Finding the Distance Between Integers On a Number Line

The purpose of the introduction is to make sure students recall the meaning of absolute value as the distance from a number to 0 on a number line.  Distances are always positive.  I will display the blank number lines on the SmartBoard.  I will cold call students to determine the absolute value of various points that I will mark on the number lines.  For the first few problems, I will show the line segment from 0 to the number over the number line. I will also make sure to label the absolute value notation |  |  for each number.  I know that students often confuse this symbol with parentheses so I will make sure they note the differences.  As students see these values represented with the absolute value symbol they are engagin in MP2  as they see a distance represented as using symbols such as | -7 | = the distance 7.  To include a bit of MP3 and MP6, I will ask some students to explain how they know their answers are valid.  While this is very basic, it will help students get into the habit of using precise language to explain and justify their reasoning.

I have presented students with 24 distance problems to solve using a given number line.  Students should be able to twork with their partners without much help from the teacher.  If students are having difficulty, I will first make sure that they are able to identify the location of the two points.  Then, I will watch them count the distances.  Students often make mistakes when counting distances.  For example, they will say the distance between 1 and 3 is 3 units because they will count 1.  To mitigate the chance of this happening, I may tell students to count the spaces between the tick marks of the number line.  Or when I see this I may ask them to count how many steps I am taking towards them.  They will see that it does not make since to start counting steps until I have made the first step.  Each distance is represented using the absolute value symbols so students should see that there are two ways to write the distance as the absolute value of their difference.  Again this presentation is here so students see that the symbols represent the distance (MP2).I am teaching this lesson before we learn to add or subtract integers, so I may allow my students to use calculators to verify that their counting on the number line matches the calculations for each problem (MP5).  For each group of problems, I have included a couple problems with numbers that do not explicitly appear on the number line or numbers that are inbetween tick marks.  I expect to hear students say "The number is not there."  Problem e, has a negative 13.  I will ask them to tell me where -13 is in relationship to -11 (the last visible number on the left of the numberline).  Problem k, has tick mark intervals in multiples of 5, but it asks students to place -46.  I will ask them to point where they think -46 is based on the values given.  If they struggle, I may ask them to identify where an easier number is like +7.  

At the end of this section, we will quickly review answers.  I will then ask students if they see a shortcut or easier way to find the distance between a positive and negative number wihtout counting each tick mark.  This is to see if they see the structure (MP7) that the sum of a negative numbers' abs value and the positive numbers' abs value is also their distance.  

This goes with pages 1-2 of the resource:

FindingTheDistanceBetweenIntegersOnANumberLine_Module.docx

The problem solving section is on pages 3-4 of the resource:

FindingTheDistanceBetweenIntegersOnANumberLine_Module.docx

Now students apply what they know in a "real-world" problem solving context.  In problem 1, students will have to figure out the appropriate scale for their number line (MP6).  Students should resolve issues with their partners, but if stuck I may ask questions like:  "How far is the scuba diver/hang glider  from sea level?  Is the scuba diver/hang glider above or below sea level?"  Part C asks students to write two different expressions to represent the distance.  My intention is for them to write these as the abs value of their distance, but it is also great if they choose to write the distance as the sum of the abs values of each altitude.  

Problem 2 is simlar, except that there are more points to find.  Students are asked to find the distances between every house.  Here is a small opportunity for students to make sense of a problem (MP1) as they must find a method to make sure they find all 6 distances.

The extension is not any more difficult than the previous problems, it may even be easier, but it puts the distances in the context of the coordinate plane.  It provides an opportunity for reviewing how to graph in all four quadrants of the the coordinate plane.

The exit ticket is a similar problem to the two problems in the problem solving section.  In fact, it is most similar to the first problem.  Students work on this independently, but it is okay for them to refer to any of their class work as an aid.