Mathematical Ideas of the Focal Points - pcmac powering 的焦点——数学思想pcmac供电.doc

Thumb-Area Student Achievement ModelFinding Focus for Mathematics Instruction Grade 4Huron Intermediate School DistrictFebruary 8, 2010IntroductionWhen teachers plan instruction, they draw on many sources such as state assessment standards, local curriculum guides, textbook materials, and supplemental assessment resources. These documents serve as useful sources of information, and it is neither necessary nor desirable to replace them.Michigans Grade-Level Content Expectations (GLCEs) describe in detail many ways in which students can demonstrate their mastery of the mathematics curriculum. The GLCEs do not, however, describe the big ideas and enduring understandings that students must develop in order to achieve these expectations. The GLCEs describe products of student learning, but they do not describe the thinking that must take place within the minds of students as they learn.It is the purpose of this document to focus on the fundamental mathematical ideas that form the basis of elementary and middle school instruction. Although a variety of research materials were used in the development of this document, several sources were relied on quite heavily.In 2006, the National Council of Teachers of Mathematics (NCTM) released Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. The Focal Points describe big topics, or focus areas, for each grade level. In May, 2009, the Michigan Department of Education published the Michigan Focal Points Core and Extended Designations. In that document, the NCTM Focal Points were adjusted to align with Michigans GLCEs. The new core and extended designations for the MEAP reflect Michigans Focal Points.This document is structured around Michigans Focal Points and supporting documents, with significant content included from two other documents:Charles, Randall I. “Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics.” NCSM Journal of Mathematics Education Leadership. Spring-Summer, 2005. vol. 8, no. 1, pp. 9 24.“Chapter 4: Curricular Content.” Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education, 2008. pp. 15 25.Particular thanks go to Ruth Anne Hodges for her contributions to this project. Additional references to research are cited throughout the document.Focusing on Mathematics at Grade 4Grade 4 Focal PointsGradeMichigan Focal PointRelated GLCE TopicsTargeted VocabularyGrade 4 #1Developing fluency with multiplication of whole numbers Use factors and multiples Multiply and divide whole numbers factor, multiple prime number distributive propertyGrade 4 #2Developing an understanding of fractions and decimals, including the connections between them Read, interpret and compare decimal fractions Understand fractions decimal / decimal fraction terminating decimal place value: tenths, hundredths improper fraction, mixed numberDecimal/fraction equivalences: decimal equivalents for halves and fourths tenths and hundredths in fraction and decimal formFraction Families halves, fourths, eighths thirds, sixths, twelfthsNational Math Panel Benchmarks for Grades 3, 4, and 5By the end of Grade 3, students should By the end of Grade 4, students should By the end of Grade 5, students should be proficient with the addition and subtraction of whole numbers.be proficient with multiplication and division of whole numbers.be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals.be proficient with comparing fractions and decimals and common percents, and with the addition and subtraction of fractions and decimals.be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids).The National Mathematics Advisory Panel Final Report, 2008, p. 20Grade 4 Focal Point #1:Developing fluency with multiplication of whole numbersGrade 3Grade 4Grade 5National Math Panel Benchmark:By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbersRepresent whole-number problems with objects, words, and mathematical statements and solveSolve measurement problemsDeveloping understandings of multiplication and division and strategies for basic multiplication facts and related division facts (NCTM-3rd) Count in steps, and understand even and odd numbersMultiply and divide whole numbersRepresent whole-number problems with objects, words, and mathematical statements and solveDeveloping fluency with multiplication of whole numbers (NCTM-4th) Use factors and multiplesMultiply and divide whole numbersDeveloping an understanding of and fluency with division of whole numbers (NCTM-5th)National Math Panel Benchmark:By the end of Grade 5, students should be proficient with the multiplication and division of whole numbersNational Math Panel Benchmark:By the end of Grade 5, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids)Understand division of whole numbersFind prime factorizations of whole numbersMultiply and divide whole numbersMultiply and divide by powers of tenFind and interpret mean and mode for a given set of dataKnow, and convert, among measurement units within a given systemNational Math Panel Benchmark:By the end of Grade 6, students should be proficient with multiplication and division of fractions and decimalsNational Math Panel Benchmark: By the end of Grade 6, students should be proficient with all operations involving positive and negative integersKey:bold, non-italic = Michigan Curriculum Focal Pointsnon-bold, non-italic = GLCE