THIRD GRADE NUMBER SENSE

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1 THIRD 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. In third grade, students continue to develop their understanding of place value. As they develop abstraction in regard to place value they are less dependent on concrete models for smaller numbers. Students develop confidence with numbers to 10,000 and are able to read, write, and work with large numbers. Students develop deeper understanding about multiplication and are able to use the commutative and associative properties. They learn multiplication facts and know how they are related to division and can use this to solve problems. KEY STANDARDS Identify the place value for each digit in numbers to 10,000. Use expanded notation to represent numbers. Find the sum or difference of two whole numbers between 0 and 10,000. Memorize to automaticity the multiplication table for numbers between 1 and 10. Use the inverse relationship of multiplication and division to compute and check results. Solve simple problems involving multiplication of multi-digit numbers by one digit numbers. Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notations. Multiply and divide money amounts in decimal notation using whole number multipliers and divisors. Add and subtract simple fractions. 1

2 Elaboration California Mathematics Framework Work with addition and subtraction problems expands in third grade to problems in which regrouping (i.e., carrying and borrowing) is required in more than one column. Particularly important and difficult for some students are subtraction problems that include zeros, such as and (VanLehn, 1990). Students should become skilled in regrouping across columns with zeroes, as these types of problems are often used with money applications, e.g., "Jerry bought an ice-cream for 62 cents and paid for it with a ten dollar bill. How much change will he receive?" As with addition and subtraction, memorizing the answers to simple multiplication problems requires the systematic introduction and practice of multiplication facts. Refer to the recommendations discussed for addition facts in first grade. Some division facts can be incorporated into the sequence for learning multiplication facts. As with addition and subtraction, inverse relationships can be used to cut down on memorization load. These related facts can be introduced together, e.g., 20 divided by 5, 5 times 4. Multiplication and division problems with multi-digit terms are introduced in third grade, e.g., 36 x 5. The basic facts used in both problem types should be ones that have already been committed to memory, e.g., students should have already memorized the answer to 6 x 5, a component of the more complex problem 36 x 5. Students should already be familiar with the basic structure of these problems, based on their understanding of how to solve the addition of a one-digit to a two-digit number, e.g., , , and As with addition and subtraction, problems that require carrying, e.g., 36 x 5, will be more difficult than the problems that do not require it (e.g., 32 x 4; Geary, 1994). The goal is to extend multiplication of whole numbers up to 10,000 by single-digit numbers, e.g., 9,345 x 2, so that students gain mastery of the standard right-to-left multiplication algorithm with the multiplier being a one-digit number. Students at this level should be able to do long-division problems in which they divide a multidigit number by a single digit. A critical component skill is being able to determine the multiple of the divisor that is just smaller than the number being divided. In 28/5, the multiple of 5 that is just smaller than 28 is 25. While the identification of remainders exceeds the third grade Standard, students need to become aware of the process for division when there is a remainder. Practice in determining multiples can be coordinated with practice of multiplication facts. Having basic multiplication facts memorized will greatly facilitate students' ability to solve these division problems. Rounding is a critical prerequisite for working estimation problems (LeFevre, et al., 1993). Below is a sequence that might be followed in introducing rounding, each of which could be introduced over a several-day period, followed by continued practice. Practice sets should include examples reviewing earlier stages as well as the current ones. 2

3 Round a 2-digit number to the nearest 10. Round a 3-digit number to the nearest 10. Round a 3-digit number to the nearest 100. Round a 4-digit number to the nearest 1,000. Round a 4-digit number to the nearest 100. The work with fractions in third grade is primarily with diagrams and concrete objects. Students should be able to compare fractions in at least two ways. First, they should be able to order proper fractions with like denominators, initially using diagrams but later realizing that if the denominators are equal, then the order depends only on the numerators. Second, they should be able to order unit fractions, perhaps only with whole number denominators less than or equal to 6. At this point students should not be expected to compare fractions with unlike denominators. They should compare particular fractions verbally and with the symbols <, =, >. With regard to multiplying and dividing decimals, care should be taken to include exercises in which students have to distinguish between adding and multiplying. Working with money can serve as the introduction of decimals. For example: "Josh had $3. He earned $2.50. How much does he have now?" This contrasts with: "Josh earned $2.50 an hour. He worked 3 hours. How much did he earn?" During third grade, the teaching of arithmetic facts can also be extended to include finding multiples and factors of whole numbers, which are critical to students' understanding of numbers and later to simplifying fractions. This is a skill that needs time to develop. Consequently it should be introduced well before the students are to be tested on it. Use only small numbers involving very few primes. As a rule, "small" means less than 30, with prime factors limited to maybe only 2, 3, or 5, e.g., 20 = 2 x 2 x 5, 18 = 3 x 3 x 2. 3

