Analyze the differences between the amount of squares of each block.
See solution.
Practice makes perfect
Consider the given squares formed by smaller blocks.
Notice that Square 3 is formed of 3* 3 blocks, while Square 4 is formed of 4* 4 blocks. Let's construct a table with the amount of blocks per square.
Square
n * n
Amount of Blocks
3
3* 3
9
4
4* 4
16
5
5* 5
25
6
6* 6
36
We can see that the number of blocks per square is written as a product of repeated factors. This means that we can rewrite them as a power. To do so, remember that the base of a power is the repeated factor. The exponent of a power indicates the number of times the base is used as a factor.
Let's rewrite the table using this information.
Square
n* n = n^2
Amount of Blocks
3
3* 3 = 3^2
9
4
4* 4 = 4^2
16
5
5* 5 = 5^2
25
6
6* 6 = 6^2
36
To predict how many blocks we will need to add to each square to make the following square in the pattern, we can analyze the difference of blocks between the given squares to see if we can find a pattern.
Square S_m and S_n
m^2 - n^2
Added Blocks
S_3 and S_4
16-9
7
S_4 and S_5
25-16
9
S_5 and S_6
36-25
11
Notice that the difference between the powers is equal to the difference between the number of blocks. This means that we can predict the amount of blocks added to the next square by using the following equation.
Added Blocks=m^2 - n^2
Here, m represents the new square and n the previous square. Let's continue the table for the next pair of squares!
