Calculating Products Using Decomposition and Division
Calculating Products Using Decomposition and Division is a Grade 5 math skill from Illustrative Mathematics Chapter 2 (Fractions as Quotients and Fraction Multiplication) that combines decomposition and division to multiply a fraction by a whole number: (a/b) × c = a × (c ÷ b). Students first find the value of one unit fractional part by dividing the whole number by the denominator, then multiply by the numerator for the total value.
Key Concepts
To multiply a fraction by a whole number, you can decompose the fraction, convert the unit fraction multiplication to division, and then multiply by the numerator. $$\frac{a}{b} \times c = a \times \left(\frac{1}{b} \times c\right) = a \times (c \div b)$$.
Common Questions
How do you calculate a fraction times a whole number using decomposition and division?
Decompose the fraction as a × (1/b). Since multiplying by 1/b equals dividing by b, compute (c ÷ b) for the unit fraction portion, then multiply by the numerator a. For example, (3/4) × 8 = 3 × (8 ÷ 4) = 3 × 2 = 6.
What does decomposing a fraction for multiplication mean?
Decomposing means rewriting the fraction as numerator × unit fraction: (a/b) = a × (1/b). This lets you replace the unit fraction multiplication with an equivalent division step, often making the calculation simpler.
What chapter covers decomposition and division for fraction multiplication in Illustrative Mathematics Grade 5?
Calculating products using decomposition and division is covered in Chapter 2 of Illustrative Mathematics Grade 5, titled Fractions as Quotients and Fraction Multiplication.
What is an example of using decomposition to multiply fractions?
To calculate (2/5) × 3: decompose as 2 × (1/5) × 3 = 2 × (3 ÷ 5) = 2 × 3/5 = 6/5. To calculate (3/4) × 8: decompose as 3 × (1/4) × 8 = 3 × (8 ÷ 4) = 3 × 2 = 6.
When does the decomposition method work especially well for fraction multiplication?
It works best when the whole number is divisible by the denominator, resulting in a whole number from the division step. This simplifies the multiplication to whole numbers only.