In grades 35, students' development of number sense should continue, with a focus on multiplication and division. Their understanding of the meanings of these operations should grow deeper as they encounter a range of representations and problem situations, learn about the properties of these operations, and develop fluency in whole-number computation. An understanding of the base-ten number system should be extended through continued work with larger numbers as well as with decimals. Through the study of various meanings and models of fractionshow fractions are related to each other and to the unit whole and how they are representedstudents can gain facility in comparing fractions, often by using benchmarks such as 1/2 or 1. They also should consider numbers less than zero through familiar models such as a thermometer or a number line.
When students leave grade 5, they should
be able to solve problems involving whole-number computation and should
recognize that each operation will help them solve many different types
of problems. They should be able to solve many problems mentally, to estimate
a reasonable result for a problem, to efficiently recall or derive the
basic number combinations for each operation, and to compute fluently
with multidigit whole numbers. They should understand the equivalence
of fractions, decimals, and percents and the information each type of
representation conveys. With these understandings and skills, they should
be able to develop strategies for computing with familiar fractions and
In grades 35, students'
study and use of numbers should be extended to include larger numbers,
fractions, and decimals. They need to develop strategies for judging the
relative sizes of numbers. They should understand more deeply the multiplicative
nature of the number system, including the structure of 786 as 7100 plus 810 plus 61. They should also
learn about the position of this number in the base-ten number system
and its relationship to benchmarks such as 500, 750, 800, and 1000. They
should explore the effects of operating on numbers with particular numbers,
such as adding or subtracting 10 or 100 and multiplying or dividing by
a power of 10. In order to develop these understandings, students should
explore whole numbers using a variety of models and contexts. For example,
a third-grade class might explore the size of 1000 by skip-counting to
1000, building a model of 1000 using ten hundred charts, gathering 1000
items such as paper clips and developing efficient ways to count them,
or using strips that are 10 or 100 centimeters long to show the length
of 1000 centimeters.
Students who understand the structure
of numbers and the relationships among numbers can work with them flexibly
(Fuson 1992). They recognize and can generate equivalent representations
for the same number. For example, 36 can be thought of as 30 + 6, 20 +
16, 94, 40 4,
three dozen, or the square of 6. Each form is useful for a particular
situation. Thinking of 36 as 30 + 6 may be useful when multiplying
» by 36, whereas thinking of it as 6 sixes or 9 fours is helpful
when considering equal shares. Students need to have many experiences
decomposing and composing numbers in order to solve problems flexibly.
During grades 35, students should build their understanding of fractions as parts of a whole and as division. They will need to see and explore a variety of models of fractions, focusing primarily on familiar fractions such as halves, thirds, fourths, fifths, sixths, eighths, and tenths. By using an area model in which part of a region is shaded, students can see how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions. They should develop strategies for ordering and comparing fractions, often using benchmarks such as 1/2 and 1. For example, fifth graders can compare fractions such as 2/5 and 5/8 by comparing each with 1/2one is a little less than 1/2, and the other is a little more. By using parallel number lines, each showing a unit fraction and its multiples (see fig. 5.1), students can see fractions as numbers, note their relationship to 1, and see relationships among fractions, including equivalence. They should also begin to understand that between any two fractions, there is always another fraction.
Students should understand the meaning of a percent as part of a whole and use common percents such as 10 percent, 33 1/3 percent, or » 50 percent as benchmarks in interpreting situations they encounter. For example, if a label indicates that 36 percent of a product is water, students can think of this as about a third of the product. By studying fractions, decimals, and percents simultaneously, students can learn to move among equivalent forms, choosing and using an appropriate and convenient form to solve problems and express quantities.
Negative integers should be introduced at this level through the use of familiar models such as temperature or owing money. The number line is also an appropriate and helpful model, and students should recognize that points to the left of 0 on a horizontal number line can be represented by numbers less than 0.
Throughout their study of numbers,
students in grades 35 should identify classes of numbers and examine
their properties. For example, integers that are divisible by 2 are called
even numbers and numbers that are produced by multiplying a
number by itself are called square numbers. Students should
recognize that different types of numbers have particular characteristics;
for example, square numbers have an odd number of factors and prime numbers
have only two factors.
