DOCSLIB.ORG

Algebra • Distributive Property

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebra • Distributive Property DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B LESSON 4.4 Algebra • Distributive Property FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards Learning Objective 3.OA.B.5 Apply properties of operations as strategies to multiply Use the Distributive Property to find products by and divide. breaking apart arrays. Also 3.OA.A.1, 3.OA.A.3, 3.OA.A.4, 3.OA.C.7 MATHEMATICAL PRACTICES Language Objective MP4 Model with mathematics. MP7 Look for and make use of structure. Students give two ways you can use the MP8 Look for and express regularity in repeated reasoning. Distributive Property to find products. F C R Coherence: Materials Standards Across the Grades MathBoard, square tiles Before Grade 3 After 2.OA.A.1 3.OA.B.5 4.OA.A.2 F C R Rigor: Level 1: Understand Concepts....................Share and Show ( Checked Items) F C R For more about how GO Math! fosters Coherence Level 2: Procedural Skills and Fluency.......On Your Own within the Content Standards and Mathematical Progressions Level 3: Applications..................................Think Smarter and Go Deeper for this chapter, see page 189J. About the Math Professional Development Model the Distributive Property 8 3 7 Models help students develop a better understanding of concepts. StudentsAbout represent the the Math concepts first with concrete materialsProfessional or drawings, Development then with numbers and symbols. A model also helps students visualize the process and form a mental image. In this lesson, a factor is partitioned using an array to show why the Distributive Property works. The Distributive Property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. 52 The goal of this lesson is to use the Distributive Property as a strategy to partition one of the factors. In this way, facts with greater products are shown as the sum of the products of two facts that students already know, making 8 × (5 + 2) multiplication faster and more flexible. (8 × 5) + (8 × 2) 40 + 16 = 56 Professional Development Videos 209A Chapter 4 1 ENGAGE Daily Routines with the Interactive Student Edition Common Core Essential Question Problem of the Day 4.4 How can you use the Distributive Property to find products? Ellis has 6 boxes. Each box holds 1 pencil. Write a number sentence to show how Making Connections many pencils Ellis has in all. Invite students to tell you what they know about multiplying by 1, 5, and 6. 6 × 1 = 6 What happens when you multiply any number by 1? The answer is the same number. Vocabulary Distributive Property What are different ways to multiply by 5? count by 5, add 5s What are different ways to multiply by 6? count by 6, add 6s Interactive Student Edition Multimedia eGlossary Learning Activity What is the problem the students are trying to solve? Connect the story to the problem. • How far does Lucia’s neighbor drive each day for work? 7 miles • How many days per week does Lucia’s neighbor drive to and from work? 5 days • What operation could you use to find the total number of miles Lucia’s neighbor drives for work per week? multiplication or addition • What multiplication expression could you use to find the total number of miles Lucia’s neighbors drives? 5 × 7 • What are different ways to solve 5 × 7? count by 7s, add 7 + 7 + 7 + 7 + 7, multiply 5 × 6 and add another 5 Literacy and Mathematics • Have students discuss the meaning of distribute in a larger context. Ask one student to distribute pieces of paper to each student to demonstrate the meaning of the word. • Have students write a short story about why Lucia wants to know how many miles his neighbor drives. How can you use the Distributive Property to find products? Lesson 4.4 209B LESSON 4.4 3.OA.B.5 Apply properties of operations as strategies to multiply and divide. 2 EXPLORE ALGEBRA Name Lesson 4.4 Unlock the Problem Distributive Property MATHEMATICAL PRACTICES Essential Question How can you use the Distributive Property to Operations and Algebraic find products? Thinking—3.OA.B.5 Also 3.OA.A.1, 3.OA.A.3, 3.OA.A.4, 3.OA.C.7 MATHEMATICAL PRACTICES Students should understand that they need MP1, MP7, MP8 to multiply the number of fish by the cost of UnlockUnlock the ProblemProblem each fish to find the total amount spent. Mark bought 6 new fish for his • Describe the groups in Have a volunteer read the definition of the aquarium. He paid $7 for each fish. this problem. How much money did he spend in all? Distributive Property. 