Muffle's Truffles: Multiplication and Division with the Array is one of five units in the Contexts for Learning Mathematics' Investigating Multiplication and Division (3 - 5) The focus of this unit is the development of the open array as a model for multiplication and division. This unit uses a series of investigations based on the context of Muffles' Truffles shop. The questions posed in the first investigation (how many boxes of ten can be made with a given quantity of truffles; how many leftovers will there be from a given quantity and how can they be combined to make assortment boxes; and what is the cost of a given quantity of truffles if they cost $1 each) give students an opportunity to explore place value - the multiplicative structure of our base-ten system and quotative division. In the second and third investigations, students build two-dimensional blueprints of one-layer boxes and use these arrays to explore some of the big ideas in multiplication (the distributive, associative, and commutative properties). In the fourth and final investigation, students work with open arrays in the context of labeling and pricing wrapped boxes of truffles. To figure out the dimensions of the wrapped boxes (or open arrays) and the cost, students need to apply a number of big ideas previously developed in this unit. There are three different kinds of minilessons for multiplication included in the unit as well: counting around the circle, strings of related problems, and quick images. The count-around is used to support the development of place value as it relates to multiplication. The strings of related problems are explicitly designed to guide learners toward computational fluency with whole number multiplication and to build automaticity with multiplication facts by focusing on relationships. The quick images use 2 x 5 and 1 x 5 arrays as units to build larger arrays. In the last days of the unit, more complex minilessons (double-digit multiplication problems) generate a wider range of student strategies that can be explored (and modeled) with the open array. To learn more visit http://www.contextsforlearning.com
The Teachers' Lounge: Place Value and Division is one of five units in the Contexts for Learning Mathematics' Investigating Multiplication and Division (3 - 5) The focus of this unit is division. It begins with the story of a teacher noticing a service person in the teachers' lounge fill two different vending machines with beverages. In the first machine, there are bottles of water only. The machine holds 156 bottles of water when full and the teacher wonders how many six-packs that might be. The second problem involves the juice machine. It also holds 156 bottles when full, but the bottles are partitioned into six columns because there are six different flavors of juice. The teacher wonders how many there are of each flavor. Although most students do not realize it at the start, the two problems are related. The first problem is a quotative division situation - the amount in each group is known, the number of groups is not. The second problem is a partitive division situation - the number of groups is known, the amount in each group is not. The problems are juxtaposed and given together to encourage students to examine the relationship between the two kinds of division. This story context of the teachers' lounge sets the stage for a series of investigations designed to support the development of a repertoire of strategies for multiplication and division, including the use of: * the ten-times strategy * partial products and partial quotients * the associative property * the distributive property of multiplication over addition - the basis for the long division algorithm Several minilessons for multiplication and division are also included in the unit. These are structured as strings of related problems designed to more explicitly guide learners toward computational fluency. Toward the end of the unit, discussion shifts to how the context of a division problem influences what to do with the remainder. Note: The context for this unit assumes that your students will have had prior experience using arrays for multiplication. If this is not the case, you might find it helpful to use the unit Muffle's Truffles first. To learn more visit http://www.contextsforlearning.com
Minilessons for Extending Multiplication and Division is one of two yearlong resource guides in Contexts for Learning Mathematics' Investigating Multiplication and Division (3 - 5) Minilessons for Extending Multiplication and Division can be helpful in grades 4 - 5 as students work with multiplication and division beyond the basic facts. This guide contains 77 minilessons structured as strings of related computation problems. They are likely to generate discussion of certain strategies or big ideas that are landmarks on the landscape of learning for multiplication and division, particularly using numbers with two and three digits. Although the emphasis is on the development of mental arithmetic strategies, this does not mean learners have to solve the problems in their heads - but it is important for them to do the problems with their heads! In other words, as you use this guide, encourage students to examine the numbers in each problem and think about clever, efficient ways to solve it. The relationships between the problems in the minilesson will support students as they progress through the string. The open array is used throughout to represent student strategies. To learn more visit http://www.contextsforlearning.com
Author: Frans Van Galen
Release Date: 2008-03
Groceries, Stamps, nd Measuring Strips: Early Multiplication is one of five units in the Contexts for Learning Mathematics' Investigating Multiplication and Division (3 - 5) The focus of this unit is the introduction and early development of multiplication. By making use of realistic contexts, the unit invites students to find ways to mathematize their lived worlds with grouping structures. The unit uses many contexts: inside the grocery store; postage stamps; city buildings, windows, and buses; tiled patios; a baker's trays; and sticker pages. Initially, formal multiplication notation is not the focus; efficient grouping is, as students are encouraged to make groups (and groups of groups) to find efficient ways to deal with repeated addition and determine totals. The unit begins with the context of a grocery store. Students view an illustration of fruits and vegetables arranged in bins, stacked packages of paper towels, and six-packs of water bottles, among other items in a grocery store. Although the objects shown can be counted by ones, the