Distance Between Points

Students complete the Think About It problems on their own.  After three minutes of work time, I ask for students to turn and talk and share their thinking with their partners.  We finish this section by having 1-2 students share out their thinking about the distance.

A common error is for students to think the distance is -9 because they might count 'backwards' to move between the points.  This is a good opportunity to reinforce that distance is always positive.  We wouldn't say that Florida is -1700 miles away.  

I want students to access what they know about absolute value in this lesson, too, which we've defined as the distance from zero.  In this problem, students can think about each point's distance from the origin, and add them to find the points' distance from one another.

To start the Intro to New Material section, I have students think about the distance between points A and B, because these are in different quadrants.  One strategy students might use is to count the units between the points.  This strategy is okay, as students master the concept of distance.  It's important that students are able to move to mastery of the more abstract, too.  I tell students in higher level math, they'll find the distance between points that are much farther apart and counting won't be efficient.  

For students who do rely on counting, I keep a careful eye on the start of their counting.  There are some students who count the starting point as 'one.'  When a student consistently makes this mistake, I coach her to say 'start' at the first point, and then begin counting as her finger moves to the next grid line. 

The numerical expression that students are to write is |4| + |-2|.

I then have students think about whether using absolute value would work when we have two points in the same quadrant.  After 45 seconds of think time, I ask the class to vote with their thumbs up or down.  I ask one student to defend his/her opinion, and allow 2-3 students to agree or disagree (with justifications).  At the end of this conversation, I want the idea to come out that when the points are in the same quadrant, we subtract the value of the coordinates that are different to find the distance.

I guide students through problems C and D, while the Visual Anchor is displayed on the document camera. 

Students work in pairs on the Partner Practice problems.  As they work, I circulate.  I am looking for:

• Are students correctly labeling all necessary components of the graph?
• Are students correctly plotting the coordinate pair on the grid?
• Are students correctly determining the distance of two points on the grid?
• Are students correctly determining the distance of two points without a grid?
• Are students correctly writing or identifying an expression that involves absolute value to describe the distance between two points?

I am asking students:

• How did you know to draw the coordinate pair in that particular place?
• How did you determine how far one point was from the other?
• Why does this expression represent the distance between the two points?
• Why might another student think distance can be negative?

This problem set is shorter than most of my Partner Practice problem sets.  After 8-10 minutes of work time, we discuss all of the problems as a class.  

Students then complete the check for understanding problem independently.  I cold call one student to provide the expression that represents this problem.  I then have students whisper the distance, so that I can quickly hear that the majority of class has the correct answer.

Students work on the Independent Practice problem set.  As they work, I am looking for and asking the same questions as I used in the Partner Practice section:

• Are students correctly labeling all necessary components of the graph?
• Are students correctly plotting the coordinate pair on the grid?
• Are students correctly determining the distance of two points on the grid?
• Are students correctly determining the distance of two points without a grid?
• Are students correctly writing or identifying an expression that involves absolute value to describe the distance between two points?

I am asking students:

• How did you know to draw the coordinate pair in that particular place?
• How did you determine how far one point was from the other?
• Why does this expression represent the distance between the two points?
• Why might another student think distance can be negative?

After independent work time, I ask students to turn to Problems 11_and_12.  I tell them that the width of this figure is 10 units, which can be found with the expression |6| + |-4|.  I then tell them that the length of this figure is 8 units, which can be found with the expression of |6| + |2|. I've intentionally made a mistake. I pause and wait for someone to disagree with me.  We stay with the mistake until a students successfully explains the mistake that I've made.  I am sure to repeat the correct answer - the length is 4 units.

Students work independently on the Exit Ticket to end the lesson.