Mathematics
Multiplicative Relationship
A multiplicative relationship refers to a connection between two or more quantities where one quantity is a multiple of the other. In this relationship, changes in one quantity result in proportional changes in the other. It is often represented using multiplication or division and is fundamental in understanding concepts such as proportions and ratios.
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7 Key excerpts on "Multiplicative Relationship"
- Frank K. Lester(Author)
- 2005(Publication Date)
- Information Age Publishing(Publisher)
The critical structural relationships, both scalar and functional, are best illustrated using Vergnaud’s (1983) framework for analyzing multiplicative struc-tures. The following example shows a simple direct proportion between two measure spaces. A linear functional relationship exists between corresponding elements of the measure spaces, and a scalar operator transforms quantities of the same type. If Maria can sew 5 team shirts with 7 1 2 yards of mate-rial, how many yards will she need to make a T-shirt for each of the 15 boys on the soccer team? Measure 1 (# shirts) 5 3 3 7 15 x Measure 2 (yards of material) scalar operator scalar operator 1 2 f( x ) = 1 x 1 2 f( x ) = 1 x 1 2 The ability to give correct answers is no guarantee that proportional reasoning is taking place. Often, proportions may be solved using mechanical knowl-edge about equivalent fractions or about numerical relationships, or by applying algorithmic procedures (for example, the cross-multiply rule) that circumvent the use of the constant of proportionality. Which is sweeter, a mixture of 2 teaspoons of sugar with 6 teaspoons of lemon juice, or a mixture of 8 tea-spoons of sugar with 24 teaspoons of lemon juice? Student: They are the same because 8 goes into 24 three times and 2 goes into 6 three times. (Karplus et al., 1983b, p. 55). In such contexts, multiplicative explanations have “counted” as proportional reasoning; however an ob-server cannot say much about these students’ under-standing of proportionality. Indeed, most adults do not associate comparison problems with trying to find out which situation has the greater constant of pro-portionality. As researchers, we must conclude that proportionality is a much larger construct than pro-portional reasoning.
- Catherine Sophian(Author)
- 2017(Publication Date)
- Routledge(Publisher)
Chapter 5Relations Among Quantities in Arithmetic: Additive and Multiplicative Reasoning
Arithmetic is the study of numerical calculations, particularly the operations of addition, subtraction, multiplication, and division. In practice, these operations consist of a set of procedures that generate numerical outcomes. Conceptually, however, they are ways of representing relations among quantities, and an understanding of those relations is critical both to determining the appropriate operation(s) to perform in problem solving and to understanding why the procedures themselves work the way they do.Two contrasting ways of thinking about relations among quantities are reflected in the operations of addition and subtraction, on one hand, and multiplication and division, on the other, and the contrast between the two hinges on differences in the way units are used. In addition and subtraction, relations between quantities are characterized in terms of a unit that is independent of either quantity. Thus, when we say, “John is six inches taller than his little sister Sarah,” the unit we are using to compare them, inches, is related to the quantities we are comparing, heights, only in that it is a linear unit and the quantities being compared are linear. We could express the height relation equally well using some other linear unit—such as 15 cm, or half a foot. In contrast, in multiplication and division, we express relations among quantities by taking one quantity as a unit against which to describe the other. For instance, when we say John is three times as tall as Sarah, we are taking Sarah’s height as a unit and using that unit to quantify John’s height. Getting John’s height, given a measure of Sarah’s height in, say, inches, is essentially a matter of converting from a measure of John’s height based on Sarah-units, 3, to a measure based on inches, which we obtain by multiplying 3 times the number of inches per Sarah-unit (i.e., Sarah’s height in inches). A number of profound differences between additive relations (addition and subtraction) and multiplicative relations (multiplication and division) follow from this basic difference in the way relations between quantities are conceptualized.