6 groups of $7 Hands On Find 6 × $7. • Circle the numbers you will use to Activity solve the problem. You can use the Distributive Property MP4 Model with mathematics. Work with to solve the problem. students to make the array and break it apart. • How can you use the smaller arrays to find The Distributive Property states that multiplying a sum by a number is the sum—the answer to an addition problem the total? Find the total for each smaller array and same as multiplying each addend by the addends—the numbers being added add them together. number and then adding the products. Hands MP6 Attend to precision. On Materials ■ square tiles • Does breaking apart a larger array into Activity 2 smaller arrays make the problem easier Make an array with tiles to show Break apart the array to make two 6 rows of 7. smaller arrays for facts you know. to solve? Explain. Yes; Possible explanation: I can use basic facts I already know to find the total for each smaller array and then add the totals to find the answer. • Why do you think this is called the Distributive Property? Possible answer: because I am distributing 6 to both 2 and 5 to multiply. 6 × 7 = ■ 6 × 5 6 × 2 6 × 7 = ■ 6 × 7 = 6 × (5 + 2) Think: 7 = 5 + 2 Math 6 × 7 = (6 × 5) + (6 × 2) Multiply each addend by 6. Talk Use Math Talk to focus on 30 12 Math students’ understanding of using a 6 × 7 = _ + _ Add the products. Talk MATHEMATICAL PRACTICES 2 Reason Quantitatively model to draw conclusions and explain reasons 6 × 7 = _42 for their conclusions. Have several students What other ways could 42 you break apart the © Houghton Mifflin Harcourt Publishing Company So, Mark spent $_ for his new fish. draw their arrays on the board and write the 6 × 7 array? Possible answers: 6 × (6 + 1); 6 × (4 + 3) multiplication problems they used to find 6 × 7. Chapter 4 209 • Can you separate the array into 2 rows of 6 and 5 rows of 6 and get the same answer? Use the Distributive Property to see if you get the same answer. Possible 3 Differentiated answer: (2 × 6) + (5 × 6) = 12 + 30, or 42. Yes, it Reteach 4.4 2 Enrich 4.4 1 Instruction works; I get the same answer. Lesson 4.4 Lesson 4.4 Name Reteach Name Enrich ELL Strategy: Algebra • Distributive Property United Arrays A garden has 4 rows of 7 corn stalks. How many corn stalks Draw an array for each clue. Then use the arrays Check students’ Rephrase in all are in the garden? to solve each problem. arrays. You can use the Distributive Property to break an array 1. Craig spent $27 to buy 3 into smaller arrays to help you fnd the answer. calendars. The next day, he spent Explain the Distributive Property using an another $18 for more calendars. Find 4 3 7. Each calendar cost the same illustrated example from the lesson as support. amount. How many calendars did Step 1 Make an array to show 4 rows of 7. Craig buy? 5 calendars • In pairs, have students draw their own 4 rows of 7, or 4 3 7 Step 2 Break apart the array to make two 2. On Monday, Mrs. Jones spent array and break it apart, or distribute it, by smaller arrays for facts you know. $32 on 4 books. On Tuesday, she spent $16 on more books. Each book cost the same amount. How drawing the array and a dotted line where Step 3 Write the multiplication for the new many books did Mrs. Jones buy? arrays. Multiply and then add the products 4 3 4 1 4 3 3 they would break it apart. to fnd the answer. 16 1 12 5 28 6 books 4 3 7 5 (4 3 4) 1 (4 3 3) 4 3 7 5 16 1 12 4 3 7 5 28 3. Hailey spent $12 to buy 2 fish. Her cousin spent double the So, there are 28 corn stalks in all in the garden. • Have them write the multiplication amount on fish. Each fish cost the same amount. How many fish expression for the array and then for the Write one way to break apart the array. Possible answers are given. do Hailey and her cousin have Then fnd the product. altogether? broken apart, or distributed, array. 1. 2. 6 f sh 4. Explain how you used the arrays to solve each problem. • Have the pairs express the property or rule Possible answer: I drew two arrays for each problem then added the number of rows to in their own words in their Math Journal. (3 3 4) 1 (3 3 5); or (6 3 1) 1 (6 3 2); or f nd the number of items.
