Getting Started 
Misconception/Error The student does not understand the order of the integers or the meaning of the inequality symbol. 
Examples of Student Work at this Level The student writes one or both inequalities incorrectly. For example, the student writes 0 < 54 or 20 < 60 and, upon questioning, indicates a lack of understanding of the order of the integers or the meaning of the inequality symbols. 
Questions Eliciting Thinking Can you read your inequalities to me? What does the inequality symbol mean?
How does the number zero compare to negative numbers? Is it larger or smaller?
How do positive numbers compare to negative numbers? 
Instructional Implications Review the structure of the positive portion of the number line. Also, review the negative integers and the kinds of quantities that they can represent. Then extend the number line to include the negative integers. Present integers in context (e.g., as a set of low temperatures for a week) and ask the student to graph the set of integers on a number line. Guide the student to use the graph to order the integers and explain their meaning in context. Assist the student in understanding that zero is greater than any negative number and any negative number is less than any positive number.
Review the meaning of the inequality symbols and provide examples of their use. Ask the student to read inequality statements and provide feedback. Then give the student a list of statements involving inequality symbols; ask the student to determine if the statements are true or false and to correct the false ones. Provide additional opportunities for the student to use inequality symbols to both read information given in problems and write responses.
Provide the student with opportunities to write inequality statements that summarize the relationship between integer quantities given in context. For example, suggest that two students are in debt due to college loans. One student owes $3000 and the other owes $2000. Ask the student to express each debt as a negative number and to relate the two quantities with an inequality. 
Making Progress 
Misconception/Error The student is unable to interpret the inequality in context. 
Examples of Student Work at this Level The student represents the first scenario with the inequality 0 > 54 or 54 < 0 and the second scenario with the inequality 20 > 60 or 60 < 20. However, the student’s explanation of the inequality 54 > 60:
 Does not reference the context.
 Is incorrect.

Questions Eliciting Thinking What do 54 and 60 represent in this context? What kinds of words do we use when we compare temperatures? Which day was warmer? Which day was colder?
Is 60 really less than 54? How do you know? 
Instructional Implications Guide the student to use the context of the inequality to explain its meaning. Expose the student to a variety of realworld situations in which integers are used to describe quantities or changes in quantities (e.g., gain/loss, increase/decrease, temperature, and altitude). Ask the student to write inequalities to describe the relationship between pairs of quantities and to interpret the inequalities in context. 
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:
 Represents the first scenario with the inequality 0 > 54 or 54 < 0.
 Represents the second scenario with the inequality 20 > 60 or 60 < 20.
 Explains the inequality 54 > 60 as indicating that 54° is warmer than 60°.

Questions Eliciting Thinking How can 54 be greater than 60 when 60 is greater than 54?
How many degrees colder is 60° than 20°? 
Instructional Implications If not done already, introduce the student to the concept of absolute value. Explain this concept in terms of the number line but also expose the student to an algebraic definition. Guide the student to distinguish between the magnitudes of integers, as expressed by their absolute values, and the order of integers. Present the student with examples of rational numbers given in context (include integers as well as both positive and negative fractions and decimals) and ask the student to interpret the meanings of the numbers in terms of order and magnitude. For example, given temperatures of 20° C and 5° C, the student should be able to explain that 5° is greater or warmer than 20° (and write the inequality 5 > 20) but 20 deviates more from zero or freezing than 5. 