Cynthia Lanius

Rectangle Pattern Challenges

Stage No. 1

Stage No. 2

Stage No. 3

Preliminary Questions

  1. Examine Stage No. 1 and Stage No. 2. Then examine Stage No. 2 and Stage No. 3. Describe what you have to do to Stage (N - 1) to create Stage N. (This problem is asking about the designs themselves, not the mathematical model.)
  2. Observe the designs looking for patterns. Use the patterns to predict Stages 0, 4 and n. Organize your information into the table below.
Stage No. 0 1 2 3 4 n
No. Blue
No. Red
No. Green
Total No.

Thought Questions

  1. Which color of squares is growing at the slowest rate? At the fastest rate? Graph the 3 formulas to get a picture of the growth of all three.
  2. How many squares of each color will be in the 8th stage of the design?
  3. Will the design use 42 blue squares in any stage? Will it use 102 red squares? Will it ever use 830 squares in all? If so, state the stage number for each answer.
  4. What are the dimensions (length and width) of the rectangular designs for each stage including n? Show that when you multiply the length and width you get the total number of squares in each rectanglular design. Organize your information in the table below.
Stage No. 1 2 3 4 n


On square grid paper create your own design, showing at least 3 stages. It must have at least two lines of symmetry, and it must follow a regular numerical growth pattern. On a separate sheet of paper, fill in the calculations in a table like the one above. Be prepared to exchange papers with the other students in class and figure out one another's patterns.

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