Load more

Recommended publications
  • An Elementary Approach to Boolean Algebra
    Eastern Illinois University The Keep Plan B Papers Student Theses & Publications 6-1-1961 An Elementary Approach to Boolean Algebra Ruth Queary Follow this and additional works at: https://thekeep.eiu.edu/plan_b Recommended Citation Queary, Ruth, "An Elementary Approach to Boolean Algebra" (1961). Plan B Papers. 142. https://thekeep.eiu.edu/plan_b/142 This Dissertation/Thesis is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Plan B Papers by an authorized administrator of The Keep. For more information, please contact [email protected]. r AN ELEr.:ENTARY APPRCACH TC BCCLF.AN ALGEBRA RUTH QUEAHY L _J AN ELE1~1ENTARY APPRCACH TC BC CLEAN ALGEBRA Submitted to the I<:athematics Department of EASTERN ILLINCIS UNIVERSITY as partial fulfillment for the degree of !•:ASTER CF SCIENCE IN EJUCATION. Date :---"'f~~-----/_,_ffo--..i.-/ _ RUTH QUEARY JUNE 1961 PURPOSE AND PLAN The purpose of this paper is to provide an elementary approach to Boolean algebra. It is designed to give an idea of what is meant by a Boclean algebra and to supply the necessary background material. The only prerequisite for this unit is one year of high school algebra and an open mind so that new concepts will be considered reason­ able even though they nay conflict with preconceived ideas. A mathematical science when put in final form consists of a set of undefined terms and unproved propositions called postulates, in terrrs of which all other concepts are defined, and from which all other propositions are proved.
    [Show full text]
  • On the Relation Between Hyperrings and Fuzzy Rings
    ON THE RELATION BETWEEN HYPERRINGS AND FUZZY RINGS JEFFREY GIANSIRACUSA, JAIUNG JUN, AND OLIVER LORSCHEID ABSTRACT. We construct a full embedding of the category of hyperfields into Dress’s category of fuzzy rings and explicitly characterize the essential image — it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker’s theory of matroids over hyperfields with Dress’s theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker’s partial demifields. 1. Introduction The important and pervasive combinatorial notion of matroids has spawned a number of variants over the years. In [Dre86] and [DW91], Dress and Wenzel developed a unified framework for these variants by introducing a generalization of rings called fuzzy rings and defining matroids with coefficients in a fuzzy ring. Various flavors of matroids, including ordinary matroids, oriented matroids, and the valuated matroids introduced in [DW92], correspond to different choices of the coefficient fuzzy ring. Roughly speaking, a fuzzy ring is a set S with single-valued unital addition and multiplication operations that satisfy a list of conditions analogous to those of a ring, such as distributivity, but only up to a tolerance prescribed by a distinguished ideal-like subset S0. Beyond the work of Dress and Wenzel, fuzzy rings have not yet received significant attention in the literature. A somewhat different generalization of rings, known as hyperrings, has been around for many decades and has been studied very broadly in the literature.
    [Show full text]
  • Exploring the Distributive Property; Patterns, Functions, and Algebra; 5.19
    Mathematics Enhanced Scope and Sequence – Grade 5 Exploring the Distributive Property Reporting Category Patterns, Functions, and Algebra Topic Investigating and recognizing the distributive property Primary SOL 5.19 The student will investigate and recognize the distributive property of multiplication over addition. Related SOL 5.7 Materials Base-10 blocks Exploring the Distributive Property Recording Sheet (attached) Colored pencils or markers Drawing paper Vocabulary distributive property Student/Teacher Actions (what students and teachers should be doing to facilitate learning) 1. Begin the lesson by asking students to compute 4 × 53 mentally. Ask volunteers to share different methods of mental computation for this problem, and write their suggested methods on the board. Spotlight any method that uses the distributive property: 4 × 53 = 4 × 50 + 4 × 3 = 200 + 12 = 212 Point out how the number 53 was broken into 50 + 3, making it easier to multiply and add mentally. Thus, this problem could be written as follows: 4 × (50 + 3) = (4 × 50) + (4 × 3) = 200 + 12 = 212 2. Have students use base-10 blocks to model 4 × 53 by showing 4 groups of 53. Guide student to see that if = 10 units and = 1 unit, they can make 4 groups of 50 and 4 groups of 3, as shown below: Using the base-10 blocks, students can see that the 4 groups of 50 = 200 (skip count the rods by 10), and the 4 groups of 3 = 12. Therefore, 200 + 12 = 212. Have students record their drawings and solution in symbols on their recording sheets. 3. Present students with another problem, 6 × 24, and ask them to model this problem with base-10 pieces.
