Sixth Grade

 

Number and Operations

 

Standard 6-2: The student will demonstrate through the mathematical processes an understanding of the concepts of whole-number percentages, integers, and ratio and rate; the addition and subtraction of fractions; accurate, efficient, and generalizable methods of multiplying and dividing fractions and decimals; and the use of exponential notation to represent whole numbers.

 

The indicators for this standard are grouped by the following major concepts:

                      

The indicators that support each of those major concepts and an explanation of the essential learning for each major concept follows.

 

 

Number Structure and Relationships – Rational Numbers

 

Indicators

6-2.1   Understand whole-number percentages through 100.

6-2.2   Understand integers.

6-2.3       Compare rational numbers and whole-number percentages through 100 by using the symbols ≤, ≥, <, >, and =.

6-2.9   Represent whole numbers in exponential form.

 

In fifth grade, students worked to analyze and compare whole numbers, decimals and fractions. In sixth grade, students enhance their number sense by working with rational numbers which is the first time students are introduced to integer numbers less than zero. As a result, by the end of sixth grade, students should be able to compare rational numbers including whole-number percentages through 100, have a contextual base of the meaning and representation of integers, and represent whole numbers in exponential form.

Also, sixth grade is the first time students are introduced to the concept of percents. Students will begin their study of whole number percentages through 100. Percent literally means per hundred. Percentages can be represented as a fraction with a denominator of 100, as a decimal, or as a whole number followed by the % symbol. Students will extend their learning from fifth grade to compare whole-number percentages through 100 as well as rational numbers by using the symbols ≤, ≥, <, >, and =. Student have previously used <, >, and =. Students will need to understand the differences between <, > and ≤, ≥. By providing experiences for students, they will see the symbols are similar but different. Students should discuss the symbolic comparison as well as the relative magnitude of the numbers being compared.

Students in sixth grade should have strong number sense with respect to whole numbers, fractions, and decimals. Now, they will begin to develop number sense for the set of integers. Students will develop a meaning of integers. Students’ knowledge of number is extended to numbers less than zero which are best represented by real world situations. Students should identify situations where numbers less than zero are used. At this stage, student exposure to numbers less than zero is limited to the set of integers. Students are not expected to perform operations on integers at this stage of their mathematical development. Students should be able to describe numbers less than zero using real world models to aid in their development of understanding integers. Students will generate strategies to compute integers in seventh grade and apply algorithms for computation in eighth grade. Because the computation of integers is abstract, students need to develop an understanding of integers in context so that they can easily connect that concrete understanding to the abstract process of computation. Students need to see that we use integers in math every day.

The concept of integers should be introduced to children using familiar models. Students may find it helpful to represent integers on a number line, with two color counters, or with everyday tools such as a thermometer. Because of the increased use of the metric system, sixth grade students may have already been informally introduced to the concept of negative numbers as they relate to temperature. Situations that can be visualized and make sense are best for introducing the concept of integers. Other models or situations that may be used to introduce this concept are distance, altitude, balances of money (quiz shows), and sports events. Horizontal number lines are useful when comparing the magnitude of numbers. Numbers to the right of zero are positive; numbers to the left of zero are negative. Vertical number lines can be used by asking students to visualize a thermometer. Students can see that numbers above zero are positive as numbers below zero are negative.

After students have experienced integers, they can begin to study rational numbers. For students to compare rational numbers, they will first need to understand what a rational number is - any number that can be written as a fraction, a ratio of two integers , where b is never zero. Students are continuing to expand in their knowledge of the number systems we use everyday. The foundation is being formed for students to continue their work with number systems in the seventh grade. 

Through fifth grade students have worked with numbers in word and standard form. The expectation now will be for sixth grade students to represent whole numbers in exponential form. Students should experience this shorthand method for writing numbers expressed as repeated multiplication such as 64 = 4 x 4 x 4 = 43. In this way students should be able to convert whole numbers into their exponential form.  Students should be able to convert whole numbers into their exponential form.

