7_1_4MA


 * Grade: 7 Unit: 1 Week: 4 Dates: 9/10-9/14 **
 * Content: Multiplication and Division of Rational Numbers **


 * __ Theme Essential Question __**** : **
 * How can you apply and extend the usage of rational numbers when dealing with real world problems? **


 * __ Essential Questions __**** : **
 * When can you apply the product of rational numbers to describing real-world contexts?
 * When can you apply the quotient of rational numbers to describing real-world contexts?


 * __ Standards __**** : **
 * 7. NS.2: ** Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
 * ** 7. NS.2a: ** Understand that multiplication is extended from fractions to rational numbers by requiring that operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
 * ** 7. NS.2b: ** Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers then – (p/q) = (–p)/q = p/ (–q). Interpret quotients of rational numbers by describing real-world contexts.
 * ** 7. NS.2c: ** Apply properties of operations as strategies to multiply and divide rational numbers.


 * __ Objectives __**** : **
 * The student will multiply and divide positive and negative rational numbers.
 * The student will solve real-world applications with multiplication and division of rational numbers.
 * The student will explore the meaning of division by zero.

By now we hope you are having a successful year implementing the Common Core Curriculum. Students should be building skills using the Eight Mathematical Practices. These practices are a big part of the process that builds student’s problem solving skills, thinking and reasoning skills while also giving students a chance to communicate and make connections with what they are learning. We hope you will continue to display and talk about these Mathematic Practices as you get a chance to incorporate them into your lessons. At this point in the curriculum, students have had an opportunity to develop strategies/algorithms which are the foundation to Mathematical Practice #3, construct viable arguments and critique the reasoning of others. Formally introduction Mathematical Practice #3:
 * __ Reflections and/or Comments from your PCSSD 7th Grade Curriculum Team __**

Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials at @http://www.azed.gov/educator-certification/
 * __ Background Information __**


 * Taken from __Ohio Dept of Education Mathematics Model Curriculum 6-28-2011)__ **

Using what students already know about positive and negative whole numbers and multiplication with its relationship to division, students should generalize rules for multiplying and dividing rational numbers. Multiply or divide the same as for positive numbers, then designate the sign according to the number of negative factors. Students should analyze and solve problems leading to the generalization of the rules for operations with integers. For example, beginning with known facts, students predict the answers for related facts, keeping in mind that the properties of operations apply (See Tables 1, 2 and 3 below).


 * ** Table 1 ** || ** Table 2 ** || ** Table 3 ** ||
 * ** 4 x 4 = 16 ** || ** 4 x 4 = 16 ** || ** -4 x -4 = 16 ** ||
 * ** 4 x 3 = 12 ** || ** 4 x 3 = 12 ** || ** -4 x -3 = 12 ** ||
 * ** 4 x 2 = 8 ** || ** 4 x 2 = 8 ** || ** -4 x -2 = 8 ** ||
 * ** 4 x 1 = 4 ** || ** 4 x 1 = 4 ** || ** -4 x -1 = 4 ** ||
 * ** 4 x 0 = 0 ** || ** 4 x 0 = 0 ** || ** -4 x 0 = 0 ** ||
 * ** 4 x -1 = ** || ** -4 x 1 = ** || ** -1 x -4 = ** ||
 * ** 4 x -2 = ** || ** -4 x 2 = ** || ** -2 x -4 = ** ||
 * ** 4 x -3 = ** || ** -4 x 3 = ** || ** -3 x -4 = ** ||
 * ** 4 x -4 = ** || ** -4 x 4 = ** || ** -4 x -4 = ** ||

Using the language of “the opposite of” helps some students understand the multiplication of negatively signed numbers (-4 x -4 = 16, the opposite of 4 groups of -4). Discussion about the tables should address the patterns in the products, the role of the signs in the products, and the commutative property of multiplication. Then students should be asked to answer these questions and prove their responses. • Is it always true that multiplying a negative factor by a positive factor results in a negative product? • Does a positive factor times a positive factor always result in a positive product? • What is the sign of the product of two negative factors? • When three factors are multiplied, how is the sign of the product determined? • How is the numerical value of the product of any two numbers found? Students can use number lines with arrows and hops, groups of colored chips or logic to explain their reasoning. When using number lines, establishing which factor will represent the length, number, and direction of the hops will facilitate understanding. Through discussion, generalization of the rules for multiplying integers would result. Division of integers is best understood by relating division to multiplication and applying the rules. In time, students will transfer the rules to division situations. (Note: In 2b, this algebraic language (–(//p/////q//) = (–//p//)///q// = //p///(–//q//)) is written for the teacher’s information, not as an expectation for students.) Ultimately, students should solve other mathematical and real-world problems requiring the application of these rules with fractions and decimals.


