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# Multiplication and Division Part B

Strand: Number
Outcomes: 6, 7

## Step 1: Identify Outcomes to Address

### Guiding Questions

• What do I want my students to learn?
• What can my students currently understand and do?
• What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies? See Sequence of Outcomes from the Program of Studies

Strand: Number  Specific Outcomes 11. Demonstrate an understanding of multiplication to 5 × 5 by: representing and explaining multiplication using equal grouping and arrays creating and solving problems in context that involve multiplication modelling multiplication using concrete and visual representations, and recording the process symbolically relating multiplication to repeated addition relating multiplication to division. 12. Demonstrate an understanding of division (limited to division related to multiplication facts up to 5 × 5) by: representing and explaining division using equal sharing and equal grouping creating and solving problems in context that involve equal sharing and equal grouping modelling equal sharing and equal grouping using concrete and visual representations, and recording the process symbolically relating division to repeated subtraction relating division to multiplication.

 Specific Outcomes 6. Demonstrate an understanding of multiplication (2- or 3‑digit by 1-digit) to solve problems by: using personal strategies for multiplication with and without concrete materials using arrays to represent multiplication connecting concrete representations to symbolic representations estimating products applying the distributive property 7. Demonstrate an understanding of division (1-digit divisor and up to 2-digit dividend) to solve problems by: using personal strategies for dividing with and without concrete materials estimating quotients relating division to multiplication.

 Specific Outcomes 2. Use estimation strategies, including: front-end rounding compensation compatible numbers in problem-solving contexts. 5. Demonstrate, with and without concrete materials, an understanding of multiplication (2‑digit by 2‑digit) to solve problems. 6. Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

### Big Ideas

Principles and Standards for School Mathematics states that "developing [computational] fluency requires a balance and connection between conceptual understanding and computational proficiency" (NCTM 2000, p. 35). Conceptual understanding requires flexibility in thinking about the structure of numbers (base ten system), the relationship among numbers and the connections between multiplication and division. The ability to generate equivalent representations of the same number provides a foundation for using personal strategies to multiply and divide, recognizing that for some problems either operation may be used. Computational proficiency includes both efficiency and accuracy. Personal strategies must be compared and evaluated to derive methods that are efficient as well as accurate.

Understanding the operations of multiplication and division and the connections between them is crucial. John Van de Walle states, "Multiplication involves counting groups of like size and determining how many are in all (multiplicative thinking). … Division names a missing factor in terms of the known factor and the product" (2001, p. 107). He goes on to say, "Division and multiplication problems presented in pairs give you an opportunity to focus on the relationships between these operations" (2001, p. 119). For example, one problem may repeat equal quantities while another problem partitions a quantity into equal parts (Willis 2006).

Problems showing addition and subtraction are similar to problems showing multiplication and division because in the former the focus is on identifying the part or the whole while in the latter, the focus is on the number of groups, quantity in each group or the whole.

The connections between addition and multiplication can be clarified by writing an addition sentence as well as a multiplication sentence for early multiplication activities. Similarly, the connections between subtraction and division can be clarified by writing a subtraction sentence (repeated subtraction) as well as a multiplication sentence for early division activities.

By using a variety of problems, the students will construct their own meaning for the inverse relationship between multiplication and division and for the following properties:

• commutative property of multiplication (numbers can be multiplied in any order)
• identity element for multiplication (any number multiplied by one remains unchanged)
• distributive property; e.g., 4 × 65 = (4 × 60) + (4 × 5) = 240 + 20 = 260.

Van de Walle and Lovin (2006) identify four different classes of multiplicative structures as follows:

• Equal-Group Problems that can be subdivided into three groups:
• Whole Unknown (Multiplication with Groups including Rates)
• Size of Group Unknown or Equal Grouping (Partition Division)
• Number of Groups Unknown or Equal Sharing (Measurement Division)
• Multiplicative Comparison Problems that can be subdivided into three groups:
• Product Unknown (Multiplication)
• Set Size Unknown (Partition Division)
• Multiplier Unknown (Measurement Division)
• Combination Problems (Cartesian Products) that can be subdivided into two groups:
• Product Unknown
• Size of Set Unknown
• Product of Measures Problems (as in finding area).

By solving problems in contexts that relate to their own lives, the students use their prior knowledge to make sense out of the problem, estimate the answer and use computational strategies that they are able to explain and justify. The students' understanding of multiplication and division is enhanced as they develop their own methods and share them with one another, explaining why their strategies work and are efficient to use (NCTM 2000, p. 220).

The use of manipulatives or models helps the students to understand the structure of the story problem and also connects the meaning of the problem to the number sentence (Van de Walle 2001, p. 108). To develop understanding of the meaning of operations, the students connect the story problem to the manipulatives, create a number sentence and then use personal strategies to solve the problem. Van de Walle states:

"It is useful to think of models, word problems, and symbolic equations as three separate languages. Each language can be used to express the relationships involved in one of the operations. Given these three languages, a powerful approach to helping children develop operation meaning is to have them make translations from one language to an other" (2001, p. 109). 