If students do not draw a separate real number line, but instead try to represent the values of that satisfy the quadratic equations as points on the coordinate plane, consider asking:

• “Can you explain how you represented your solutions.”

• “What is the same and what is different about the solutions to and ?”

Select students to share their explanations. Display a graph of , and draw the lines , , and for all to see. Then make sure the real number line that shows the solutions is drawn separately.

Tell students that whenever we square a number, we multiply it by itself. When squaring a number on the number line, otherwise known as a real number, the result is one of the following:

• Positive, because a positive times a positive is positive.
• 0, because 0 times 0 is 0.
• Positive, because a negative times a negative is positive.

So squaring a real number never results in a negative number. We can see this in the graph of because none of the points on the graph are below the -axis. That tells us that the equation does not have any real solutions, which means none of the numbers on the number line make this equation true.

Let’s invent a new number that is not on the number line that does satisfy this equation. Let’s write it as and draw a point to represent this number. Let’s put it here, one unit above 0 on the real number line. Display this image for all to see:

Explain that is defined to be a solution to the equation , so

Lastly, tell students that even though it isn’t on the real number line and therefore isn’t a real number, it really is a number—it is just a different kind of number called an imaginary number. It could have been named a “blue number” or a “fish number.” The word “imaginary” shouldn’t be taken literally.

If students are unsure of how to represent multiples of on the imaginary number line, consider saying:

• “Can you explain how you drew an arrow that represents 3 on the real number line. How does that compare to how you would draw an arrow that represents 1?”

• “What is similar about going from 1 to 3 and going from to ?”

Tell students that they have just drawn points on the imaginary number line. The imaginary number line can be thought of as a vertical line that intersects the real number line at the point 0. Points on the imaginary number line are numbers that are a real number times the imaginary number . Positive multiples of (like ) are drawn above 0 on the imaginary number line, and negative multiples (like ) are drawn below 0 on the imaginary number line.

Here are some questions for discussion:

• “How are the real number line and imaginary number line the same? How are they different?” (They are the same because they are both lines, and points on the lines represent different numbers. Both number lines have the number 0 on them. They are different because all imaginary numbers have a factor that is .)
• “What happens when you square an imaginary number? Choose an imaginary number and square it. How is that different from what happens with real numbers?” (For example, . Imaginary numbers square to make negative numbers. Real numbers are always positive when they are squared.)