FOURTH GRADE NUMBER SENSE

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1 FOURTH GRADE NUMBER SENSE Number sense is a way of thinking about number and quantity that is flexible, intuitive, and very individualistic. It grows as students are exposed to activities that cause them to think about numbers in many ways and in different contexts. Number sense includes the ability to compute accurately, to self correct by detecting errors, and to recognize results as reasonable. According to the California Framework, a person has Number Sense if he or she has an intuitive feel for number size and combinations as well as the ability and facility to work with numbers in problem situations in order to make sound decision and reasonable judgments. The mathematics curriculum enables students to work with numbers to develop number sense traits that include a thorough understanding of number meanings, abilities to represent quantities in multiple ways, recognize the magnitude of number, to know the relative effects of operating on numbers, and to estimate and judge the reasonableness of quantitative results. Numbers enable students to count, to measure, to compare, and to make predictions. Helping students to develop number sense requires appropriate modeling, posing process questions, encouraging thinking about numbers, and in general creating a classroom environment that nurtures number sense. The Number Sense strand in fourth grade continues the development of whole numbers to millions and then moves on to ordering and comparing numbers to two decimal places. By the end of fourth grade, students understand large numbers and addition, subtraction, multiplication, and division of whole numbers. Students describe and compare simple fractions and decimals. Students learn to round whole numbers and decimals. They also understand what a prime number is and know the prime numbers up to 50. Key concepts: 1. The real number line supplies a model for locating numbers, whether they are decimals or fractions. 2. The fraction m/n represents the equal division of m into n equal pieces. 3. In a decimal numeral, a one in each place is worth ten in the place to the right of it. 4. Division can be thought of as repeated subtraction. KEY STANDARDS Read and write whole numbers in the millions. Order and compare whole numbers and decimals to two decimal places. Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand or hundred thousand. Decide when a rounded solution is called for, and explain why this is the case. 1

2 Use concepts of negative numbers. Identify the relative position of fractions, mixed numbers, and decimals to two decimal places on the number line. Demonstrate understanding of, and the ability to use standard algorithms for addition and subtraction of multi-digit numbers. Demonstrate understanding of, and ability to use standard algorithms for multiplying a multidigit number by a two digit number and long division for dividing a multi-digit number by a one digit number; use relationships between them to simplify computations and to check results. Solve problems involving multiplication of multi-digit numbers by two-digit numbers. Solve problems involving division of multi-digit numbers by one-digit numbers. Know that numbers such as 2, 3, 5, 7, 11 do not have any factors except 1 and themselves, and that such numbers are called prime numbers. Elaboration California Mathematics Framework Knowledge of multiplication and division facts should be re-assessed at the beginning of the school year and systematic instruction and practice should be provided to enable students to reach high degrees of automaticity in recalling these facts, as described for addition in grade 2. Reading and writing thousands and millions numbers with one or more zeroes in the middle can be particularly troublesome for students (Seron and Fayol, 1994). Thus, assessment and teaching should be thorough so that students are able to read and write difficult numbers such as 300,200 and 320,000. Students need to understand that 0s in different positions represent different place values--tens, hundreds, thousands, and so forth--and they need practice working with these type of numbers, e.g., determining which is larger, 320,000 or 300,200 and translating verbal labels "one million two hundred thousand to Arabic representations, 1,200,000. For "real world" applications of mathematics, for understanding the way numbers distribute on the number line, and ultimately for more advanced topics in mathematics, it is essential that students begin to understand the concept of "closeness" in rounding numbers. It is probably not wise to push too hard on the notion of "close enough" while students are still struggling with the abstract idea of a number itself. Nonetheless, by now they should be ready for this next step. It should be emphasized in the discussion of rounding that one only rounds off if the result of rounding is "close enough." It is essential that students understand fraction equivalencies as related to the ordering and comparison of decimals. Students must understand, for instance, that 2/10 = 20/100 and then equate these fractions to decimals. The Standards in fourth grade call for an understanding of 2

