Roadway Inequalities

Getting Started

The student does not write an inequality statement that represents the condition described in the problem.

The student:

• Graphs all possible values on a number line.
  
  
  
  
• Writes an inequality statement involving two values (e.g., writes, “15 12”).
  
  
  
  
• Writes the variable, a number, and an inequality symbol in a meaningless way (e.g., writes, “12 t ”).
  
  
  
  
• Writes an equation (e.g., writes, “a = 12”). 

What is an inequality statement? How is an inequality statement different from a graph?

Is there only one number that would make the given statement true? Can you think of others? How can we describe all the numbers that will make the statement true?

Can you read your statement to me? What are the two things being compared?

What is the difference between an inequality and an equation? What part of the description tells you which one you should write?

Provide direct instruction on writing and interpreting inequality statements. Make explicit the difference between an equation and an inequality. Show the student an inequality and model a verbal translation. Given an inequality statement such as x 5, prompt the student to identify values that satisfy the inequality. Then guide the student to graph the inequality on a number line so that the student can relate the inequality statement to a visual representation of the numbers it represents. Clarify the meaning of the variable, that is, that it can represent any of a number of values or solutions.

Review the differences between statements, equations, expressions and graphs with the student. Explain the need for variables when representing a wide range of numbers that makes a statement true. Discuss the importance of order in an inequality statement.

Making Progress

The student uses an incorrect inequality symbol.

In one or both statements, the student:

• Reverses the direction of the inequality symbol.
  
  
  
  
• Does not use a strict inequality when one should be used or uses a strict inequality when one should not be used. 
  
  

Can you read your statement to me? How do you read the symbol that you wrote?

Why did you choose the symbol you did? What key words help you make the decision?

Does it matter if you include an equal bar or not? Will it change which values make the statement true? If so, which values?

Provide instruction on the meaning of each inequality symbol (<, =, >, and =) and how each symbol is read. Guide the student to write statements relating specific numbers using each inequality symbol (e.g., 5 > 2, 4 5, or 4 4). Allow the student to refer to the examples until he or she can confidently read and use each symbol.

Have the student work with a “Got It” level student to develop a list of key words/phrases that lead to the use of each inequality symbol. This list might include:

• > More than, exceeding, above, greater than. 
• At least, no fewer than, not under, no less than. 
• No more than, not above, no greater than, does not exceed, at most.
• < Fewer than, below, less than.

Consider having the student use the MFAS task Acres and Altitudes for additional practice.

Got It

The student provides complete and correct responses to all components of the task.

The student correctly writes both inequality statements as:

1. w  12 or 12  w (any variable may be used)
2. g < 4.06 or 4.06 > g (any variable may be used)

Are there any other limits that should be placed on these solutions, based on the context of the problem (e.g., is it possible to have a 100 foot wide road? Is there a lower limit that should be placed on the gasoline prices)?

How many values does each of your inequalities represent?

Introduce the student to graphing inequalities on a number line diagram. Consider using the MFAS tasks Rational Number Lines and Transportation Number Lines.

Ask the student to write the same inequality in two different but equivalent ways (e.g., write h 12 and 12 h).

Introduce the student to compound inequalities. Give the student a statement such as, “The heights of the students in our class range from 52 inches to 68 inches” and ask the student to represent the heights, h, as a compound inequality (e.g., in the form 52 h 68 or in the form h 52 and h 68).