Working With Rational Numbers

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Compare and order positive and negative numbers and place them on a number line.
Understand the concepts of opposites absolute value.

Lesson Flow

The unit begins with students using a balloon model to informally explore adding and subtracting integers. With the model, adding or removing heat represents adding or subtracting positive integers, and adding or removing weight represents adding or subtracting negative integers.

Students then move from the balloon model to a number line model for adding and subtracting integers, eventually extending the addition and subtraction rules from integers to all rational numbers. Number lines and multiplication patterns are used to find products of rational numbers. The relationship between multiplication and division is used to understand how to divide rational numbers. Properties of addition are briefly reviewed, then used to prove rules for addition, subtraction, multiplication, and division.

This unit includes problems with real-world contexts, formative assessment lessons, and Gallery problems.

Students use the Hot Air Balloon interactive to model integer addition. They then move to modeling addition on horizontal number lines. They look for patterns in their work and their answers to understand general addition methods.Key ConceptsTo add two numbers on a number line, start at 0. Move to the first addend. Then, move in the positive direction (up or right) to add a positive integer or in the negative direction (down or left) to add a negative integer.Here is −6 + 4 on a number line: The rule for integer addition (which extends to addition of rational numbers) is easiest to state if it is broken into two cases:If both addends have the same sign, add their absolute values and give the result the same sign as the addends. For example, to find −5 + (−9), first find |−5|  +|−9| = 14. Because both addends are negative the result is negative. So, −5 + (−9) = −14.If the addends have different signs, subtract the lesser absolute value from the greater absolute value. Give the answer the same sign as the addend with the greater absolute value. For example, to find 5 + (−9), find |−9| − |5| = 9 − 5 = 4. Because −9 has the greater absolute value, the result is negative. So, 5 + (−9) = −4.Goals and Learning ObjectivesModel integer addition on a number line.Learn general methods for adding integers.

Students use number lines to solve addition and subtraction problems involving positive and negative fractions and decimals. They then verify that the same rules they found for integers apply to fractions and decimals as well. Finally, they solve some real-world problems.Key ConceptsThe first four lessons of this unit focused on adding and subtracting integers. Using only integers made it easier for students to create models and visualize the addition and subtraction process. In this lesson, those concepts are extended to positive and negative fractions and decimals. Students will see that the number line model and rules work for these numbers as well.Note that rational number will be formally defined in Lesson 15.Goals and Learning ObjectivesExtend models and rules for adding and subtracting integers to positive and negative fractions and decimals.Solve real-world problems involving addition and subtraction of positive and negative fractions and decimals.

What is a number line? How can you use one to compare large and small numbers? Find the answers to these questions, play with an interactive number line, and follow links to information on adding, subtracting, and understanding absolute value using a number line.

This site provides a detailed overview of integers, including positive and negative integers, number lines, absolute values of integers, and comparing integers.

This Math Page tutorial explains how percents are ratios and then features a practice exercise.

This website from The Math Page explains some helpful hints about how to do mental math.

If you are looking for a website that explains and defines the basics of mental multiplication with helpful hints then you have arrived at an excellent resource. There are even practice problems at the very end of the lesson.

This website explains and defines basic subtraction. There are practice problems at the very end of this lesson.

Whole numbers are no better than any others! Practice plotting values on the number line as a passionate activist rises up and demands equity for all numbers, including fractions and decimals. A teaching guide with printable resources is included. [3:09]

In this Cyberchase video segment, the CyberSquad helps resolve a dispute about the squares on the Gollywood Walk of Fame.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

En este módulo de 35 días de grado 3, los estudiantes extienden y profundizan la práctica de segundo grado con "acciones iguales" para comprender las fracciones como particiones iguales de un todo. Su conocimiento se vuelve más formal a medida que trabajan con los modelos de área y la línea numérica.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
In this 35-day Grade 3 module, students extend and deepen second grade practice with "equal shares" to understanding fractions as equal partitions of a whole. Their knowledge becomes more formal as they work with area models and the number line.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

Given a, b, and c shown on the number line, Sal determines if statements like -b < c are true. [4:54]

Khan Academy learning modules include a Community space where users can ask questions and seek help from community members. Educators should consult with their Technology administrators to determine the use of Khan Academy learning modules in their classroom. Please review materials from external sites before sharing with students.

Oscar looks at patterns by using a number line, and makes predictions about a giant sighting in the area.

Learn about using a number line with the Odd Squad. Agents Olive and Otto help Oscar figure out which Oscarbots are missing by using a number line.

Play a fast-paced video game that involves finding rational numbers on the number line. This game from Math Snacks focuses on using what you know about positive and negative numbers and fractions to place them correctly on the variable number line and retrieve pearls.

Create your own fractions using fun interactive objects. Match shapes and numbers to earn stars in the fractions games. Challenge yourself on any level you like. Try to collect lots of stars!

The virtual simulation allows users to explore the features of fractions while playing this fun interactive game. Learn about numerators, denominators, number lines, and picture patterns.

Students build on their understanding of the structure of the place value system, start to use rounding as a way to estimate quantities, and develop fluency with the standard algorithm of addition and subtraction. Students focus on the precision of their calculations, and use them to solve real-world problems.