MULTIPLYING AND DIVIDING:
AN ALTERNATE DESTINATION
In Chapter 2, the authors discuss multiplication and division. Their main focus is on understanding the operations conceptually. Their point is that a student needs to understand the concepts behind the operations to gain operation sense.
Operation sense is gained through the use of word problems, prior to introducing any standard algorithms or requiring rote memorization of the facts. This conceptual knowledge helps a student’s algebraic reasoning and allows them to use what they already know about numbers to solve the problems.
In multiplication, a student is able to visualize equal amounts of objects into a certain amount of groups to reach an answer. It also allows students to invent their own strategies to reach the solution to the problems. The progression to multiplication begins in Kindergarten, by learning to count by 10's, 5's and 2's. This basis is furthered by learning to add doubles in First Grade. Finally in Second Grade, the student progresses to repeated addition. These foundations allow a student to reason through arrays and models of the word problems to discover their own way to solve the equation.
PROBLEM
I have decided to redecorate the classroom. How many desks will we need to have 6 rows of 5 desks?
EXAMPLE 1
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This dot array allows the students to see the equal number of desks in each group and reach the answer of 30 in a multitude of ways.
1: 5 + 5 + 5 + 5 + 5 + 5 = 30
2: Group the rows into 3 tens and repeat the addition. 10 + 10 + 10 = 30
3: Simply count by 5's to reach the answer.
These arrays can be helpful. The students are given the opportunity to group the number of objects together to discover their own way to reach the answer, prior to learning and memorizing 5 x 6 = 30. These methods and their knowledge of place value will also help them in solving multi-digit problems, as well. Students are apt to use the same strategies for multi-digit multiplication that they used in multi-digit addition to answer the problems. The ability to decompose the numbers into their respective parts and then solve the problem will strengthen their number and operation sense.
PROBLEM
There are 25 desks in our classroom at the moment. We need to have 30. How many more do we need? Also, how many rows of 6 desks can we make, once you have the required number of desks?
EXAMPLE
The children are able to solve multi-step problems by this stage, so reaching the answer of needed desks being 5 is easily reached.
The final question is to divide the total number of desks by the known number of desks in a row. Again, the students’ number and emerging operation sense will give multiple methods for solutions. Creating a model is just one way to answer the problem.
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Students, as they count to 30 and created the rows of 6 desks, can easily see that there will be 5 rows of desks. This will allow them to come to the conclusion that 30 / 6 = 5.
Students can truly learn their basic facts, when they are given the chance to invent their own strategies to solve problems. Further, by sharing their invented strategies with their classmates, they are vested in each others educations.
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