arrangements naturally invite repeated addition, skip-counting, and doubling strategies as well as the language of grouping - for example, 8 groups of 6 is equivalent to 4 groups of 12 which is equivalent to 4 groups of 6 plus 4 groups of 6. The stamp context used next eliminates objects that can be counted by ones. Now the value printed on the stamp is the focus. This context thus supports the development of unitizing by providing the value (e.g., seven cents) as a unit that can be counted. Providing the value also offers a built-in-constraint to counting by ones, and supports repeated addition and efficient grouping employing doubling, doubling and halving, and the addition of partial products. This context promotes a natural shift in students' language to "5 sevens" (5 seven-cent stamps). Formal notation (the use of x to indicate multiplication) is introduced halfway through the unit with the context of measurement. Students view an illustration of a cityscape with high buildings, large windows, tall trees, and a school bus. A four-foot tall, eight-year-old boy, Antonio, is shown on the street; Antonio wonders how much taller everything is than he. Because his height is used for a unit of measurement, the natural language that evolves is "times, for example, "eight times the size of Antonio". Formal notation is introduced to match the language - 8 x 4. As the unit progresses, students make a set of measurement strips (for the multiplication tables) and explore the relationships between the products on them. In the last few days of the unit, the measurement strips are represented as number lines that students use to determine missing products from the expressions and products provided. Here the five- and ten-structures are emphasized, supporting students in using five-times to help with four-times and six-times, and ten-times to help with nine-times. Several minilessons are also included in this unit. Quick images, count-around-the-circle activities, and pictures with built-in constraints support the construction of efficient strategies - strategies that over time will help students automatize the basic facts. Note: This unit also incorporates aspects of the measurement strand as students measure the height of various objects in the illustration of the city. They use the height of Antonio as an iterated unit, and make measurement strips for the lengths of various groups of connecting cubes. To learn more visit http://www.contextsforlearning.com
The first in a three-volume set, Young Mathematicians at Work focuses on young children between the ages of four and eight as they construct a deep understanding of number and the operations of addition and subtraction.
In The Big Dinner the preparation of a turkey dinner introduces early multiplication strategies and supports automatizing the facts, using the ratio table, and developing the distributive property with large numbers. Strings of problems guide learners toward computational fluency with whole-number multiplication and build automaticity with multiplication facts by focusing on relationships.
Contexts for Learning consists of: Investigations and Resource Guides - workshop structure involves students in inquiring, investigating, discussing, and constructing mathematical solutions and strategies - investigations encourage emergent learning and highlight the developmental landmarks in mathematical thinking - strings of related problems develop students' deep number sense and expand their strategies for mental arithmetic Read-Aloud Books and Posters - create rich, imaginable contexts--realistic and fictional--for mathematics investigations - are carefully crafted to support the development of the big ideas, strategies, and models - encourage children to explore and generate patterns, generalize, and develop the ability to mathematize their worlds Resources for Contexts for Learning CD-ROM - author videos describe the series' philosophy and organization - video overviews show classroom footage of a math workshop, including minilessons, investigations, and a math congress - print resources include research base, posters, and templates
Anyone who grooved to the counterculture vibe of Doctorow's young-adult novels Little Brother (2008) and For the Win (2010) will embrace these stories heartily-no one can dole out technological cautionary tales while simultaneously celebrating technology as cunningly as Doctorow. This volume's single never-before-published story, "Epoch," is the standout, an ethically thorny but heartfelt update on the classic sf conceit of an AI that becomes too self-aware. Never one to avoid the jugular, Doctorow doesn't bother to assign Google an alias in "Scroogled"; the depiction of a world where we're all "Googlestalked" until we're "guilty of something" feels chillingly immediate. It's not always easy to warm up to Doctorow's purposeful characters, but it's easy to be swept up in their just-barely-futuristic travails of surveillance gone wrong and privacy shattered. Reading this on your iPhone? Then these stories are probably for you. -- Booklist
The topics explored in each chapter are based on hundreds of discussions the author has led with adult science learners over many years – people who came from all walks of life and had no scientific training, but had developed a burning curiosity to understand the world around them. This book encourages us to reflect on our own relationship with science and serves as an important reminder of why we should continue learning as adults. Praise for Why Icebergs Float 'Asking questions is an important scientific skill and sometimes we can only understand something when we can find the language to ask the right questions; books like this can be really helpful in this respect....This book is one of UCL’s open access books. This means that it can be downloaded as a free PDF from the UCL Press website. The commitment to making scientific works such as this freely available is very welcome. This book is very accessible and deserves to reach a wide audience.' - School Science Review 'Morris says in the prologue: ‘If you come away from this book with a greater interest in science and enhanced confidence about tackling it, the book will have served its purpose.’ So, don’t be afraid of science and give Why Icebergs Float a chance. You will absolutely enjoy it.' - Chemistry World '[Why Icebergs Float] draws on experiences and first-person narratives of adult learners who – out of genuine curiosity or embarrassment at their levels of scientific ignorance – have sought to catch-up on lost school science and get a better understanding of their surroundings as a result.' - Education Journal '‘The approach illustrates beautifully the influence of language on understanding. The author makes clear how common language can be misleading when scientists have used everyday words but given them very specific meanings.’ Physics Education
