Multiplying and Dividing Whole Numbers

Strand: Number
Outcomes: 5, 6

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

Grade 4

Grade 5

Grade 6

Specific Outcomes

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.

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
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.

Specific Outcomes
2.	Solve problems involving whole numbers and decimal numbers.
6.	Demonstrate an understanding of multiplication and division of decimals (1-digit whole number multipliers and 1-digit natural number divisors).
7.	Explain and apply the order of operations, excluding exponents, with and without technology (limited to whole numbers).

Big Ideas

To successfully solve problems using operations, students must understand what each number in the problem represents and their relation to the answer. Then students require computational fluency to carry out the appropriate operation.

Principles and Standards for School Mathematics describes the development of computational fluency:

"Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently" (NCTM 2000, p. 152).

Van de Walle and Lovin (2006) present the following big ideas related to personal, flexible, invented strategies in computation:

Flexible methods of computation involve taking apart and combining numbers in a wide variety of ways.
Invented strategies are flexible methods of computing that vary with the numbers and the situation. . . . [These strategies] must be constructed by the student.
Flexible methods for computation require a good understanding of the operations and properties of the operations, especially the commutative property and the distributive property for multiplication. How the operations are related – addition to subtraction, addition to multiplication, and multiplication to division – is also an important ingredient.

Reproduced from John A. Van de Walle, LouAnn H. Lovin, Teaching Student-Centered Mathematics: Grades 5–8, 1e (p. 37). Published by Allyn and Bacon, Boston, MA. Copyright Ó 2006 by Pearson Education. Reprinted by permission of the publisher.

Students should solve a variety of problems involving multiplication and division so they have abundant experience in realizing that numbers have different meanings, depending on the context. Numbers can represent the number of groups, the number of items in each group or the total number of items. As students make sense of the problem, they will be able to communicate the meaning of each number, perform the appropriate calculation and correctly answer the question asked in the problem. Principles and Standards for School Mathematics makes the following suggestions related to problem solving:

Modelling multiplication problems with pictures, diagrams, or concrete materials helps students learn what the factors and their product represent in various contexts (NCTM 2000, p. 151).
[Students] should learn the meaning of a remainder by modeling division problems and exploring the size of remainders given a particular divisor (NCTM 2000, p. 151).
As students develop methods to solve multidigit computation problems, they should be encouraged to record and share their methods. As they do so, they can learn from one another, analyze the efficiency and generalizability of various approaches, and try one another's methods (NCTM 2000, p. 153).

The role of the teacher is as a guide at the side, probing students' thinking with thought provoking questions, providing meaningful problem contexts and differentiating instruction to accommodate students' needs; i.e., encouraging the use of personal strategies with the aid of manipulatives, diagrams, and/or mental calculation. The goal is to develop student success in problem solving with operations as they invent strategies with understanding. Teachers must provide guidance to ensure that the strategy is:

efficient enough to be used regularly
mathematically valid
generalizable.