 * __ Assessment __**** : **
 * Product **
 * This will be an ongoing product that will be developed during each week of the unit. Continue foldables this week with multiplication and division. **
 * Foldable **
 * Make a foldable with five divided sections (3 pieces of paper needed). Students will create a section for each of the following:
 * Addition- Have written rule and examples using integers, fractions, and decimals.
 * Subtraction- Have written rule and examples using integers, fractions, and decimals.
 * Multiplication- Have written rule and examples using integers, fractions, and decimals.
 * Division- Have written rule and examples using integers, fractions, and decimals.
 * Long Division- Have written rule and examples.


 * Key Questions **
 * How can you model multiplication and division of positive and negative numbers?
 * What do you notice about the product/quotient of two rational numbers with different signs?
 * What do you notice about the product/quotient of two rational numbers with the same sign?


 * Observable Student Behaviors **
 * The student can apply and extend their previous understanding of multiplication and division to rational numbers.
 * The student can apply properties of operations as strategies to multiply and divide rational numbers:
 * They can use the distributive property to explain the products such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
 * They understand that every quotient of integers (with non-zero divisor) is a rational number.
 * They can represent a quotient of integers p and q in the forms – (p/q) = (–p)/q = p/ (–q).
 * They can interpret quotients of rational numbers by describing real-world contexts.

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||
 * ** Mathematical Practices **


 * __ Vocabulary __**** : **

Undefined ||
 * ** Math ** ||
 * Fractions Integers Negative Numbers Positive Numbers Multiplication Division Quotient Product

Nothing Available at this Time (NS.2) The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
 * __ Suggested Activities __** :
 * **// On Core Mathematics Grade 7 //** Workbook 1-4 and 1-5, pp. 15-22
 * **// Mastering the COMMON CORE in Mathematics Grade 7 //** by American Book Company. Chapter 4 Pp 34-42
 * **// Teaching the Common Core Math Standards with Hands-On Activities //**
 * NS.2 Activity 1 p. 93, Activity 2 p. 96
 * Gizmo Correlation
 * None
 * ** Highly Recommended **
 * ** JBHM **
 * **// Glencoe Mathematics Application and Concepts Course 2 //** 6.4 (Multiplying Fractions)**//,//** pp. 254-257 and 6.6 (Dividing Fractions), pp. 264-266
 * **// Glencoe Pre-Algebra //** 5-3 and 5-4, pp. 210-219
 * **// Glencoe Algebra 1 //** 2-3 (Multiplying/Dividing Rational Numbers) and 2-4, p. 79-87


 * __ Diverse Learners __** :
 * Odyssey (teacher discretion)
 * Skill Tutor (teacher discretion)
 * Math’s Cool Unit A Mod 5 and 6
 * Algebra’sCool Unit A Mod 1
 * @http://www.mathgoodies.com/lessons/vol5/division.html Website designed for you to explore division of integers.


 * __ Homework: __**
 * ** @http://www.kutasoftware.com/free.html ** to print assignments on variety of topics
 * See appropriate Glencoe, OnCore, JBHM, and ABC materials under Suggested Activities


 * __ Terminology for Teachers: __**

** E ** thnicity/**C**ulture | **I**mmigration/**M**igration | **I**ntercultural **C**ompetence | **S**ocialization | **R**acism/**D**iscrimination ** High Yield Strategies ** ** S ** imilarities/**D**ifferences | **S**ummarizing/**N**otetaking | **R**einforcing/**R**ecognition | **H**omework/**P**ractice | ** N ** on-**L**inguistic representation | **C**ooperative **L**earning | **O**bjectives/**F**eedback | ** G ** enerating-**T**esting **H**ypothesis | **C**ues, **Q**uestions, **O**rganizers ||   || [|Lesson Plan] in Word Format (Click Cancel if asked to Log In)
 * || ** Multicultural Concepts **


