Fifth Grade

 

Number and Operations

 

Standard 5-2:      The student will demonstrate through the mathematical processes an understanding of the place value system; the division of whole numbers; the addition and subtraction of decimals; the relationships among whole numbers, fractions, and decimals; and accurate, efficient, and generalizable methods of adding and subtracting fractions.

 

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 - Whole Numbers

 

Indicators

5-2.7   Generate strategies to find the greatest common factor and the least   

            common multiple of two whole numbers.

5-2.6    Classify numbers as prime, composite, or neither.

 

 

          As the verb “Generate” implies in Indicator 5-2.7, students should be given opportunities to generate and share strategies as they develop a conceptual understanding of the greatest common factor and the least common multiple of two whole numbers. Students should be familiar with the terms factor and multiple since the concept of multiplication was introduced in third grade. In addition, fourth grade students were expected to explain the effect on the product when one of the factors were changed. As a result, fifth grade students should build on that knowledge when generating strategies to find the greatest common factor and the least common multiple of two whole numbers.

Experiences involving least common multiples and greatest common factors provide opportunities for students to work with rational numbers in a variety of problem solving situations. This will later help fifth grade students to begin generating strategies to add and subtract fractions with like and unlike denominators (indicator 5-2.8) as well as simplifying fractions. The emphasis is on student understanding, not memorizing a process.  Continuing to use models and pictorial representations in fifth grade will help students connect to the symbolic representation of the concept of applying algorithms for simplifying fractions and adding and subtracting fractions with unlike denominators in later grades.

 

Fifth grade is the first year students classify whole numbers as prime, composite, or neither. In order to do so, many opportunities must be provided for students to conceptually understand these classifications. Experiences such as constructing arrays for whole numbers and categorizing the arrays into 2 groups of arrays with “Factors of 1 and Itself”   (The number 11 only has 2 factors, 1 and 11)  and arrays with “More than 1 Factor and Itself”  (The number 10 has 4 factors of 1, 2, 5, 10) will enhance students’ conceptual understanding.  The number 1 is neither prime nor composite because it has only one factor - itself.

Initially using concrete or pictorial representations of multiplication arrays will enable students to concretely see and begin to classify numbers as prime, composite, or neither as the chart below indicates.

 

EXAMPLE:

 

Composite

Prime

 

 

More than 1 Factor and Itself

1 Factor and Itself

 

Numbers

 

 

 

 

Factors

Factors

 

2

 

1,2

 

3

 

1,3

 

4

1,2,4

 

 

5

 

1,5

 

6

1,2,3,6

 

 

7

 

1,7

 

8

1,2,4,8

 

 

9

1,3,9

 

 

10

1,2,5,10

 

 

 

Teacher Note: Students typically confuse the concepts of greatest common factor and multiples. Therefore, when engaging in classroom discussion, require students to use those terms in their explanations. Also, as teacher pose student questions that help students make the connection between “factor” and the “parts of a multiplication problem”. Students experiences should help them see that multiplies are derived from multiplying or using repeated addition.

 

Number Structure and Relationships - Whole Numbers, Fractions, and Decimals

 

Indicators

5-2.1   Analyze the magnitude of a digit on the basis of its place value, using whole numbers and decimal numbers through thousandths.

5-2.4   Compare whole numbers, decimals, and fractions by using the symbols <, >, and =.

 

Students have analyzed the magnitude of a digit based on its place value since kindergarten. Therefore, this concept is not new. The only change and where emphasis should be placed is on the decimal portion. Decimals were introduced for the first time in fourth grade and students analyzed digits through hundredths. Fourth grade students also generated strategies to add and subtract decimals through hundredths. Now in fifth grade students should have an understanding through thousandths. Fifth grade students should also examine the relationship between the place value structure of whole numbers and the place value structure of decimals through thousandths. 

          Students in fifth grade should build on prior concrete experiences and learn to move fluently and confidently among and between the representations of whole numbers, fractions and decimals using symbols for comparison. Students should be able to decompose whole numbers and extend this notion to decimal numbers. (Principles and Standards for School Mathematics, p. 150) Again, however, student work with decimals should be limited to thousandths and there is no limit on the magnitude of fractions. Sound educational practice dictates that fractions should be of reasonable size that emphasis is on understanding the relative magnitude NOT on applying some memorized pneumonic device when making comparisons.

