FIFTH GRADE NUMBER SENSE

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1 FIFTH 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. By the end of fifth grade, students increase their facility with the four basic arithmetic operations, as applied to positive and negative numbers, fractions, and decimals. Students will continue to learn about the relative positions of numbers on the number line especially negative numbers. The latter is especially important, since for the first time negative numbers begin to play a major part in core number sense expectations. The most important aspect of students work with negative numbers is to learn the rules for doing the basic operations of arithmetic with them. Mastery of the general division algorithm is also important. Students should be gradually moved to a stage where they are comfortable with the algorithm in carefully selected cases in which the numbers needed at each step are clear. Essential number sense skills students should learn in fifth grade are the addition and multiplication of fractions. Here two main skills are involved: factoring whole numbers in order to put fractions into reduced forms and the basic arithmetic skills involved in this factoring. Key Concepts : 1. A whole number can be factored into a product of prime numbers. 2. Fractions can be represented as decimals and as percents. 3. The basic properties of addition, subtraction, multiplication, and division for whole numbers also hold for these operations on fractions or decimals. 1

2 KEY STANDARDS Interpret percents as part of a hundred; find decimal and percent equivalents for common fractions; explain why they represent the same value; and compute a given percent of a whole number. Determine the prime factors of all numbers through 50 and write numbers as the product of their prime factors using exponents to show multiples of a factor. Identify and represent positive and negative integers, decimals, fractions, and mixed numbers on a number line. Add, subtract, multiply, and divide with decimals and negative numbers and verify the reasonableness of the results. Are proficient with division including division with positive decimals and long division with multiple digit divisors. Solve simple problems including ones arising in concrete situations involving the addition and subtraction of fractions and mixed numbs (like and unlike denominators of 20 or less) and express answers in simplest form. Elaboration California Mathematics Framework By the time students have finished 4th grade they should have a basic understanding of whole numbers, some understanding of fractions, and some understanding of decimals. They should also have had some exposure to negative numbers. These skills will be enhanced in the 5th grade. The fact that a fraction c/d is both c parts of a whole consisting of d equal parts and the quotient of the number c by the number d must be carefully explained. The importance of providing logical explanations for all aspects of the teaching of fractions cannot be overstated because the fear of fractions as well as the mistakes related to fractions appear to underlie the failure of mathematics education. Once c/d is clearly understood to be the division of c by d, then the conversion of fractions to decimals can be explained logically. Identifying numbers as points on the real line is an important step in relating students arithmetic concept of numbers to geometry. The fusion of arithmetic and geometry adds a new dimension to students understanding of numbers and is ubiquitous in mathematics. The most important aspect of students work with negative numbers is to learn the rules for doing the basic operations of arithmetic with them. This is the beginning of a three-year process of familiarizing students with the full arithmetic of integers. At this point, students should find it profitable to interpret, geometrically, both addition and subtraction of positive and negative 2

3 whole numbers on the number line: adding a positive number b shifts the number line to the right by b units, and adding a negative number -b shifts the number line to the left by b units, etc. The introduction of the general division algorithm is also important, but it can be very complicated and consequently difficult for many students to master. In particular, the skills needed to find the largest product of the divisor with an integer between 0 and 9 that is less than the remainder at the current step are likely to be very demanding for 5th grade students. Students should become comfortable with the algorithm in carefully selected cases where the numbers needed at each step are clear. Putting such a problem in context may help. For instance, imagine dividing 153 by 25 as packing 153 students into a fleet of buses for a field trip, with each bus carrying a maximum of 25 passengers. Drawing pictures to help with the reasoning if necessary, one can see that it takes 6 buses with 3 left over; now the 3 students get to enjoy being in the 7th bus with room to spare. But it seems both unnecessary and unwise to attempt to achieve greater fluency than this. The most essential number-sense skills that students should learn in 5th grade are the addition and multiplication of fractions. In this regard it should be emphasized that the primary definition of the addition of two fractions a/b and c/d is ad/bd + bc/bd, and this should be carefully explained to students rather than be imposed on them without explanation. Otherwise, the common mistakes that students make of believing that a/b + c/d = (a+c)/(b + c) and a/(b+c) = a/b + a/c would be the result. From this definition, the usual formula for addition involving the least common multiple of b and d can be easily deduced, but the latter rule should not be used as the definition of adding fractions in general. Grade Level Readiness Considerations for Grade 5 At the beginning of grade five, students need to be assessed carefully for their knowledge of core content taught in the lower grades, particularly: Knowledge and fluency of basic fact recall including addition, subtraction, multiplication, and division facts. Students should know all basic facts and be able to recall them instantly by this level. Mental addition - the ability to add a single-digit number to a two-digit number mentally. Rounding off hundreds and thousands numbers to the nearest ten, hundred, or thousand and rounding off two-place decimals to the nearest whole number of tenth. Place value - the ability to read and write numbers through the millions. Knowledge of measurement equivalencies, both customary and metric, for time, length, weight, and liquid capacity. Prime numbers and the ability to determine prime factors of numbers up to 50. The ability to use algorithms to add and subtract whole numbers, multiply a two-digit number and a multi-digit number, and divide a multi-digit number by a single-digit number. 3

