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.

Adapted from mathematicsvisionproject.com’s Material Overview:
The Mathematics Vision Project (MVP) was created as a resource for teachers to implement the Common Core State Standards (CCSS) using a task-based approach that leads to skill and efficiency in mathematics by first developing understanding. The MVP approach develops the Standards of Mathematical Practice through experiential learning. Students engage in mathematical problem solving, guided by skilled teachers, in order to achieve mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. The MVP authors created a curriculum where students do not learn solely by either “internalizing what a teacher or book says or, on the other hand, solely by inventing mathematics on their own.”
The MVP classroom experience begins by confronting students with an engaging problem and allows them to grapple with solving it. As students’ ideas emerge, take form, and are shared, the teacher deliberately orchestrates the student discussions and explorations toward a focused math goal. Students justify their own thinking while clarifying, describing, comparing, and questioning the thinking of others leading to refined thinking and mathematical fluency. What begin as ideas become concepts that lead to formal, traditional math definitions and properties. Strategies become algorithms that lead to procedures supporting efficiency and consistency. Representations become tools of communication which are formalized as mathematical models. Students learn by doing mathematics.

Students are provided with a scenario and asked to determine possible outcomes. The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task.

This lesson unit is intended to help teachers assess how well students are able to: recognize the differences between equations and identities; substitute numbers into algebraic statements in order to test their validity in special cases; resist common errors when manipulating expressions such as 2(x Đ 3) = 2x Đ 3; (x + 3)_ = x_ + 3_; and carry out correct algebraic manipulations. It also aims to encourage discussion on some common misconceptions about algebra.

Spiral Review on expressions, equations, consecutive integers, etc.We use Spirals to review past content. They are ususally given out once a week.

This task is from Tools 4 NC Teachers. Students write expressions using parentheses that equal a "target number." This is remixable.

This task is from Tools 4 NC Teachers. Students record expressions based on a real life context. This is remixable.

In this lesson, students encounter a situation in which they must solve for the value of a variable. The equations are in the form of p = qx, in which q, x, and p are all whole number values. This CYBERCHASE activity is motivated by an episode in which the CyberSquad plays a game against Hacker to catch enough "gleamers" to power up the Cyberspace ship. If they win, they get the Encryptor Chip. But first, they must figure out the relationship between gleamers and power glows.

In this task, students are provided with a scenario and asked to write an expression to represent the number of guests that can be seated at 57 tables. The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task.