In grades 35, students
should focus on the meanings of, and relationship between, multiplication
and division. It is important that students understand what each number
in a multiplication or division expression represents. For example, in
multiplication, unlike addition, the factors in the problem can refer
to different units. If students are solving the problem 294 to find out how
many legs there are on 29 cats, 29 is the number of cats (or number of
groups), 4 is the number of legs on each cat (or number of items in each
group), and 116 is the total number of legs on all the cats. Modeling
multiplication problems with pictures, diagrams, or concrete materials
helps students learn what the factors and their product represent in various
Students should consider and discuss different types of problems that can be solved using multiplication and division. For example, if there are 112 people traveling by bus and each bus can hold 28 people, how many buses are needed? In this case, 11228 indicates the number of groups (buses), where the total number of people (112) and the size of each group (28 people in each bus) are known. In a different problem, students might know the number of groups and need to find how many items are in each group. If 112 people divide themselves evenly among four buses, how many people are on each bus? In this case, 1124 indicates the number of people on each bus, where the total number of people and the number of groups (buses) are known. Students need to recognize both types of problems as division situations, should be able to model and solve each type of problem, and should know the units of the result: Is it 28 buses or 28 people per bus? Students in these grades will also encounter situations where the result of division includes a remainder. They should learn the meaning of a remainder by modeling division problems and exploring the size of remainders given a particular divisor. For example, when dividing groups of counters into sets of 4, what remainders could there be for groups of different sizes? »
Students can extend their understanding of multiplication and division as they consider the inverse relationship between the two operations. Another way their knowledge can grow is through new multiplicative situations such as rates (3 candy bars for 59 cents each), comparisons (the book weighs 4 times as much as the tablet), and combinations (the number of outfits possible from 3 shirts and 2 pairs of shorts). Examining the effect of multiplying or dividing numbers can also lead to a deeper understanding of these operations. For example, dividing 28 by 14 and comparing the result to dividing 28 by 7 can lead to the conjecture that the smaller the divisor, the larger the quotient. With models or calculators, students can explore dividing by numbers between 0 and 1, such as 1/2, and find that the quotient is larger than the original number. Explorations such as these help dispel common, but incorrect, generalizations such as "division always makes things smaller."
Further meaning for multiplication should develop as students build and describe area models, showing how a product is related to its factors. The area model is important because it helps students develop an understanding of multiplication properties (Graeber and Campbell 1993). Using area models, properties of operations such as the commutativity of multiplication become more apparent. Other relationships can be seen by decomposing and composing area models. For example, a model for 206 can be split in half and the halves rearranged to form a 1012 rectangle, showing the equivalence of 1012 and 206. The distributive property is particularly powerful as the basis of many efficient multiplication algorithms. For example, figure 5.2 shows the strategies three students might use to compute 728all involving the distributive property.
By the end of this grade band, students should be computing fluently with whole numbers. Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships.
A significant amount of instructional
time should be devoted to rational numbers in grades 35. The focus
should be on developing students' conceptual understanding of fractions
and decimalswhat they are, how they are represented, and how they
are related to whole numbersrather than on developing computational
fluency with rational numbers. Fluency in rational-number computation
will be a major focus of grades 68.
Fluency with whole-number computation depends, in large part, on fluency with basic number combinationsthe single-digit addition and multiplication pairs and their counterparts for subtraction and division. Fluency with the basic number combinations develops from well-understood meanings for the four operations and from a focus on » thinking strategies (Thornton 1990; Isaacs and Carroll 1999). By working on many multiplication problems with a variety of models for multiplication, students should initially learn and become fluent with some of the "easier" combinations. For example, many students will readily learn basic number combinations such as 32 or 45 or the squares of numbers, such as 44 or 55. Through skip-counting, using area models, and relating unknown combinations to known ones, students will learn and become fluent with unfamiliar combinations. For example, 34 is the same as 43; 65 is 5 more than 55; 68 is double 38. Because division is the inverse of multiplication, students can use the multiplication combinations to learn division combinations. For example, 246 can be thought of as 6? = 24. If by the end of the fourth grade, students are not able to use multiplication and division strategies efficiently, then they must either develop strategies so that they are fluent with these combinations or memorize the remaining "harder" combinations. Students should also learn to apply the single-digit basic number combinations to related problems, for example, using 56 to compute 506 or 5000600.
