Learning Macmillan
ningarLeanillmacM12 Sowder 93330 Ch12 254-302.indd 25422/09/16 12:22 PM
Part IIningReasoning About Algebraand ChangeYMacmillanLearour experience with algebra in school likely was in one or two courses devotedto the subject. So you may be surprised to learn that algebra is increasingly recognized as a continuing topic throughout the curriculum, with ideas from algebra occurring as early as the first grade and with an increased use of symbols and equations in the K–6 mathematics curriculum, often under the guise of algebraicreasoning.Algebra has acquired a somewhat negative image, and many cartoons play on thebelief that algebra is difficult to understand and not too useful. This conception is farfrom the truth. When algebra is taught with sense-making always a goal, its powerin solving many types of problems becomes apparent. Of course, this approach toteaching algebra is easier when the teacher has the background to help studentsmake sense of the subject, including algebra topics now found in the elementarygrades’ curriculum.The goal of the following chapters is to provide you with the ability and confidenceto teach the ideas and skills that underlie algebra. This is not an algebra course devoted to building skills by manipulating numbers and variables. We do in fact assumethat you have been exposed to such skills in earlier algebra classes, but, nonetheless,we have built in some review.Chapter 12 in this part of Reconceptualizing Mathematics first centers on thinkingabout algebra in elementary school and what algebraic thinking is for elementaryschool children. This focus is integrated with a discussion of different facets of algebra,which lays the groundwork for the remaining chapters in Part II. We continue the discussion of quantitative reasoning as a way of understanding problem situations andour discussion of problem-solving heuristics. Chapter 13 focuses on using graphs tostudy quantitative relationships. Chapter 14 naturally follows this work by describingmathematical change and emphasizing what graphs tell us. Chapter 15 uses algebrato approach additional types of problems.In each chapter we take opportunities to help you better understand the Common Core State Standards (CCSS) Standards for Mathematical Practice (SMP). TheSMP are process standards that describe the ways students are expected to engagewith mathematics. You can develop a better understanding as you reflect on the waysin which you engage in the mathematical practices as you work through the activitiesin Part II.12 Sowder 93330 Ch12 254-302.indd 25522/09/16 12:22 PM
ningarLeanillmacM12 Sowder 93330 Ch12 254-302.indd 25622/09/16 12:22 PM
ChapterWhat Is Algebra?ning12arThere are many ways of thinking about algebra—as a symbolic language, as a study of pat terns and functions, as generalized arithmetic, as reasoning about quantities, and as a power ful tool for solving problems.1 This chapter introduces these different ways of thinking aboutalgebra and provides a foundation for the next three chapters. We begin by thinking aboutalgebra in elementary school. What is algebraic thinking for elementary school children?Le12.1 Algebraic Reasoning in Elementary SchoolExample 1millanLong before children are introduced to formal algebra, they can engage in a level of gen erality that can be considered algebraic. Early algebra involves using symbols to expressrelationships, identifying patterns and making and expressing generalizations. We willdiscuss what algebra looks like in elementary grades in the first section, and then in sub sequent sections, we will go deeper and revisit algebraic topics more formally.Algebra provides a symbolic language that can be used to represent quantitative rela tionships, such as those in story problems. We use symbols in algebra in three primaryways: to stand for a particular value in an expression or sentence, to state formulas, and toexpress arithmetic and algebraic properties. The symbols most often used are letters of thealphabet, but in the primary grades other symbols, such as boxes, are sometimes used.MacJaime has 16 baseball cards but lost 4 of them. How many does he have left? We canrepresent this problem as 16 2 4 5 . Although you can substitute different numbersfor the box, only one number will make this a true statement.NotesThe Instructor ResourceManual contains several itemsof interest for you to readbefore beginning this chapter.First, information aboutChapters 12, 13, 14, and 15 canbe found