3 fraction equivalencies for halves, fourths, tenths, and hundredths, but it is advisable to teach a general procedure that could apply to any fraction. Teaching of the conversion of fractions to decimals and improper fractions to decimals should be structured so students see relationships, e.g., the fraction 7/4 can be converted to 4/4 + 3/4, which in turn equals 1 and 3/4. The fourth-grade Standards do not require any arithmetic with fractions; however, practice with addition and subtraction of fractions (like denominator) must be continued in this grade, because they become very important in fifth grade. Students could also be introduced to the concept of unlike denominators in preparation for the following year. The Standards require that students know the definition of prime numbers and know that many whole numbers decompose in different ways. For example, with the number 150, they should realize that 150 = 5 x 30 and 30 = 5 x 6, so 150 = 5 x 5 x 6, which can be decomposed to 5 x 5 x 3 x 2. Students will be using these factoring skills extensively in the later grades. Even though determining the prime factors of all numbers through 50 is a fifth-grade Standard, practice on finding prime factors can begin in fourth grade. Students should be given extensive practice over an extended period of time with finding prime factors so that they can develop automaticity in the factoring process. By the end of fifth grade, students should be able to determine with relative ease if any of the prime numbers 2, 3, 5, 7, or 11 are factors of a number less than 200. Multiplication and division problems with multi-digit problems are expanded at this grade level. Division problems with a zero in the quotient (e.g., 5102/5 = 1020 R2) can be particularly difficult for students and requires systematic instruction. The Standards call for "understanding of the standard algorithm." What is expected here is that the teacher will sketch the reasons why the algorithm works and will carefully show the students how to use it. It is not expected that the students will be able to reproduce this discussion in any detail. But it is expected that they will have a general idea of why it works and can expand it out in detail for small numbers. As students grow older, this experience should lead to increased confidence in these and similar algorithms. They should also gain a grasp of how to construct algorithms in other situations where they might be useful, as well as some idea of the importance of verifying the correctness of such algorithms before beginning to rely on them. For example, the process of writing any kind of program for a computer amounts first to creating algorithms for automating a task, then implementing them on the machine. Without experience of the type above, students will be illequipped to construct correct programs. 3

4 Grade Level Readiness Considerations for Grade 4 Multiplication and division facts. Students who enter fourth grade without multiplication facts committed to memory are at risk of having difficulty as more complex math is taught. An assessment of basic fact knowledge needs to be undertaken at the beginning of the school year. Systematic daily fact practice with multiplication and division facts needs to be provided for students who have not learned the facts. Addition and subtraction. Mentally adding a two-digit number and a one-digit number is a component skill for working multiplication problems that was targeted for second grade. Students have to add the carried number to the product of two factors. They should be assessed for the ability to add numbers mentally at the beginning of the school year, and systematic practice should be provided for students who are not able to do this. Reading and writing numbers. Reading and writing thousands and millions numbers with one or more zeroes in the middle can be particularly troublesome for students. Assessment at the beginning of fourth grade should test students on reading and writing the more difficult thousand numbers, such as 4002 and When teaching students to read five- and six-digit numbers, teaching should be thorough enough so that students are able to read and write difficult numbers such as 300,200 and 320,000. Fractions equal to one. Understanding fractions equal to one (8/8, 4/4) is important in understanding the procedure for working with equivalent fractions. Students should have an indepth understanding of how to construct a fraction that equals one. When students work equivalent fraction problems, the teacher should prompt students to find the equivalent fraction or missing number in the equivalent fraction by finding the fraction of one by which you multiply or divide to determine the equivalent fraction. Multiplication and division problems. Multiplication problems in which either factor has a zero are likely to cause difficulties. Provide extra practice on problems, such as 20 x 315 and 24 x 308. Division problems in which there will be a zero in the answer tend to be difficult for students, e.g., 152/3 and 5115/5. Students will need prompting on how to determine whether they have completed the problem with enough digits in the answer. 4