 * __ Resources __**


 * Professional Texts **

** Literary Texts ** __http://sci.tamucc.edu/~eyoung/fractions_literature.html__ Perry the Penguin needs 9 clams to buy an ice scooter -- but he's not very good at saving. As Perry earns, spends, finds, loses, and borrows clams, a simple line graph demonstrates the concept of negative numbers. __http://www.amazon.com/exec/obidos/tg/detail/-/0060001267/qid=1088704854/sr=1-2/ref=sr_1_2/002-0537282-1976045?v=glance&s=books__ > Many children's books include math concepts and can be used to help teach them in a fun way. This website includes several annotated Lists of Children's Literature including the math concepts and grade levels. Click on the following link, __@http://libguides.nl.edu/mathinchildrenslit__, then look under __Math and Literature Bibliographies__. >>> After the cousins outwit a plague of punctuation, grateful villagers beg them to find Lexicon's missing children, who have been enticed away by dancing lights in the sky. Trekking between villages in search of clues, the cousins encounter a talking thesaurus, a fog of forgetting, the Mistress of Metaphor, a panel of poets, feuding parts of speech, and the illogical mathematicians of Irrationality. When a careless Mathemystical reflects them across the border into the ominous Land of Night, their peril deepens. Kidnapped, imprisoned, and mesmerized—with time running out—will Ivan and Daphne find a way to solve the mystery of the lights in the sky and restore the lost children of Lexicon to their homes? >>> //Lost in Lexicon// will whisk children away into an interactive and magical world of learning. Many books include websites with __lesson ideas__. __http://sci.tamucc.edu/~eyoung/middle_school_literature.html__ __http://sci.tamucc.edu/~eyoung/Literature%20webpages/Leonor/index.html__ __http://sci.tamucc.edu/~eyoung/literature.html__
 * Fractions
 * __ Less Than Zero __ (MathStart 3) __Stuart J. Murphy__
 * Mathematics in Children's Literature:
 * **__Lost in Lexicon: An Adventure in Words and Numbers__** by __Pendred Noyce__ and Joan Charles (Jul 26, 2011)
 * "If this is an adventure, we should just plunge in..."
 * When thirteen-year-old cousins Ivan and Daphne go on a treasure hunt in the rain one summer day, they never expect to stumble into a whole new world where words and numbers run wild.
 * __101 Things Everyone Should Know About Math__ by __Marc Zev__, __Kevin Segal__ and Nathan Levy (Mar 16, 2010) (real world)
 * Math is a critical part of our everyday lives; we use it dozens of times daily. and wish we understood it better. The second title in the 101 Things Everyone Should Know series, this book makes understanding math easy and fun! Using an appealing question and answer format, this book is perfect for kids, grown-ups and anyone interested in the difference between an Olympic event score of 9.0 and Richter scale score of 9.0.
 * Middle & High School: Literature in Mathematics
 * Lesson Plans for Using Literature in Middle and High School Mathematics (developed by Leonor and edited by Elaine)
 * Miscellaneous Math and Children's Literature

** Informational Texts **

** Art, Music, and Media **


 * Manipulatives **
 * Calculators
 * Versa Tiles
 * Two-sided Counters
 * Number Lines
 * Money


 * Games **
 * @http://www.mathplayground.com/SMP_index.html
 * @http://www.sumdog.com/ A variety of games covering many math topics

** Videos **
 * Discovery Learning
 * @http://www.discoveryeducation.com/
 * The teaching Channel currently offers videos of K-12 mathematics teaching aligned with the Common Core Sate Standards, which would be perfect for professional development with teacher teams.
 * https://www.teachingchannel.org/videos?categories=topics common-core

__@http://exchange.smarttech.com/details.html?id=1764f052-3804-448f-9f12-95bc5646a2ea__
 * SMART Board Lessons, Promethean Lessons **
 * Smart Board rational number lessons
 * Rational numbers Smart Board lessons

http://www.shodor.org/interactivate/guide/ __@http://www.shodor.org/interactivate/activities/__ @http://www.uen.org/Lessonplan/preview.cgi?LPid=19833 @http://illuminations.nctm.org/LessonDetail.aspx?ID=L257 The following grades 6-8 activities allow students to explore statistics surrounding baseball. They are exposed to connections between various mathematical concepts and see where this mathematics is used in areas with which they are familiar. This lesson plan is adapted from the May 1996 edition of Mathematics Teaching in the Middle School. @http://illuminations.nctm.org/LessonDetail.aspx?ID=U78 The following lessons focus on student organization, preparation, and presentation of some simple foods as a way of applying various mathematical concepts, with problem-solving techniques being central to almost all the activities. This unit was adapted from the "Ideas" column in the February 1994 issue of The Arithmetic Teacher, Vol. 41, No. 6, pp. 309. @http://www.nctm.org/standards/content.aspx?id=25090 Students can learn to visualize the effects of multiplying a fixed positive number by positive numbers greater than 1 and less than 1 with this tool. Using interactive figures, students can investigate how changing the height of a rectangle with a fixed width changes its area. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The e-examples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math investigations. @http://illuminations.nctm.org/LessonDetail.aspx?ID=L285 In this lesson, students experience beginning-algebra concepts through discussion, exploration, and videotaping. The concept of multiplication of integers is presented in a format which encourages understanding, not simply rote memorization of facts. This lesson plan is adapted from the article, "A Videotaping Project to Explore the Multiplication of Integers", by Marcia B. Cooke, which appeared in Arithmetic Teacher, Vol. 41, No. 3 (November 1993) pp. 170-171. @http://illuminations.nctm.org/LessonDetail.aspx?ID=L252 In this lesson, students develop number sense through a series of three hands-on activities. Students explore the following concepts: the magnitude of a million, fractions between 0 and 1, and the effect of decimal operations.
 * Other Activities, etc **.
 * // Interactivate // is a set of free, online courseware for exploration in science and mathematics. It is comprised of activities, lessons, and discussions. The site is structured around collections of activities, lessons, and discussions.
 * Activities and tools:
 * Utah Education Network
 * ** Fun with Baseball Stats **
 * ** Ideas with Food **
 * ** Learning about Multiplication Using Dynamic Sketches of an Area Model **
 * ** Multiplying Integers Using Videotape **
 * ** Too Big or Too Small? **

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