Understanding The Pros & Cons of the Number Line: Rational Numbers

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Objective

SWBAT demonstrate how positive and negative numbers are positioned on a horizontal and vertical number line

Big Idea

Are you positive that's negative? Understanding positive and negative rational numbers

Curriculum Reinforcer

10 minutes

Students will complete 3 problems to review units 1 through 6. The curriculum reinforcer, is a daily practice piece that is incorporated into almost every lesson to help my students to retain skills and conceptual understanding from earlier lessons. My strategy is to use Spiraled Review to help my students retain what they learned during the earlier part of the year. This will help me to keep mathematical concepts fresh in the students mind so that the knowledge of these concepts become a part of students' long term memories.

Engagement

10 minutes

To start off today’s lesson we will begin by having a discussion about the number line. I will ask my students what they can tell me about a number line. As they give me the information that they know, I will draw the number line on the white board and label it as the students provide me with information about it.

More than likely, the students will mention positive and negative integers but, they probably will not mention the positive and negative rational numbers. For this reason, I will ensure to ask them specifically about where fractions and decimal quantities fall on the number line. 

to deepen their conceptual understanding of fractional and decimal quantities on the number line, I will say to my students, "Did you know that the same principles used in measurement apply to the number line?" To demonstrate my meaning, behind asking this question, I will display a ruler using my digital projector and show my students how you can have a fraction of an inch. I will then demonstrate that the same is true on the number line, that you can have a fraction of a whole number displayed on the number line.

At this time, I will also ask the students where they see negative numbers in the real world. I should get answers such as:

  • Temperature
  • Debits & Credits (Bank Account)
  • Golf
  • Above & Below Sea Level

So that my students can keep all of this information being presented organized, they will have a foldable that they can fill-in with information as we discuss rational numbers on the number line.

Attached to this section of this lesson is a short video showing what this foldable looks like.

Instruction & Teacher Modeling

10 minutes

As we are discussing the number line, which we will do during the opening of the lesson, I will pass out sentence strips to each of my students. Once we have concluded the opening discussion, I will then have my students to fold their sentence strip in half and place the number 0 on that fold. I will then have my students fold each half in eighths by having them fold each half in half three times.

 

When my students open up their folded sentence strip, they should now have 16 even spaces all the way across the sentence strip. I will then have them number the folds on the sentence strip by starting at zero, skipping two spaces, and then placing a number on one on the fold. They will then skip two more spaces and place a number two, skip two more and place a number three. Then, the students will start back at 0 and do the same thing on the other side creating a number line that has positive and negative integers. Then the students will put in positive and negative rational numbers in the form of halves and fourths.

After having the students create their number lines, I will use this number line to teach students about the following:

  • Where positive and negative integers are located on the number line in relation to each other (mirror image).
  • Where positive an negative rational numbers are located on the number line in relation to each other (mirror image).
  • The concept of infinity in both the positive and negative direction.
  • Develop the understanding that the more negative a number is the less it is worth (the further left you are on the number line the less you are worth).
  • Developing the understanding of distance between two numbers that are on the same side of zero on the number line and distance between numbers that are on different sides of zero on the number line (this will help to prepare the students for absolute value and distance between two points on  a coordinate plane).

As I am teaching these concepts my students will be taking notes and making the appropriate labels on their number line (i.e. marking the positive and negative sides of the number line).

After we are finished labeling the number line, I will then tell my students to take a look at the number line. Once I have given them ample time to study it, I will ask them, are the numbers that we placed on the number line in order? The students should give a response of “yes.” After they do, I will ask them to raise their hand to tell me in what manner they are in order. The student that I select should provide the answer that, “the numbers are ordered from least to greatest.”

Using this information I will show students how the number line is ALWAYS in order from least to greatest and if I have to place the following numbers {-2, 1 ½ , - ¾ , - 1 ¼ , ½ } in order from least to greatest or from the greatest to the least, all I have to do is mark them on the number line and they will already be in order. At this point, I will actually mark these points on the number line to show the students the proper ordering of the numbers.

 

Try It Out

5 minutes

To deepen student understanding of rational numbers on the number line, I will pass out a fraction circle with eight sections and a card with either a positive rational number or a negative rational number. The cards that have rational numbers on them will not surpass the whole number two or negative.

The number line on the board will be labeled so that there are enough spaces to show eighths and the whole number markings will be more pronounced that the fractional markings. The number line should contain NO numbers of any sort. It will be the students’ job to figure out how the number line should be numbered.

 

Try It Out Activity Directions:

I will have my students engage in an instructional activity where they will color in their fraction circle or circles according to what was given to them on their card. However, if they were given a negative number, then they must color in the fractional circle using red. If they were given a positive number, they must use green.

After coloring in their fraction circle, I will call each rational number given. If they were given that rational number, they will come up to the board and place their fraction circle on the right spot on the number line on the board.

Once we are finished labeling the number line properly with the fraction circles, I will go back and put the rational numbers that correspond with the fraction circles. Using this number line, I will demonstrate the following:

  • The larger amount of red you see the less the number is worth
  • The conversion from fraction to decimal
  • The “backwards” placement of the numbers on the negative side of the number line.

Independent Exploration

20 minutes

During the independent practice portion of this lesson, the students will complete the following activities:

1.) First, my students will number a blank number line and plot points that will be given to them.

2.) Then, my students will determine how far apart one plotted point is from another.

3.) After this, my students will then write a brief essay describing all that they have learned today.            Their essay should include the following important lesson points.

  • Integers are positive and negative whole numbers and zero.
  • Positive numbers are on the right side of zero on the number line and negative numbers are on the left side of zero on the number line.
  • Negative numbers are infinite just as positive numbers are infinite.
  • The bigger the negative number, the less it is worth. In other words, the further to the left a number is on the number line, the less it is worth.
  • A description as to how to find the distance between two points on the number line.

 

 

Closing Summary

15 minutes

To close out the lesson, I will select several students to present their work. One student will be selected to show how they numbered their number line and plotted the given integers on the number line. Another student will be selected to show how they determined the distance between two points. And a third student will be chosen to share what they wrote in their essay.