topics associated with that focal pointnon-bold, italic = Cross-over GLCE topics associated with another focal pointGrade 4 Focal Point #1:Developing fluency with multiplication of whole numbersBIG MATHEMATICAL IDEAS AND UNDERSTANDINGSBig Idea #2 (The Base Ten Numeration System)The base ten numeration system is a scheme for recording numbers using digits 0-9, groups of ten, and place value. Digits in each place are worth ten times as much as digits in the place to the right and a tenth as much as digits to the left.Big Idea #5 (Operation Meanings & Relationships)The same number sentence . . . e.g. 12 4 = 3 can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation. The real-world actions for multiplication and division of whole numbers are the same for operations with fractions, decimals, and integers. Any division calculation can be solved by using multiplication, and multiplication can be used to check division. Any two numbers can be multiplied or divided. The product or quotient has a new unit. o 12 cookies 3 people = 4 cookies per persono 4 pages per person x 6 people = 24 total pages The same division problem can be represented with different models, depending on what information is known. o Partitive division: 12 cookies are shared equally among 3 children. How many cookies does each child receive? (12 3 = 4)o Measurement division: 12 cookies are bundled into groups of 3. How many bundles can be made? (12 3 = 4)o Product-factor division: There are 12 cookies on a tray today. That is 3 times the number of cookies on the tray yesterday. How many cookies were on the tray yesterday? (12 3 asks the same question as “3 x what = 12?”) The standard multiplication algorithm works because of the distributive property and powers of 10.o 32 x 23 = (2 x 23) + (30 x 23) = (2 x 23) + (3 x 23 x 10) = 46 + 690 Big Idea #6 (Properties)For a given set of numbers there are relationships that are always true, and these are the rules that govern arithmetic and algebra. One way to represent multiplication is as the area of a rectangle. Because the area does not change when the rectangle is rotated, multiplication is commutative.o 3 x 5 = 5 x 3 = 15 Division is not commutative, but each division fact has a related division fact:o 12 4 4 12, but 12 4 = 3, and 12 3 = 4. The distributive property (of multiplication over addition) relates addition and multiplication. The distributive property can be shown numerically and visually, using arrays and area models (CCSI Common Core Standards Initiative DRAFT 1/13/2010).Big Idea #7 (Basic Facts & Algorithms)Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. Any unknown multiplication fact can be found by using known facts.o 6 x 8 is equivalent to 5 x 8 plus one more 8. (6 x 8 = 40 + 8)o 9 x 3 is 3 less than 10 x 3. (9 x 3 = 30 3)o I dont know how much 8 groups of 2 is, but I know that 2 groups of 8 is 16. Since multiplication is commutative, 8 x 2 = 16.o 3 x 5 x 2 = 5 x 2 x 3 = 10 x 3 = 30 (associative property applies because multiplication is commutative) Multiplication and division are inverses of one another and can be used to check each other. Any multiplication problem has related division problems, and any division problem has related multiplication problems. o 6 x 3 = 18 AND 3 x 6 = 18 AND 18 6 = 3 AND 18 3 = 6Grade 4 Focal Point #1:Developing fluency with multiplication of whole numbersINSTRUCTIONAL IMPLICATIONSMODELS OF MULTIPLICATIONMultiplication can be represented as repeated addition or as the area of a rectangle. It is important that students understand and be able to use both models, but the area model has particular applications in Grade 4, where students are expected to use the area formulas for squares and rectangles.Area models can represent whole numbers, fractions, decimals, or polynomials. Area models can be represented visually, such as with grid paper, or tactilely, using tools such as Base 10 blocks or Algebra Tiles.Using area models at a variety of levels 2 x 3 = 66 2 = 33 x 2 = 66 3 = 21/41/3 x 1/4 = 1/1221 x 12 = 200 + 10 + 40 + 2 = 252(2x + 1) (x + 2) = 2x2 + 5x + 2For more information about how to use the distribute property to explain the standard multiplication algorithm (lower left, above), see the Michigan Mathematics Program Improvement Project (MMPI) (, Chapter 4, pp. 9-10)Assessing the Meanings of MultiplicationOne way to assess if students understand the meaning of multiplication is to ask students to draw pictures to represent story problems, making sure to include both repeated addition and rectangular area problems. The diagnostic assessments from the Michigan Mathematics Program Improvement Project (MMPI) contain good examples (). COMMUTATIVITY OF MULTIPLICATIONLike addition, multiplication is commutative, which means that multiplication facts can be turned around: 3 x 4 = 4 x 3. One way to illustrate this for students is by using the area or array model of multiplication. Since the area of a shape does not change when the shape is turned or moved, a rectangle built with three rows of four tiles each has the same area as a rectangle built with four rows of three tiles each.3 x 4 = 124 x 3 = 12DEVELOPING FLUENCY WITH MULTIPLICATION AND DIVISION FACTSWhen learning basic facts, students should first understand the meanings of multiplication and division and represent with models. For exampleShow five groups of six using a picture and a number