in the Overview ofPart II: Reasoning AboutAlgebra and Change. You mayfind this information helpful inplanning your course. Second,if you have not already done so,please read the Introduction tothe ReconceptualizingMathematics Program in themanual. Third, INSTRUCTORNOTE 12 provides an overviewof this chapter. AdditionalINSTRUCTOR NOTES in thismanual are referencedthroughout these chapters.The first major topic of thissection is that algebra is asymbolic language used torepresent quantitativerelationships.Elementary teachers should be concerned with the concept of equality as soon asstudents start writing symbols for number operations. Because elementary children usuallyencounter number sentences with the structure in Example 1, they often see the equals signas a prompt to do the operation that precedes it. When elementary students solve 3 1 7 51 2, most put 10 or 12 in the box. They do not generally see the equal sign as “same as”or relational, but rather as an instruction to carry out the preceding procedure. They see theequal sign as separating the problem from the answer. Open number sentences, which in clude unknown starting points and unknown change problems, can encourage students toreason flexibly. We read in Chapter 10 about children as young as first and second gradewho had a good understanding of equality, easily able to reason when shown equations withsigned numbers and open boxes.25712 Sowder 93330 Ch12 254-302.indd 25722/09/16 12:22 PM
258EXAMPLE 2ningKaren is running a 5-mile race. If we let x represent the number of miles she has run atany time during the race, we can represent the number of miles remaining as 5 2 x. Inthis case, x can take on a variety of values between and including 0 and 5, depending onthe distance Karen has run at any point in time.These discussions of relationships and equality can lead children to create mathemati cal expressions and equations that use symbols to express relationships in preparation formore formal algebra. For example, consider the earlier example of candies in boxes. Chil dren could reason about who has more. They understood that the child who had the secondbox had three more than the child who had the first box and could write an equation:M 5 J 1 3. Children can then eventually write more complex algebraic equations.EXAMPLE 3arMany prospective elementaryteachers have negative feelingsabout the algebra course(s)they took. If they experiencedalgebra primarily as symbolmanipulations, having suchfeelings is not surprising.We suggest that you exploretheir experiences by askingstudents a question such as“What do you remember aboutalgebra?” Provide them withsome hope that algebra canmake sense and that theysurely would not want theirown students to have troublelearning algebra.Discussions grounded in relationships provide another context for using symbols torepresent quantitative relationships. In research, children were shown two boxes and weretold that each box contained the same number of candies. Though the number of candies ineach box was unknown, when three more candies were placed in the second box, childrenas young as 8 or 9 could express the number of candies in the boxes as ? and ? 1 3, andeasily adopted N and N 1 3 for the number of candies in the boxes.2 In other words, theycould conceptualize symbols as standing in for a number of values.LeNotesChapter 12: What Is Algebra?anSuppose a child (or child’s parent) pays 3.00 for each superhero figurine and 0.50 foreach Superhero sticker. We can use symbols to tell us how much money was paid forfigurines and stickers: 3f 1 0.5s, where f represents the number of figurines purchasedand s represents the number of stickers purchased.illIn these three examples, symbols are used to represent the values of certain quantities.These symbols are called variables.mA variable is a symbol used to stand for a value from a particular set of values.MacAs we have seen, children do not always begin by using conventional algebraic sym bols. They can be encouraged to represent and reason about mathematics in their own way.Many people believe children benefit from opportunities that begin with their own intu ition, and then they gradually adopt more formal representations and terminology.Children also benefit from the study of patterns in numbers, shapes or objects, andeven systems. Exercises in recognizing, describing, and extending patterns occur early inthe curriculum. The primary focus of the next section (12.2) will be numerical patterns.However, you should be aware that repeating patterns of blocks or shapes, as in the follow ing diagram, can appear in first-grade textbooks.?Children should be encouraged to notice and describe patterns, structure, and regular ity in arithmetic operations. Number properties are the foundation of algebra. As you willsee, these properties are used to evaluate and simplify mathematical expressions. Numberproperties (e.g., commutative property of multiplication or associative property of addi tion) must be understood and used in mental computation, though you may not be awareyou are using them.12 Sowder 93330 Ch12 254-302.indd 25822/09/16 12:22 PM
