5.NBT.7

Standard

Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Student Language:

"I can add, subtract, multiply, and divide decimals to hundredths."
"I can use concrete models or drawings to explain the method used."

Explanation

About the Math, Learning Targets, and Rigor

This standard builds on the work from fourth grade where students are introduced to decimals and compare them. In fifth grade, students begin adding, subtracting, multiplying and dividing decimals. This work should focus on concrete models and pictorial representations, rather than relying solely on the algorithm.

Note when adding/subtracting decimals: Students should line up the place values as they would whole number.

Before students are asked to give exact answers, they should estimate answers based on their understanding of operations and the value of the numbers.

Examples

3.6 + 1.7 =?
5.4 − 0.8 =?
6 × 2.4 =?

Resources

Videos

Instructional Video (14:31 - 26:27)
Khan Academy
Learn Zillion
Virtual Nerd

EngageNY Lessons

Extra Practice

Released Assessment Questions

PARCC

Common Core State Standards

5.NBT.7-1

Add two decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Tasks do not have a context.
Only the sum is required. For the explain aspect of 5.NBT.7-1, see 5.C.1-2, 5.C.2-2, and 5.C.4-4 explanations are not assessed here.
Prompts may include visual models, but prompts must also present the addends as numbers, and the answer sought is a number, not a picture.
Each addend is greater than or equal to 0.01 and less than or equal to 99.99.
20% of cases involve a whole number—either the sum is a whole number, or else one of the addends is a whole number presented without a decimal point. (The addends cannot both be whole numbers.)

5.NBT.7-2

Subtract two decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Tasks do not have a context.
Only the difference is required. For the explain aspect of 5.NBT.7-2, see 5.C.1-2, 5.C.2-2, and 5.C.4-4.
Prompts may include visual models, but prompts must also present the subtrahend and minuend as numbers, and the answer sought is a number, not a picture.
The subtrahend and minuend are each greater than or equal to 0.01 and less than or equal to 99.99. Positive differences only. (Every included subtraction problem is an unknown-addend problem included in 5.NBT.7-1.)
20% of cases involve a whole number—either the difference is a whole number, or the subtrahend is a whole number presented without a decimal point, or the minuend is a whole number presented without a decimal point. (The subtrahend and minuend cannot both be whole numbers.)

5.NBT.7-3

Multiply tenths with tenths or tenths with hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Tasks do not have a context.
Only the product is required. For the explain aspect of 5.NBT.7-3, see 5.C.1-2, 5.C.2-2, and 5.C.4-4.
Prompts may include visual models, but prompts must also present the factors as numbers, and the answer sought is a number, not a picture.
Each factor is greater than or equal to 0.01 and less than or equal to 99.99.The product must not have any non-zero digits beyond the thousandths place. (For example, 1.67 x 0.34 = 0.5678 is excluded because the product has an 8 beyond the thousandths place; cf. 5.NBT.3, and see p. 17 of the Number and Operations in Base Ten Progression document.)
Problems are 2-digit x 2-digit or 1-digit by 3- or 4-digit. (For example, 7.8 x 5.3 or 0.3 x 18.24.)
20% of cases involve a whole number—either the product is a whole number, or else one factor is a whole number presented without a decimal point. (Both factors cannot both be whole numbers.)

5.NBT.7-4

Divide in problems involving tenths and/or hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Tasks do not have a context.
Only the quotient is required. For the explain aspect of 5.NBT.7-4, see 5.C.1-2, 5.C.2-2, and 5.C.4-4.
Prompts may include visual models, but prompts must also present the dividend and divisor as numbers, and the answer sought is a number, not a picture.
Divisors are of the form XY, X0, X, X.Y, 0.XY, 0.X, or 0.0X (cf. 5.NBT.6), where X and Y represent non-zero digits. Dividends are of the form XY, X0, X, XYZ.W, XY0.Z, X00.Y, XY.Z, X0.Y, X.YZ, X.Y, X.0Y, 0.XY, or 0.0X, where X, Y, Z, and W represent non-zero digits.
Quotients are either whole numbers or else decimals terminating at the tenths or hundredths place. (Every included division problem is an unknown-factor problem included in 5.NBT.7-3.)
20% of cases involve a whole number—either the quotient is a whole number, or the dividend is a whole number presented without a decimal point, or the divisor is a whole number presented without a decimal point. (If the quotient is a whole number, then neither the divisor nor the dividend can be a whole number.)

