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.

Gallery OverviewAllow students who have a clear understanding of the content in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit’s concepts or to assist students who may have fallen behind on work.Problem DescriptionsMultiplication WebsStudents fill in the blanks to create expressions equal to the number in the center of the web.Number TreesStudents complete number tree puzzles.Are They Equivalent?Students decide when given expressions will have the same value as ab.Squaring and CubingStudents find solutions to simple equations and inequalities involving squares and cubes.Transforming TrianglesStudents investigate how a triangle changes when they multiply the coordinates of its vertices by positive and negative numbers.True or False?Students determine whether given statements about positive and negative numbers are true or false.Saving MoneyStudents use positive and negative numbers to make sense of changes to Lucy’s savings account.Altitude and TemperatureStudents explore how the air temperature changes as the height of an airplane changes.

Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit’s concepts or to assist students who may have fallen behind on work.Gallery DescriptionsTemperature ChangesStudents solve a puzzle by using clues about the temperatures and temperature changes between several cities.Time ZonesStudents use integers to solve problems about times in different world time zones.Build ExpressionsPairs of students play a game in which they use cards to build two expressions that are as close in value as possible.Hexagon PuzzleStudents assemble triangular puzzle pieces by matching the problems and answers on their sides. When the puzzle is complete, the pieces will form a large hexagon.Equivalent ExpressionsStudents sort expressions into groups that have the same value.Are They Equivalent?Students decide when given expressions will have the same value as a − b.Graphical Addition and SubtractionThe locations of a and b on a number line are shown, and students must graph −a, −b, a − b, b − a, a + b,and −a − b.p and nStudents decide whether statements about a positive number p and a negative number n are true for all, some, or no values of p and n.

Cover sheet and example for math notes for 1st grade and middle school

A checklist used by teachers to assess students as they work on mathematical problems. It can be modified to be used by students as a peer- or self-assessment.

Students enter our math classrooms with anxiety about performance, misconceptions about what math is, and a lack of confidence that can limit their ability to have meaningful learning experiences. In response to this challenge, Stanford researcher Jo Boaler has focused on some key tenants to help students transform their mindset to find more success with math teaching and learning. Some of these mindset shifts include recognizing that: (1) anyone can learn math, (2) making mistakes is essential to learning, (3) math is about fluency and not speed, (4) math is visual, (5) being successful in math requires creativity, flexibility, problem solving, and number sense.

In order to start building these mindsets, Boaler advocates, among other strategies, that students build a habit of being mathematical through common routines, tasks, and puzzles.

This guide will introduce 3 of those routines/puzzles including tips on how to successfully implement these tasks in a face to face, blended, or distance learning setting.

The Need
Many adult education students had difficult (and often negative) experiences with math teaching and learning during their time in the K-12 system. Without addressing their math trauma and helping them to build a mathematical mindset, our students may continue to struggle and be limited in their ability to succeed in math class, on the equivalency exam, and in college and career settings. So our program views math mindsets as the greatest challenge and largest opportunity for transforming the experience our students have when returning to school. Without this shift, we could share the best lesson plans, the most engaging OERs, and the most transformative teachers, and students will continue to be held back by self-limiting perceptions about math and about their ability to succeed.

This course is designed to promote reasoning, problem-solving and modeling through thematic units focused on mathematical practices while reinforcing and extending content in Number and Quantity, Algebra, Functions, Statistics and Probability, and Geometry. It is a yearlong course taught using student-centered pedagogy.

A rubric in student language used by elementary students to meet high standards of quality as they problem solve using math.

This course was originally developed for the Open Course Library project.  The text used is Math in Society, edited by David Lippman, Pierce College Ft Steilacoom.  Development of this book was supported, in part, by the Transition Math Project and the Open Course Library Project. Topics covered in the course include problem solving, voting theory, graph theory, growth models, finance, data collection and description, and probability.

Matrix Algebra with Computational Applications is a collection of Open Educational Resource (OER) materials designed to introduce students to the use of Linear Algebra to solve real-world problems. These materials were developed specifically for students and instructors working in a "flipped classroom" model that emphasizes hands-on problem-solving activities during class meetings, with students watching lectures and completing readings and assignments outside of the classroom. To access the Matrix Algebra with Computational Applications website, please go to http://colbrydi.github.io/MatrixAlgebra

This activity is a lab investigation where students gather data about the masses of various solid objects found in a classroom. The students graph their data, compare their data, and draw conclusions about what kinds of materials contain more matter than others.

This course introduces the fundamentals of machine tool and computer tool use. Students work with a variety of machine tools including the bandsaw, milling machine, and lathe. Instruction given on MATLAB®, MAPLE®, XESS™, and CAD. Emphasis is on problem solving, not programming or algorithmic development. Assignments are project-oriented relating to mechanical engineering topics. It is recommended that students take this subject in the first IAP after declaring the major in Mechanical Engineering.
This course was co-created by Prof. Douglas Hart and Dr. Kevin Otto.

This outdoor investigation involves students observing, recording, comparing and pondering the differing landscapes and rocks located along a river. Follow-up class sessions involve student generation of investigable questions, student-generated studies with required write-up and a mapping activity.

Decision-Making often refers to a multi-stage process that starts with some form of introspection or reflection about a situation in which a person or group of people find themselves. These ruminations usually lead to series of questions that need to be answered, or to a set of data that needs to be collected and analyzed, or to some calculations that need to be performed before someone can be in a position to make informed decisions and take appropriate actions.

In this document, we provide some simple examples of Quantitative Models, which are often found in a decision-making situation. We focus on the use of algebraic equations, probability models, the “Payoff Table” and “Decision Tree” models, to represent situations involving a sequence one or more of decisions over time. Concepts are illustrated with a large set of examples that can be presented during classroom instruction and can be practiced by the students, either individually or in groups, through homework or lab exercises.

This activity investigates the molar conversions used in a single replacement reaction between copper and iron.

My Math GPS: Elementary Algebra Guided Problem Solving is a textbook that aligns to the CUNY Elementary Algebra Learning Objectives that are tested on the CUNY Elementary Algebra Final Exam (CEAFE). This book contextualizes arithmetic skills into Elementary Algebra content using a problem-solving pedagogy. Classroom assessments and online homework are available from the authors.

This activity is a longer project that involves using inquiry to find out what kinds of aquatic animals live in our school pond. Students use direct observation, journaling, outdoor classroom techniques, research skills, collaboration, and service learning to learn about their question.