eBook - ePubThe Problem with Math Is English
A Language-Focused Approach to Helping All Students Develop a Deeper Understanding of Mathematics
- Concepcion Molina(Author)
- 2012(Publication Date)
- Jossey-Bass(Publisher)
Chapter Eleven Mathematics: It's All About Relationships!Arithmetic is about computation. Mathematics is about relationships.—UnknownIn a whodunit, discovering one key relationship can sometimes cause all the clues to fall into place, revealing the identity of the perpetrator and solving the case. Mathematics is similar. Determining the key relationships in a problem often reveals the solution or at least the best way to obtain it.Learning to approach mathematics from a relationships perspective can be a powerful tool. Yet, despite the critical role of relationships in math, few problem-solving strategies specifically include this as a critical step of the process. As a result, teachers need to ensure they help students find and understand the relationships in mathematical contexts as an integral part of instruction.The focus on relationships was saved as the last major topic for discussion because it builds on much of the previous information. Unlike human relationships, which are based on emotion, mathematical relationships are based on reason. However, similar to relationships among humans, relationships in math are not independent entities. Uncovering and understanding relationships in math is dependent on content expertise as well as a deep understanding of the language and symbolism. In essence, relationships are the by-product of making connections among concepts, with all these factors being interdependent and supportive of each other. Because of the complexity of these relationships, illustrating how they can manifest in various contexts can be helpful. These contexts involve factors such as language and proportional reasoning, which in turn are influenced by thought processes such as interpretation, assumptions, and multiple perspectives. Among humans, some relationships last even though contexts may change. Math is no different. Included in this chapter are example problems where teachers can see how the same relationships in fundamental math emerge later in more sophisticated scenarios.
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
The first two representations below are equivalent, and we call them both additive comparisons . The last three are equivalent, and we call them multiplicative comparisons . Additive comparisons b 5 a 1 5000 b 2 a 5 5000 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 250 CHAPTER 5 Proportional Reasoning Multiplicative comparisons b 5 1 1 2 a b a 5 3 2 b : a 5 3 :2 We tend to use multiplicative comparisons more than additive ones. For example, we hear on television that a paper towel absorbs “50% more liquid than the leading brand” as opposed to “in tests it absorbed 4 more ounces of water than the other brand.” Multiplicative comparisons provide a context that helps us to know how much more. THE UNIT CONCEPT MATURES A noted educator and scientist once wrote that “it seems odd to refer to a relationship as a quantity.” 1 For example, if we say that a “large” drink is 20 ounces, we can see or visualize that amount using our knowledge of measurement. However, when we say that a car gets “35 miles per gallon,” the 35 actually expresses a relationship between two amounts (miles traveled and gallons consumed). As Schwartz notes, we “refer to [this] relationship as a quantity.” However, 35 miles per gallon is much more abstract than 35 ounces or 35 people. In the primary grades, children mostly work with whole numbers and with the op-erations of addition and subtraction. As their understanding of mathematical ideas grows, they move from counting physical objects to counting numbers themselves.
eBook - PDFPrimary Mathematics
Capitalising on ICT for Today and Tomorrow
- Penelope Serow, Rosemary Callingham, Tracey Muir(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
Despite using this thinking informally quite regularly, it is surprising how many people have trouble with this concept. Doubling or trebling a quantity is one thing, but what about wanting one-and-a-half times, or only needing one-fifth of something? These calculations can become very tricky. Often we make some kind of estimate, and either end up with too little or too much of something. Proportional reasoning is used widely to solve a range of everyday problems from ‘best buys’ to understanding data presented in tables. It underpins scal- ing problems such as scale drawings of house plans and currency conversions, and appears in many other situations. ACTIVITY 6.1 Think about what you do each day. Identify times when you may have need- ed to use proportional reasoning. Examples might be in cooking, making something, house decorating, shopping, going on a journey … there are many possibilities. Make a list and identify as many different situations as you can. For example, if you have to increase a recipe serving four people to feed 10 people, each of the ingredients will need to be increased 2 1 2 times because 10 = 2 1 2 x 4. Compare your list of everyday activities using proportional reasoning with that of another person, and comment on the similarities and differences. Developing multiplicative thinking is fundamental to proportional reason- ing, and the shift from additive to multiplicative thinking is a focus for the middle years of schooling. In this chapter, the underpinning conceptual under- standing of numbers such as fractions, decimals and percentages is explored as an important precursor to reasoning proportionally. Key ideas are devel- oped about the appropriate pedagogy for working with primary-age children and using technology effectively to enhance understanding. 142 Primary Mathematics Some background Moving from additive to multiplicative thinking is one of the great challenges in the primary years of schooling.
