Standard

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Student language:

• "I can explain in words or pictures how two fractions can sometimes be equal."
• I can compare fractions by reasoning about their size."
• "I can show whole numbers as fractions. (3 = 3/1)"
• "I can recognize fractions that are equal to one whole. (1 = 4/4)."

Note: Grade 3 expectations are limited to fractions with denominators 2, 3, 4, 6, and 8

Explanation

About the Math, Learning Targets, and Rigor

An important concept when comparing fractions is to look at the size of the parts and the number of the parts. For example, 1/8 is smaller than 1/2 because when 1 whole is cut into 8 pieces, the pieces are much smaller than when 1 whole is cut into 2 pieces. (Students can SEE this by modeling----folding paper in half, in half again, and so on.)

3.NF.3a and 3.NF.3b

Students use visual fraction models (area models) and number lines to explore the idea of equivalent fractions. Note: Students should only explore equivalent fractions using models, rather than using algorithms or procedures.

3.NF.3d

This standard involves comparing fractions with or without visual fraction models including number lines. Experiences should encourage students to reason about the size of pieces, the fact that 1/3 of a cake is larger than 1/4 of the same cake. Since the same cake (the whole) is split into equal pieces, thirds are larger than fourths.

(In this standard, students should also reason that comparisons are only valid if the wholes are identical. For example, 1/2 of a large pizza is a different amount than 1/2 of a small pizza.)

Students recognize when examining fractions with common denominators, the wholes have been divided into the same number of equal parts. So the fraction with the larger numerator has the larger number of equal parts.

2/6 < 5/6

To compare fractions that have the same numerator but different denominators, students understand that each fraction has the same number of equal parts but the size of the parts are different. They can infer that the same number of smaller pieces is less than the same number of bigger pieces.

3/8 < 3/4

Resources

Videos

• Instructional Video (13:46 - 17:01)
• Khan Academy
  • scroll to 3.NF.3, click on a skill, watch videos on left side in each skill
• Learn Zillion
  • 3.NF.3a
  • 3.NF.3b
  • 3.NF.3c
  • 3.NF.3d

EngageNY Lessons

Extra Practice

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

Released Assessment Questions

PARCC

3.NF.3a-1

Explain equivalence of fractions in special cases and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same size.

• Tasks do not involve the number line.
• Fractions equivalent to whole numbers are limited to 0 through 5.
• Tasks are limited to fractions with denominators 2, 3, 4, 6, and 8.
• The explanation aspect of 3.NF.3 is not assessed here.

STUDENT SAMPLE RESPONSE

3.NF.3a-2

Explain equivalence of fractions in special cases and compare fractions by reasoning about their size. a. Understand two fractions as equivalent (equal) if they are the same point on a number line.

• Tasks are limited to fractions with denominators 2, 3, 4, 6, and 8.
• Fractions equivalent to whole numbers are limited to 0 through 5.
• The explanation aspect of 3.NF.3 is not assessed here.

STUDENT SAMPLE RESPONSE

3.NF.3b-1

Explain equivalence of fractions in special cases and compare fractions by reasoning about their size. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3).

• Tasks are limited to fractions with denominators 2, 3, 4, 6, and 8.
• Fractions equivalent to whole numbers are limited to 0 through 5.
• The explanation aspect of 3.NF.3 is not assessed here.

STUDENT SAMPLE RESPONSE

3.NF.3c

Explain equivalence of fractions in special cases and compare fractions by reasoning about their size. c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

• Tasks are limited to fractions with denominators 2, 3, 4, 6, and 8.
• Fractions equivalent to whole numbers are limited to 0 through 5.
• The explanation aspect of 3.NF.3 is not assessed here.

STUDENT SAMPLE RESPONSE

3.NF.3d

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

• Tasks are limited to fractions with denominators 2, 3, 4, 6, and 8.
• Fractions equivalent to whole numbers are limited to 0 through 5.
• Justifying is not assessed here. For this aspect of 3.NF.3d, see 3.C.3-1 and 3.C.4-4.
• Prompts do not provide visual fraction models; students may at their discretion draw visual fraction models as a strategy.

STUDENT SAMPLE RESPONSE

3.NF.A.Int.1

In a contextual situation involving a whole number and two fractions not equal to a whole number, represent all three numbers on a number line diagram, then choose the fraction closest in value to the whole number.

• Fractions equivalent to whole numbers are limited to 0 through 5.
• Fraction denominators are limited to 2, 3, 4, 6 and 8.

STUDENT SAMPLE RESPONSE

Performance Indicators: 3.NF.3a-1, 3.NF.3a-2, 3.NF.3b-1, 3.NF-3c, 3.NF-3d, 3.NF.A.Int.1

Level 5: Exceeds Expectations

• Understands, recognizes and generates equivalent fractions with denominators of 2, 3, 4, 6 and 8.
• Expresses whole numbers as fractions and recognize fractions that are equivalent to whole numbers.
• Compares two fractions that have the same numerator or same denominator using symbols to justify conclusions. Plots the location of equivalent fractions on a number line. The student must recognize that two fractions must refer to the same whole in order to compare.
• Given a whole number and two fractions in a real-world situation, plots all three numbers on a number line and determines which fraction is closest to the whole number. Justifies the comparison by plotting points on a number line.

Level 4: Meets Expectations

• Understands, recognizes and generates equivalent fractions using denominators of 2, 4, and 8.
• Expresses whole numbers as fractions.
• Compares two fractions that have the same numerator or same denominator using symbols and justifies conclusions by using a visual model. The student must recognize that two fractions must refer to the same whole in order to compare.

Critiquing Standards

3.C.3-1

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 3.NF.3b, 3.NF.3d)

• Tasks may present realistic or quasi-realistic images of a contextual situation (e.g., a drawing of a partially filled graduated cylinder). However, tasks do not provide the sort of abstract drawings that help the student to represent the situation mathematically (e.g., a number line diagram or other visual fraction model).
• Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
• For fractions equal to a whole number, values are limited to 0 through 5.

3.C.4-4

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 3.NF.3b, 3.NF.3d)

• Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
• For fractions equal to a whole number, values are limited to 0 through 5.

Performance Indicators: 3.C.3-1

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
• determination 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
• evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning.

Performance Indicators: 3.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 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 counter-example 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
• evaluating, interpreting and critiquing the validity of other’s responses, approaches and reasoning.