# MA.6.NSO.1.1

Extend previous understanding of numbers to define rational numbers. Plot, order and compare rational numbers.

### Clarifications

Clarification 1: Within this benchmark, the expectation is to plot, order and compare positive and negative rational numbers when given in the same form and to plot, order and compare positive rational numbers when given in different forms (fraction, decimal, percentage).

Clarification 2: Within this benchmark, the expectation is to use symbols (<, > or =).

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Integers
• Number Line
• Rational Number
• Whole Number

### Vertical Alignment

Previous Benchmarks

http://flbt5.floridaearlylearning.com/standards.html

Next Benchmarks

### Purpose and Instructional Strategies

In grade 4, students plotted, ordered and compared fractions, including mixed numbers and fractions greater than one, with different numerators and different denominators. In grade 5, students plotted, ordered and compared multi-digit numbers with decimals up to the thousandths. In grade 6, students plot, order and compare on both sides of zero on the number line with all forms of rational numbers. Additionally, students will rewrite positive rational numbers in different but equivalent forms, which will extend to all rational numbers, including repeating decimals, in grade 7. In grade 8, students will plot, order and compare rational and irrational numbers.
• Instruction builds on the understanding of numbers from elementary grades to include rational numbers. Students should understand that rational numbers are part of a larger number system, the real numbers, and that numbers from previous learning are all rational numbers. The graphic below explains the types of numbers that make up the rational number system.

• Inequality is a comparison between two values. Students should connect the words to the symbols, for example, the symbol < reads “is less than” and the symbol > reads “is greater than.”
• It can be helpful for students to think about placement of rational numbers on a number line using benchmark comparison values.
• Examples of this could include the following questions: Between what two whole numbers does the rational number fall? Is it more than half or less than half way to the next whole number?
• Within the grade 6 benchmarks, students are only expected to convert between forms of positive rational numbers. When working with negative rational numbers, students are only expected to determine their relationship based on relative location to each other, as can be demonstrated on a number line.
• Having students read the comparison relationships from right to left and from left to right can help students develop flexibility of thinking and fluency with using the inequality symbols.
• When comparing negative rational numbers, it is helpful to reason through the comparisons using real-world scenarios (MTR.7.1)
• For example, if a person owes \$12.47 and another person owes \$6.50, most students can quickly see and understand that −12.47 < −6.50 because the person owes more money would have less money.
• Students should be given opportunities to order rational numbers by notating as a list (as is often done with data sets) and by notating using inequality symbols.
• For example, when ordering from least to greatest, the set could be written as –$\frac{\text{5}}{\text{8}}$,  −0.33, 2, 5.8 or as –$\frac{\text{5}}{\text{8}}$ < −0.33 < 2 < 5.8.
• Instruction includes the use of technology to help define and explore rational numbers.

### Common Misconceptions or Errors

• Students may think if a positive number is larger than a second positive number that the relationship will hold true given the negative or opposite of the given numbers.
• For example, if given 6.75 > 6.71, then a student may think that – 6.75 > −6.71. Using a number line (vertical or horizontal) can help students to see the relationship with whole numbers, meaning the greater value is always to the right (horizontal) or above (vertical).
• This relationship also holds true with rational numbers. If students accurately plot the points they can determine which value is larger by looking at the position.
• Students may think that rational numbers cannot be compared unless they are in the same form (i.e., only fractions can be compared to fractions). Instruction should showcase how to make comparisons using benchmark values for plotting numbers without converting between forms.

### Strategies to Support Tiered Instruction

• Instruction includes using a physical or digital number line where students can plot the values and see that the further to the left they move on the number line, the lesser the number.
• When comparing positive numbers in different forms, students may think they can order them just by looking, or that they cannot be compared since they are in different forms. Reviewing converting between fractions, decimals, and percent will help students in the comparing process.
• Teacher provides explanation of negative numbers (perhaps starting with integers) in terms of currency to help bring real-world connection to the standard. Students should know they would rather have someone owe them \$34 over someone owing them \$21. Similarly, they understand that if they owed someone \$6.25 (−6.25), that is less than owing them \$11.84 (−11.84). Then, using the number lines, students can begin to recognize patterns that will bridge their understanding from conceptual to procedural, as well as apply their understanding in real world situations.
• When plotting rational numbers, students should start by making assumptions and reasonable estimations.
• For example, if a student is plotting −4$\frac{\text{3}}{\text{5}}$ on a number line, they need to know that $\frac{\text{3}}{\text{5}}$ is more than half. So, to graph this value, they should plot a point between −4 and −5, with the point closer to the −5. Then, the student can more precisely adjust the point if necessary if any other value is between −4 and −5.
• Instruction showcases how to make comparisons using benchmark values for plotting numbers without needing to convert between forms.