Performance Indicators: 5.NBT.7-1, 5.NBT.7-2

Level 5: Exceeds Expectations

Adds or subtracts two decimals to hundredths using concrete models, drawings or strategies based on place value, properties of operations and/or the relationship between addition and subtraction.

Applies this concept to a real-world context, and relates the strategy to a written method and explain the reasoning used.

Level 4: Meets Expectations

Adds or subtracts two decimals to hundredths using concrete models, drawings or strategies based on place value, properties of operations and/or the relationship between addition and subtraction.

Performance Indicators: 5.NBT.7-3, 5.NBT.7-4

Level 5: Exceeds Expectations

Performs exact and approximate multiplications and divisions by mentally applying place value strategies when appropriate.

Relates the strategy to a written method.

Level 4: Meets Expectations

Multiplies tenths by tenths or tenths by hundredths and divides in problems involving tenths and/or hundredths using concrete models or drawings and strategies based on place value, properties of operations and/or the relationship between addition and subtraction.

Relates the strategy to a written method.

Integrated Standards

5.NBT.Int.1

Perform exact or approximate multiplications and/or divisions that are best done mentally by applying concepts of place value, rather than by applying multi-digit algorithms or written strategies.

Performance Indicators: 5.NBT.Int.1

Level 5: Exceeds Expectations

Multiplies tenths by tenths or tenths by hundredths and divides in problems involving tenths and/or hundredths using concrete models or drawings and strategies based on place value, properties of operations and/or the relationship between addition and subtraction. Performs exact and approximate multiplications and divisions by mentally applying place value strategies when appropriate. Relates the strategy to a written method.

Level 4: Meets Expectations

Multiplies tenths by tenths or tenths by hundredths and divides in problems involving tenths and/or hundredths using concrete models or drawings and strategies based on place value, properties of operations and/or the relationship between addition and subtraction. Relates the strategy to a written method.

Critiquing/Reasoning Standards

5.C.1-2

Base explanations/reasoning on the properties of operations.

(Content Scope: Knowledge and skills articulated in 5.NBT.7)

Tasks do not have a context.
Students need not use technical terms such as commutative, associative, distributive, or property.
Unneeded parentheses should not be used. For example, use 4 + 3 x 2 rather than 4 + (3 x 2).

5.C.2-2

Base explanations/reasoning on the relationship between addition and subtraction or the relationship between multiplication and division.

(Content Scope: Knowledge and skills articulated in 5.NBT.7)

5.C.4-4

Base arithmetic explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt or constructed by the student in her response), connecting the diagrams to a written (symbolic) method.

(Content Scope: Knowledge and skills articulated in 5.NBT.7)

Performance Indicators: 5.C.1-2, 5.C.2-2

Level 5: Exceeds Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student constructs and communicates a well-organized and complete written response based on explanations/reasoning using the:

properties of operations
relationship between addition and subtraction
relationship between multiplication and division

Response may include:

a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
an efficient and logical progression of steps with appropriate justification
precision of calculation
correct use of grade-level vocabulary, symbols and labels
justification of a conclusion
evaluation of whether an argument or conclusion is generalizable
evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate). Provides a counter-example where applicable.

Level 4: Meets Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student constructs and communicates a well-organized and complete written response based on explanations/reasoning using the:

properties of operations
relationship between addition and subtraction
relationship between multiplication and division

Response may include:

a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
a logical progression of steps
precision of calculation
correct use of grade-level vocabulary, symbols and labels
justification of a conclusion
evaluation of whether an argument or conclusion is generalizable
evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate).

Performance Indicators: 5.C.4-4

Level 5: Exceeds Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams--including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:

a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
an efficient and logical progression of steps with appropriate justification
precision of calculation
correct use of grade-level vocabulary, symbols and labels
justification of a conclusion
evaluation of whether an argument or conclusion is generalizable
evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning, and providing a counterexample where applicable

Level 4: Meets Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams--including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:

a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
a logical progression of steps
precision of calculation
correct use of grade-level vocabulary, symbols and labels
justification of a conclusion
evaluation of whether an argument or conclusion is generalizable
evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning.

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