    [Show full text]
  • “The Distributive Property: the Core of Multiplication” Appendix (Page 1 of 5)
    » “The Distributive Property: The Core of Multiplication” appendix (page 1 of 5) Activity 1: Go fi sh This game is like the traditional Go Fish game, except that the goal is to put down sets of three cards that make an addition equation. The game may be played in grades 1–5. Materials • A standard deck of cards for each group of 2–4 students. (Be sure to remove the jokers.) Other kinds of numbered decks (like Uno cards) will also work. • Counters, such as coins or tokens for younger children Purpose • To gain fl uency in decomposing the numbers from 0–10 • To form a foundation for mental addition of a one-digit number to a two-digit number Rules 1. The game is played by a group of 2–4 players. The fi rst player to eliminate all his or her cards wins. If the deck runs out, the player with the fewest cards in his or her hand wins. 2. Shuffl e the cards, and deal seven cards to each player. Place the remainder of the deck in a pile, facedown. 3. Players may look at their own cards but not at each others’ cards. 4. At the beginning or the end of their turn, players with three cards in their hand that make an addition sentence may place those three cards in a group, faceup on the table in front of them, as in the picture above. 5. On your turn, choose another player, and ask for a card. For example, “Janice, do you have a king?” If Janice has a king, she must give it to you.
    [Show full text]
  • DISTRIBUTIVE PROPERTY What Is the Distributive Property?
    DISTRIBUTIVE PROPERTY What Is the Distributive Property? • The distributive property offers a choice in multiplication of two ways to treat the addends in the equation. We are multiplying a sum by a factor which results in the same product as multiplying each addend by the factor and then adding the products. 4 x (5 + 6) = 4 x 11 = 44 4 x (5 + 6) = 20 + 24 = 44 • When applying the distributive property, you can add the numbers inside the parentheses first and then multiply, for instance: 8 x (2 + 7) = 8 x 9 = 72 • When applying the distributive property, you can multiply each of the numbers inside the parentheses by the same factor and then add the products, for instance:www.newpathlearning.com 8 x (2 + 7) = 16 +56 = 72 • Whichever method you choose to apply the distributive property, the result (answer) will be the same. Distributing the multiplication factor over each addend or over the sum of the addends is your choice. Sometimes the two numbers to be added are easy to solve: 15 + 25 or 17 + 20. This is when it is more efficient to add first. When the factor belongs to a familiar set, multiply first. This would be true if the factor was 2, 3, 4, 5, 10, 20, 100, etc. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com. Examples: Here are a variety of examples in which the distributive law has been applied correctly: 7 x (6 + 7) = 42 +49 = 91 10 x (3+ 17) = 30 +170 = 200 8 x (6 + 5) = 8 x 11 = 88 12 x (2 + 7) = 12 x 9 = 108 Note: The distributive law can not be applied to subtraction or division.
    [Show full text]
  • Commutative Property
    Properties of Addition & Multiplication for Ms. Davis’s 5th-Grade Math Classes © Deanne Davis, 2012 Before We Begin… • What do these symbols mean? ( ) = multiply: 6(2) or group: (6 + 2) * = multiply = multiply ÷ = divide / = divide Before We Begin… • The numbers in number sentences have names. • ADDITION: 3 + 4 = 7 addend addend sum Before We Begin… • The numbers in number sentences have names. • MULTIPLICATION: 3 x 4 = 12 factor factor product (multiplier) (multiplicand) Before We Begin… • So, FACTORS are numbers that are multiplied together. • Factors of 12 are: 1, 2, 3, 4, 6, & 12 1 x 12 = 12 2 x 6 = 12 3 x 4 = 12 • To find factors of a number, just think of all the different numbers we can multiply together to get that number as a product. Before We Begin… • MULTIPLES are products of given whole numbers. • Multiples of 5 are: 5, 10, 15, 20, 25, 30… 5x1 5x2 5x3 5x4 5x5 5x6 • To find a multiple of a number, just take that number, and multiply it by any other whole number. Commutative Property • To COMMUTATE is to reverse the direction of something. • The COMMUTATIVE property says that the order of numbers in a number sentence can be reversed. • Addition & multiplication have COMMUTATIVE properties. Commutative Property Examples: 7 + 5 = 5 + 7 9 x 3 = 3 x 9 Note: subtraction & division DO NOT have commutative properties! Commutative Property Practice: Show the commutative property of each number sentence. 1. 13 + 18 = 2. 42 x 77 = 3. 5 + 4 = 4. 7(3) = 5. 137 48 = Commutative Property ANSWERS: Show the commutative property of each number sentence.
