Understanding Distance & Coordinates

Understanding Distance & Coordinates

Distances on Long Street

Opening

Distances on Long Street

Sophie and her friends Karen and Marcus all live on Long Street. This number line shows the locations of their houses. Each unit on the number line represents 1 block. Sophie’s house is located at 0. The house of the friend who lives to the west of Sophie is a negative number.

Karen’s house is at point K and Marcus’s house is at point M.

Karen says, “From my house to your house is 9 blocks.” She wrote the expression this way:

8 − (−1)

Marcus says, “Then from my house to your house, the distance is 9 blocks." He wrote the expression this way:

(−1) − 8

  • Do you agree with Karen’s and Marcus’s statements? Explain why or why not.
  • A distance is always a positive number. If you wanted to describe how to get from Marcus’s house to Karen’s house, what would you say?
  • When subtraction is represented as distance, what would you need to do after you find the distance, in order to find the answer to the subtraction problem?

Math Mission

Opening

Model integer subtraction by finding distances between points on a number line.

Find Distances on a Number Line

Work Time

Find Distances on a Number Line

  • Use the Number Line Tool to model each subtraction problem as a distance on a number line. Write the difference as an equation.
    • 17 − 3
    • 3 − 15
    • −7 − (−9)
    • 1 − (−5)
  • Create a problem in the form: positive number – negative number = 7

  • Create another problem in the form: positive number – negative number = 7

  • How do the number lines for the two problems you created compare? What is similar and what is different? Explain.

INTERACTIVE: Number Line Tool

Hint:

A distance is always a positive number. So, how can you determine whether the answer to the subtraction problem is positive or negative?

Differences in Temperature

Work Time

Differences in Temperature

This number line shows very high and very low temperatures for different states in the United States.

  • Find the distance between the point for the high temperature for California and the point for the high temperature for Florida. What does this distance represent?
  • How much higher is the high temperature for Florida than the low temperature for Florida?
  • How much lower is the low temperature for Alaska than the low temperature for Alabama?
  • What is the difference in temperature between the high temperature for California and the low temperature for Alaska?

Hint:

The method you used to subtract on the horizontal number also works for a vertical number line.

Prepare a Presentation

Work Time

Prepare a Presentation

Explain the method you used to subtract integers by looking at subtraction as the distance between two points. Give examples—both equations and number line representations—to support your explanation.

Make Connections

Performance Task

Ways of Thinking: Make Connections

Take notes about how your classmates used distance to subtract positive and negative numbers.

Hint:

As your classmates present, ask questions such as:

  • How does your example support your method?
  • Why can’t you just write ab to find the distance?
  • Can you show us again how you got your answer?

Subtraction as Distance

Formative Assessment

Summary of the Math: Subtraction as Distance

Read and Discuss

Subtraction can be looked at as the distance between two numbers.

You can use the number line to model the subtraction ab as distance, where a and b are any two numbers (positive or negative), as follows.

  • Locate points a and b.
  • Find the distance between points a and b. This value will be a positive number, because a distance is always positive.
  • Determine the sign of the distance value in order to find the answer to the subtraction problem. The value of a − b is:
    • Positive if a > b (if a is to the right of b on the number line).
    • Negative if a < b (if a is to the left of b on the number line).

Hint:

Can you:

  • Explain the method for subtracting integers by finding the distance between two points on a number line?

Reflect On Your Work

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

Something new I learned today is …