5 FOURTH GRADE ALGEBRA AND FUNCTIONS Algebra Learning algebra is important in a student s mathematical development. It opens the door to organized abstract thinking and supplies a tool for logical reasoning. Algebra embodies the construction and representation of patterns and generalization, and active exploration and conjecture. By itself algebra is the language of variables, operations, and symbol manipulation. Every mathematical strand uses algebra to symbolize, clarify, and communicate. According to the California Framework, algebra is the fundamental language of mathematics. It enables students to create a mathematical model of a situation, provides the mathematical structure necessary to use the model to solve problems, and links numerical and graphical representatives of data. Algebra is the vehicle for condensing large amounts of data into efficient algebraic statements. The use of symbols greatly enhances the understanding of mathematics. Familiarity with symbols and with algebraic ideas provides a basis of learning to translate between a naturally occurring problem situation and an algebra expression and vice versa. This process by which we transform a problem from the natural world into an equation to be solved enables us to think abstractly and to tie together apparently different situations through generalization. Functions Functions are a means to explore the many kinds of relationships among quantities and the manner in which those relationships can be made explicit. The basic idea of a function, according to the state framework, is that two quantities are related in some way. The value of one quantity may depend on the value of the other quantity. A function from set A to set B is a special relationship which is a correspondence from A to B in a special relationship which is a correspondence from A to B in which each element of A is paired with one and only one element of B. A function can be represented as a rule (function machine) that makes clear how pairs of numbers are related. Functions appear in all the strands to describe relationships. Key Concepts: In fourth grade the Algebra strand continues to grow in importance. Students begin to see that formulas can express precise relationships among quantities. All five of the sub-topics under the first Standard are important. But the degree to which students need to understand these strands differs. Standards 1.2 and 1.3--"interpret and evaluate mathematical expressions involving parentheses" and "use parentheses to indicate which operation to perform first when writing expressions containing more than two terms and different operations"--do not need much emphasis since these are notational issues. The real skill here is simply learning how to write expressions so that others can understand, unambiguously, what they mean. However, it would 5

6 be appropriate at this point to carefully explain to students the significance of the associative and commutative laws, and explain carefully why arbitrary sums or products, such as (-6) or 113 x 212 x 31 x 11 do not have to be ordered in any particular way, nor do they have to be done in any particular order. Standards 1.4 and 1.5 relate to functional relationships and are much more important theoretically. In particular, students should understand that an equation, such as y = 3x + 5 is a prescription for determining a second number when a first number is given, as it takes the mystery out of the topic. One way to understand an equation, such as y = ex + 5 is to work through many pairs of numbers (x,y) to see if they satisfy this equation. For example, (1,8) and (0,5) do, but (-1,3) and (2,10) do not. The second Algebra Standard is basic--"students know how to manipulate equations." If understood now, it and the two basic rules--"equals added to equals are equal" and "equals multiplied by equals are equal"-- will make much of what happens in mathematics and other subjects from fifth grade through high school easier for students to grasp. If these concepts are not clear to students, it is a virtual guarantee of difficulties in later grades, so careful assessment of students understanding of these principles should be done here. KEY STANDARDS Interpret and evaluate mathematical expressions that use parentheses. Use parentheses to indicate which operations to perform first when writing expressions containing more than two terms and different operations. Understand that an equation such as y = 3x + 5 is a prescription for determining a second number when a first number is given. Know and understand that equal added to equals are equal. Know and understand that equals multiplied by equals are equal. 6

7 FOURTH GRADE MEASUREMENT AND GEOMETRY Measurement Measuring is a process by which a number is assigned to an attribute of an object or event. Length, capacity, weight, area, volume, time, and temperature are measurable attributes in the elementary math curriculum. Measurement can be used to help students learn other topics in mathematics. For example, students count the number of grams it takes to balance a scale or add to find the perimeter of a triangle. Measurement can help teach about other operations. Many of the numeration models used have a measurement base. For example, the number line is based on length. Measurement is of central importance to the curriculum because it provides the critical link between mathematics and objects and events in everyday life. Measurement leads to geometry through the measurement of angles, perimeters, areas, and volumes. Students learn to identify plane and solid geometric objects, such as lines, squares, rectangles, triangles, circles, cubes, and spheres, and then to determine their mathematical properties. Geometry Geometry is the study of sets of points and the relationships between them. Through the study of geometry, students link mathematics to space and form in the world around them and in the abstract. Students are exposed to and investigate two-dimensional and three-dimensional space by exploring shape, area, and volume; studying lines, angles, points, and surfaces; and engaging in other visual and concrete experiences. In the early grades this process is informal and highly experiential; students explore many objects and discover and discuss the attributes of different shapes and figures. Students gradually build on their foundation and become more familiar with the properties of geometrical figures and get better at using them to solve problems. They explore symmetry and proportion and begin to relate geometry to other areas of mathematics. For example, graphical representations of functions can help explain and generalize geometric relationships while geometrical insights inform the study of functions. Fourth grade contains an enormous amount of material for this strand. The first Standard relates to perimeter and area. Here it is basic that students understand that area of a rectangle is obtained by multiplying length by width and that perimeter is given by a linear measurement. The intent of most of this standard is that students know the reasons behind the formulas for the perimeter and area of a rectangle and can see how these formulas work when perimeter and area vary as the rectangles vary. 7