25912.1 Algebraic Reasoning in Elementary SchoolSimilarly, children have some understanding of these properties, though they do not yetknow their formal names. Consider this fourth-grader’s logic for mentally obtaining theproduct of 4 3 20: “Four times twenty is double four times ten because twenty is doubleten.”3 We can use symbols and properties to record this child’s thinking. The property statesthat, in general, if we switch the order of the numbers being multiplied, the product isunaffected.4 3 20 5 2 3 (4 3 10) because 20 5 2 3 104 3 (2 3 10) 5 2 3 (4 3 10) associativity and commutativity of multiplicationACTIVITY 1 Listen Carefully NowarHere are some children’s mental and written computations. If the child is using correctlogic, discuss what properties he or she may be recognizing and describing. If the child ismaking an error, describe what property he or she appears to not understand.Lea. 45 1 15 is the same as 50 1 10 because I borrow 5 from the 15 to get to 50 and thatleaves 10 more to add.23 1 5 5. I think the answer must be 8 because I add 3 to get to 0 and 5 more.That’s 8.ille.anb. 5 3 6 is 26 because 5 3 5 is 25 and 6 is one more than 5. Twenty-six is 1 more than 25.82 12 because 2 3 4 5 8 and 3 3 4 5 12. I have to multiply by the same thing inc. 3 5the numerator as the denominator.d. 23 3 9 is 23 3 10 minus 23.mACTIVITY 2 Finding PropertiesTell what property or properties have been used:ac1 144a. a 1 (n 1 49) is rewritten as (a 1 n) 1 49. b. 6 5 3 2 5 2 3 6 5 Mc. (2b)c 5 2(bc) (or 2bc)d. z 1 2z 5 07 87 8e. 29 3 8 3 7 5 29 3 8 3 7 5 29 3 1 5 2941114f. 2 6 1 5 can be calculated by 2 6 1 2 5 .g. xy2 1 0 5 xy2h. 1 x 5 x (Note: x is commonly used in place of 1 x.)Use the properties to change the following expressions to make them easy to do mentally,and tell what properties you used:i. (7 1 40) 1 3 j. 25 3 (4 3 72.7) k. 24 3 38 1 24 3 12Some representations (e.g., rectangular area, number lines, unit cubes) can help us seethe structure and regularities of the properties. The distributive property of multiplicationover addition can be represented with a rectangular area model; the area model is some times coupled with a number line. Commutativity and associativity of addition can bemodeled on a number line or with unit cubes. The model is used to show that the result ofthe operations is the same.12 Sowder 93330 Ch12 254-302.indd 259See INSTRUCTOR NOTE 12.1Afor information on studentexpectations on the topic ofproperties of arithmeticoperations. The Common CoreState Standards emphasizestudents’ understanding anduse of mathematicalproperties.Activity 1Answersa. Associative property ofadditionb. The child does notunderstand that multiplicationis different from addition in that“1 more” means one moregroup of 5: 5(5 1 1) 5 25 1 5.ningChildren sometimes understand and state their observations more generally, as in thefollowing explanation: “When you add zero to any number, you get the number you startedwith.” This assertion is a partial statement of zero as an additive identity. It can be recordedwith variables: a 1 0 5 a, where a is any number.These properties are so important in both arithmetic and algebra that we restate themat the end of this section for future reference.Notesc. 1 is the multiplicativeidentity, but the child may alsobe parroting a rule.d. Distributive propertye. Additive inversesActivity 2Answersa. Associativity of additionb. Commutativity ofmultiplicationc. Associativity of multiplicationd. Additive inversese. First, associativity ofmultiplication; thenmultiplicative inverses; andfinally, 1 is the identity formultiplication.f. Distributivity of multiplicationover additiong. Zero is the identity foraddition.h. Identity property formultiplicationi. (40 1 7) 1 3 5 40 1 (7 1 3)5 50; commutativity thenassociativity of additionj. (25 3 4) 3 72.7 5 100 372.7 5 7270; associativityof multiplicationk. 24(38 1 12) 5 24 3 50 510024 3 2 5 12 3 100 5 1200;distributive property22/09/16 12:22 PM
