Teaching Multiplication(and Division) ConceptuallyJenny C. Ray, Mathematics SpecialistKentucky Department of EducationDiane Culbertson, Mathematics ConsultantNorthern Kentucky Cooperative for Educational Services

Todays Targets1. I can describe what it means and what it looks like to teach multiplication (and division) conceptually for my grade level.2. I can explain how and when I will implement a formative assessment lesson for my grade level.3. I can describe at least one component of a formative assessment lesson that can be integrated into everyday teaching.www.JennyRay.net*

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Prior UnderstandingsGrades K-2Counting numbers in a set (K)Counting by tens (K)Understanding the numbers 10, 20, 30, 40, , 90 refer to one, two, three, four, , nine tens (1)Counting by fives (2)www.JennyRay.net*

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Prior Understandings2.G.2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

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Grade 3 IntroductionIn Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

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Commutative PropertyIt is not intuitively obvious that 3 x 8 = 8 x 3. A picture of 3 sets of 8 objects cannot immediately be seen as 8 piles of 3 objects. Eight hops of 3 land at 24, but it is not clear that 3 hops of 8 will land at 24.

The array, however, can be quite powerful in illustrating the commutative property.

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Grade 3Represent and solve problems involving multiplication and division (Glossary-Table 2)

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Grade 33.NBT.3: Multiply one-digit whole numbers by multiples of 10 in the range 1090 (e.g., 9 80, 5 60) using strategies based on place value and properties of operations.

9 x 80: 80 is ten 8s. So, if I know that 8x9 is 72, then I have ten 72s. That equals 720.

Or..80 is 8 tens. So, if 10 x 9 = 90, then I know I have 8 of those (90s). 90 + 90 + 90 + 90 + 90 + 90 + 90 + 90 = (800-80) = 720www.JennyRay.net*

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Grade 3

3.MD.7. Relate area to the operations of multiplication and addition. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a b and a c. Use area models to represent the distributive property in mathematical reasoning.

Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

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Sample Activity: Finding Factors from Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, Bay-WilliamsStart by assigning a number that has several factors12, 18, 24, 30, 36.. Have students find as many multiplication expressions for their assigned number as possible, using equal sets, arrays (square tiles, cubes, or grid paper), and number lines. For each, both an addition and a multiplication equation should be written.

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Grade 4A focus on teaching multiplication (and division) conceptually

Grade 4 IntroductionIn Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividendsThey apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context.www.JennyRay.net*

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Selected Standards4.NBT.5.Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

(Area models for this standard are directly linked to the understanding of partitioning a rectangle into equal parts and 3.MD.7c)www.JennyRay.net*

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Partitioning Strategies for Multiplication27 x 4 27 x 4 4 x 20 = 80 27 + 27 + 27 + 274 x 7 = 28 108 54 54 108----------------------------------- 267 x 77 x 200 = 1400 1820 7 x 60 = 420 18767 x 8 = 56www.JennyRay.net*

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Invented Strategies 35 x 12www.JennyRay.net*

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Selected Standards4.NBT.6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

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Selected Standards4.OA.3.Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

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Formative Assessment Lesson4th Grade FAL Pre-AssessmentDo not grade the pre-assessmentOrganize according to common misconceptionsMake a list of questions based on those misconceptionsPair students who have the same or similar misconceptions for the collaborative activity the next day

Framing the lesson for the Collaborative ActivityExplain the importance of working together so that both partners understand and can explain why cards are matched

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Formative Assessment LessonCollaborative Activity

Formative Assessment LessonDuring the Collaborative Activity the teacher should circulate and make notes of groups progressions and where groups are when time is up (after one or more groups have completed the final layer of the card sort)The notes during the collaboration time should also include which group should present a match first for the class, using the document camera, for example.Full Class Discussion After the Collaborative ActivityEngineer effective classroom discussion by ordering the way students using the notes from the collaboration time. Post-Assessment (2nd try at the Pre-Assessment)Students should work alone and consider the questions that the teacher posed to move their thinking forward.The importance is in seeing GROWTH in what the student has learned.www.JennyRay.net*

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Formative Assessment Components:1. Clarifying, sharing, and understanding goals for learning and criteria for success with learners.2. Engineering effective classroom discussions, questions, activities, and tasks that elicit evidence of students learning3. Providing feedback that moves learners forward.4. Activating students as owners of their own learning.5. Activating students as learning resources for one another.

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2 Types of Formative Assessment LessonsConcept-Focused LessonsPre-AssessmentFrame the lessonCollaborative Pairs with same/similar misconceptionsWhole class discussionFeedback QuestionsPost-Assessment to determine individual student growthProblem-Solving LessonsIndividual plan for solving the problemCollaborative Pairs with different solution methodsAnalysis of sample student workWhole class discussionFinal plan for solving the problem, better than the original individual planswww.JennyRay.net*

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2 Types of Formative Assessment LessonsConcept-Focused LessonsOccurs about 2/3 of the way through a unitFocuses on specific cluster of standards in the unit; makes connectionsSecondary focus on the 8 Standards for Mathematical PracticeIndividual growth after working/learning together as a team.Problem-Solving LessonsCan occur any time in a unit or after/before a unit.Many entry points or solution methodsFocuses on the 8 Standards for Mathematical PracticeCollaborative solution process is better than original, individual planwww.JennyRay.net*

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Grade 5A focus on teaching multiplication (and division) conceptually

Grade 5 Introduction(2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations.They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

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Grade 5 Selected Standards5.NBT.2 and 5.NBT.5Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Fluently multiply multi-digit whole numbers using the standard algorithm.*www.JennyRay.net

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Gr. 5 Selected Standards5.NF.5Interpret multiplication as scaling (resizing), by:a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b =(na)/(nb) to the effect of multiplying a/b by 1.*www.JennyRay.net

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Gr. 5 Selected Standards5.NBT.6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area modelswww.JennyRay.net*There must be a clear connection between multiplication and division using manipulatives, before this understanding takes place.

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Zero and Identity PropertiesRules with no reasons?Noask students to reason.Ex: How many grams of fat are there in 7 servings of celery? Celery has 0 grams of fat.Ex: Note that on a number line, 5 hops of 0 land at 0. Also, 0 hops of 20 also stays at 0.

Arrays with factors of 1 are also worth investigation to determine the identity propertywww.JennyRay.net*

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Distributive Property ActivitySlice it UpEach pair, please use the grid paper to make a rectangle that has a total area greater than 10 square units. Make a slice through the rectangle and write an equation that matches using the lengths and widths of the smaller rectangles created.Continue this process until you have found all the ways to slice it up.www.JennyRay.net*

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Assessing Todays TargetsChart Paper Responses1. I can describe what it means and what it looks like to teach multiplication (and division) conceptually for my grade level.2. I can explain how and when I will implement a formative assessment lesson for my grade level.3. I can describe at least one component of a formative assessment lesson that can be integrated into everyday teaching.www.JennyRay.net*

www.JennyRay.net

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How may we continue to support you?

Jenny.ray@education.ky.govDiane.culbertson@nkces.edu

NOTE: Information was obtained from the following sources:Common Core State Standards for MathematicsElementary and Middle School Mathematics (Van de Walle, Karp, Bay-Williams)

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