8 Key concepts: 1. The area of plane figures in general are not determined by their perimeters. 2. Points in a plane can be identified and located by their coordinates. 3. Be sure to introduce the symbol for perpendicular. 4. It is suggested to introduce the abbreviated notation ab in place of a x b. KEY STANDARDS Students use two-dimensional coordinate grids to represent points and graph lines and simple figures. Understand that the length of a horizontal line segment equals the difference of the x- coordinates. Understand that the length of a vertical line segment equals the difference of the y- coordinates. 8

9 FOURTH GRADE STATISTICS, DATA ANALYSIS, AND PROBABILITY Statistics Statistics is collecting, organizing, representing, and interpreting data. Probability and statistics are now highly visible topics in elementary school. According to the California Framework, the rapid evolution in information processing has greatly stimulated the use of data analysis throughout modern society. The techniques of data analysis help us in two basic ways to deal with the ever-increasing volume of available data. Data analysis is used to summarize and describe the features in a set of data so that we may understand and make use of the information. Its techniques are also useful in making inferences, including forming conclusions, answering questions, and making predictions based on data. Decision making in business, industry, and government is increasingly based on the understandings and conclusions derived from data. The processes that link our interpretations and conclusions to data are part of mathematics. Data analysis is important because of its use of information to reach conclusions and make predictions, thus guiding decision making. When we use data to make inferences, we may use inferential statistics, but the ability to draw conclusions based on data follows a special form of mathematical reasoning. In fourth grade students begin descriptive statistics. Pictures of data are ways to describe data, however, it is common to measure the data in some way to produce a number that describes the data. These numbers are measures of the data that quantify some attribute of the data. For example, students begin to use measures of central tendency to describe average. In third grade probability was emphasized. Students may need to review statistical material from second grade. Probability Probability is the mathematical study of uncertain events. The ability to describe events, model processes, analyze data, and predict involves the mathematics of probability. From a basic understanding that one event can be more or less likely than another, fourth grade students begin to represent all possible outcomes for simple probability situations. They begin to conduct experiments and record the results. 9

10 FOURTH GRADE MATHEMATICAL REASONING A major goal of math instruction is to help children develop autonomy and confidence in their ability to reason and justify their thinking. Making conjectures, gathering evidence, and building an argument to support ideas are fundamental to doing mathematics. Mathematical reasoning is synonymous with sense making. It is how we discern truth. This is generally done through the application of deductive, inductive, spatial, or algebraic reasoning. According to the California Framework, mathematics provides an opportunity to encounter reasoning in one of its purest forms and to establish mathematical truths with a certainty that is rare in other disciplines. The importance of reasoning to mathematics cannot be overstated. Mathematics makes unique and indispensable contributions to the development of the students ability to think and communicate in a logical manner, a major goal of mathematical study. At fourth grade, mathematical reasoning should involve the kind of informal thinking, conjecturing, and validating that helps children see that mathematics makes sense. Manipulatives and other physical models help children relate processes to their conceptual learning and gives them concrete objects to talk about in explaining and justifying their thinking. Mathematical reasoning does not develop in isolation. It shows up in many strands and characterizes the thinking skills that students can carry from mathematics into other disciplines. Constructing valid arguments and criticizing invalid ones is part and parcel of doing mathematics. The development of mathematical reasoning is thus a principal objective in the curriculum. Students need to develop strategies to use and questions to ask themselves when they do not know what to do. They need to become independent learners and this takes effort and perseverance. Mathematical reasoning should be a part of every standard. 10

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