Getting Started

The student does not correctly represent the solutions to the inequality on a number line diagram.

On the graph, the student:

• Shades in the wrong direction to indicate values that satisfy the inequality (or does not shade at all).
  
  
  
  
• Incorrectly uses an open circle or closed circle (or uses no circle at all).
  
  

What does “>” mean? Does it include numbers that are greater than or less than the given number? Does it also include the exact value of the given number?

What does an open or closed circle indicate on a graph? How do you choose which one to use?

How do you decide which direction to shade and draw the arrow? What does the arrow mean on the graph of an inequality? Is there a way to use a specific value to check if your shading is correct?

Provide instruction on graphing inequalities on the number line. Be sure the student understands the conventions in graphing inequalities and their meaning (e.g., the use of an open versus closed circle, the direction of shading, and the arrow). If necessary, provide instruction on the meaning of the inequality symbols. Have the student graph a variety of inequalities (including some with the variable written to the right of the inequality symbol) and write inequalities to match given graphs.

Consider implementing the MFAS task Transportation Number Lines (6.EE.2.8).

Moving Forward

The student does not correctly identify values corresponding to the given constraints.

The student:

• Only chooses values equal to the number in the inequality statement, regardless of the symbol or constraints given (e.g., only chooses values equal to 48 for h > 48).
  
  
  
  
• Chooses numbers that are both greater than and less than the given number.
• Cannot correctly compare rational numbers or convert improper fractions.

What does your graph indicate about the values that satisfy the inequality?

Does the statement include numbers that are greater than or less than the given number? Will that also include the exact value of the given number? What are some numbers that would make the inequality true?

How do you decide if a decimal (or fraction) is greater or less than a given number? That number should be placed where on the number line? How does that compare to the given value?

How can you find an equivalent decimal or mixed number for the improper fraction? Does that equivalent value meet the given constraint?

Looking at the values you circled, do they match the constraints of the original problem?

Review the meaning of each inequality statement by relating them to the graphs that the student produced. Guide the student to use his or her graphs to identify several values that satisfy each inequality. Prompt the student to identify both integer and rational number values.

Give the student instruction comparing rational numbers written in different formats. Include review of transforming numbers from fractions and mixed numbers to decimals.

Consider having the student use the MFAS task Transportation Number Lines (6.EE.2.8) for additional practice.

Almost There

The student is unable to explain that there are other possible values that satisfy the inequality.

The student:

• Simply says “yes” or “no” without any explanation.
• Gives one or more specific solution values without an explanation of whether there are other possibilities or not.
  
  

Why do you think there are other possible values? Can you explain what those might be?

You have given good example(s) of other values that satisfy the inequality. Are those the only other values that will work?

What does the arrow on your graph mean?

Have the student consider an interval bounded by two consecutive integers, such as zero and one, and list as many rational values as possible between these integers. Then have the student make a conjecture about the number of rational values within this interval on the number line. Guide the student to an understanding that there are infinitely many rational values between any two consecutive integers. Then ask the student to apply this notion to the question, “Are there other possible values that will also satisfy this inequality?”

Expose the student to the responses of students at the Got It level.

Got It

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

For the first problem, the student:

• Correctly graphs the inequality using an open circle and shading to the right of 48, 
• Identifies 50 and 48 as values that satisfy the inequality, and 
• Explains that any value greater than 48 inches satisfies the inequality. Additionally, the student might explain that there are an infinite number of values greater than 48 inches.

For the second problem, the student:

• Correctly graphs the inequality using a closed circle and shading to the left of 12.5, 
• Identifies 1.25, 12, 12.05, 12, and  as values that satisfy the inequality, and 
• Explains that any value less than or equal to 12.5 mph (or any value less than or equal to 12.5 mph but greater than zero) satisfies the inequality. Additionally, the student might explain that there are an infinite number of values less than or equal to 12.5 mph but greater than zero.

Are there any other limits that could be placed on these solutions, based on the context of the problem (e.g., is it possible to have a negative speed)? If so, how could you show that on the graph? How could you write another inequality statement to show all necessary constraints?

Introduce the student to compound inequalities. Give the student a statement such as, “The temperature ranged from a low of -2° to a high of 28º” and ask the student to represent the temperature, t, as a compound inequality (e.g., in the form -2 = t = 28 or in the form t = -2 and t = 28). Guide the student to graph the inequality on the number line and verbally describe the values that satisfy it.