Identifying Positive Rational Numbers on Number Lines

Students work independently on the Think About It problem.  After two minutes, I have students show me on their fingers how many answer choices they've selected.  

I then have students explain why each answer choice is (or isn't) a correct answer choice.

I frame the lesson by letting students know that in this lesson students will plot positive decimals and fractions.  This will get us ready to plot negative decimals and fractions, and then we'll compare all rational numbers.

Teacher's Note: this lesson assumes students have fluency with decimal/fraction conversions.  If your students need practice with this skill, either build in extra time to this lesson or pre-teach/review conversions before this lesson.

To start the lesson, I have students create their own number lines on a blank piece of paper. Students draw a straight line across the width of the paper.  They add and label endpoints as 0 and 1.  Students then fold the paper in half, vertically, and draw a line there.  I ask students what the line represents, and we label it with 1/2.  I let students know that 1 can also be represented as 2/2 because it is made up of two ½ sized pieces.  We go through the same process with fourths and eights.  The idea here is to have students conclude that the same number can be expressed in a variety of ways.  It is also important that students understand that we created the number line by splitting it into equal sized parts so that there is the same amount of space between each number on the number line. 

After we construct the number line, we move to the first example in the Intro to New Material section.  To plot the numbers on the number line, these are the steps students will follow:

1. Determine which two integers the rational number is between.
2. Convert fractions and decimals to the same representation.
3. Partition the whole. Note: each whole can be partitioned differently.
4. Plot the numbers, using a point!

Once we've plotted the points, I have students fill in the notes:  Rational numbers are all positive and negative integers, fractions, terminating and repeating decimals. Another way that we say this is that a rational number is any positive or negative number expressed as a/b.

Students work in pairs on the Partner Practice problem set.  As students work, I circulate around the room and check in with each group.  I am looking for:

• Are students choosing the correct intervals?
• Are students converting all rational numbers to the same representation?
• Are students creating/labeling their number lines correctly?
• Are students plotting both a point and labeling with a letter to represent points on number lines?
• Are students rewriting fractions and decimals in the same representation when necessary?
• Are students answering in complete sentences? 

I'm asking:

• How did you know how to label the number line with those fractions/decimals?
• How did you know that the point goes there?
• How did you know that the number falls between those two integers?

After partner work time, I have the class come back together for a conversation about Problem C.  I ask students how they partitioned their number line.  Some students may have plotted 0.15 to the right of .7.  We talk about .1 and .2 being equivalent to .10 and .20.  Once students have this piece of information, I have them articulate that .15 is in the middle of .10 and .20.

Students complete the Independent Practice problem set.  

In addition to the questions I used during the Partner Practice section, I am also asking students if there are answer choices they can eliminate for the multiple choice questions.  I want students to reason about the magnitude of numbers and use logic to eliminate choices.

As I circulate around the room, I also as students for alternate ways to express various numbers in the problem set.  I want students to be able to fluently name and re-name rational numbers.  Students may give me decimals for fraction numbers, improper fractions for mixed numbers, etc.  My goal is have students comfortable with expressing numbers in a variety of ways.

After independent work time, I have students compare their responses to Problem 3 with their partners.  Students have the chance to check their work, ask clarifying questions, and give one another feedback on their work.

Once students have had 2 minutes of discussion time, I have a student explain to the class what to do with the absolute value in this problem.  I open up the discussion for any questions students might have.   

Students then independently complete the Exit Ticket to close the lesson.