 

Connections to:

 

Technology

Several lessons used to teach exponential form and interactive practice provided.

http://www.studyzone.org/mtestprep/topic6.cfm?TopicID=219

 

 

Operations - Addition and Subtraction

 

Indicator

6-2.4 Apply an algorithm to add and subtract fractions.

 

In fourth grade, students had experience writing equivalent fractions and representing improper fractions and mixed numbers. Students were given sufficient experiences with concrete and pictorial models to fully grasp these concepts which are extremely important prerequisites to adding and subtracting fractions. In fifth grade, students generated strategies to add and subtract fractions. As a result of sharing those generated strategies, students developed an understanding of the concepts. In sixth grade the emphasis is on applying an algorithm. As a result, by the end of sixth grade students should exhibit fluency when solving a wide range of addition and subtraction problems involving fractions. 

Sixth grade is the first time students are required to perform addition and subtraction of fractions symbolically (numbers only). As a result, children need experiences that will enable them to make the link between those concrete and pictorial models used in fifth grade and the new symbolic operations. Students often have difficulty with the algorithm to add and subtract fractions and will forget that a common denominator is a necessity. Students should work with fractions in all forms including mixed fractions. Students should also be given opportunities to apply addition and subtraction of fractions in context – not merely perform the operation for the sake of adding and subtracting fractions. That means a problem situation should be introduced and students should explore possible solution strategies. Students should then share their strategies with the whole class as the teacher facilitates. The students’ understanding of computation increases when they develop their own methods and discuss those methods with others.  Finally, when the traditional algorithm is shown, some students will begin to use it while others will revert back to an invented algorithm until they are comfortable enough to move on.

          Sixth grade students should build on their prior experiences with fraction models, benchmark fractions, and fraction equivalencies when they begin to estimate sums and differences of fractions. Emphasis should be placed on students explaining the method they used to estimate.  Estimation should be used as a tool to check the reasonableness of answers to contextual problems. 

 

Connections to:

 

Other Sixth Grade Indicators

6-3.3       Represent algebraic relationships with variables in expressions, simple equations, and simple inequalities.

6-3.4       Use the commutative, associative and distributed properties to show that two expressions are equivalent.

6-3.2       Apply order of operations to simplify whole-number expressions.

6-3.5       Use inverse operations to solve one-step equations that have whole-number solutions and variables with whole-number coefficients.

 

For an explanation of essential learning for these above related Indicators see the Algebra strand.

 

 

 

Operations - Multiplication and Division

 

Indicators

6-2.5   Generate strategies to multiply and divide fractions and decimals.

6-2.6   Understand the relationship between ratio/rate and multiplication/division.

6-2.7       Apply strategies and procedures to determine values of powers of 10, up to

           106.

6-2.8   Represent the prime factorization of numbers by using exponents.

 

           Sixth grade is the first time students have been introduced to the concept of multiplying and dividing fractions and decimals with the emphasis on generating strategies. Students in the sixth grade should not multiply or divide fractions and decimals symbolically. It is essential that students develop an understanding of the concepts of multiplication and division of fractions and decimals by sharing their generated strategies. Students are encouraged to explore and discover various methods. In seventh grade students will apply an algorithm for multiplication and division of fractions and decimals. In order to do so, students need opportunities to investigate contextual problems without first being shown an algorithm. That means a problem situation should be introduced and students should explore possible solution strategies. Students should then share their strategies with the whole class as the teacher facilitates. The students’ understanding of computation increases when they develop their own methods and discuss those methods with others. Finally in seventh grade, when the expectation is to apply an algorithm, some students will begin to use the traditional algorithms while others will revert back to an invented algorithm until they are comfortable enough to move on.

