This picture could be used in a problem concerning geometric series, such as, find a formula to represent the area of the first 3 sequences in this diagram. The Common Core State Standard that aligns with this type of problem is CCSS.Math.A-SSE.4, which involves students using the formula to sum a finite geometric series.
The mathematical concept is related to mathematical process because the students need to know how to find the area of a triangle and then apply this information to find a formula to represent the area of a triangle that is being halved in each consecutive sequence. This is also connected to mathematical reasoning because the students must be able to think about what is happening to the dimensions of the triangle if the area is being cut in half with each new term; therefore, the students must be able to use mathematical reasoning to conclude what is happening in order to derive a mathematical equation that represents the geometric series.
The picture and task suggested for this is great for algebra. I like that it is clearly a series problem, but you can trick kids into thinking about it and finding a solution through geometry.
The use of this picture could be extended further by asking students to describe the series as a function. This would be for standard CCSS.Math.HS.F-BF.1, “write a function that describes a relationship between two quantities”.
Another way to use this same image for a different problem would be to superimpose it on top of a coordinate plane. Then you would have students use the coordinate to compute the perimeter of the square and each triangle as well as the area of the square and the area of each triangle within the square. This relates to the following CCSS standard:
CCSS.Math.HSG-GPE.B.7
Use coordinates to prove simple geometric theorems algebraically
7. Use coordinates to compute perimeters of polygons and areas of
triangles and rectangles, e.g., using the distance formula.★