260Chapter 12: What Is Algebra?EXAMPLE 4Notes5 3 (10 1 7) 5 5 3 10 1 5 3 751075035 EXAMPLE 513 1 17 5 17 1 13 17Think About . . .AnswerConstruct an array or areamodel and show that themodel for the commutednumbers is the same rectanglerotated.0515Think About 202530How can we model the commutative property of multiplication?These same properties justify some of our routine algebraic manipulations, whichhelps make sense of these manipulations.EXAMPLE 6We can replace 5x 1 3x with 8x because of the distributive property: 5x 1 3x 5(5 1 3)x.anSee INSTRUCTOR NOTE 12.1Babout using properties tocompute mentally.10ning 13Expressing Regularity inReasoningar Le MathClipsACTIVITY 3 To Help Explainill1. Use properties to explain why 5d 2 d Þ 5.m2. Use a model or properties to explain why 3a 1 2 Þ 5a.MacAs you can see, children in elementary school have very powerful ways of reasoningwhen they are encouraged to describe patterns and regularity in arithmetic operations.Early algebra also includes an emphasis on teaching children how to make generalizationsabout “growing patterns.” Often this begins by asking them to describe how geometricpatterns change. Children can be invited to describe how to draw more members of thesame sequence. They use that context to anticipate or predict how many segments orcorners will be in a particular picture in the sequence.This context provides a space for thinking about variables, recognizing numerical pat terns, describing relationships among variables, and linking multiple representations.ACTIVITY 4There are 41 toothpicks inShape 10.ACTIVITY 4 Leah Made a PatternLook at the following sequence of shapes. Build Shape 5. How many toothpicks would bein Shape 10? Describe how you thought about building the shapes.Shape 112 Sowder 93330 Ch12 254-302.indd 260Shape 2Shape 3Shape 422/09/16 12:22 PM
26112.1 Algebraic Reasoning in Elementary SchoolNotesarningChildren initially express the pattern iteratively; that is, they describe how many tooth picks are added to Shape 1 to get Shape 2 and how many are added to Shape 2 to getShape 3, and so forth. It is far more complex to be able to say how many toothpicks are usedto generate a particular shape without information about how many toothpicks form theshape before it in the sequence. When solving this problem, did you think about whatchanges and what is constant? Did you reflect on whether the change was constant or dif fered as the pattern grew? This involves powerful algebraic reasoning.There is also mathematical richness in saying how the pattern grows. If children seethe pattern growing in different ways, they can describe the growth with different rules. Achild might say that four toothpicks are added to Shape 1 to get Shape 2 and four more toget Shape 3, focusing on the four toothpicks added to the right of the figure each time.Alternatively, a child might say that they see another figure being added, with overlappingtoothpicks “erased” (i.e., Shape 2 is two shapes of five toothpicks minus the one toothpickin the middle). Rich conversations can ensue in thinking about what is the same about therepresentations: 5 1 4 and (2 3 5) 2 1. In describing growing patterns, children learn howto make generalizations and express correspondence. The geometric arrangement of theobjects in the pattern can help them bridge to generalizing for n objects and writing alge braic expressions. This is an early introduction to function. We will learn more aboutfunctions in later sections.DISCUSSION 1 Algebra in Some Elementary ClassroomsLeCan you solve the following two problems? Are you surprised that these types of problemsare being solved in elementary classrooms?c How many hexagonal tables would it take to seat 25 guests? banAssume that 6 guests could be seated at one hexagonal table, and ifthere are two or more hexagonal tables, they are arranged as shown inthe diagram. Guests can be seated at each side of a hexagonal tableexcept where they are joined.illc What are some mathematically complex ways to arrive at 9 (e.g., 350 2 341)? bmRead this excerpt from a report on mathematics in classrooms in the Lebanon, Oregon,school district, published December 29, 2008, in the Portland newspaper The Oregonian.Discussion 1The yellow hexagons usedin this classroom were from aset of pattern blocks. Here, itappears that the tables are setup so that every