eBook - ePubA Focus on Multiplication and Division
Bringing Research to the Classroom
- Elizabeth T. Hulbert, Marjorie M. Petit, Caroline B. Ebby, Elizabeth P. Cunningham, Robert E. Laird(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
143). In elementary school, for example, students engage in a range of multiplication and division contexts, including equal groups, equal measures, unit rates, measurement conversions, multiplicative comparisons, scaling, and area and volume. The types of quantities involved and how the quantities interact are key to understanding multiplication and division in these different contexts. In this chapter the general concept of initial multiplicative understanding is examined. Chapter 5 provides an in-depth discussion of each of these contexts and how they affect student strategies and reasoning. This section focuses on the following: Why multiplication and division are not simply an extension of addition and subtraction: How the number relationships in addition are different from those in multiplication. How the actions in addition are different from those in multiplication. The difference between additive (absolute) reasoning and multiplicative (relative) reasoning. There is a commonly held belief among many educators that multiplication and division are just extensions of addition and subtraction. This belief arises because it is possible to solve whole number multiplication and division problems using repeated addition and subtraction, respectively. However, multiplication and division involve a different set of number relationships and different actions than addition and subtraction, which are described next. Number Relationships and Actions in Addition Additive reasoning involves situations in which sets of objects are joined, separated, or compared. For example, 3 apples + 6 apples = 9 apples. Each apple is a separate entity, and the sum is the union of all the apples as shown in Figure 1.1. It is also important to remember that in additive situations the numbers tell the actual size of each set. So in the case of 3 apples + 6 apples, the 3 means how many in one set and the 6 means how many in another set
eBook - PDF- Tom Bassarear, Meg Moss(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
SECTION 5.1 Ratio and Proportion SECTION 5.2 Percents In one respect, this chapter represents the culmination of the first four chapters. Over the first four chapters, we have explored fundamental mathematical concepts and we have investigated appli- cations of those concepts in real-life settings. In this chapter, we will focus explicitly on the idea of proportional reasoning. Ratios and Proportional Relationships is a separate content strand in the Common Core State Standards, with reasoning and problem solving with ratios in sixth grade and with proportional relationships in seventh grade. The NCTM asserts that “the ability to rea- son proportionally . . . is of such great importance that it merits whatever time and effort must be expended to [ensure] its careful development” (Curriculum Standards, p. 82). Let us examine why. n n Additive Versus Multiplicative Comparisons In this chapter, we will examine two fundamentally different ways in which we can compare amounts and describe changes. For example, let’s say we are comparing the cost of two used cars, one of which costs $10,000 and the other $15,000. We can say that the second car costs $5,000 more than the first car, $10,000 $5,000 $15,000 1 5 or that the second car costs 1 1 2 times as much as the first car, $15,000 1 $10,000 1 2 ( ) 5 The first description of the relationship is expressed in additive terms, while the second is expressed in multiplicative terms. To make the mathematics more visible, let the cost of the second car b 5 and a 5 the cost of the first car. The first two representations below are equivalent, and we call them both addi- tive comparisons. The last three are equivalent, and we call them multiplicative comparisons. Additive Comparisons Abstractly we write as: $15,000 $10,000 $5000 or $15,000 $10,000 $5, 000 5000 or 5000 b a b a 5 1 2 5 5 1 2 5 5 Proportional Reasoning 247
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