4 Grade Level Readiness Considerations for Grade 3 Addition and subtraction facts. Students who enter third grade without addition and subtraction facts to 20 committed to memory are at risk of having difficulty as mathematics becomes more complex. An assessment of basic fact knowledge needs to be undertaken at the beginning of the school year. Systematic daily practice with addition and subtraction facts needs to provided for students who have not learned the facts. Reading and writing numbers. Thousands numbers with zeroes in the hundreds and/or tens place (4006, 4060, 4600) can be particularly troublesome for at-risk students. Systematic presentations focusing on reading and writing thousands numbers with one or two zeroes needs to be presented until students can read and write these more difficult numbers. Rounding. Rounding a thousands number to the nearest ten, hundred, and thousand requires a good understanding of the rounding off process. Students need to learn which digit to cue off of when rounding to a particular unit. For example, when rounding off to the nearest hundred, the student needs to look at the current digit in the tens column to determine whether the digit in the hundreds column will remain the same or be increased when rounded off. Practice items should include a variety of types, e.g., round off 2,375 to the nearest hundred and then to the nearest thousand. 4

5 THIRD 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 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. The algebra and function strand grows in importance in third grade. Students use the relationship of numbers and move between operations easily. They know that there is a relationship between addition and subtraction and between multiplication and division and use this to solve problems in context. They are able to use skip counting and simple functions to solve problems by identifying the rules of the given pattern and functional relationship. 5

6 Representing relationships of quantities is very important and must be treated very carefully. This can be a very difficult step for students, and many examples should be given: 3x12 inches in 3 feet, 4x 11 legs in 11 cats, 2 x 15 wheels in 15 bicycles, 3 x 15 wheels in 15 tricycles, the number of students in the classroom <50, the number of days in a year >300 etc. Be sure that students are aware of the power of commutativity and associativity in multiplication as a simplifying mechanism and as a means of avoiding overemphasis on pure memorization of the formulas without understanding. KEY STANDARDS Represent relationship of quantities in the form of mathematical expressions, equations, and inequalities. Solve simple problems involving functional relationships between two quantities. 6

7 THIRD 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. In third grade students, continue building understandings about shapes in the plane and in space. They identify more complex shapes and solids and are learning that there are various groups of shapes (triangles). Third graders use the attributes of shapes to classify and work with shapes and solids. They extend understanding about putting shapes together to form new shapes in the plane to solid objects. Key Concepts: Analogy should be constantly drawn between length and area. For example, a line segment having length 3 means that, compared with the segment L that has been declared to be of length 1, it can be covered exactly by 3 non-overlapping copies of L. Likewise, rectangle with sides of lengths 3 and 1 has area equal to 3 because it can be exactly covered by three non-overlapping copies of the square declared to have length 1. 7

8 KEY STANDARDS Estimate or determine the area or volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. Find the perimeter of a polygon with integer sides. Identify, describe, and classify polygons. Identify attributes of triangles. Identify attributes of quadrilaterals. Grade Level Readiness Considerations for Grade 3 All of the standards have the potential for being difficult to master if they are made too general. Only the topics mentioned explicitly in the Standards (convex) polygons throughout octagons and only regular polygons, isosceles and equilateral triangles, squares, rectangles, and a few parallelograms should be discussed. Geometry. While many of the geometric concepts for this grade level are not difficult in themselves, students typically have difficulty, becoming confused as new concepts and terms are introduced. This can be solved through a cumulative manner of introduction in which previously introduced concepts are reviewed as new concepts are introduced. Measurement. The Standards call for students to learn a significant number of measurement equivalencies. These should be introduced in a manner that does not overwhelm students with too much information at one time. Adequate practice and review should be provided so that students can readily recall all equivalencies. 8

9 3RD GRADE STATISTICS, DATA ANALYSIS, AND PROBABILITY Probability Probability is the mathematical study of uncertain events. The ability to describe events, model processes, analyze data, and predict events involves the mathematics of probability. From a basic understanding that one event can be more or less likely than another, third grade students begin to predict specific outcomes of simple events. Before students have worked with part-to-whole ratios, teachers should use languages as 1 out of 3 instead of fractional probabilities. Third grade students will learn to conduct experiments, record results, and predict future events. Probability is closely tied to collecting, organizing, and representing data. In 3rd grade, teachers should build on developing vocabulary for example, It will probably rain this afternoon or its unlikely to rain today. Students see how probability can be used to describe the likelihood of events. KEY STANDARDS Record the possible outcomes of a simple event (e.g., tossing a coin) and systematically keep track of the outcomes when the event is repeated many times. Summarize and display the results of probability experiments in a clear, organized way. 9

10 THIRD 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 third 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. This means that in addition to memorizing to automaticity multiplication facts, students should be expected to explain their reasoning as to why 4 x 5 is 20. 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