Comparing Rational Numbers

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SWBAT compare and order two or more rational numbers using the number line and by reasoning about absolute value.

Big Idea

Rational numbers and number lines can be used to represent real world situations.

Think About It

7 minutes

Students complete the Think About It problems with their partners.  After 2-3 minutes, I call on 2-3 students to share their answers and explain how they were able to compare the two numbers.

I'll then ask students to share why was it very easy to compare the two numbers in the first problem even though they are mixed numbers. Students might say that it was easy because one was positive and the other was negative and positive numbers are always greater than negative numbers.

For the second problem, I want students to talk about the number's position on the number line, and their distances from zero (the absolute value) for each of the numbers. These are the skills students will use throughout the lesson.

Intro to New Material

15 minutes

To start the Intro to New Material section, students independently plot the numbers from the first example on the number line.  This is a skill that is relatively new for students, so I call on a student to summarize how to plot the numbers on the number line, as a way to review what they learned in the previous lesson.

I then ask for a volunteer to order the numbers and explain how they know their response is correct.  The rationale should include that the numbers decrease in value as you move from the right to the left on the number line.   

Together, we fill in the key points in the notes.  Because these are review, I'll ask a student to fill them in: Numbers increase in value as you move to the right/up on a number line and numbers decrease in value as you move to the left/down on a number line.

Problem Two asks students to work with and reason about fractions.  After reading the problem, I ask students to describe the meaning of the numbers.  Students need to be able to make sense of the problem and the given context.  In this instance, that means understanding that the team lost yards on these plays.  

I'll also ask what we need to determine about the numbers, if we are trying to figure out the play for which they lost the fewest yards. The key idea is that we need to figure out which number is closest to where they started with the ball or 0 since that would mean that they move the shortest distance from where they started.

Students plot the numbers on the number line.  For this problem, I guide students to use fractions with their number lines, rather than converting to decimals.  Students will have the chance to work with decimals and fractions throughout the lesson and they'll be able to decide how/when to convert numbers.  I want my students to be fluent working with either, and I want them to have a fraction example to reference later as they work.

If I feel students need a bit more support from me before they begin partner practice, I will use the first problem from the Partner Practice problem set as guided practice.


Partner Practice

20 minutes

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

  • Are students interpreting the questions correctly?
  • Are students choosing the correct intervals on their number lines?
  • Are students converting all rational numbers to the same representation (either decimals or fractions)?
  • 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 creating/labeling their number lines correctly?
  • Are students using the correct symbols to compare and order values?
  • Are students explaining how they used a number line to compare/order values correctly?
  • Are students answering in complete sentences? 


I'm asking:

  • How did you know to use the >, < or = sign?
  • How did you use the number line to compare the numbers?
  • How did you use the absolute value of the numbers to compare the numbers?
  • How did you know what the question was asking you to compare?


After partner work time, students complete the Check for Understanding problem independently. Once students have completed the problem, I'll gather some quick data from them.  I'll ask students to snap when I point to the number that is the greatest.  Asking students to snap is an engaging way for me to hear from all students, and gives me a sense of mastery.  If a student snaps for -5.5 being the largest, for example, I'll know to check in with him at the start of independent practice to help clear up the misunderstanding.


Independent Practice

15 minutes

Students work on the Independent Practice problem set.  As they're working, I am looking for students to compare the signs first, whole numbers second and fractions/decimals third.

Problem 4 gives students a mix of both decimals and fractions to work with, as they compare rational numbers.  Problem 5 takes away the support of a provided number line.  Students can always create their own number lines - and I'll praise them when I see it.  I want students to always seek out ways to represent the problems they're given.

Problem 8 can be difficult for students.  It asks for the greater depth, which will actually be the smaller number.  The question references absolute value, which is a scaffold.  Students can also draw a vertical number line to help make sense of the problem. 

Closing and Exit Ticket

10 minutes

After independent work time, I have students turn back to Problem 11.  First, I'll display one student's number line on the document camera.  I'll have the class check the displayed work, and suggest any edits to the number line that might be needed.  Then, the class will show me with their thumbs about the truth (up) or falseness (down) of each statement in this problem.  I'll ask one student to pick any of the problems and explain her thinking about the problem.

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