two tablesare linked along one edge ofeach; thus, one table wouldseat 6 people, two tableswould seat 10, three tableswould seat 14, etc.Make sure students notethe grade levels here. Thisdiscussion should help promotethe earlier claim that algebraicproblems are now appearing inelementary school classrooms.MacLEBANON—Lori Haley and Mya Corbett hunch over a pile of yellow hexagons,trying to figure out how many hexagonal tables it would take to seat 25 guests. Thepair want to get the answer, but what they’re really itching to do is to come up witha formula that will tell them how many people they could seat for any given numberof tables.Suddenly, the girls detect a pattern, and one shouts: “(t 3 4) 1 2 5 s!” They tryit on one table, two tables, eight tables—it works. They beam, flashing smiles. . . .Lori and Mya just started third grade.Visit a Lebanon elementary math class, and you will see: First-graders set up and solve formulas such as 9 2 x 5 5, as they did whenRaylene Sell talked with her class about “some teddy bears” walking awayfrom the classroom rug, leaving five behind.Third-graders suggest mathematically complex ways to arrive at9: 2219 1 228 or (10 3 5) 2 40 2 1, or even (3 3 3) 1 (8 3 8) 2((4 3 4) 1 (4 3 4)) 2 32.One researcher observed that arithmetic is not about obtaining results, but rather it isabout learning the means by which one obtains particular solutions. Similarly, algebra isnot about the manipulation of symbols for particular results but about expressing generalityabout patterns, the properties of numbers, and the rules of arithmetic. What is algebraic12 Sowder 93330 Ch12 254-302.indd 26122/09/16 12:22 PM
262Chapter 12: What Is Algebra?thinking for elementary children? We need to encourage children to focus on relationshipsand to describe patterns and structure; in short, we need to support them
Macmillan Learning. 12_Sowder_93330_Ch12_254-302.indd 256 22/09/16 12:22 PM Macmillan Learning. What Is Algebra? There are many ways of thinking about algebra—as a symbolic language, as a study of pat .
Your Macmillan Practice Online course Joining a class Opening resources Completing a resource Submitting answers Viewing your scores Sending and receiving messages My Profile Using Macmillan English Dictionary online Macmillan Practice Online guide for students Macmillan Practice Online is an online practice environment for learners of English.
Macmillan is based in the United States. Our representative in the EEA is: Macmillan Publishers International Limited Company number: 02063302 . Pan Macmillan T. he Smithson 6 Briset Street . London, EC1M 5NR. Att: Legal Department . Contact Point for inquiries: Helaine Ohl, VP Global HR Director, Macmillan, 120 Broadway, 22nd Floor, New York, New
Rocket Science, in partnership with Consilium Research and Consultancy, was commissioned in early 2016 to evaluate Phase 2 of Macmillan @ Glasgow Libraries. Macmillan @ Glasgow Libraries is a tiered model of Macmillan Cancer Information and Support Services, which a
Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, ls,Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin's Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies
MACMILLAN MODERN DRAMATISTS J . . Synge Bugene Benson Professor of English, College of Arts University of Guelph, Ontario Macmillan Macmillan Education . THE MACMILLAN PRESS LTD London and Basingstoke Companies and representatives throughout the world ISBN 978--333-28922-8 ISBN 978-1-349-16915-3 (eBook)
Little Women Retold by Anne Collins CLASSICS MACMILLAN _pqg. Macmillan Education The Macmillan Building 4 Crinan Street London N1 9XW A division of Macmillan Publishers Limited Companies and representatives throughout the world ISBN 978--230-03500-3 ISBN 978-1-4050-7620-3 (with CD edition)
Macmillan Practice Online guide for teachers Macmillan Practice Online is an online practice environment for learners of English. Students login to complete activities from an online course that can match the syllabus of a Macmillan book, offer preparation for a specific exam or practice for general and business English.
About this report 5 Patient stories 7 The link between patient and staff experience 28 Listening to staff stories 29 Empowering and valuing staff 31 How Macmillan supports empowering staff 37 Staff Wellbeing 38 How Macmillan is supporting wellbeing 43 Training for Staff 44 How Macmillan is supporting training 46 Afterword 48 References 49. 4 Every person diagnosed with cancer should have a .