    [Show full text]
  • Commutative, Associative & Distributive
    READING MATERIAL Read About Multiplication Properties (Commutative, Associative & Distributive) WHAT ARE MULTIPLICATION PROPERTIES: COMMUTATIVE, ASSOCIATIVE, AND DISTRIBUTIVE? Properties of multiplication are special facts about multiplication that you can use to find products. These properties can help you break down seemingly difficult multiplication problems into simpler ones. To better understand multiplication properties… LET’S BREAK IT DOWN! Basketball April and Marcos are playing basketball. April scored two 3-point shots and Marcos scored three 2-point shots. Who scored more points? April’s score can be expressed as 2 × 3, which equals 6. Marcos’ score can be expressed as 3 × 2, which also equals 6. Their scores are the same, and we have discovered the commutative property of multiplication: The order in which we multiply two factors does not change the product. Try this one yourself: You have 3 boxes that each contain 5 cookies and your friend has 5 boxes that each contain 3 cookies. Show that you and your friend have the same number of cookies. Page 1 Postcards Adesina, April, and Marcos have postcards arranged in a 12 by 6 grid. How many postcards do they have? The distributive property will help to solve this multiplication problem more easily than multiplying 12 by 6. is the grid can be divided into two smaller rectangles, so that one rectangle measures 10 by 6 postcards and the other measures 2 by 6. These calculations are easier than the original multiplication, and once we have the products, we can add them together to find the total number of postcards. This expression can be written as 10 × 6 + 2 × 6, and we can use parentheses to make the expression easier to read: (10 × 6) + (2 × 6) = 60 + 12 = 72.
    [Show full text]
  • De Morgan's Laws Revisited: to Be AND/OR NOT to Be
    Paper PO25 De Morgan’s Laws Revisited: To Be AND/OR NOT To Be Raoul A. Bernal, Amgen, Inc., Thousand Oaks, CA ABSTRACT De Morgan's Laws, named for the nineteenth century British mathematician and logician Augustus De Morgan (1806- 1871), are powerful rules of Boolean algebra and set theory that relate the three basic set operations (union, intersection and complement) to each other. If A and B are subsets of a universal set U, de Morgan’s laws state that (A ∪ B) ' = A' ∩ B' (A ∩ B) ' = A' ∪ B' where ∪ denotes the union (OR), ∩ denotes the intersection (AND) and A' denotes the set complement (NOT) of A in U, i.e., A' = U\A. The first law simply states that an element not in A ∪ B is not in A' and not in B'. Conversely, it also states that an element not in A' and not in B' is not in A ∪ B. The second law simply states that an element not in A ∩ B is not in A' or not in B'. Conversely, it also states that an element not in A' or not in B' is not in A ∩ B. This paper will demonstrate how the de Morgan’s Laws can be used to simplify complicated Boolean IF and WHERE expressions in SAS code. Using a specific example, the correctness of the simplified SAS code is verified using direct proof and tautology table. An actual SAS example with simple clinical data will be executed to show the equivalence and correctness of the results. INTRODUCTION In general, for any collection of subsets, de Morgan’s Laws are as follows: Theorem.
    [Show full text]
  • Intermediate Algebra This Course Covers the Topics Shown Below
    Intermediate Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs. Curriculum (495 topics + 307 additional topics) Real Numbers (54 topics) Plotting and Ordering (5 topics) Plotting integers on a number line Ordering integers Square root of a perfect square Using a calculator to approximate a square root Absolute value of a number Operations with Rational Numbers (24 topics) Integer addition: Problem type 1 Integer addition: Problem type 2 Integer subtraction: Problem type 1 Integer subtraction: Problem type 2 Integer subtraction: Problem type 3 Addition and subtraction with 3 integers Addition and subtraction with 4 or 5 integers Word problem with addition or subtraction of integers Integer multiplication and division Multiplication of 3 or 4 integers Division involving zero Identifying numbers as integers or non-integers Identifying numbers as rational or irrational Least common multiple of 2 numbers Signed fraction addition or subtraction: Basic Signed fraction subtraction involving double negation Signed fraction addition or subtraction: Advanced Addition and subtraction of 3 fractions involving signs Signed fraction multiplication: Basic Signed fraction multiplication: Advanced Signed fraction division Signed decimal addition and subtraction Signed decimal addition and subtraction with 3 numbers Operations with absolute value: Problem type 2 Exponents and Order of Operations (5 topics)
    [Show full text]
  • 2.2 Set Operations 127
    P1: 1 CH02-7T Rosen-2311T MHIA017-Rosen-v5.cls May 13, 2011 10:24 2.2 Set Operations 127 2.2 Set Operations Introduction Two, or more, sets can be combined in many different ways. For instance, starting with the set of mathematics majors at your school and the set of computer science majors at your school, we can form the set of students who are mathematics majors or computer science majors, the set of students who are joint majors in mathematics and computer science, the set of all students not majoring in mathematics, and so on. DEFINITION 1 Let A and B be sets. The union of the sets A and B, denoted by A ∪ B, is the set that contains those elements that are either in A or in B, or in both. An element x belongs to the union of the sets A and B if and only if x belongs to A or x belongs to B. This tells us that A ∪ B ={x | x ∈ A ∨ x ∈ B}. The Venn diagram shown in Figure 1 represents the union of two sets A and B. The area that represents A ∪ B is the shaded area within either the circle representing A or the circle representing B. We will give some examples of the union of sets. EXAMPLE 1 The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, {1, 3, 5}∪{1, 2, 3}={1, 2, 3, 5}. ▲ EXAMPLE 2 The union of the set of all computer science majors at your school and the set of all mathe- matics majors at your school is the set of students at your school who are majoring either in mathematics or in computer science (or in both).
