Submarines

Getting Started

The student is unable to use negative numbers to represent quantities or locations.

The student represents the depths of the submarines with positive numbers.

What number best represents sea level?

If a submarine is 40 feet below sea level, what kind of number best represents its depth?

Why did you choose to represent these depths with positive numbers?

Have the student draw a vertical number line that represents altitude. Then give the student a list of items and locations along with their altitudes and have the student place them on the number line. Be sure to include altitudes that are below and at sea level.

Expose the student to a variety of real-world situations in which integers are used to represent quantities such as gain/loss, increase/decrease, and above/below sea level. Guide the student to represent integer quantities in the context of problems. Also ask the student to describe a quantity that can be represented by given integers.

Consider using MFAS task Relative Integers.

Moving Forward

The student is unable to interpret the order of the integers in context.

The student reverses the order of the negative integers in his or her graph or shows Nautilus above Sea Wolf.

   

 

Can you explain the structure of your graph? Can you explain how it relates to the inequality you wrote?

Can you draw a vertical number line that includes positive numbers, zero, and negative numbers? How should this number line be oriented?

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). 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.

If necessary, 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 and 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 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. Then ask the student to graph the quantities on a number line.

Almost There

The student errs in writing an inequality to express the relationship between the integers.

The student correctly describes the depths using negative numbers and draws a diagram that indicates that he or she understands the relative position of the two submarines. But, the student writes the inequality as -100 > -40.

Which number is greater: -40 or -100?

Can you read your inequality to me? What does the inequality symbol mean?

Provide direct feedback on the student’s error with regard to the inequality symbol. 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 to write responses.

Got It

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

The student represents the locations of Sea Wolf and Nautilus with -40 and -100 respectively, and writes the inequality -40 > -100 (or -100 < -40) to compare these values. The student draws a number line diagram with the two submarines correctly positioned.

What made you decide to orient your number line vertically (horizontally)? How would the positions of the submarines compare if the number line was horizontal (vertical)?

Where would sea level be on your number line? How far is each submarine from sea level?

Extend the student’s understanding of rational numbers and the number line to the coordinate plane. Provide a context in which both positive and negative rational numbers have meaning and present the student with a table of values (e.g., the ages, in months, of a sample of sixth grade students along with the positive and negative deviations of their heights from an average) and have the student graph the data as ordered pairs in the coordinate plane.