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Grade 6
Domain The Number System
Cluster Apply and extend previous understandings of numbers to the system of rational numbers.
Standard Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
Task Extending the Number Line

Extending the Number Line


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Alignments to Content Standards: 6.NS.C.6
Student View

Task

  1. Draw a line on graph paper. Make a tick mark in the middle of the line and label it 0. Mark and label 1, 2, 3, ... 10. Since 6+2 is 2 units to the right of 6 on the number line, we can represent 6+2 like this:

    1_10865a822b91510e2a84b4c2281abca5

    Describe the location of 3+4 on the number line in terms of 3 and 4. Draw a picture like the one above.

  2. 6-2 is 2 units to the left of 6 on the number line, which we can represent like this:

    2_affb3474dfb97f31cf9105c069677b08

    Describe the location of 3-4 on the number line in terms of 3 and 4. Draw a picture like the one above.

IM Commentary

The purpose of this task is to understand that there are natural mathematical questions to ask for which there are no answers if we restrict ourselves to the positive numbers. The idea is to motivate the need for negative numbers and to see that there is a natural representation of them on the number line. In 7th grade, students will formally explore using a number line to add and subtract signed numbers.  This task does not aim to teach students to operate with integers on a number line, but simply to understand why we extend our number system beyond whole numbers. As students learn more about integer arithmetic and the number line, they will see, for part (b), that 3-4=-(-3+4)=-1.

This task could be used as a quick warm-up before a more formal introduction to negative numbers on the number line. In the solution, the negative numbers have not been marked on the number line: because this is an introductory task, students are not assumed to know how to label the negative numbers and the task serves as motivation for a discussion which can culminate in labeling the negative numbers.   

Solution

  1. Since 3+4 is 4 units to the right of 3 on the number line, we can find and represent 3+4 like this:

    3_d9385603114059ba52634930fc3932da

    We can see from the number line that 3+4=7.

  2. Since 3-4 is 4 units to the left of 3 on the number line, we can find and represent 3-4 like this:

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    The difference 3-4 has a well determined place on the number line as shown in the picture. This place, however, is not marked and is not a whole number. The difference 3-4 is one unit to the left of 0.

Extending the Number Line

  1. Draw a line on graph paper. Make a tick mark in the middle of the line and label it 0. Mark and label 1, 2, 3, ... 10. Since 6+2 is 2 units to the right of 6 on the number line, we can represent 6+2 like this:

    1_10865a822b91510e2a84b4c2281abca5

    Describe the location of 3+4 on the number line in terms of 3 and 4. Draw a picture like the one above.

  2. 6-2 is 2 units to the left of 6 on the number line, which we can represent like this:

    2_affb3474dfb97f31cf9105c069677b08

    Describe the location of 3-4 on the number line in terms of 3 and 4. Draw a picture like the one above.

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