3.NF.3 - About the Math, Learning Targets, and Rigor

Grade 3 Fractions

3.NF.A.3

Full Standard

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. Note: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. It includes fractions greater than 1.

  1. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
  2. 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.
  3. 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.
  4. 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.

 

Measurement Topic

This standard is reported on the report card in these quarters as follows:

3rd Grade Students Learning 3rd Grade Standards
Quarter 1 Quarter 2 Quarter 3 Quarter 4

 

 

Report Card Measurement Topic: Demonstrates understanding of number and fraction concepts.

 

 

Learning Targets

  • Represent whole numbers as fractions and fractions equivalent to one whole.
  • Explain why equivalent fractions must describe the same-size whole.
  • Explain equivalent fractions with representations including color tiles, pattern blocks, Cuisenaire rods, and fraction tiles. 
  • Locate equivalent fractions on a number line.
  • Generate equivalent fractions.
  • Explain that when comparing fractions one must compare same-size wholes.
  • Use benchmarks of 0, 1/2, or 1 to compare fractions.
  • Use the number of same-size parts to compare fractions (without calling them common denominators).
  • Use the same number of parts to compare fractions (without calling them common numerators).
  • Record comparisons using <, >, or =.

 

About the Math

  • Represent whole numbers as fractions.

Any whole number can be expressed as a fraction. First students should show understanding that a fraction is equivalent to one whole. They should be able to explain it. From there, students can begin to work with other whole numbers expressed as a fraction. It is important that this learning target is approached conceptually instead of simply through computation.

  • Explain why equivalent fractions must describe the same-size whole.

There is a difference between a half of a small pizza and a large pizza. That notion is captured in this learning target. Essentially, students must understand that equivalent fractions must refer to the same-sized whole. 

  • Explain equivalent fractions with representations.

Students should be able to represent equivalent fractions in a variety of ways including color tiles, pattern blocks, Cuisenaire rods, and fraction tiles. 

  • Locate equivalent fractions on a number line.

Two fractions are equivalent if they occupy the same location on a number line. Students should work with equivalent fractions on a number line after showing understanding of equivalence with other models. 

  • Generate equivalent fractions.

Students should first be able to explain if two fractions are equivalent. With that understanding in place, they should be asked to generate equivalent fractions. Students should use representations to model their thinking. Students should generate equivalent fractions through understanding rather than procedure and simple multiplication. 

  • Explain that when comparing fractions one must compare same-size wholes.

As noted above, the size of a pizza affects how much is in a half. Students should explore the comparison of fractions with like and unlike-sized wholes to develop an understanding that the same-sized whole must be compared. 

  • Use benchmarks of 0, 1/2, or 1 to compare fractions.

Fractions can be compared by using benchmarks. A fraction less than 1/2 is less than a fraction greater than 1/2. Benchmark comparisons are efficient for comparing when the benchmarks are more clear. 

  • Use the number of same-size parts to compare fractions (without calling them common denominators).

If two fractions have the same number of parts, the fraction that has more parts will be greater. For example, 3/8 is greater than 1/8.

  • Use the same number of parts to compare fractions (without calling them common numerators).

Two fractions can be compared by thinking about relative size. For example, fourths are larger than eighths. Therefore, we can conclude that the 3/4 is more than 3/8 because we have 3 of each.

  • Record comparisons using <, >, or =.

We can compare fractions using the same symbols we use to compare whole numbers.

Essential vocabulary for this standard includes fraction, numerator, denominator, and number line. 

Math in Minutes:  The video links below show strategies for comparing fractions.

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Progression of Standard within Grade 3

This progression informs how to develop the standard within the grade level. This progression is provided by HCPSS Elementary Mathematics.

Progression Throughout Year
Quarter 1 Quarter 2 Quarter 3 Quarter 4
 

 
  • Represent whole numbers as fractions.
  • Explain why equivalent fractions must describe the same-size whole.
  • Explain equivalent fractions with representations.
  • Locate equivalent fractions on a number line.
  • Generate equivalent fractions.
  • Explain that when comparing fractions one must compare same-size wholes.
  • Use benchmarks of 0, 1/2, or 1 to compare fractions.
  • Use the number of same-size parts to compare fractions (without calling them common denominators).
  • Use the same number of parts to compare fractions (without calling them common numerators).
  • Record comparisons using <, >, or =.
  

 

Progression of this Standard Across Grades

This progression is informed by the Achieve the Core Coherence Map Links to an external site.. Information is not the complete standard.

Progression Across Grades
Grade 2 Grade 4
  • Partition circles and rectangles into two, three, or four equal shares (2.G.A.3)
  • Compare measurements of different units (2.MD.A.2) 
  • Recognize and generate equivalent fractions (4.NF.1)

 

 

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TASKS

These tasks can be used with small group or whole group instruction.

