If we put all integers between minus one thousand three hundred and twenty and positive two thousand and sixteen on one axis, how many 3 cm long line segments can be named with these points? (Each unit equals 1 cm)

To solve this problem, we need to calculate the number of 3 cm line segments that can be named using integer points between \(-1320\) and \(2016\) on a number line, where each unit represents 1 cm.

### Step-by-Step Solution:

1. **Determine the Range of Points:**

   The points on the axis range from \(-1320\) to \(2016\). The total number of integer points \(P\) between \(-1320\) and \(2016\) is calculated as follows:
   \[
   P = 2016 - (-1320) + 1 = 2016 + 1320 + 1 = 3337
   \]

2. **Calculate the Number of Possible 3 cm Segments:**

   A 3 cm segment consists of 4 points on this axis (since each unit is 1 cm). If we name the starting point as \(x\), the segment will span from \(x\) to \(x+3\).

   For a valid 3 cm segment starting at point \(x\), \(x+3\) must also lie within the range from \(-1320\) to \(2016\).

   The first possible starting point is \(-1320\), and the last possible starting point is \(2016 - 3 = 2013\).

3. **Count the Number of Valid Segments:**

   The number of valid starting points \(x\) is:
   \[
   2013 - (-1320) + 1 = 2013 + 1320 + 1 = 3334
   \]

### Conclusion:

There are **3334** possible 3 cm long line segments that can be named using integer points between \(-1320\) and \(2016\) on the axis.

Would you like more details or have any questions? Here are 5 related questions:

1. What if the line segments were 5 cm long?
2. How would the answer change if the range was from -1000 to 1000?
3. What is the formula for finding the number of \(n\)-cm segments on a given range?
4. How would the solution differ if the spacing between points was 2 cm instead of 1 cm?
5. What if negative points were not allowed in naming segments?

**Tip:** When working with ranges and intervals, always remember to include the endpoints in your calculations where appropriate!

Find out how to calculate the number of 3 cm long line segments between integers from -1320 to 2016 on a number line. This problem involves basic arithmetic and number line concepts, making it suitable for students in Grades 6-8.

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