# I have 15 apples to share between four friends and myself. We each get 3. So I have 5 groups of 3 apples. Dina Mendola CCLM^2 Project Summer 2012

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1 Dina Mendola CCLM^2 Project Summer 2012 DRAFT DOCUMENT. This material was developed as part of the Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee. Grade: 3rd Grade Common Core Domain: Operations and Algebraic Thinking Common Core Cluster: Represent and solve problems involving multiplication and division Common Core Standard: 3.OA.1 Interpret products of whole numbers, e.g., interprets 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7. Explanation of 3.OA.1: Often times, instructional practice tends to jump right into the memorizing of multiplication facts without creating that vital connection to the meaning and representation of multiplication concepts. Standard 3.OA.1 is developing that real meaning of multiplication that we are actually talking about a specific amount of equal groups. More specifically they must interpret which are the factors (parts) versus the products (wholes) within a context. OR I have 15 apples to share between four friends and myself. We each get 3. So I have 5 groups of 3 apples.

2 Above are two different ways in which we can represent 3x5 as groups of. We can use an array (apple picture), which can be looked at with rows or columns. Arrays have the ability illustrate the use of the commutative property; 3 rows of 5, which represents 3x5 or 5 columns of 3, which represents 5x3; each illustrating the product of 15. The second illustration is also a visualization to portray multiplications as groups of. This is a very visual illustration that clearly demonstrates groups by circling quantities. However, we must be aware that this specifically represents 5 groups of 3. We cannot use the commutative property with this illustration as it is and state that it also represents 3 equal groups of 5. This is a misconception that is inaccurate and we must be aware of when working with students. Our class went on a fieldtrip to the zoo. During our fieldtrip, our teacher placed us into 3 equal groups with 8 students in each group. How many students are in our class? Tape diagrams (model based drawings) can be used to represent groups of as well; these diagrams would be used with Level 3 thinking. Progression of Operations and Algebraic Thinking Involving the Concept of Multiplication From 2-4 Grade: The progression from this standard extends all the way down into Kindergarten when we discuss the decomposition of numbers and begin to view them as parts and wholes. As we move into second grade, the basis for multiplication and division begin. Students are required to see objects as groups. They work with and express these solutions in groups of 2 and later as groups of 5. Illustrations using drawing and arrays begin to foster this development. As they progress into third grade, these connections are developed even further contextually. Students are required to know more than just memorizing their multiplication facts, but rather they must also have the conceptual understanding of the representation in context. As they develop further into grade 4, students are expanding on this concept to further develop their fluency and representations.

3 Textbook Development: Investigations Textbook Program Alignment and Progression with 3.0A.1 Grade 2 Grade 3 Grade 4 Standards 2.0A.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as the sum of two equal addends. 2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 3.OA.1 Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7. 4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. Location in Investigations Series U3 Sessions 3.1, 3.2, 3.3 U5 Sessions 2.2 U6 Sessions 3.1 U8 Sessions 1.1, 1.2, 1.3, 1.4 U5 Sessions 1.1, 1.2, 1.3, 1.4, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 3.1, 3.2, 3.3, 3.4, 3.6, 4.7 U1 Sessions 1.6A, 3.2, 3.3 U3 Sessions 3.1, 3.2, 3.3, 4.1, 4.3 U1 Sessions 3.2, 3.3, 3.4, 4.5, 4.7, 4.8, 4.9 U2 Sessions 1.2, 1.3, 2.4, 2.5, 2.6, 2.10a U3 Sessions 1.2, 1.6, 2.2, 2.4, 3.3, 4.1 U5 Sessions 1.1, 1.2, 1.3, 1.4

4 Activities -Partners and Teams -More Partners and Teams -Even or Odd? -Counting by 2 s -Get to 100 -Partners and Teams with Two Groups -More Partners and Teams with Two Groups -Adding Even and Odd Numbers -More Adding Even and Odd Numbers -Tens Go Fish -Strategies for Finding Combinations of 10 -More Than Two to Make 10 -Solving Story Problems -Games About Doubling -How Many Cans? -Working With Geoblocks -Sorting Geoblocks -Building Rectangles -Drawing Rectangles -How Many Rectangles -Number Strings -Number Strings -Revisiting Subtraction Story Problems -Problems With Unknown Change -Even or Odd -Groups of 2, 5, and 10 -How Many Floors? How Many Rooms? -Using Tables to Record -Comparing Tables -Understanding Tables -Many Things Come in Groups -How Many in Several Groups -Solving Multiplication Problems -Solving Problems About Our Pictures -Highlighting Multiples on 100 Charts -More Multiples -Solving Related Story Problems -Solving Related Story Problems -Patterns and Relationships -Counting Around the Class -Using Multiplication Combinations -Arranging Chairs -Investigating Arrays -Finding the Number of Squares in an Array -Array Games -Multiplicative Comparisons -Factors of the Multiples of 100 -Factors of Related Numbers -Building Multiple Towers -Multiplying Groups of 10 -Multiplying 2-Digit Numbers -Doubles and Halves -Strategies for Multiplication

5 Conclusions and Suggestions: Overall, Investigations does a good job aligning with the CCSSM 3.OA.1. It uses visuals, manipulatives, and questioning within its lessons to reinforce the concept of multiplication being represented as groups of where those groups represent a total amount, the product. The exposure of groups of being used to represent multiplication as well as applying this knowledge in a contextual setting is also sufficiently represented. There is still the caution that at times Investigations tends to make the assumption that students will make these connections along the way. We need to be aware and cautious of that, as some of our students will not automatically connect the concepts. Therefore, we need to be mindful of this and provide further opportunities and dialogue to support this standard. When looking at the lessons, we must keep in mind when this standard is being taught so that it can reinforced with further questioning and exposures to reinforce the concept thoroughly so that students will actually make these connections and carry them onto the next progression in order to fully build the understanding of multiplication representation and meaning. In addition, we must be readily aware of misconceptions that we are teaching in regards to how this is illustrated and verbalized. These forms of misconceptions are mentioned in the example above. Conceptual understanding that helps students truly know what kinds of contexts are applicable must be fully developed, as 3x5 cannot always look the same as 5x3 even though it yields the same product. By being aware of these misconceptions, creating dialogue and exposure to these methods, we will build conceptual understanding that will improve fluency. When these facts are connected with meaning, they are easier to recall and use correctly.

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