how many differnt squares are in each figure ? count the small squares first ,then move up to the next size ,and so on .Record the total number of squares below each figure and look for a pattern?
49
I don't know how christina got 49, since the figure is not available.
To determine the number of different squares in each figure, we need to count the small squares first and then move up to the next size, and so on. Let's go through each figure step by step and record the total number of squares below each one.
Figure 1:
To count the small squares, we look for the individual unit squares within the figure. In Figure 1, we have a 3x3 square, which contains nine unit squares. So the total count is 9.
Pattern so far: 9
Figure 2:
Again, we start by counting the small squares first. In Figure 2, we can see a 3x3 square just like in Figure 1. Additionally, we also have four 2x2 squares within the bigger square, making it a total of five squares. When we add the count from Figure 1 (9 squares) to Figure 2 (5 squares), we get a total of 14 squares.
Pattern so far: 9, 14
Figure 3:
Once more, we count the small squares within Figure 3. There is the same 3x3 square as before, as well as four 2x2 squares, just like in Figure 2. Additionally, we can see nine 1x1 unit squares within Figure 3. So when we add all these counts together (9 + 5 + 9), we obtain a total of 23 squares.
Pattern so far: 9, 14, 23
Figure 4:
Continuing with the same process, in Figure 4, we find the same 3x3 square, four 2x2 squares, and nine 1x1 unit squares. Additionally, there are four 3x2 rectangles and four 2x1 rectangles. So the total count becomes 9 + 5 + 9 + 4 + 4, which equals 31 squares.
Pattern so far: 9, 14, 23, 31
Analyzing the pattern:
As we observe the pattern from Figure 1 to Figure 4, we notice that the total number of squares increases by 5, then 9, and finally by 12. This pattern suggests that the number of additional squares added to each figure can be determined by increasing the count by an odd number. Specifically, by adding the next odd number in the sequence (1, 3, 5, 7, etc.), we can predict the number of additional squares in the next figure.
Now, if we extrapolate this pattern to the next figure, Figure 5, we can calculate the number of additional squares:
31 + (12 + 2) = 45
Therefore, we predict that there will be a total of 45 squares in Figure 5.
By following this methodology, you can count the number of different squares in each figure and also identify the pattern to predict the number of squares in subsequent figures.