Getting Started 
Misconception/Error The student does not recognize the change in scale on the number lines and does not adapt his or her approximation to the change in scale. 
Examples of Student Work at this Level The student approximates and plots it similarly on all three number lines.

Questions Eliciting Thinking How are the three number lines different?
How did you decide where to graph on each number line?
Can you write in the missing coordinates on each number line? Will that help you graph ?
What do you know about the decimal representation of irrational numbers? Why do you have to approximate irrational numbers?
How will you decide which place value to use if you want to round an irrational number and plot it on a number line? 
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 in understanding 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).
Show the student how the same irrational number can be approximated (rounded) using different place values. Explain why estimating irrational numbers is helpful in a realworld context.
Provide the student with opportunities to approximate irrational numbers in context. 
Moving Forward 
Misconception/Error The student adapts his or her approximation to only some of the number lines. 
Examples of Student Work at this Level The student approximates and plots it more precisely on one or two number lines but not on all.

Questions Eliciting Thinking How are the three number lines different?
Why did you decide to plot between two values on the first number line, but on an exact value for the second and third number lines?
Is equal to 2.8? Can you approximate it and graph it more precisely?
How will you know when to round an irrational number? 
Instructional Implications Review the definition of an irrational number and discuss the purpose of 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 accurate the approximation can be.
Consider using MFAS task Locating Irrational Numbers (8.EE.1.2). 
Almost There 
Misconception/Error The student adapts his or her approximation appropriately for each number line, but has trouble explaining. 
Examples of Student Work at this Level The student plots between 2 and 3 on the number line scaled in whole numbers; between 2.8 and 2.9 on the number line scaled in tenths; and between 2.82 and 2.83 on the number line scaled in hundredths.
The student could not explain except to say “I estimated.”

Questions Eliciting Thinking Why did you plot in three different places?
Why did you plot using different place values?
What was your approximation of for each of the graphs?
Why do you have to approximate an irrational number?
What made you decide to approximate? 
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 approximate form.
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 (e.g., say “the square root of eight is about 2.83 when it is approximated to the hundredths place” or “this number line is scaled to tenths so I approximated to the hundredths place so that I could locate it between 2.8 and 2.9”).
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 adapts his or her approximation of to each number line according to its scale.
The student plots between 2 and 3; between 2.8 and 2.9; and between 2.82 and 2.83.
The student explains that an irrational number can be approximated and plotted between two points.
The student explains that an irrational number can be plotted more accurately by using more of its place values. 
Questions Eliciting Thinking Why did you plot between 2.82 and 2.83 instead of rounding off and plotting on 2.83?
What do you know about the decimal representation of irrational numbers?
Why does an irrational number always lie between two points? 
Instructional Implications Pair the student with a Moving Forward partner. Have the student model approximating and plotting irrational numbers for the Moving Forward partner.
Challenge the student to evaluate expressions involving irrational numbers (e.g., ) to a given decimal place.
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. 