Getting Started 
Misconception/Error The student is unable to correctly estimate values of irrational numbers using a calculator. 
Examples of Student Work at this Level The student:
 Attempts an estimation strategy that does not involve a calculator.
 Disregards the order of operations conventions when using a calculator and incorrectly calculates as 5 or 5.4.
 Finds an interval bounded by two consecutive whole numbers in which the irrational number falls and then guesses the value of an estimate.
 Makes several errors including calculating as .

Questions Eliciting Thinking How might you use your calculator to estimate these values?
How would you apply the order of operations rules to calculating ? Is that what you “asked” your calculator to compute?
What is ? Is the same as 9 2? 
Instructional Implications Provide instruction on and practice using a calculator to evaluate numerical expressions involving fractions and square roots. Review the definitions of rational number and irrational number. Discuss with the student the need to estimate irrational numbers in order to graph and order them. Ask the student to approximate irrational numbers by rounding to different decimal places and to consider the number of decimal places needed in order to adequately compare two values.
Provide the student with additional opportunities to find rational approximations of irrational numbers using a calculator. 
Moving Forward 
Misconception/Error The student makes errors in ordering or graphing the set of numbers. 
Examples of Student Work at this Level The student correctly estimates each irrational number but:
 Graphs and in the same location or in reverse order.
 Graphs numbers in the correct order but locates several incorrectly on the number line.

Questions Eliciting Thinking Do and have the same value? Which is larger? How can you tell?
By what interval is this number line scaled? How can you tell if numbers are graphed in the right places on this number line? 
Instructional Implications Assist the student in identifying the scale on the number line and then correctly graphing each value. Provide additional sets of irrational numbers for the student to estimate and graph. Discuss with the student considerations in determining a scale that is appropriate for graphing a given set of numbers. Provide opportunities for independent practice in scaling and graphing irrational numbers on a number line. 
Almost There 
Misconception/Error The student does not demonstrate an understanding of the difference between an irrational number and its rational approximation. 
Examples of Student Work at this Level The student labels the graphed points on the number line with the approximations rather than the original numbers to be graphed.
Upon questioning, the student does not appear to understand that the value produced by the calculator is a rational approximation of the irrational number. 
Questions Eliciting Thinking What kind of number is ? What kind of number is 1.41? Are these numbers equal?
Could you have ordered the original numbers without first approximating them? What was the point of using the calculator?
What numbers were you asked to graph – the original irrational numbers or their decimal approximations? 
Instructional Implications Review the definitions of rational number and irrational number. Discuss with the student the need to estimate irrational numbers in order to graph and order them. Ask the student to label his or her graphed points with the corresponding irrational numbers. Provide the student with additional opportunities to find rational approximations of irrational numbers in realworld contexts and in the context of graphing. 
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 approximates each irrational number and accurately graphs each number on the number line. The positions of and are clearly differentiated and the student labels the graphed points with their corresponding irrational representations.

Questions Eliciting Thinking What is the difference between an irrational number like and a number like 3.14 or ?
Is it possible to find a fraction that represents the exact value of ? Why or why not?
Could you have ordered the original numbers without first approximating them? 
Instructional Implications Ask the student to compare a common decimal approximation (e.g., 3.14) to the common fraction approximation of to determine which is closer to its actual value and by about how much. Ask the student to consider when one might use each of the two approximations. Challenge the student to find a fraction that is a better approximation of than . 