## High order questions

**Below are questions students can be asked at different points throughout the lesson, before and after viewing the video.**

**Prior to Presentation**(These questions should be given before any information is presented in order to direct students thinking and connect new information with what is already known.)

What does the term rational mean in your own words? How do you think this applies to the term rational numbers? What does this tell us about the properties of this number set?

*When I think of rational, I think of a type of conduct. A rational person will react in ways that are not extreme and are reasonable; they have a sound mind and make good decisions. Mathematically, rational numbers are plain, basic numbers, they can be described easily.*

What does the term irrational mean in your own words? How do you think this applies to the term irrational numbers? What does this tell us about the properties of this number set?

*Irrational means to be unreasonable and lacking sense. In mathematics this could describe numbers that cannot be written simply. (Students may even predict that these are numbers that do not make sense- be sure by the end of the lesson they gain the understanding that the values they represent are defined, but they cannot be written as simply as rational numbers).*

**Throughout**(Present these questions throughout the lesson, after materials/activities are presented.)

What is the significance of using pi to find the area of a circle? Is this a beneficial method in comparison to using the circumference? If so, how?

*We cannot calculate the area of a circle as easily as we can the area of a rectangle. Pi allows us to find the area of a circle without knowing its circumference. There are some objects which may be too large to measure its circumference, or that it may be too difficult to measure.*

What is the significance of pi in relation to the circumference of a circle?

*Pi allows us to find the circumference of a circle without having to measure it. When we find the perimeter of a rectangle we add the lengths of all sides. However, a circle is made of one continuous line, which cannot be measured with a ruler. We can, however, measure the distance across a circle (its diameter) and using pi can easily find its circumference. This also allows us to calculate the circumference of much larger objects.*

What is the relationship between rational and irrational numbers?

*They are both numbers, but they each have their own definition. A rational number can be put in a fraction, whereas an irrational number cannot. A rational number has a continuous pattern, and an irrational number does not. An irrational number never ends.*

What would be the effects of using a number system in which decimals could only be extended to a finite amount, not infinitely, for example, a system in which they could only extend 5 decimal places?

*If our number system did not allow us to extend decimal places infinitely, our solutions and calculations would not be as accurate. There are objects in our world in which measurements have to be precise and objects which are microscopic in which measurements have to extend to very small decimals. We would not be able to measure such objects accurately.*

Are there times when it would be sufficient to use a rounded value of an irrational number, if so, when? Are there times when it is not, and when?

*We can round an irrational number when we are trying to tell someone its value or when we are solving some mathematical problems. If it is a problem in which it doesn't matter if we are one ten-thousandths of a decimal off, then it would be okay to round. However, if the problem and solution are more important, such as the work of an engineer, then it would be best to use the most accurate form of the number, such as root 2, instead of 1.414. This is because their work needs to be more precise.*

**Reflective Questions**(Questions can be asked after material has been introduced and video has been viewed. Reflection helps students to understand and remember information. It encourages them to further investigation.)

How could you explain the meaning of an irrational number to someone else, possibly a peer that does not know what an irrational number is? What specific information would be important to include?

*An irrational number is a number that, when in decimal form, the decimal does not end or repeat. There are some significant irrational numbers which occur naturally, such as pi, which is the ratio of the circumference of a circle to its diameter. There are infinite amount of irrational numbers.*

Why are irrational numbers significant?

*It is important to know that there are numbers that exist whose decimal does not end; that cannot be written with integers as a fraction. This extends our understanding of real numbers and the amount of numbers that exist. Some irrational numbers are more significant because they are naturally occurring.*

In your opinion, what is the most important concept of irrational numbers? Why?

*Specific irrational numbers that represent specific natural ratios are one of the most important concepts of irrational numbers. They allow us to perform calculations more efficiently and give us a greater understanding of the problem it is involved in.*