sentence. If 36 candies are shared equally among four children, how many candies does each child receive? Draw a picture and write a number sentence.When a student understands what multiplication means, he can begin to develop strategies for finding unknown facts based on known facts. Here are some examples: Identify facts related to special numbers like 1 or 0. For example, use pictures and models to help students understand that one group of any number is that number: 1 x 3 = 3; 1 x 4 = 4; 1 x 92 = 92; etc. and that zero groups is always 0: 0 x 5 = 0; 0 x 1,965 = 0; etc. Use the commutative property of multiplication. When faced with “5 x 1,” a student should think “5 x 1 is the same as 1 x 5, and one group of five is five.” Use fact families. When over exposed to flash cards or math fact sheets, students might see 3 x 4, 4 x 3, 12 4, and 12 3 as four unrelated problems rather than a single fact family. This can also be taught as related facts: When asked “What is 12 3 ?” a student might think “what times three equals 12?” Students often find doubles the easiest to learn, both in addition and subtraction. Explicit discussion of the relationship between adding a number to itself and multiplying that number by two can help students to learn the multiples of two. Multiplying a number by three is the same as doubling the number then adding the number again. Multiplying a number by four is the same as doubling the double. To multiply by five, multiply by 10, then cut it in half. Multiplying a number by six is the same as multiplying by five then adding the number again. Many students also find the square products (2 x 2 = 4; 3 x 3 = 9; 4 x 4 = 16; etc.) easy to learn.Students will find some strategies easier than others, but regardless of the strategies used, the most successful students are those who use numeric reasoning to find unknown facts. Students should not rely solely on inefficient strategies such as repeated addition.When a student thoroughly understands the meanings of multiplication and division and is able to use a variety of strategies to determine unknown facts, he is ready for repeated practice to develop fluency. Practice should be distributed over time in short chunks. The time allotted for a given set of problems should be long enough to discourage wild guessing but not long enough to allow the student to resort to inefficient means. MULTIPLICATION, DIVISION, AND FRACTIONSMultiplication, division, and fractions all depend on equal-sized groups. Many student misconceptions can be traced back to this critical concept.Before becoming fluent with “naked” numbers, it is important that students understand the meanings of multiplication, division, and fractions through context. Consider this basic situation: A total number of objects (t) are shared equally among a number of groups (g); each group has the same number of objects (n).The operation this situation represents depends on the information that is known or unknown:KnownUnknownOperation number of groups number of objects in each grouptotal number of objects (t)Multiplicationn x g = t; g x n = t total number of objects number of groupsnumber of objects in each group (g)Division (partitive / fair shares)t n = g total number of objects number of objects in each groupnumber of groups (n)Division (quota or measurement)t g = n total number of objects number of objects in each group OR number of groupsnumber of objects in each group OR number of groupsDivision (product-factor)n x g = t; g x n = tFor examples of each problem type, see “Big Mathematical Ideas” previously in this section. While it is not critical that students be able to name the different types of division problem, they need to be able to recognize them in context. A variety of activities with concrete manipulatives and pictorial representations is essential to building strong mental images. Students will later be able to draw on these images when faced with abstract problems such as this one:Write a story problem for 1 .Many teachers struggle with this problem, writing scenarios that actually represent 1 2 or 1 x 2. For many people, this problem becomes meaningful in the context of quota division:I am making bows to decorate packages. I have 1 yards of ribbon. Each bow needs yard of ribbon. How many bows can I make? Unfortunately, quota or measurement situations like the one above are not presented nearly as often as partitive or fair share situations. As a result, many students have a limited understanding of division. For concrete manipulative activities designed to help students deepen their understanding, see the Michigan Mathematics Program Improvement Project (MMPI) at (Chapter 4, Pages 5-6). To complete the cycle of concrete manipulatives, pictorial representations, and abstract computation, consider having students engage in the following types of activities: given a story problem, draw a picture given a number sentence, write a story problem given a picture, write a number sentence When students can express equivalent problems in different ways as shown below, they truly understand the operation.story problempicturenumber sentenceThe MMPI Diagnostic Inventories include story problems, pictures, and number sentences that can be used to assess students understanding of multiplication and division and pinpoint instructional needs: (Chapter 4).Grade 4 Focal Point #1:Developing fluency wit