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Skills Tested: Common Core Standard(s)

 Fluently divide multi-digit numbers using the standard algorithm. (6.NS.2)

 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. (6.NS.3)


Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express ). (6.NS.4)



Vocabulary: sum, addends, difference, divisor, dividend, quotient, product, factor, algorithm, greatest common factor, least common multiple, prime number, prime factorization, distributive property,


Appendix Overview:

It is important to note that the standards addressed in this appendix are often best taught within a specific context or standard where they are used. For example, finding greatest common factors or least common multiples are often an important component in simplifying fractions, finding least common denominators, or rewriting an algebraic expression by factoring. It would be appropriate to address these standards within the unit that focuses on fractions or algebraic expressions. For this purpose we offer these standards not only as a stand- alone unit but encourage teachers to address these standards where appropriate and maintained as a subtext throughout the whole grade.


In 6 grade students consolidate the work done in previous grades on operations with whole numbers and decimals to become fluent with all four operations with these numbers. It is important to understand that in order to be fluent in these operations a student can not only implore a standard algorithm to execute an operation but they can choose methods of computation that help them arrive at their desired outcome most efficiently. This might be estimating or using mental math or they might decide to change decimals to fractions or vice versa. In addition, a student can reflect upon their answer and determine if it makes sense given the context. Throughout this chapter students will begin to see whole numbers, fractions, and decimals all as numbers that belong to the same number system, just represented in different ways. This understanding is crucial for their fluency with arithmetic operations to grow.


The chapter begins by reviewing arithmetic operation with whole numbers. In previous grades students have used models, place value charts, and properties of operations to find sums, differences, products, and quotients. They have then connected these models to the standard algorithms for all operations except division. It is likely that your students have some experience with the standard long division algorithm previous to 6 grade but they have not yet reached fluency. After students have reviewed the standard algorithms for addition, subtraction, and multiplication students will connect place value models with the standard long division algorithm. They  will draw on their knowledge of place value as they aim for fluency in dividing multi-digit numbers with the algorithm, addressing standard 6.NS.2


The next section focuses on finding the least common multiple, greatest common factor, and using these skills

to rewrite a the sum of two whole numbers using the distributive property as multiple of a sum of two whole numbers with no common factor. This prepares them for writing equivalent algebraic expressions later on in 6 grade and in future grades.

In the last section of this appendix students return to working with arithmetic operations. They apply the knowledge and skills learned in previous grades and from the first section of this appendix to add, subtract, multiply, and divide multi-digit decimals. They practice each of these operations and aim to obtain fluency not only in mastering algorithms but through estimating and using mental math.



Connections to Content:

Prior Knowledge:

In previous grades students have achieved fluency in adding, subtracting, and multiplying multi-digit whole numbers. They have founds whole-number quotients with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, modeling, and/or the relationship between multiplication and division. It is not until 6 grade that they connect these strategies to a division algorithm for multi-digit numbers and extend their work to numbers with any given number of digits. In 5 grade students add, subtract, multiply, and divided decimals to hundredths, using concrete models or drawing and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. They relate these strategies to a written method but have not yet solidified a connection to the standard algorithm. Again, it is in 6 grade that students solidify this connection and extend their knowledge to work with decimals beyond hundredths and ultimately gain fluency.


In 4 grade students study factors and multiples and the dual relationship that they hold. They have also learned how to determine whether a given whole number in the range 1-100 is prime or composite. In 3 grade students relate multiplication to the area of a rectangle and use area models to represent the distributive property.


Future Knowledge:

Fluency in the arithmetic operations for multi-digit whole numbers will help students to perform these operations with integers and rational numbers in 7 grade. Operations with decimals are particularly helpful when students learn to convert a rational number to a decimal using long division and in turn determine whether or not a number is rational. It is true that in the future many decimal operations will be performed with a calculator; however fluency will help a student reflect upon whether the answer the calculator gives them makes sense. Fluency also helps them to use strategies of mental math and estimation to obtain answers with ease and efficiency rather than always having to reach for a calculator. Also if students are fluent in changing decimals to fractions and vise versa they add to another great strategy for problem solving to their “tool box”


Using the distributive property to “factor out” a greatest common factor from a sum two whole numbers prepares students to write equivalent expressions for polynomials. This in turn lays a foundation for simplifying rational expressions with polynomials. In addition students will use a greatest common factor to simplify fractions and the least common multiple to find a least common denominator.



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