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# 6-N-4-3

last edited by 2 years, 10 months ago

6.N.4.3 Multiply and divide fractions and decimals using efficient and generalizable procedures.

In a Nutshell

A solid understanding of the multiplication and division models of fractions, decimals, and mixed numbers from 6.N.4.2 is key in solving problems involving rational numbers. With this foundation students will be able to understand and use efficient and generalizable procedures such as the invert-and-multiply algorithm for dividing fractions or finding the correct placement of the decimal in the product of decimals.

## Teacher Actions

• Develop accurate and appropriate procedural fluency by exploring multiplication and division of fractions, decimals and mixed numbers in real-world and mathematical situations.

• Develop the ability to communicate mathematically with others through discussion or writing about procedures used to multiply and divide fractions, decimals and mixed numbers.

• Build procedural fluency by facilitating student exploration with multiplication and division of fractions, decimals and mixed numbers using students’ conceptual understanding gained through modeling. (See 6.N.4.2)

• Pose purposeful questions to assess a student’s understanding of the procedures used to multiply and divide fractions, decimals and mixed numbers. (Ex. Why is multiplying a number by 1/4 equivalent to dividing the same number by 4?)

## Misconceptions

• The product for real-world and mathematical problems involving multiplication of fractions, decimals and mixed numbers can be smaller than the numbers being multiplied. For example, 0.75  0.4 = 0.3.

• The quotient for real-world and mathematical problems involving division of fractions, decimals and mixed numbers can be larger than the dividend and the divisor. For example, 2 ÷ ¼ = 8.

• That a product of fractions can be found by multiplying the numerators together and then the denominators.

• How to multiply mixed numbers by changing mixed numbers into improper fractions before multiplying or using partial products.  For example, 3 2/3  2 1/4 is equivalent to (3 + 2/3)(2 + 1/4) which means the product can be found using the distributive property, (3  2) + (3  1/4) + (2/3  2) + (2/3  1/4).
• How to use different algorithms for dividing fractions and mixed numbers including the common-denominator algorithm and invert-and-multiply algorithm.

• That the placement of the decimal in the product of decimals is determined by the combined number of decimals place in the decimals being multiplied together.

• Have trouble understanding that when you multiply numbers the product can be smaller than the numbers being multiplied.

• Have trouble understanding that the quotient of two numbers can be larger than the dividend and the divisor.

• Forget that a whole number can be written as a fraction with a denominator of 1.

• Think the product of mixed numbers is determined by multiplying the whole numbers and then multiplying the fractions.

• Misapply the invert-and multiply algorithm for fraction division by inverting the first fraction instead of the second fraction or inverting both fractions.
• Determine the number of decimals places in a problem involving multiplication of decimals by counting the decimal places to the left of the decimal instead of the right.  For example, students may believe that 18.6 x 5.9 = 10.974.

• Move the decimal in both the divisor and dividend to make both numbers whole numbers when dividing a decimal by another decimal.  For example, a student may think 4.567  0.25 is equivalent to 4567  25.

OKMath Framework Introduction