## Standards

Below are the Common Core State Standards for Mathematics that are addressed through the lesson.

1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Curricular Connections: The concepts and information presented through this lesson will engage students by teaching through technology and participation in active activities. Students will gain an understanding of rational and irrational numbers. The Xtranormal video presents concepts of irrational numbers through the topic of circles and pi. In the video students are also presented with the use of pi in the formulas for a circles circumference and area. Although the definition of a rational number is not stated and the video is specifically speaking of pi and circles, students will learn that a class of numbers called irrational numbers exists. They will be presented with a generic explanation of the importance of pi, it is not explained thoroughly or extensively, but students are given an idea. Through the activities students will be able to take learning off of the paper. When they participate in the number line activity, they will be able to "locate them approximately on a number line diagram," (Common Core, 8.NS.2). When students complete the first activity, of categorizing the numbers as rational or irrational, converting all fractions into decimal form, they will, "show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number," (Common Core, 8.NS.1). Although they will not be directly working with formulas, through the video they are presented with two formulas, the circumference and the area of a circle. Students will also be graphing in one of their activities and noting that the slope of the line is approximately pi. These experiences will indirectly complement their learning and understanding of formulas.

**The Number System 8.NS****Know that there are numbers that are not rational, and approximate them by rational numbers.**1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

**Functions 8.F**

**Define, evaluate, and compare functions.**1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

*For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.*3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

*For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.***Use functions to model relationships between quantities.**4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Curricular Connections: The concepts and information presented through this lesson will engage students by teaching through technology and participation in active activities. Students will gain an understanding of rational and irrational numbers. The Xtranormal video presents concepts of irrational numbers through the topic of circles and pi. In the video students are also presented with the use of pi in the formulas for a circles circumference and area. Although the definition of a rational number is not stated and the video is specifically speaking of pi and circles, students will learn that a class of numbers called irrational numbers exists. They will be presented with a generic explanation of the importance of pi, it is not explained thoroughly or extensively, but students are given an idea. Through the activities students will be able to take learning off of the paper. When they participate in the number line activity, they will be able to "locate them approximately on a number line diagram," (Common Core, 8.NS.2). When students complete the first activity, of categorizing the numbers as rational or irrational, converting all fractions into decimal form, they will, "show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number," (Common Core, 8.NS.1). Although they will not be directly working with formulas, through the video they are presented with two formulas, the circumference and the area of a circle. Students will also be graphing in one of their activities and noting that the slope of the line is approximately pi. These experiences will indirectly complement their learning and understanding of formulas.