Growing Squares

Investigation: Growing Squares

There is evidence from ancient Babylon (2000 BC -500 BC) that they
developed basic ideas in algebra and geometry. One piece of evidence can
be found in the Yale Babylonian Collection. It is a round clay tablet with a
picture of a square and two diagonals. Further analysis of the tablet reveals
that the Babylonians had an understanding of the ratio between side length
and diagonal.

In this investigation you will explore relationships within squares. As you work, look for answers to the
following questions:

What are the relationships between area and side length of squares?
How can this relationship be described and represented?

Area and Side Length

1. The areas of the squares are growing by 1 square unit.
a. Determine the side length of each square.

b. Create a table, graph, and function rule that represents the relationship (area of square, side
length) for squares with an area up to 25 square units.

c. Describe the following key features of the function.
intercepts:
intervals of increase / decrease:
intervals of positive / negative:
rate of change:
domain:
range:

d. How is this function different than other functions you have studied? How is it similar?

Developed by K McPherson 1

The Square Root Function

The function used to represent the side length of a square as a
function of its area is called a square root function. The graph of
( ) = √ is has distinct key features.

2. The domain is restricted to ≥ 0.
a. Explain how this is shown in the graph?

b. Why is the domain restricted?

3. The function is increasing at a decreasing rate.
a. Explain how this is shown in the graph?

b. What does it mean to have a decreasing rate? Support your answer with examples from a table
of values.

4. Graph each of the following functions and compare it to the function ( ) = √ . Describe any

similarities and any differences.

a. ( ) = √ + 5 b. ( ) = √ − 3 c. ( ) = 2√

Summarize the Mathematics
a. Sketch a graph of the function ( ) = √ and label the key features.

b. Explain why the function ( ) = √ has a domain of ≥ 0 and the function ( ) = √ + has a
domain of ≥ .

Developed by K McPherson 2