Georgia Department of Education

Georgia Performance Standards of Excellence

Framework Student Edition

Sixth Grade Mathematics

** ****Unit 1 **

** Number System Fluency**** **

**TABLE OF CONTENTS **

Overview

Key Standards

Enduring Understandings

Concepts & Skills to Maintain

Selected Terms and Symbols

**OVERVIEW**

In this unit students will:

• Find the greatest common factor of two whole numbers less than or equal to 100

• Find 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.

• Interpret and compute quotients of fractions

• Solve word problems involving division of fractions by fractions using visual fraction models and equations to represent the problem.

• Fluently divide multi-digit numbers using the standard algorithm

• Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

At each grade level in the standards, one or two fluencies are expected. For sixth graders the expected fluencies are multi-digit whole number division and multi-digit decimal operations. Procedural fluency is defined by the Common Core as “skill in carrying out procedures flexibly, accurately, efficiently and appropriately”. Students may not achieve fluency within the scope of one unit but it is expected the fluency will be obtained by the conclusion of the course.

In the past fraction and decimal computation have been dominated by rules but research based best practices have proven that students who are taught to focus on the pencil-and-paper rules for decimal computation do not even consider the actual values of the numbers. Therefore a good place to begin decimal computation is with estimation. It helps children to look at answers in terms of a reasonable range.

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight practice standards should be addressed constantly as well. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement this unit. This unit provides much needed content information, but excellent learning activities as well. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources.

**STANDARDS ADDRESSED IN THIS UNIT **

Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics especially with respect to fluency.

Apply and extend previous understandings of multiplication and division to divide fractions by

fractions.

**MGSE6.NS.1**:

INTERPRET and COMPUTE quotients of fractions, and SOLVE word problems involving division of fractions by fractions, INCLUDING reasoning strategies such as USING visual fraction models and equations to represent the problem.

**MGSE6.NS.2**:

FLUENTLY DIVIDE multi-digit numbers USING the standard algorithm.

**MGSE6.NS.3**:

FLUENTLY ADD, SUBTRACT, MULTIPLY, and DIVIDE multi-digit decimals USING the standard algorithm for each operation.

### Supporting Standards

Apply and extend previous understandings of multiplication and division to divide

fractions by fractions.

**MGSE6.NS.4**

Find the common multiples of two whole numbers less than or equal to 12 and common factors of two whole numbers less than or equal to 100.

### Standards for Mathematical Practices

- Make sense of problems and persevere in solving them. (Daily)
- Reason abstractly and quantitatively. (Daily)
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision. (Daily)
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

ENDURING UNDERSTANDINGS

•The meanings of each operation on fractions are consistent with the meanings of the operations on whole numbers.

•When diving by a fraction, there are two ways of thinking about the operation – partition and measurement which will lead to two different thought processes

for division.

•When we divide one number by another, we may get a quotient that is bigger than the original number, smaller than the original number or equal to the original

number.

**CONCEPTS & SKILLS TO MAINTAIN**

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

•number sense

•computation with multi-digit whole numbers and decimals (to hundredths), including application of order of operations

•addition, subtraction, multiplication, and division of common fractions

•familiarity with factors and multiples

• data usage and representations

**SELECTED TERMS AND SYMBOLS **

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

The websites below are interactive and include a math glossary suitable for middle school children.

Note – At the middle school level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks.

http://www.amathsdictionaryforkids.com/

This web site has activities to help students more fully understand and retain new vocabulary

http://intermath.coe.uga.edu/dictnary/homepg.asp

Definitions and activities for these and other terms can be found on the Intermath website. Intermath is geared towards middle and high school students.

**• Algorithm:** a step-by-step solution to a problem.

Decimal: A number system based on 10.

**• Difference:** The amount left after one number is subtracted from another number.

**• Distributive Property: **The sum of two addends multiplied by a number is the sum of the product of each addend and the number.

**• Dividend:** A number that is divided by another number.

**• Divisor:** A number by which another number is to be divided.

Equation: A mathematical statement containing an equals sign, to show that two expressions are equal.

**• Factor: **When two or more integers are multiplied, each number is a factor of the product. "To factor" means to write the number or term as a product of its

factors.

Fraction: Any part of a group, number or whole.

**• Greatest Common Factor: **The largest factor that two or more numbers have in common.

**Least Common Multiple:** The smallest multiple (other than zero) that two or more numbers have in common.

**• Minuend:** The number that is to be subtracted from.

**• Multiple:** The product of a given whole number and an integer.

**• Quotient:** A number that is the result of division.

**• Reciprocal: **Two numbers whose product is 1.

**• Sum: **The number you get by adding two or more numbers together

**• Subtrahend:** The number that is to be subtracted.

**• Product:** A number that is the result of multiplication.