Getting Started 
Misconception/Error The student is unable to correctly estimate the value of an irrational expression. 
Examples of Student Work at this Level The student:
 Indicates that he or she does not know how to use a calculator to evaluate the expression.
 Does not follow order of operations when using a calculator.
 Estimates using the calculator and then does not follow the order of operations rules for subsequent calculations.

Questions Eliciting Thinking How did you use your calculator to estimate this value? Did you consider the order of operations rules?
What does mean? Can this number be written as a decimal in exact form? 
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 and . 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 irrational numbers.
Review the order of operations rules. Then provide instruction on calculator use, particularly on how to adhere to the order of operations rules when using a calculator. Be sure the student understands how to take a square root on the calculator and that when inputting an irrational number (in the form of a square root of a nonperfect square), the calculator is displaying a rational approximation.
Provide additional practice in calculating rational approximations of irrational numbers and expressions. 
Moving Forward 
Misconception/Error The student is unable to generate an appropriate high and low approximation for the irrational expression with a calculator. 
Examples of Student Work at this Level The student:
 Bases calculations on two lower bounds of .
 Provides a single estimate of the rectangle’s length rather than bounds.
 Provides bounds for the value of the golden ratio but not for the rectangle’s length.

Questions Eliciting Thinking How did you calculate your estimates? What makes one low and the other high?
Is it possible to write the exact value of this number? Why or why not?
What two whole numbers is this (the number shown on the calculator) between? What two tenths is it between? 
Instructional Implications Review the definition of an irrational number and discuss why rational numbers are used to approximate irrational numbers. Ask the student to use a calculator to approximate irrational numbers and guide the student to describe lower and upper bounds on them. Use a number line to illustrate the relationship between the irrational number and a set of bounds. Model finding lower and upper bounds to increasingly greater precision. Provide the student with additional opportunities to find rational approximations of irrational numbers and find increasingly more precise (whole number, tenth, hundredth) lower and upper bounds. 
Almost There 
Misconception/Error The student provides insufficient reasoning or is imprecise when relating the actual length of the rectangle to the rational approximations. 
Examples of Student Work at this Level The student correctly estimates the value of and provides appropriate lower and upper bounds. However, when asked to explain the significance of the lower and upper bound, the student:
 Is unsure.
 Provides a vague or incomplete answer such as “the real answer will be close to the estimates,” “there is not an exact number,” or “the length is around” the given lower and upper bound.

Questions Eliciting Thinking What do your low and high estimates indicate? What do they tell you about the length of the rectangle?
Could you have calculated the exact length using your calculator? 
Instructional Implications Ask the student to use a calculator to approximate irrational numbers to varying degrees of precision and to provide lower and upper bounds on irrational numbers. Model using the terminology of place value to assist the student in describing each approximation, for example, say, “The square root of eight is about 2.83 when it is approximated to the hundredths place.” Or, say, “The square root of eight is between 2.82 and 2.83.” Provide the student with additional opportunities to find rational approximations of irrational numbers and to describe the degree of precision by using the terminology of place value and providing lower and upper bounds. 
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 correctly estimates the value of and provides appropriate lower and upper bounds. The student states that the true value of the rectangle’s length is between the low and high estimates.

Questions Eliciting Thinking You said the value is between 8 and 9, but can you narrow that range even more? 
Instructional Implications Ask the student to use guessandcheck approximation with or without a calculator to find increasingly narrow bounds for . Ask the student to provide written responses to the following prompts:
 Is the value generated by a calculator for an exact value? Why or why not? Could a powerful computer find the exact value?
 Compare and contrast the use of bounds to describe the value of an irrational number to the use of a single number approximation.