Research suggests that by solving problems that require calculation, students develop methods for computing and also learn more about operations and properties (McClain, Cobb, and Bowers 1998; Schifter 1999). As students develop methods to solve multidigit computation problems, they should be encouraged to record and share their methods. As they do so, they can learn from one another, analyze the efficiency and generalizability of various approaches, and try one another's methods. In the past, common school practice has been to present a single algorithm for each operation. However, more than one efficient and accurate computational algorithm exists for each arithmetic operation. In addition, if given the opportunity, students naturally invent methods to compute that make sense to them (Fuson forthcoming; Madell 1985). The following episode, drawn from unpublished classroom observation notes, illustrates how one teacher helped students analyze and compare their computational procedures for division:
Students in Ms. Spark's fifth-grade class were sharing their solutions to a homework problem, 72834. Ms. Sparks asked several students to put their work on the board to be discussed. She deliberately chose students who had approached the problem in several different ways. As the students put their work on the board, Ms. Sparks circulated among the other students, checking their homework.
Henry had written his solution:
Henry explained to the class, "Twenty 34s plus one more is 21. I knew I was pretty close. I didn't think I could add any more 34s, so I subtracted 714 from 728 and got 14. Then I had 21 remainder 14." »
Michaela showed her solution:
Michaela says, "34 goes into 72 two times and that's 68. You gotta minus that, bring down the 8, then 34 goes into 48 one time."
Ms. Sparks saw relationships between the two methods described by students, but she doubted that any of her students would initially see these relationships. Through her questioning, she helped students focus on the ways in which both Michaela's and Henry's methods used multiplication to find the total number of 34s in 728 and helped students » clarify what quantities were represented by the notation in Michaela's solution. As the class continues their study of division, Ms. Sparks should encourage this type of explanation and discussion in order to help the students understand, explain, and justify their computational strategies.
As students move from third to fifth grade, they should consolidate and practice a small number of computational algorithms for addition, subtraction, multiplication, and division that they understand well and can use routinely. Many students enter grade 3 with methods for adding and subtracting numbers. In grades 35 they should extend these methods to adding and subtracting larger numbers and learn to record their work systematically and clearly. Having access to more than one method for each operation allows students to choose an approach that best fits the numbers in a particular problem. For example, 29842 can be thought of as (30042) (242), whereas 4116 can be computed by multiplying 418 to get 328 and then doubling 328 to get 656. Although the expectation is that students develop fluency in computing with whole numbers, frequently they should use calculators to solve complex computations involving large numbers or as part of an extended problem.
Many students are likely to develop and use methods that are not the same as the conventional algorithms (those widely taught in the United States). For example, many students and adults use multiplication to solve division problems or add starting with the largest place rather than with the smallest. The conventional algorithms for multiplication and division should be investigated in grades 35 as one efficient way to calculate. Regardless of the particular algorithm used, students should be able to explain their method and should understand that many methods exist. They should also recognize the need to develop efficient and accurate methods.
As students acquire conceptual grounding related to rational numbers, they should begin to solve problems using strategies they develop or adapt from their whole-number work. At these grades, the emphasis should not be on developing general procedures to solve all decimal and fraction problems. Rather, students should generate solutions that are based on number sense and properties of the operations and that use a variety of models or representations. For example, in a fourth-grade class, students might work on this problem:
Jamal invited seven of his friends to lunch on Saturday. He thinks that each of the eight people (his seven guests and himself) will eat one and a half sandwiches. How many sandwiches should he make?
Students might draw a
picture and count up the number of sandwiches, or they might use reasoning
based on their knowledge of number and operationsfor example, "That
would be eight whole sandwiches and eight half sandwiches; since two halves
make a whole sandwich, the eight halves will make four more sandwiches,
so Jamal needs to make twelve sandwiches."
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