Lesson 2

How Does it Change?

  • Let’s describe some patterns of change.

2.1: Squares in a Figure

How does each expression represent the number of small squares in the figure?

An image of a block of squares. There are eight squares on rows 1, 2, 3 and 4 and five squares on rows 5 and 6. There are six squares missing from the upper right corner of the block.
  • Expression A: \(6 \boldcdot 8 - 2 \boldcdot 3\)
  • Expression B: \(4 \boldcdot 8 + 2 \boldcdot 5\)
  • Expression C: \(8+8+8+8+5+5\)
  • Expression D: \(5 \boldcdot 6 + 3 \boldcdot 4\)

2.2: Patterns of Dots

Pattern 1

Pattern. Step 0,1 dot. Step 1, 3 dots, 2 by 2 with upper left dot removed. Step 2, 5 dots, 3 by 2 with upper left dot removed. Step 3, 7 dots, 4 by 2 with upper left dot removed.

Pattern 2

Pattern. Step 0, 0 dots. Step 1, 1 dot. Step 2, 4 dots, arranged 2 by 2. Step 3, 9 dots, arranged 3 by 3.
  1. Study the 2 patterns of dots.
    1. How are the number of dots in each pattern changing?
    2. How would you find the number of dots in the 5th step in each pattern?

  2. Complete the table with the number of dots in each pattern.

    step number of dots in Pattern 1 number of dots in Pattern 2
    0    
    1    
    2    
    3    
    4    
    5    
    10    
    \(n\)    

  3. Plot the number of dots at each step number.

    Pattern 1

    Blank coordinate plane with grid, origin O. Horizontal axis from 0 to 6 by 1’s, labeled “step number”. Vertical axis from 0 to 30 by 5’s, labeled “number of dots”.

    Pattern 2

    Blank coordinate plane with grid, origin O. Horizontal axis from 0 to 6 by 1’s, labeled “step number”. Vertical axis from 0 to 30 by 5’s, labeled “number of dots”.
  4. Explain why the graphs of the 2 patterns look the way they do.

2.3: Expressing a Growth Pattern

Three steps of a growing pattern.

Here is a pattern of squares.

  1. Is the number of small squares growing linearly? Explain how you know.
  2. Complete the table.
    step number of small squares
    1  
    2  
    3  
    4  
    5  
    10  
    12  
    \(n\)  
  3. Is the number of small squares growing exponentially? Explain how you know.


Han wrote \(n(n+2) - 2(n-1)\) for the number of small squares in the design at Step \(n\).

  1. Explain why Han is correct.
  2. Label the picture in a way that shows how Han saw the pattern when writing his expression.

Summary

In this lesson, we saw some quantities that change in a particular way, but the change is neither linear nor exponential. Here is a pattern of shapes, followed by a table showing the relationship between the step number and the number of small squares.

Three steps of a growing pattern.
step total number of small squares
1 2
2 5
3 10
\(n\) \(n^2+1\)

The number of small squares increases by 3, and then by 5, so we know that the growth is not linear. It is also not exponential because it is not changing by the same factor each time. From Step 1 to Step 2, the number of small squares grows by a factor of \(\frac{5}{2}\), while from Step 2 to Step 3, it grows by a factor of 2.

From the diagram, we can see that in Step 2, there is a 2-by-2 square plus 1 small square added on top. Likewise, in Step 3, there is a 3-by-3 square with 1 small square added. We can reason that the \(n\)th step is an \(n\)-by-\(n\) arrangement of small squares with an additional small square on top, giving the expression \(n^2 + 1\) for the number of small squares.

The relationship between the step number and the number of small squares is a quadratic relationship, because it is given by the expression \(n^2 + 1\), which is an example of a quadratic expression. We will investigate quadratic expressions in depth in future lessons.

Glossary Entries

  • quadratic expression

    A quadratic expression in \(x\) is one that is equivalent to an expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).