Math Strand: Number sense and numeration
Location on the Continuum:
Overview:
“Decomposing a Factor” can be described as a multiplication strategy with legs.
This strategy builds on the student’s understanding of number and leverages that knowledge to be able to work with numbers with increased flexibility and efficiency.
We often observe students using Repeated Addition or Skip Counting strategies to solve multiplication problems with smaller numbers or friendly numbers. These strategies support Additive Thinking.
3 x 7
7 + 7 + 7
=21
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25 x 6
25, 50,75,100,125,150
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When students engage with multiplication problems with larger numbers and use these strategies, we often notice that accuracy and efficiency is negatively impacted.
When a student becomes comfortable with a strategy, it is time to challenge their thinking.
Moving from Additive Thinking to Multiplicative Thinking.
We are looking for ways to move students when they may be stuck/overly reliant on these strategies. Our challenge is to help shift students from additive thinking into multiplicative thinking. Introducing and developing the “Decomposing the Factor” will help students to make the shift.
Additive Thinking
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Multiplicative Thinking
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Addition - action of joining or putting things together
Addition - one to one relationship
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Multiplication - repeated action, iterative (Repetitive) action
Multiplication - many to one relationship (or one to many)
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Decomposing a Factor (number) means to break apart the number to uncover the numbers within the factor.
7 = 5 + 2
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7 = 3 + 3 +1
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9 = 5 + 4
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9 = 3 + 3 + 3
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Supporting Students: Representing on Number Lines
A student may decompose connecting to known facts (doubles).
14 = 7+7
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22 = 11 + 11
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44 = 22 + 22
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When working with larger numbers, students may decompose the number by place value.This a specific type of decomposing along place value lines is called SPLITTING.
14 = 10 + 4
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44 = 40 + 4
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125 = 100 + 20 +5
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Supporting Students: Representing with Base Ten, Place Value Chart
When students use the Decomposing a Factor strategy to solve multiplication problems, how they choose to decompose the number will reflect how they plan to work with the numbers. The goal is for students to be increasingly flexible, strategic, and efficient as they solve basic fact problems.
Using this strategy also hinges on accurately applying the distributive property when multiplying.
Students need to trust
That 7 decomposes into a 5 & 2,
They can work with 5 and 2 separately and then add them back together to get to the solution.
Ontario Math Curriculum Glossary
Distributive Property.
The property that allows a number in a multiplication expression to be decomposed into two or more numbers;
for example, 51 x 12 = 51 x 10 + 51 x 2.
More formally, the distributive property holds that, for three numbers, a, b, and c,
a x (b + c) = (a x b) + (a x c)
and a x (b – c) = (a x b) - (a x c);
for example,
2 x (4 + 1) = 2 x 4 + 2 x 1
2 x (4 – 1) = 2 x 4 – 2 x 1.
Multiplication is said to be distributed over addition and subtraction
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Examples
Develop Decomposing Skills:
Arrays: The array model is a tool that helps students represent their math thinking.
Concretely building arrays with manipulatives, pictorially representing closed arrays on grid paper, or drawing open arrays helps students to visualize and verify the decomposed factor and how it can be used to determine the product (answer) The Key Idea of Part Part Whole is developing.
.Encourage students to “Break apart” the array using the decomposed numbers.
“The numbers within the factor”
Where to Next?
Strengthen student skills by working with intentional sequences of numbers (a.k.a. strings of numbers).
Number Talks/Strings/Sequences
Challenge students to explain what numbers they find within a factor. Students can leverage their “Friendly number” knowledge.
The Decomposing a Factor strategy is a strategy that helps students access other strategies on the continuum. (Using 10X, Using Familiar Facts,)
When students flexibly and efficiently Decompose Factors, they will have the knowledge and skills to develop and access the Using Partial Products strategy.
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