Building learning around rich, instructionally sound contexts was an overarching goal during the development of the Contexts for Learning Mathematics series. Throughout the series context is used to set the stage for learning. It establishes a terrain that will intrigue children and ignite their imaginations. The contexts are situations children can imagine - either realistic or fictional - that enable them to reflect on what they are doing and apply mathematical thinking to their own world. Contexts for investigations are typically developed with stories and pictures. These are carefully crafted to involve students in meaningful investigations of the big ideas, strategies, and models that shape mathematical thinking. • The images and texts are engaging and include age-appropriate children using mathematics to solve real-world problems. • The numbers referenced represent landmark numbers or number relationships that are significant and telling. • The models and metaphors within a context make relationships and strategies more tangible and explicit. The contexts for the five units in Investigating Multiplication and Division (Grades 3 - 5) are established through 17 vibrant posters (15” x 24”) that meld humor, intrigue, and good math sense. To learn more visit www.contextsforlearning.com
Author: Peter F. Bulmer
Publisher: Strategic Book Publishing
Release Date: 2014-03
Do you want to learn The Lost Secret of Speaking Perfect English? The Moving Mouth Dictionary technique provides a very simple approach to perfecting English speaking and pronunciation. The book takes a down-to-earth approach for speaking clear English, as it breathes some fresh air into the stuffy corridors of academic learning. It is geared to help students and business people speak impressive and naturally clear English, taking much of the guessing out of pronunciation and spelling. English will become more of a physical activity, rather than a cerebral academic subject. The key is in identifying and improving specific types of reverse and forward mouth movements, actions based on using simple vertical mouth movement notations that have simple associations with key phonetics sounds for specific letters. The technique's forward and reverse mouth movements combined with a natural English rhythm also helps trigger and access vocabulary and verbs, while aiding in word retention, fluency and auto correcting. The book features a dictionary of over 11,000 words, including some of the most difficult words in the English language, which have been broken down, putting these notations into "mouthables." The process draws heavily on early humans' natural ability to howl and growl, using their mouths vertically. Hence, the lost connection between our near ancestors can aid our ability to speak clear English, an ability we have lost and need to rediscover. About the Author Originally from the Yorkshire Dales in England, Peter F. Bulmer developed his presentational skills as a marketing and export director traveling and selling to different cultures throughout the world. Now retired and based in Europe, he still coaches bankers, consultants, and marketing people in perfecting their English presentation and communication skills. Publisher's website: http: //sbprabooks.com/PeterFBulmer
The Box Factory: Extending Multiplication with the Array is one of five units in the Contexts for Learning Mathematics' Investigating Multiplication and Division (3 - 5) The focus of this unit is the deepening and extending of students' understanding of multiplication, specifically the associative and commutative properties and their use with computation, systematic factoring, and the extension of students' understanding of two-dimensional rectangular arrays to three-dimensional arrays within rectangular prisms. The unit includes a series of investigations based on the context of a cardboard box factory. Initially students design a variety of boxes (rectangular prisms) that hold 24 items arranged in rows, columns, and layers. The questions posed in the first investigation (how many box arrangements are possible, and how do we know for certain that we have found all the possibilities) give students an opportunity to explore the associative and commutative properties, factor pairs, doubling and halving strategies, and systematic ways of organizing their work to determine all possible cases. In the second and third investigations, students analyze the amount and cost of the cardboard needed for their boxes, deepening their understanding of the associative property, examining congruency vs. equivalency, and exploring the relationship of surface area to the shape of the box. Subsequent investigations involve using two different cubic boxes as units of measurement, and determining the volume of a shipping box that measures 4 feet by 6 feet by 4 feet. By the end of the unit, formulas for surface area and volume of rectangular prisms are the focus. Several minilessons for multiplication are included in the unit as well - these are structures as strings of related problems explicitly designed to guide learners toward computational fluency with whole-number multiplication, by focusing on factors and efficient grouping. To learn more visit http://www.contextsforlearning.com
Author: A. Furnham
Release Date: 2004-02-02
Genre: Business & Economics
This book takes a very critical look at management fads and fashions and shows that many do not stand up to scrutiny. It is believed for example that coaching is successful. What is the evidence for this? Selection procedures at interviews result in the right candidate being selected. Again what is the evidence? The author shows that many fads and fashions in management are short lived and based upon flimsy evidence, yet enjoy a period of support. His book will be an important tool for managers who want to understand the substance and rigour or lack of it associated with modern management ideas and concepts.
Author: William Marling
Publisher: JHU Press
Release Date: 2006-05-31
Genre: Business & Economics
In this provocative work, Marling analyzes--in specific terms--the impacts of American technology and culture on foreign societies. Deconstructing the myth of global Americanization, he argues that despite the typically American belief that the United States dominates foreign countries, the practical effects of "Americanization" amount to less than one might suppose.