Patterns #2

Introduction (continued)

Exercise

Consider the sequence of figures below, and assume the pattern continues.

Figure 1
Figure 2
Figure 3

  1. What does Figure 4 look like? Draw it out. How many filled-in squares are in Figure 4? Illustrate on your picture how you are counting them, and write down the mathematical work you used to get the answer. Avoid counting one-by one.






  2. Start by filling in the first 4 rows of the table below. In the column for Work, show the arithmetic for how you are computing the number of squares. Then look for a pattern in the table, and use it to determine the number of squares in Figure 10 without actually drawing Figure 10.

    Figure Number Work Number of Squares
    1                       
    2    
    3    
    4    
    \(\vdots\) \(\vdots\) \(\vdots\)
    10    

  3. What is the relationship between number of squares in Figures 1 and 2? What about Figures 2 and 3? Figures 3 and 4? What is the pattern?


Exercise

Consider the sequence of figures below, and assume the pattern continues.

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6

  1. What does Figure 7 look like? Draw it out. How many squares are in Figure 7?







  2. What is the relationship between the number of squares in Figures 1, 2, and 3? What about Figures 2, 3, and 4? Figures 3, 4, and 5? What is the pattern?






  3. Use the pattern you found to fill in the table below without drawing any more of the figures.

    Figure Number   1     2     3     4     5     6     7     8     9     10  
    Number of Squares