Instructional Task 1 (MTR.1.1, MTR.4.1, MTR.5.1)
Provide students with an open number line and cards labeled with different rational numbers. Students may work together in groups or individual; if individual be sure to provide opportunities for students to discuss with a partner and as a whole group.
• Part A. Arrange the cards on the number line.
• Part B. Discuss your rationale for placement of rational numbers in your group or with a partner.
• Part C. Compare your work and discuss whether you agree or disagree on rational number placement on the number line.

Sharon was being paid to paint a mural on a wall in the gym. The mural was supposed to be at least 6.2 feet tall and 12.8 feet wide. When she measured her completed mural, the tape measure showed the height as 6$\frac{\text{1}}{\text{4}}$ feet and the width as 12$\frac{\text{7}}{\text{8}}$ feet. Did the dimensions of Sharon’s mural meet the given requirements? How do you know?

Cameron wrote on a quiz that $\frac{\text{2}}{\text{3}}$, −$\frac{\text{6}}{\text{8}}$, and $\frac{\text{1}}{\text{9}}$  are all rational numbers because they are written as fractions and that –6, 12% and 4.333 are not rational numbers because they are not written as fractions. Is Cameron’s answer correct? Explain your reasoning using a model to represent the situation.

### Instructional Items

Instructional Item 1
Plot and label each of the following numbers on a number line: 3.4, $\frac{\text{2}}{\text{3}}$ , 2.7, 7, $\frac{\text{3}}{\text{4}}$ and 1.

Instructional Item 2
Write the comparisons of $\frac{\text{1}}{\text{6}}$ and $\frac{\text{1}}{\text{3}}$ in more than one way using words and inequalities.

Instructional Item 3
• Part A. Plot −5 and 5 on a number line.
• Part B. Write a comparison that describes the relationship.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205010: M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812015: Access M/J Grade 6 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.6.NSO.1.AP.1: Plot, order and compare rational numbers (positive and negative integers within 10 from 0, fractions with common denominators, decimals up to the hundredths and percentages) in the same form.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

South Pole:

Students are asked to interpret an inequality relating two temperatures.

Type: Formative Assessment

Submarines:

Students are asked to write integers to represent quantities given in context and to relate the integers with an inequality.

Type: Formative Assessment

Positions of Numbers:

Students are asked to describe the positions of numbers relative to each other on a number line.

Type: Formative Assessment

Graphing Points on the Number Line:

Students are asked to find the coordinates of graphed points and graph points with rational coordinates on a number line.

Type: Formative Assessment

## Lesson Plans

The Layers of the Atmosphere, Guest Starring the Integers! :

Students will learn the functions and characteristics of the four main layers of Earth's atmosphere. They will also determine the thickness of each layer and display them to scale. Students will plot the layers' temperatures, noting the change in temperature from the bottom to the top on a number line.

Type: Lesson Plan

Where's The POINT? What's The POINT? The Point is... a DECIMAL. "Multiply with Decimals":

Multiply efficiently and fluently with multi-digit decimals using a standard algorithm for the operation.

Type: Lesson Plan

Mapping the School:

This project is used to help students enhance their ability to use and understand the coordinate plane by creating a map of their school.

Type: Lesson Plan

Modern Math Target Practice:

The lesson uses the classroom as a coordinate plane then moves into plotting points on a graph. It culminates with a target-practice game.

Type: Lesson Plan

Understanding Integers:

This lesson is an introduction to integers. Students will compare, order, and describe real-life situations using positive and negative whole numbers. The concepts of opposites and vertical as well as horizontal number lines are addressed.