    [Show full text]
  • Thermodynamic Semirings
    J. Noncommut. Geom. 8 (2014), 337–392 Journal of Noncommutative Geometry DOI 10.4171/JNCG/159 © European Mathematical Society Thermodynamic semirings Matilde Marcolli and Ryan Thorngren Abstract. The Witt construction describes a functor from the category of Rings to the category of characteristic 0 rings. It is uniquely determined by a few associativity constraints which do not depend on the types of the variables considered, in other words, by integer polynomials. This universality allowed Alain Connes and Caterina Consani to devise an analogue of the Witt ring for characteristic one, an attractive endeavour since we know very little about the arithmetic in this exotic characteristic and its corresponding field with one element. Interestingly, they found that in characteristic one, theWitt construction depends critically on the Shannon entropy. In the current work, we examine this surprising occurrence, defining a Witt operad for an arbitrary information measure and a corresponding algebra we call a thermodynamic semiring. This object exhibits algebraically many of the familiar properties of information measures, and we examine in particular the Tsallis and Renyi entropy functions and applications to non- extensive thermodynamics and multifractals. We find that the arithmetic of the thermodynamic semiring is exactly that of a certain guessing game played using the given information measure. Mathematics Subject Classification (2010). 94A17, 13F35, 28D20. Keywords. Entropy (Shannon, Renyi, Tsallis, Kullback–Leibler divergence), semiring, Witt construction, multifractals, operads, binary guessing games, entropy algebras. Contents 1 Introduction ...................................... 338 2 Preliminary notions .................................. 340 3 Axioms for entropy functions ............................. 345 4 Thermodynamic semirings .............................. 348 5 Statistical mechanics ................................. 351 6 The Rényi entropy ................................... 353 7 The Tsallis entropy .................................
    [Show full text]
  • Properties Multiplicative Property of Zero Associative Property
    Properties Multiplicative Property of Zero What do you multiply a number by Statements that are true to get zero? for any number of variables. a • 0 = 0 (If you multiply by 0, the answer is 0.) Commutative Property Associative Property Commutative means that the order does not Associative means that the grouping does not make any difference. make any difference. a + b = b + a a • b = b • a (a + b) + c = a + (b + c) (ab) c = a (bc) Examples Examples 4 + 5 = 5 + 4 (1 + 2) + 3 = 1 + (2 + 3) 2 • 3 = 3 • 2 (2 • 3) • 4 = 2 • (3 • 4) The associative property does not work for The commutative property does not work for subtraction or division. subtraction or division. 1 Identity Properties 1) Additive Identity What do you add to a number to get Distributive the same number? Property a + 0 = a 2) Multiplicative Identity What do you multiply a number by to get the same number? a • 1 = a Name the property Inverse Properties 1) 5a + (6 + 2a) = 5a + (2a + 6) 1) Additive Inverse (Opposite) commutative (switching order) a + (-a) = 0 2) 5a + (2a + 6) = (5a + 2a) + 6 2) Multiplicative Inverse associative (switching groups) (Reciprocal) 1 3) 2(3 + a) = 6 + 2a a 1 a distributive 2 1. 4. 0 12 = 0 Multiplicative Prop. Of Zero 5. 6 + (-6) = 0 Additive Inverse 6. 1 m = m Multiplicative Identity (2 + 1) + 4 = 2 + (1 + 4) 7. x + 0 = x Additive Identity Associative Property 1 8. 11 1 Multiplicative Inverse 11 of Addition 2. 3. 3 + 7 = 7 + 3 8 + 0 = 8 Commutative Identity Property of Property of Addition Addition 3 5.
    [Show full text]