Getting Started 
Misconception/Error The student rounds or truncates each value so that it can be graphed at a point to which the number line has been scaled. 
Examples of Student Work at this Level The student:
 Approximates and graphs as 2 since the number line is scaled in whole numbers.
 Approximates and graphs as 3.1 since the number line is scaled in tenths.
 Approximates and graphs 3.428571… as – 3.42 (truncated value) or 3.43 (rounded value) since the number line is scaled to hundredths.

Questions Eliciting Thinking Why did you graph at two?
Is equal to two? Is equal to three?
What does equal?
How did you determine to what place to round ?
What do you know about the decimal representation of irrational numbers? Why do you have to approximate irrational numbers? 
Instructional Implications Provide instruction on irrational numbers. Describe an irrational number as a number that cannot be written in the form where a and b are integers. Guide the student to understand that when irrational numbers are written in decimal form, they neither terminate nor repeat. Describe taking the square root of a nonperfect square as a way of generating examples of some irrational numbers. Compare and contrast irrational and rational numbers. Consider using MFAS task Rational Numbers (8.NS.1.1).
Discuss the need for approximating irrational numbers in realworld contexts, and model how to graph irrational numbers using rational approximations. Provide the student with opportunities to approximate irrational numbers in realworld contexts. 
Moving Forward 
Misconception/Error The student graphs correctly on the number line scaled to whole numbers and tenths, but graphs incorrectly on the number line scaled to hundredths. 
Examples of Student Work at this Level The student graphs between two and three and graphs between 3.1 and 3.2 but graphs 3.428571… at 3.42 or 3.43 (not between).

Questions Eliciting Thinking How did you determine where to graph and ?
How did you determine where to graph 3.428571...? Is this number closer to 3.42 or 3.43? 
Instructional Implications Be sure the student understands that irrational numbers can be approximated with rational decimal numbers to varying degrees of precision. Explain that an irrational number can be approximated more precisely using more place values. Make explicit the difference between rounding rational numbers and approximating irrational numbers. Model how an irrational number can be approximated more precisely by using more of its place values. Show the student the approximation of an irrational number on number lines scaled differently, so the student can understand that the more he or she “zooms in” on the number line, the more precise the approximation.
Consider using MFAS task Approximating Irrational Numbers (8.NS.1.2). 
Making Progress 
Misconception/Error The student graphs negative values incorrectly. 
Examples of Student Work at this Level The student graphs and correctly but graphs 3.428571… slightly to the right of 3.42 instead of slightly to the right of 3.43.

Questions Eliciting Thinking How did you decide where to graph 3.428571…?
What happens to positive numbers as you move to the left on a number line? What happens to negative numbers as you move to the left on a number line?
Did you approximate 3.428571...? Is 3.428571... greater than or less than 3.42? Greater than or less than 3.43? 
Instructional Implications Review ordering negative numbers reminding the student that the more negative a number, the smaller it is. Use the negative portion of the number line to explain how values increase as you move to the right and decrease as you move to the left. Consider using MFAS task Positions of Numbers (6.NS.3.7) and Illustrative Math task Integers on the Number Line 2 (6.NS). Provide the student with additional opportunities to graph negative values on a number line.
If the student finds and corrects his or her own mistake, consider the student to be at Almost There or Got It, depending on his or her explanation. 
Almost There 
Misconception/Error The student does not appear to understand the difference between an irrational number and its decimal approximation. 
Examples of Student Work at this Level The student graphs each irrational number by referring to an appropriate digit in the decimal representation of the number but the student accepts the calculator approximation as an exact value. The student says that equals 2.4 or that equals 3.14.

Questions Eliciting Thinking What value did your calculator display for ? Is this exactly what equals or is this an approximation?
What does equal? If you wrote as a decimal number, what would happen to the decimal part?
Can you ever write the decimal part of an irrational number completely? 
Instructional Implications Review the definition of an irrational number and discuss why a decimal approximation is used to graph an irrational number on a number line. Introduce the student to the “approximately equal to” symbol (i.e., ) and guide the student to use it when appropriate. Discuss the distinction between writing a number in exact form, such as , and in an approximate form, such as 2.4.
Ask the student to use a calculator to approximate an irrational number such as to varying degrees of precision. Model using the terminology of place value to assist the student in describing each approximation. For example, say, “The square root of six is about 2.45 when it is approximated to the hundredths place.” Or say, “This number line is scaled to whole numbers so I approximated to the tenths place so that I could locate it between 2.0 and 3.0.”
Provide the student with additional opportunities to approximate and graph irrational numbers to varying degrees of precision. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student graphs:
 about midway between two and three,
 about midway between 3.1 and 3.2, and
 3.428571… between 3.42 and 3.43 but closer to 3.43.
Upon questioning, the student explains that the decimal part of an irrational number is infinite and does not repeat so he or she used an approximation of each irrational number in order to graph it.

Questions Eliciting Thinking How do the scales of each number line differ?
How does the scale of the number line help you determine the location of each number?
What value did you get from your calculator for ? Is this exactly what equals or is this an approximation?
What kind of number is ? If you wrote as a decimal number, what would happen to the decimal part?
Can you ever write the decimal part of an irrational number completely? 
Instructional Implications Give the student a set of irrational numbers presented in various formats and ask him or her to put them in order from least to greatest. Ask the student to explain his or her strategy.
Introduce the student to the “approximately equal to” symbol (i.e., ) and guide the student to use it when appropriate. Discuss the distinction between writing a number in exact form, such as , and in approximate form.
Introduce evaluating expressions involving irrational numbers (e.g., or ). 