Getting Started

The student is unable to use inequality symbols to compare positive and negative integers.

The student:

• Makes a statement about the two values without using inequality symbols to compare them.
• Makes an incorrect comparison (e.g., -282 > 203).
  
     

What does the symbol < mean? What does the symbol > mean?

Can a negative number be greater than a positive number?

If two points were plotted on a number line, how could you tell which is greater?

Using a number line, review the order of the integers. Be sure the student understands that numbers increase in value as one progresses from left to right on the number line so that:

• Any positive number is greater than any negative number.
• Zero is greater than any negative number.
• Zero is less than any positive number.

If needed, review the meaning of the inequality symbols >, <, etc. and how they are used. Provide practice problems for the student to compare integers using inequality symbols and order integers from least to greatest. Have the student use a number line to illustrate the answer and provide a written explanation.

Help the student recognize the distinction between the magnitude and the direction of a number. Explain that a number’s magnitude is given by its absolute value and represents its distance from zero on the number line. Its direction is given by its sign (either positive or negative) and indicates to what side of zero the number is located on the number line. For example, -3 has a magnitude of three and is located to the left of zero on the number line indicating that it is less than zero. Expose the student to additional problem contexts that provide an opportunity to distinguish between a number’s magnitude and its order or direction. For example, “Yesterday, the predicted high temperature for both St. Louis and Chicago was 0°C. The recorded high temperature in St. Louis was 2°C while the recorded high temperature in Chicago was -4°C.” Ask the student to determine which city was warmer but to also describe which actual temperature deviated the most from the predicted high.

Moving Forward

The student does not demonstrate an understanding of absolute value.

The student correctly writes -282 < 203 or 203 > -282 for question #1 but does not appropriately compare the absolute values of these two numbers. The student:

• Explains the meaning of -282 < 203 within the context of the problem but does not refer to absolute value.
  
  
  
  
• Confuses absolute value with opposites and writes 282 > -203.
• Eliminates the negative sign from -282 but does not correctly compare 282 and 203. 

What does absolute value mean? Can you explain in terms of the number line?

What do absolute value symbols look like?

Is absolute value the same as opposite?

What is the absolute value of -282? What is the absolute value of 203?

Present the concept of absolute value as distance from zero on a number line. Provide practice questions in which the student must find an absolute value and illustrate it on a number line. For example, given |-12|, ask the student to plot the position of -12 on a number line, highlight the number line between 0 and -12, and identify this distance (12) as signifying the absolute value.

To further reinforce the concept of absolute value as distance, the student can do the above activity using number lines with 1 cm spacing between tick marks. The student can use a centimeter ruler to directly measure the distance between a given point and zero.

Help the student recognize the distinction between the magnitude and the direction of a number. Explain that a number’s magnitude is given by its absolute value while its direction is given by its sign (either positive or negative). Ask the student to compare numbers in terms of their values and in terms of their magnitudes. For example, -4 > -7 can be translated to “-4 is further to the right on a number line” whereas |-7| >|-4| can be translated as “-7 is further from zero on the number line than -4.”

Ask the student to complete the MFAS task Visualizing Absolute Value (6.NS.3.7).

Almost There

The student is unable to appropriately interpret absolute value within a real-world context.

The student provides a correct response to question #1 and a correct comparison of the absolute values of 203 and -282 but does not appropriately explain the meaning of the absolute value comparison within the problem context. The student:

• Interprets the integer comparison instead of the absolute value comparison, stating “Tallahassee is higher than Death Valley.”
• Provides a mathematical description of the absolute value comparison without referring to the problem context (e.g., “The absolute value of 203 is less than the absolute value of -282.”).
• References the problem context but does not interpret the meaning of absolute value within the context (e.g., “Death Valley has a greater absolute value.”).
• Refers to the problem context but provides an incorrect or insufficient interpretation.
  
     
  
  
• Describes the elevation of Death Valley as being positive.
  
     
  

What does absolute value mean?

Can you explain what your absolute value comparison means in terms of Tallahassee and Death Valley?

If absolute value is the distance from zero on a number line, what does zero represent in this context?

Expose the student to additional problem contexts that provide an opportunity to distinguish between a number’s magnitude and its order or direction. For example, “Yesterday, the predicted high temperature for both St. Louis and Chicago was 0°C. The recorded high temperature in St. Louis was 2°C, while the recorded high temperature in Chicago was -4°C.” Ask the student to determine which city was warmer and which actual temperature deviated the most from the predicted high. Ask the student to use absolute value symbols as part of the explanation.

Provide the student with practice problems that use real number measurements in real-world contexts. Ask the student to identify the significance of zero and provide a written interpretation of the absolute value of the given real number measurements.

Review the concept in real-world contexts, where absolute value should be seen as the distance from a zero point rather than an inversion of sign from negative to positive. For example, taking the absolute value of Death Valley’s elevation refers to measuring the distance between sea level and the valley floor, not turning the valley “inside out.” Provide the student with practice problems similar to the problem given in the Absolute Altitudes worksheet. Ask the student to relate the responses to the definition of absolute value.

Got It

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

The student correctly uses inequality symbols to compare -282 and 203, writing either -282 < 203 or 203 > -282.

The student correctly uses inequality symbols to compare the absolute values of -282 and 203, writing:

• 282 > 203
• 203 < 282
• |-282|>|203|
• |203|<|-282|

The student accurately explains the significance of the absolute value comparison of the two altitudes, writing:

• Death Valley is deeper than Tallahassee is high.
  
  
  
  
• Death Valley is further from sea level than Tallahassee.
  
  
  
  
• Tallahassee is closer to sea level than Death Valley.

What is the definition of absolute value? How is it different from taking the opposite of a number?

Review the use of absolute value to describe distances between two points on the number line (e.g., the distance of point A, whose coordinate is -3, from point B, whose coordinate is 17, can be described as |-3 – 17|). Ask the student to describe and compute distances using absolute value symbols. Extend this idea to the coordinate plane. Have the student describe and compute the distance between two points with the same x-coordinate or the same y-coordinate.