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**Teaching Multiplication (and Division) Conceptually**

Jenny C. Ray, Mathematics Specialist Kentucky Department of Education

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**Professional Learning Targets…**

1. I can describe what it means and what it looks like to teach multiplication (and division) conceptually. 2. I can describe how standards progress across grade levels, giving details for the grade span in which I teach. 3. I can match assessment/learning tasks with the appropriate Common Core Standard, and determine it’s congruency to the standard.

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**Prior Understandings— Grades K-2**

Counting 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)

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Prior Understandings 2.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 Introduction In 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 Property It 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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**Distributive Property & Area Models**

3 x 7 =__ 3 15 + 6 3 x 7 = 3 x (5 + 2) = (3 x 5) + (3 x 2)= = 21

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

Not until 4th Grade

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**Connections to other 3rd Grade Standards**

3.NBT.3: Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. 9 x 80: 80 is ten 8’s. So, if I know that 8x9 is 72, then I have ten 72’s. That equals 720. Or..80 is 8 tens. So, if 10 x 9 = 90, then I know I have 8 of those (90s) = (800-80) = 720

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**Resources are available**

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Sample Activity: Finding Factors (Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, Bay-Williams) With a partner, choose one of the following numbers: 12, 18, 24, 30, 36, or 48 Use equal sets, arrays, or number lines to find as many multiplication expressions as possible to represent your number. For each multiplication expression, write its equivalent addition expression showing your groupings.

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Sample Activity: Finding Factors from Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, Bay-Williams

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**A focus on teaching multiplication (and division) conceptually**

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

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Grade 4 Introduction In 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 dividends They 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.

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**Selected Standards… 4.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)

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Area Models 25 x 38= 950 30 + 8 150 40 20 + 5 160 600 = 950

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**Partitioning Strategies for Multiplication**

27 x x 4 4 x 20 = 4 x 7 = 108 267 x 7 7 x 200 = 7 x 60 = 7 x 8 = 56

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**Invented Strategies– 35 x 12**

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**Selected Standards… 4.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 Standards… 4.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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**A focus on teaching multiplication (and division) conceptually**

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

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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 Standards**

5.NBT.2 and 5.NBT.5 Explain 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.

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**Gr. 5 Selected Standards 5.NF.5**

Interpret 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 =(n×a)/(n×b) to the effect of multiplying a/b by 1.

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**Scaling Recognize that 3 x (25,421 + 376) is 3 times larger than**

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**Gr. 5 Selected Standards 5.NBT.6.**

There must be a clear connection between multiplication and division using manipulatives, before this understanding takes place. 5.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 models

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**Zero and Identity Properties**

Rules with no reasons? No…ask 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 property

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**Distributive Property Activity**

Slice it Up… Each 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.”

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14 x 25: An Area Model *Sketch is not drawn to scale. 80 20 50 200

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**Algebra 1: Multiplying Binomials**

*Sketch is not drawn to scale. x x 4x 20 5x x2

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**Vertical Team Activity**

As a vertical team, determine the grade level appropriate for each task given. After making a determination, look in the standards and determine which standard(s) the task is illustrating.

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**Assessing Today’s Targets… Chart Paper Responses**

I can describe what it means and what it looks like to teach multiplication (and division) conceptually. I can describe how standards progress across grade levels, giving details for the grade span in which I teach. I can match assessment/learning tasks with the appropriate Common Core Standard, and determine it’s congruency to the standard.

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**How may we continue to support you?**

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

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