4 Knowledge of customary and metric units and equivalencies for time, length, weight, and capacity. All of the topics above require teaching over an extended period of time. A systematic program must be established to enable students to reach high rates of accuracy and fluency with these skills. Long division. Long division requires the application of a number of component skills. Students must be able to round tens and hundreds numbers and work estimation problems dividing a two-digit number into a two or three-digit number with paper and pencil and also mentally do the steps in the division algorithm. This realm contains a number of distinct problem types. Problems in which the estimations give the correct numbers in the quotient are easier than problems in which the estimate is too high or too low. The rule about what to do if the estimation gives a quotient that is not correct needs to be carefully explained and modeled and examples presented to allow students to develop proficiency. Cumulative review integrating all problem types presented to date needs to occur. Adding and subtracting fractions with unlike denominators. See the instructional profile on adding and subtracting fractions with unlike denominators. Working with negative numbers. The Standards call for students to add, subtract, multiply, and divide negative numbers. Students often become confused by operations with negative numbers, because too much is introduced at once and they do not have the opportunity to master one type before another type is introduced. Ordering fractions and decimal numbers. Students can use fraction equivalence skills in order to compare fractions and convert fractions to decimals. Students need to know that 3/4 = 75/100 =.75 = 75% Working with percents. In order to compute a given percent of a number, students can covert the percent to a decimal and then multiply. Students must know that 6 percent translates to.06 (percents under ten percent can be troublesome). Students should be assessed on multiplying decimals times whole numbers before work begins on this type problem. 4

5 FIFTH 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. In the 5th grade Algebra and Functions strand, according to the Framework, we come to one of the defining steps in moving from simply learning arithmetic to learning mathematics, the replacement of numbers by variables. The importance of this step in terms of reasoning rather than simple manipulative facility mandates that particular care be taken. the basic idea that, for example, 3x + 5 is a shorthand for an infinite number of sums 3(1) + 5, 3(2.4) + 5, 3(11) + 5, etc., must be thoroughly presented and understood by students and they must practice solving simple algebraic expressions. But it is probably a mistake to push too hard here. Check for student understanding of concepts, perhaps providing students with some simple puzzle 5

6 problems to give them practice in writing an equation for an unknown from data in a word problem. Students must understand how to evaluate simple expressions. The ability to graph functions is an essential fundamental skill and linear functions are the most important concept for applications of mathematics. The importance of these topics can hardly be overstated. KEY STANDARDS Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution. Identify and graph ordered pairs in the four quadrants of the coordinate plane. Solve problems involving linear functions with integer values, write the equation, and graph the resulting ordered pairs of integers on a grid. 6

7 FIFTH 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. By the end of fifth grade students know and use common measuring units to determine length and area; they know and use formulas to determine the volume of simple figures. Students know the concept of angle measurement and use a protractor and compass in solving problems. Key concepts for fifth grade include the following: Areas of geometric figures can often be found by dividing and combining them into figures whose areas are already known; and in any triangle the sum of the angles is the same as the angle in a straight line, 180 degrees. Make sure students commit to memory the formulas for the area of a triangle, parallelogram, and a rectangle as well as the formula for the circumference of a circle. 7

8 KEY STANDARDS Derive and use the formula for the area of right triangles and of parallelograms by comparing with the area of rectangles. Measure identify and draw angles, perpendicular and parallel lines, rectangles and triangles, using appropriate tools. Know that the sum of the angles of any triangle is 180 degrees and the sum of the angles of any quadrilateral is 360 degrees and use this information to solve problems. Construct cube and rectangular boxes from two-dimensional patterns and use this to compute the surface area for these objects. Understand the concept of volume and use the appropriate units in common measuring systems (cubic centimeters etc.) to compute the volume of rectangular solids. 8

9 FIFTH 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 fifth grade, students use grids, tables, graphs, and charts to record and analyze data. The ability to graph functions is an essential fundamental skill especially linear functions which are the most important for applications of mathematics. The standards related to graphing indicate ways in which the skills involved in the Algebra and Functions strand can be reinforced and applied. KEY STANDARDS Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph. Know how to write ordered pairs correctly (e.g., (x,y)). 9

10 FIFTH GRADE MATHEMATICAL REASONING 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 fifth grade, mathematical reasoning is involved in explaining arithmetic facts, in solving problems and puzzles at all levels, in understanding algorithms and formulas, and in justifying basic results in all areas of mathematics. Students should develop the habits of logical thinking and recognize and critically question all assumptions. Students should learn to generate examples to test conjectures and learn to search for possible counterexamples. A focus is for students to create mathematical ideas themselves and to articulate, examine, and evaluate these ideas. 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. Students need to know that posing conjectures and trying to validate them is an expected part of their mathematical activity. 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. 10