Type: Lesson Plan

Rational vs Irrational:

Students will organize the set of real numbers and be able to identify when a number is rational or irrational. They will also learn the process of how to change a repeating decimal to its equivalent fraction.

Type: Lesson Plan

Capture the Boat - Sink the Teacher's Fleet!:

In this lesson, students learn about the four quadrants of a coordinate plane and how to plot points in those quadrants. Students also learn how to use linear equations to predict future input and output pairs. Students work together to try to sink the teacher's fleet in a Battleship-type game while the teacher tries to sink theirs first.

Type: Lesson Plan

Positive or Negative, It's All About Shopping!:

This lesson introduces students to the concept of negative and positive integers as opposites and as indicators of movement, beginning with elevation and ending with real-world application to money.

Type: Lesson Plan

Positive, Zero, or Negative?:

This lesson involves students using positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of zero in each situation. Students will understand the positive and negative numbers are used together to describe quantities having opposite values.

Type: Lesson Plan

Absolutely Integers:

Students will review how to graph positive numbers and then negative numbers on a number line. The students will review absolute value and apply this to different integers. They will then play a fun game to check their understanding.

Type: Lesson Plan

## Original Student Tutorial

Golf: Where Negative Numbers are a Positive Thing:

Learn how to create and use number lines with positive and negative numbers, graph positive and negative numbers, find their distance from zero, find a number’s opposite using a number line and signs, and recognize that zero is its own opposite with this interactive, golf-themed tutorial.

Type: Original Student Tutorial

Comparing Temperatures:

The purpose of the task is for students to compare signed numbers in a real-world context.

Fractions on the Number Line:

The purpose of this task is to help students get a better understanding of fractions on a number line.

Integers on the Number Line 2:

The purpose of this task is for students to get a better understanding of the relative positions and values of positive and negative numbers.

## Tutorials

Comparing Rational Numbers:

In this tutorial, you will compare rational numbers using a number line.

Type: Tutorial

Decimals and Fractions on a Number Line:

Locate fractions and decimals on the same number line in this tutorial.

Type: Tutorial

Ordering Negative Numbers:

Let's order negative numbers from least to greatest in this video.

Type: Tutorial

Ordering Rational Numbers:

In this tutorial, you will learn how to order rational numbers using a number line.

Type: Tutorial

Sorting Values on Number Line:

This video demonstrates sorting values including absolute value from least to greatest using a number line.

Type: Tutorial

## MFAS Formative Assessments

Graphing Points on the Number Line:

Students are asked to find the coordinates of graphed points and graph points with rational coordinates on a number line.

Positions of Numbers:

Students are asked to describe the positions of numbers relative to each other on a number line.

South Pole:

Students are asked to interpret an inequality relating two temperatures.

Submarines:

Students are asked to write integers to represent quantities given in context and to relate the integers with an inequality.

## Original Student Tutorials Mathematics - Grades 6-8

Golf: Where Negative Numbers are a Positive Thing:

Learn how to create and use number lines with positive and negative numbers, graph positive and negative numbers, find their distance from zero, find a number’s opposite using a number line and signs, and recognize that zero is its own opposite with this interactive, golf-themed tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorial

Golf: Where Negative Numbers are a Positive Thing:

Learn how to create and use number lines with positive and negative numbers, graph positive and negative numbers, find their distance from zero, find a number’s opposite using a number line and signs, and recognize that zero is its own opposite with this interactive, golf-themed tutorial.

Type: Original Student Tutorial

Comparing Temperatures:

The purpose of the task is for students to compare signed numbers in a real-world context.

Integers on the Number Line 2:

The purpose of this task is for students to get a better understanding of the relative positions and values of positive and negative numbers.

## Tutorials

Comparing Rational Numbers:

In this tutorial, you will compare rational numbers using a number line.

Type: Tutorial

Decimals and Fractions on a Number Line:

Locate fractions and decimals on the same number line in this tutorial.

Type: Tutorial

Ordering Negative Numbers:

Let's order negative numbers from least to greatest in this video.

Type: Tutorial

Ordering Rational Numbers:

In this tutorial, you will learn how to order rational numbers using a number line.

Type: Tutorial

Sorting Values on Number Line:

This video demonstrates sorting values including absolute value from least to greatest using a number line.

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.