Answer
Explain how to locate the point $\sqrt 2$ on the number line using the figure provided:
We can locate $\sqrt 2$ by marking a point at 1 on the number line, and drawing a perpendicular line from that point with length 1. Place the pin of a protractor on 0 and the pencil on the end of the perpendicular line that's not on the number line. Using the protractor draw an arc (towards the right) until you hit the number line. The point where the arc meets the number line is $\sqrt 2$.
Can you locate $\sqrt 5$ by a similar method?
Yes we can locate $\sqrt 5$ using a similar method. Rather than relying on the properties of a 1, 1, $\sqrt 2$ triangle, we would instead find 2 on the number line and create a perpendicular line of length 1 thereby creating a 1,2,$\sqrt 5$ right triangle. Using the protractor-arc method, we would locate $\sqrt 5$ on the number line.
How can the circle figure shown help us to locate $\pi$ on the number line?
Because the radius of this circle is one, we know it is the unit circle. The circumference of the unit circle is $\pi$.
Thus, if we rotate the unit circle on the number line, after one full rotation it will land on $\pi$.
List some other irrational numbers that you could locate on the number line:
Using the method involving properties of right triangles that we used to locate $\sqrt 2$ we can find other irrational numbers so long as they are the hypotenuse of a right triangle. For instance, the hypotenuse of the 2, 3, $\sqrt 13$ right triangle is $\sqrt 13$, an irrational number. Using the same arc-protractor method, we can find $\sqrt 13$ on the number line. We could use this method to find an infinite amount of irrational numbers that are hypotenuses to right triangles.
We can also use the method of circle circumference. For example, the circumference of a circle with radius 2 is 4$\pi$, an irrational number. Thus if we rotate that particular circle on the number line, after one full cycle it will land on 4$\pi$. We could use this method to find an infinite amount of irrational numbers that are circumferences of circles.
Work Step by Step
Steps to locate $\sqrt 2$
1. Mark the point 1 on the number line
2. Draw perpendicular line from 1 of length 1 at that point
3. Put the pin of a protractor at 0 on the number line, and the pencil at the end of the perpendicular line that's not on the number line
4. Then draw an arc going towards the right until you reach the number line
5. The point where the arc meets the number line is $\sqrt 2$ based on the properties of the 1, 1, $\sqrt 2$ right triangle.
Steps to locate $\sqrt 5$
1. Mark the point 2 on the number line
2. Draw perpendicular line from 2 of length 1 at that point
3. Put the pin of a protractor at 0 on the number line, and the pencil at the end of the perpendicular line that's not on the number line
4. Then draw an arc going towards the right until you reach the number line
5. The point where the arc meets the number line is $\sqrt 5$ based on the properties of the 1, 2, $\sqrt 5$ right triangle.
Steps to explain how the circle can locate $\pi$:
1. Recognize that the circle in the diagram is the unit circle because it has a radius of 1.
2. Recognize that the circumference of the unit circle is $\pi$.
3. Since the circumference is $\pi$, then 1 full cycle of the unit circle will measure $\pi$.
4. Thus, by rotating the unit circle on the number line, the point where it lands after 1 full cycle is $\pi$
List some other irrational numbers that you can locate on the number line:
1. Recognize the method we used to locate $\sqrt 2$ and $\sqrt 5$ can be expanded to an infinite amount of right triangles with irrational hypotenuses.
2. Thus, by using the same protractor arc method we can measure irrational numbers such as $\sqrt 13$ based on the 2, 3, $\sqrt 13$ right triangle.
3. Recognize the circle method we used to locate $\pi$ can also be expanded to an infinite amount of circles with irrational circumferences.
4. Thus by using the same circle-rotation method we can measure irrational numbers such as 4$\pi$ by rotating a circle with radius 2 on the number line.
