Getting Started |
| Misconception/Error The student does not correctly represent the solutions to the inequality on a number line diagram. |
| Examples of Student Work at this Level On the graph, the student:
- Incorrectly uses an open circle or closed circle (or uses no circle at all).
- Shades in the wrong direction to indicate values that satisfy the inequality (or does not shade at all).
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| Questions Eliciting Thinking What does this inequality symbol 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? |
| Instructional Implications 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 having the student use the MFAS task Rational Number Lines (6.EE.2.8) for additional practice. |
Moving Forward |
| Misconception/Error The student does not correctly identify values corresponding to the given constraints. |
| Examples of Student Work at this Level The student:
- Chooses numbers that are both greater than and less than the given number.
- Includes the value of 15,000 as an example for p > 15,000.
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| Questions Eliciting Thinking What does your graph indicate about the values that satisfy the inequality?
What does this inequality symbol 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?
Why did you include numbers that are greater than and less than the given number? |
| Instructional Implications 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.
Have the student label each of their number examples on the given graph. Have them decide if the location of their value meets the given criteria.
Consider having the student use the MFAS task Rational Number Lines (6.EE.2.8) for additional practice. |
Making Progress |
| Misconception/Error The student does not graph rational numbers on the number line correctly. |
| Examples of Student Work at this Level The student graphs 67.86 at 67 or 68, rather than approximating its location between these two values.
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| Questions Eliciting Thinking How did you determine where to graph 67.86?
Is 67.86 the same as 67 (or the same as 68)? |
| Instructional Implications Have the student consider the interval bounded by 67 and 68. Ask the student to create a new number line just for this interval scaled by tenths and to determine between which two tenths 67.86 falls. Then ask the student to create a new number line just for this interval of tenths scaled by hundredths and to determine where 67.86 should be graphed.
Give the student experience with number lines with varying scales. Have the student place integer and rational number values appropriately on each graph. |
Almost There |
| Misconception/Error The student is unable to clearly explain how many values are represented by the given inequality. |
| Examples of Student Work at this Level The student:
- Answers “a lot” or “many” without further explanation.
- Gives more examples of values that meet the given criteria or repeats the inequality statement, without explanation.
- Lists a specific finite number.
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| Questions Eliciting Thinking 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? |
| Instructional Implications 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, “How many values does this inequality represent in all?”
Expose the student to the responses of students at the Got It level. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level For the first problem, the student:
- Correctly graphs the inequality using an open circle and shading to the right of 15,000,
- Lists three examples of numbers that are greater than 15,000, and
- Says there are an infinite number of values greater than 15,000.
For the second problem, the student:
- Correctly graphs the inequality using a closed circle and shading to the left of 67.86,
- Lists three examples of numbers that are less than or equal to 67.86, and
- Says there are infinitely many values less than or equal to 67.86. But the student understands that in the context of money, only positive numbers to the hundredths place make sense.
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| Questions Eliciting Thinking Are there any other constraints based on the contexts of the inequalities that might limit the values that each represents (e.g., is it possible for the plane to go to any altitude greater than 15,000; is it possible for Edin to have $58.457)?
Exactly how many amounts of money are there less than or equal to $67.86? |
| Instructional Implications Introduce the student to compound inequalities. Give the student a statement such as, “The various routes that Kelvin can drive to work range from 8.2 miles to 9.7 miles in length” and ask the student to represent the lengths, m, as a compound inequality (e.g., in the form 8.2  m 9.7 or in the form m  8.2 and m 9.7). Guide the student to graph the inequality on the number line and verbally describe the values that satisfy it. |
