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.

Zooming In On Figures

Unit Overview

Type of Unit: Concept; Project

Length of Unit: 18 days and 5 days for project

Prior Knowledge

Students should be able to:

Find the area of triangles and special quadrilaterals.
Use nets composed of triangles and rectangles in order to find the surface area of solids.
Find the volume of right rectangular prisms.
Solve proportions.

Lesson Flow

After an initial exploratory lesson that gets students thinking in general about geometry and its application in real-world contexts, the unit is divided into two concept development sections: the first focuses on two-dimensional (2-D) figures and measures, and the second looks at three-dimensional (3-D) figures and measures.
The first set of conceptual lessons looks at 2-D figures and area and length calculations. Students explore finding the area of polygons by deconstructing them into known figures. This exploration will lead to looking at regular polygons and deriving a general formula. The general formula for polygons leads to the formula for the area of a circle. Students will also investigate the ratio of circumference to diameter ( pi ). All of this will be applied toward looking at scale and the way that length and area are affected. All the lessons noted above will feature examples of real-world contexts.
The second set of conceptual development lessons focuses on 3-D figures and surface area and volume calculations. Students will revisit nets to arrive at a general formula for finding the surface area of any right prism. Students will extend their knowledge of area of polygons to surface area calculations as well as a general formula for the volume of any right prism. Students will explore the 3-D surface that results from a plane slicing through a rectangular prism or pyramid. Students will also explore 3-D figures composed of cubes, finding the surface area and volume by looking at 3-D views.
The unit ends with a unit examination and project presentations.

Mathematically Productive Instructional Routines (MPIR) are short (10ish minutes), daily exercises aimed at building number sense. These six different MPIR are part of the Mathematically Productive Instructional Routines collection from the Washington Office of Public Instruction and the Washington Association of Educational Service Districts.

In this unit, students will read and interpret primary sources to address the question “How do we measure the attainment of human rights?” By exploring the Universal Declaration of Human Rights, the UN’s Guide to Indicators of Human Rights, and data about development indicators from multiple databases, students will unpack the complexities of using indicators to measure human rights.

Monitor the temperature of a melting ice cube and use temperature probes to electronically plot the data on graphs. Investigate what temperature the ice is as it melts in addition to monitoring the temperature of liquid the ice is submerged in.

This lesson unit is intended to help teachers assess how well students are able to: interpret a situation and represent the constraints and variables mathematically; select appropriate mathematical methods to use; make sensible estimates and assumptions; investigate an exponentially increasing sequence; and communicate their reasoning clearly.

Study the motion of a toy car on a ramp and use motion sensors to digitally graph the position data and then analyze it. Make predictions about what the graphs will look like, and consider what the corresponding velocity graphs would look like.

(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 el módulo 3, la comprensión de los estudiantes de la adición y la resta de las fracciones se extiende desde el trabajo anterior con equivalencia de fracción y decimales. Este módulo marca un cambio significativo lejos de la centralidad de los grados elementales de las diez unidades de base al estudio y el uso del conjunto completo de unidades fraccionarias desde el avance de grado 5, especialmente como se aplica al álgebra.

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

English Description:
In Module 3, students' understanding of addition and subtraction of fractions extends from earlier work with fraction equivalence and decimals. This module marks a significant shift away from the elementary grades' centrality of base ten units to the study and use of the full set of fractional units from Grade 5 forward, especially as applied to algebra.

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

Take a breath — where does the oxygen you inhaled come from? In our changing world, will we always have enough oxygen? What is in water that supports life? What is known? How do we know what we know about our vast oceans? These are just a few of the driving questions explored in this interactive STEAM high school curriculum module.

Students in marine science, environmental science, physics, chemistry, biology, integrated science, biotechnology and/or STEAM courses can use this curriculum module in order to use real-world, big data to investigate how our “invisible forest” influences ocean and Earth systems. Students build an art project to represent their new understanding and share this with the broader community.

This 4-week set of lessons is based on the oceanographic research of Dr. Anne Thompson of Portland State University in Oregon, which focuses on the abundant ocean phytoplankton Prochlorococcus. These interdisciplinary STEAM lessons were inspired by Dr. Thompson’s lab and fieldwork as well as many beautiful visualizations of Prochlorococcus, the ocean, and Earth. Students learn about the impact and importance of Prochlorococcus as the smallest and most abundant photosynthetic organism on our planet. Through the lessons, students act as both scientists and artists as they explore where breathable oxygen comes from and consider how to communicate the importance of tiny cells to human survival.

This module is written as a phenomenon-based, Next Generation Science Standards (NGSS) three-dimensional learning unit. Each of the lessons below also has an integrated, optional Project-Based Learning component that guides students as they complete the PBL process. Students learn to model a system and also design and evaluate questions to investigate phenomena. Students ultimately learn what is in a drop of ocean water and showcase how their drop contributes to our health and the stability and dynamics of global systems.

This lesson unit is intended to help teachers assess how well students are able to: recognize and visualize transformations of 2D shapes; and translate, reflect and rotate shapes, and combine these transformations. It also aims to encourage discussion on some common misconceptions about transformations.

Explore your own straight-line motion using a motion sensor to generate distance versus time graphs of your own motion. Learn how changes in speed and direction affect the graph, and gain an understanding of how motion can be represented on a graph.

This lesson unit is intended to help teachers assess how well students are able to identify and use geometrical knowledge to solve a problem. In particular, this unit aims to identify and help students who have difficulty in: making a mathematical model of a geometrical situation; drawing diagrams to help with solving a problem; identifying similar triangles and using their properties to solve problems; and tracking and reviewing strategic decisions when problem-solving.

This unit consists of five lessons encouraging younger learners to engineer increasingly better towers using blocks and recycled materials. Each 30 minute lesson ("phase") includes goals, discussion, activity instructions, extensions, and student worksheets.

Phase 1: Paper Cut-Outs Activity
Phase 2: Building Blocks Activity
Phase 3: Number of Blocks Activity
Phase 4: Building within a Space Activity
Phase 5: Recycled Tower Activity

NGSS: K-2-ETS1-1, K-2-ETS1-2, K-2-ETS1-3

Common Core ELA: RI.2.1, W.2.6, W.2.8, SL.2.5

Common Core Math: MP.2, MP.4, MP.5, 2.MD.D.10