uses? Are there multiple options that could be a "best fit" for the crazy pattern arrangement the person desires? Is there a problem to the person picking 4 different colors to make up the small triangles? An extension would be to offer that each square tile is 2x2 square and the entire room is 12 x 12. How many square units of each color would the person need to buy to complete this project (assuming no mistakes were ever made in the cutting and placing of the tile )? 

Anxious to hear others thoughts and extensions! I love to see problems that get me thinking in other ways and I love to hear the different ways people approach those problems!