Standard

Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

Explanation

About the Math, Learning Targets, & Rigor

A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than unit fractions, such as 2/3, they should be able to decompose the non-unit fraction into a combination of several unit fractions.

Being able to visualize this decomposition into unit fractions helps students when adding or subtracting fractions. Students need multiple opportunities to work with mixed numbers and be able to decompose them in more than one way. Students may use visual models to help develop this understanding.

Students can solve 3 3/4 + 2 1/4 in multiple ways:

Resources

Videos

• Teaching Video (7:46-10:16)
• Khan Academy (scroll to find 4.NF.3 - click on a skill)
  • Videos (on left side in each skill)
• Learn Zillion
  • Videos
• Virtual Nerd 1
• Virtual Nerd 2

EngageNY Lessons

• 4.NF.3a
  • Module 3, Topic A, Lesson 2: Make equivalent fractions with sums of fractions with like denominators.
  • Module 5, Topic D, Lesson 17: Use visual models to add and subtract two fractions with the same units, including subtracting from one whole.
  • Module 5, Topic D, Lesson 18: Add and subtract more than two fractions.
• 4.NF.3b
  • Module 5, Topic A, Lesson 1: Decompose fractions as a sum of unit fractions using tape diagrams.
  • Module 5, Topic A, Lesson 2: Decompose fractions as a sum of unit fractions using tape diagrams.
  • Module 5, Topic A, Lesson 3: Decompose non-unit fractions and represent them as a whole number times a unit fraction using tape diagrams.
  • Module 5, Topic A, Lesson 4: Decompose fractions into sums of smaller unit fractions using tape diagrams.
• 4.NF.3c
  • Module 5, Topic F, Lesson 29: Estimate sums and differences using benchmark numbers.
  • Module 5, Topic F, Lesson 30: Add a mixed number and a fraction.
  • Module 5, Topic F, Lesson 31: Add mixed numbers.
  • Module 5, Topic F, Lesson 32: Subtract a fraction from a mixed number.
  • Module 5, Topic F, Lesson 33: Subtract a mixed number from a mixed number.
  • Module 5, Topic F, Lesson 34: Subtract mixed numbers.
• 4.NF.3d
  • Module 5, Topic D, Lesson 19: Solve word problems involving addition and subtraction of fractions.

Extra Practice

• North Carolina Tasks
• Illustrative Math Tasks
• Howard County Tasks
  • Click on 4th grade at bottom, then 4.NF.3
• Common Core Sheets
  • 4.NF.3a
  • 4.NF.3b
  • 4.NF.3c
  • 4.NF.3d

Released Assessment Questions

PARCC

Common Core State Standards

4.NF.3a

Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

STUDENT SAMPLE RESPONSE

4.NF.3b

Understand a fraction a/b with a > 1 as a sum of fractions 1/b. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

• Only the answer is required.
• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Tasks may include fractions that equal whole numbers. Whole numbers are limited to 0 through 5.

STUDENT SAMPLE RESPONSE

4.NF.3c

Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

• Tasks do not have a context.
• Denominators are limited to grade 3 possibilities (2, 3, 4, 6, 8) so as to keep computational difficulty lower.

STUDENT SAMPLE RESPONSE

4.NF.3d

Understand a fraction a/b with a>1 as a sum of fractions 1/b. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Addition and subtraction situations are limited to the dark- or medium-shaded types in Table 2, p. 9 of the OA Progression document; these situations are sampled equally.
• Prompts do not provide visual fraction models; students may at their discretion draw visual fraction models as a strategy.

STUDENT SAMPLE RESPONSE

Performance Indicators: 4.NF.3a, 4.NF.3b, 4.NF.3c, 4.NF.3d

Level 5: Exceeds Expectations

• Understands and solves mathematical and real-world problems involving the addition and subtraction of fractions and mixed numbers with like denominators by joining and separating parts referring to the same whole, and justifying the solution by using a visual model.
• Decomposes a fraction into a sum of fractions with the same denominator in more than one way and records the decomposition using an equation.

Level 4: Meets Expectations

• Using visual models and/or manipulatives, solves mathematical and word problems involving the addition and subtraction of fractions and mixed numbers with like denominators by joining and separating parts referring to the same whole.
• Decomposes a fraction into a sum of fractions with the same denominator in more than one way and records the decomposition using an equation.

Integrated Standards

4.INT.6

Solve real-world and mathematical problems about perimeter involving grade-level addition and subtraction of fractions, such as finding an unknown side of a rectangle.

• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Tasks must be aligned to both standards listed in the content scope. (4.NF.3, 4.MD.3)

Critiquing & Reasoning Standards

4.C.4-2

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 4.NF.3a, 4.NF.3b only)

• Tasks have “thin context” or no context.
• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Tasks may include fractions that equal whole numbers. Whole numbers are limited to 0 through 5.

4.C.5-4

Distinguish correct explanation/reasoning from that which is flawed, and – if there is a flaw in the argument – present corrected reasoning. (For example, some flawed ‘student’ reasoning is presented and the task is to correct and improve it.)

(Content Scope: Knowledge and skills articulated in 4.NF.B)

• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Results may equal fractions greater than 1 (including fractions equal to whole numbers limited to 0 through 5).

4.C.6-2

Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1 + 4 = 5 + 7 = 12, even if the final answer is correct), or identify or describe errors in solutions to multi-step problems and present corrected solutions.

(Content Scope: Knowledge and skills articulated in 4.NF.3c only)

• Tasks have “thin context” or no context.
• Denominators are limited to grade 3 possibilities (2, 3, 4, 6, 8) so as to keep computational difficulty lower.
• Multi-step problems must have at least 3 steps

4.C.6-3

Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1 + 4 = 5 + 7 = 12, even if the final answer is correct), or identify or describe errors in solutions to multi-step problems and present corrected solutions.

(Content Scope: Knowledge and skills articulated in 4.NF.3d,4.NF.4c only)

• Denominators are limited to grade 3 possibilities (2, 3, 4, 6, 8) so as to keep computational difficulty lower.
• Multi-step problems must have at least 3 steps

4.C.7-3

Base explanations/reasoning on a number line diagram (whether provided in the prompt or constructed by the student in her response)

Content Scope: Knowledge and skills articulated in 4.NF.3a only)

Performance Indicators: 4.C.4-2 & 4.C.7-3

"Concrete diagrams"

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.

Performance Indicators: 4.C.5-4

"Distinguish correct explanation/reasoning from that which is flawed"

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 by:

• presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
• evaluating explanation/reasoning; if there is a flaw in the argument
• presenting and defending corrected reasoning

Response 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 by:

• presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
• distinguishing correct explanation/reasoning from that which is flawed
• identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems
• presenting corrected reasoning

Response 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.

Performance Indicators: 4.C.6-2 & 4.C.6-3

"Distinguish correct explanation/reasoning from that which is flawed"

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 by:

• presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
• evaluating explanation/reasoning; if there is a flaw in the argument
• presenting and defending corrected reasoning

Response 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 by:

• presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
• distinguishing correct explanation/reasoning from that which is flawed
• identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems
• presenting corrected reasoning

Response 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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