          In fourth grade students had opportunities to analyze decimal numbers as a part of a whole using concrete and pictorial models. The focus was on conceptual understanding of decimals through hundredths.  Fifth grade builds on that knowledge and extends decimal place value through thousandths.

 

Operations - Addition and Subtraction

 

Indicators

5-2.5   Apply an algorithm to add and subtract decimals through thousandths.

5-2.8   Generate strategies to add and subtract fractions with like and unlike

          denominators.

 

Fourth grade students were introduced to the concept of decimals for the first time. Besides creating concrete and pictorial models to gain an understanding of decimals through hundredths, they generated strategies to add and subtract decimals through hundredths. As discussed in the major concept above, fifth grade students extend that knowledge to thousandths. Also, fifth grade students should be able to fluently add and subtract decimals through thousandths. While fourth grade work was limited to concrete and pictorial models in an effort to develop an in-depth conceptual understanding, fifth grade should place an emphasis on symbolic manipulation (numbers only when adding and subtracting through thousandths). Of course, addition and subtraction experiences should be in context.

Fifth grade is the first year students begin to develop computational

strategies for adding and subtracting fractions with like and unlike denominators.  In fourth grade students generated equivalent fractions which laid the foundation for their work this year.  As the verb “Generate” implies in indicator 5-2.5, students should generate and share their own strategies for adding and subtracting fractions. That means that all addition and subtraction work with fractions during fifth grade should be with concrete and pictorial models. Also, problems should be posed in context – not adding and subtracting for the sake of doing so – but having a reason, a problem to solve that requires addition or subtraction of fractions. Ample time should be provided for students to share their strategies and learn from each other. Such sharing and discussing leads students to discover an efficient algorithm that they understand – not just memorize a strategy that they soon forget.

 

Connections To:

 

Other Fifth Grade Indicators:

5.2.7   Generate strategies to find the greatest common factor and the least            

           common multiple of two whole numbers

5-3.4       Identify applications of commutative, associative, and distributive       

           properties with whole numbers.

 

Since these connections are self-explanatory please see the essential learning explanation for 5.27 under “Number Structure and Relationships – Whole Numbers” above and for 5-3.4 under the Algebra Standard.

 

 

Operations - Division

 

5-2.2 Apply an algorithm to divide whole numbers fluently. 

5-2.3 Understand the relationship among the divisor, dividend, and quotient.

5-2.9 Apply divisibility rules for 3, 6, and 9.

 

          Fourth grade students generated strategies to divide whole numbers by a single digit divisor with no remainders. That means their learning experiences involved strictly concrete and pictorial models for division – an emphasis on understanding division. In fifth grade student work should link those previous concrete and pictorial experiences to the symbolic. While fifth grade students should become fluent with division, sound educational practice dictates that the magnitude of the divisor and dividend should be reasonable. Division should not be a laborious task to be dreaded by students. In the contrary, students should see and understand division as a means to problem solving. Fifth grade learning experiences should involve quotients both with and without remainders. If students have a conceptual understanding of division, they have an understanding of the relationship among the divisor, dividend, and quotient.

          By applying an algorithm to divide whole numbers fluently, students

should be able to explain what each number in a division algorithm means

and understand the relationship among the divisor, dividend and quotient.

For example, how the quotient becomes larger when the divisor is changed

to a smaller digit or how the quotient becomes smaller when the divisor is

changed to a larger digit. Students should also understand that if they are

unable to efficiently find the answer to a problem such as 39 ÷ 3,

they can decompose the dividend 39 to 30 + 9 then divide each easily by 3

so that 30 ÷ 3 = 10 and 9 ÷ 3 = 3 so that the quotients of

10 and 3 can be added to get 13. Again, the emphasis is on understanding and dividing fluently – not on pages of symbolic manipulation.

          In fourth grade, students were introduced to the concept of divisibility rules for 2, 5, and 10.  After fifth grade students are comfortable with applying an algorithm to divide whole numbers, the divisibility rules for 3, 6, and 9 should be introduced as a way of quickly determining by what numbers a whole number may be evenly divided.

 

Teacher Note:  Multiplication is not mentioned in the fifth grade standards, however, students should maintain multiplication fluency as part of the division algorithm.

 

 

Connections to:

 

Other Fifth Grade Indicators

5-3.4          Identify applications of commutative, associative, and distributive

            properties with whole numbers.

 

Since the connection is self-explanatory please see the essential learning explanation 5-3.4 under the Algebra Standard.