“Students use simple reasoning about multiplication and division to solve ratio and rate problems (e.g., ‘If 5 items cost $3.75 and all items are the same price, then I can find the cost of 12 items by first dividing $3.75 by 5 to find our how much one item costs and then multiplying the cost of a single item by 12’). By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative sizes of quantities, students extend whole number multiplication and division to ratios and rates. Thus, they expand the repertoire of problems that they can solve by using multiplication and division, and they build on their understanding of fractions to understand ratios. Students solve a wide variety of problems involving ratios and rates.” (NCTM Curriculum Focal Points, 2006, p.18) Another example to show this relationship would be the formula distance = rate x time. When given the distance and time, students will divide distance by time to find the rate.

By the end of the elementary years, students should be comfortable with place value structure and should be able to work with large numbers. Elementary experiences include patterns of tens and computing multiplication problems that involve multiples of ten. Sixth grade is the first time students will be introduced to the concept of applying strategies and procedures to determine values of powers of 10, up to 106 with a goal of student fluency. Students work to represent multiples of ten in exponential form as an extension of the understanding of place value in both standard and expanded form. Tying together the place value experiences from elementary school with the new knowledge of powers of ten from sixth grade will enable students to gain confidence and represent the same number in several different ways: word form, standard form, numerals and word form, expanded form, and exponential form.

           In fifth grade, students classified numbers as prime, composite, or neither. In sixth grade, instruction should deepen students’ understanding of these characteristics of whole numbers. Students will write a composite number as the product of prime numbers. With an understanding of exponential notation, students will rewrite the product so that a base is not repeated in the final prime factorization. Understanding how to write prime factorization will help students simplify fractions and find common denominators; it can also help students find the least common multiple and greatest common factor of numbers. Common multiples and factors can help students add, subtract, multiply, and divide fractions, in addition to solving everyday problems that do not involve fractions.

 

Representations, Properties, and Proportional Reasoning

 

6-3.3   Represent algebraic relationships with variables in expressions, simple

           equations, and simple inequalities.

6-3.4   Use the commutative, associative, and distributive properties to show that

            two expressions are equivalent.

 

Solve Mathematical Situations

 

6-3.2   Apply order of operations to simplify whole-number expressions.

6-3.5     Use inverse operations to solve one-step equations that have whole-number

            solutions and variables with whole-number coefficients

 

Teacher Notes:

Remediation Strategies

“Part-Part-Whole” (PPW) is an effective strategy for solving addition and subtraction problems. Students may have been exposed to this approach in previous grades. Therefore using refocusing questions that help students identify the “Parts” versus the “Whole” may be sufficient to move the student forward. For students who have not used or are not comfortable with a PPW strategy, more assistance may be needed.

 

There are many variations to the PPW strategy – “Join”, “Separate”, “Compare” to name a few. However, for a student that is experiencing difficulty, using multiple variations may not be the best approach. In the PPW strategy, the parts equal the whole. All addition and subtraction type problems are approached by asking what the parts are and what the whole is. Identification of the parts and the whole is where a student may experience the most difficulty. If that is the case, substituting his/her final answer in the PPW “formula” will enable the student to quickly see their error. In the following examples, addition or subtraction may be used to solve the problem depending on what is unknown. But in all examples, a simple PPW strategy was used. Some examples:

 

1.                           Cameron and Kellen are eating one pizza. Cameron ate 2/3 of the pizza. How much pizza is left for Kellen to eat?

 

Part = 2/3   Part = unknown     Whole = 1 = 3/3  

The student knows that the two parts equal the whole. Therefore, to find the unknown simply subtract the known part from the whole.

 

2.                           Cameron ran 1½ miles on Monday. He ran 4/5 of a mile on Tuesday. What is the total number of miles Cameron ran?

 

Part = 1½    Part = 4/5             Whole = Unknown

The student knows that the two parts equal the whole. Therefore, to find the unknown whole simply add the two known parts.

 

3.                           Cameron ate 2/3 of the pizza. Kellen ate 1/6 of the pizza. How much more pizza did Cameron eat than Kellen?

 

Part = 1/6   Part = Unknown    Whole = 2/3

The student knows that the two parts equal the whole. Therefore, to find the unknown part simply subtract the known part from the whole.