  • Distribute fraction manipulatives to each student. You can use circles, rectangles or square materials. Explain that two students each have a pizza that is the same size. One student cuts their pizza in 4 equal pieces and the other students cuts their pizza into 8 equal pieces. Ask students to use the materials to show the two pizzas. Now explain that each of the students ate half of their pizza? What pieces in their model could show this? What is the fraction that represents the eaten pizza? Have a discussion about equivalent fractions and establish that 1/2=2/4=4/8. This can be difficult because students see different numbers, but yet the different fractions represent the same value. Repeat with different equivalencies.
  • Distribute fraction manipulatives to each student. Ask students to work in pairs to find all ways to show 2/3. Discuss all found solutions, including justification of why 4/6 is the same as 2/3, etc. Record equivalent fractions on the board/chart paper as they are discussed. Repeat with other fractions. In looking at the list of equivalent fractions, ask students discuss any patterns they see. Students may notice the "multiplying the numerator and denominator by the same" type thing. This opens the door to discuss you are actually multiplying by 1 as 2/2, 3/3, etc are equivalent to one. This is the application of the Identity of Multiplication Property, when multiplying by 1, we get the same value. So, 2/3 multiplied by 2/2 results in 4/6, which makes sense because 2/3=4/6 and represent the same value.
  • Use Cuisenaire Rods to find equivalent fractions on a number line. Find the Beyond Pizzas & Pies book in your building and see activity page numbers below on this tab.
  • Distribute fraction manipulatives to each student. Write the following fractions on the board: 1/6, 1/3, 3/4. Ask students to use their materials to represent the three fractions. The discussion should then be about how these fractions compare. Which is the largest and why? Could you order the fractions least to greatest? Repeat with other fractions.  This is an introduction and provides students an opportunity to see the physical size of the fraction pieces and why one fraction is larger than another even when the digit is opposite, for example 1/6 is less than 1/3 even though 6 is bigger than 3.
  • Distribute fraction manipulatives to each student and ask them to build 3/8 and 5/8. Discuss which is greater/least and why. Guide the conversation to same size parts (denominator) but one fraction has more of those parts. Ask students to build 3/8 and 3/4. Discuss which is greater/least and why. Guide the conversation to same number of parts under consideration (numerator) but different sized parts and what that means. (fourths are larger than eighths so if I have the same number of each, 3/4 will be larger)  The goal is for students to reason about fractions using these two strategies without the manipulatives, but you start here with them and then remove them.
  • Show ½ and 4/8. Have students compare the two fractions using different representations. Ask students to explain why these two fractions are equal and have them brainstorm other equivalent fractions to ½. (3.NF.3a)
  • Have students use a number line to convince someone that ¾ = 6/8. (3.NF.3a)
  • Use manipulatives such as Cuisinaire rods, fraction strips/circles/towers, pattern blocks, counters, etc. for students to show various representations of ½. (3.NF.3.b)
  • Find a fraction that is equal to ⅔. Equal to ¾. Justify your answers. (3.NF.3b)
  • Draw a number line and count the jumps to show how whole numbers are formed. For example, jump 1, 2, 3 to show 3/1 and then jump ¼, 2/4, ¾, 4/4 to show reaching 1. (3.NF.3c)
  • How many different ways can you show 1 whole? (3.NF.3c)
  • Choose two fractions with different denominators, but the same numerator (for example 3/4 and 3/8). Explain which fraction is smaller or larger and why that is. (3.NF.3d)
  • Have students use linking cubes to show why ⅔ = 4/6. (3.NF.3d)
  • Without drawing a picture or using a number line, Danielle says that ⅜ is not equal to 3/6 ? Is she right or wrong? Explain why or why not. (3.NF.3d)
  • Find a fraction that is greater than three fourths? Greater than ⅓, but less than ½? Greater than one? Justify your answers. (3.NF.3d)
  • What if a friend offered you ¾ of a cookie or ⅝ of a cookie. Which fraction of the cookie would be greater? (3.NF.3d)
  • Which is greater?  1/2 or 2/3?  Prove it using two different representations.
  • Benny says that 2/4, 3/4 and 4/8 are all equivalent to 1/2.  Do you agree?  If so, prove using a number line (or another tool)  If you disagree, which fraction(s) are not equivalent to 1/2 and how do you know?  

 

Slide-Based Tasks 

These links are HCPSS created instructional tasks. These tasks are provided in Google slides.  These tasks should be used for inspiration and resources, but instruction should start with students having the opportunity to engage with the math first (often involving physical and/or visual models) followed by discussion and explicit instruction to ensure student understanding.

 

 

Additional Tasks 

These links provide instructional ideas connected to this standard. These Cuisinaire rods have long been utilised to teach and learn Te Reo Māori.

 ***These lessons are all contained in the same file. ***

 

Tasks From Print Resources

These publications have been provided for each school. They are typically stored in team closets or the media center. Check with your team leader if you cannot find them. 

 

Print Resources
Book Thumbnail Book Title Grade Pages
 Teaching Student-Centered Mathematics  K-3   

264 (Ordering Unit Fractions, Activity 9.5)
266 (Choose, Explain, Test, Activity 9.9)
266 (Line 'Em Up, Activity 9.10)
269 (Different Fillers, Activity 9.12)
269 (Dot Paper Equivalencies. Activity 9.13)
271 (Missing-Number Equivalencies, Activity 9.15)
  Hands-On Standards,
Common Core Fractions
 		 3 40-54 (Equivalent Fractions Lessons 1-3)
56-70 (Comparing Fractions Lessons 1-3)

  Hands-On Standards
 		 3-4 48 (Equivalent Fractions)
50 (Comparing and Ordering Fractions)
  Brain-Compatible Activities for Mathematics

2-3

4-5

65-69

47-50
  Math Intervention: Building Number Power  3-5 166-172
  20 Thinking Questions for Pattern Blocks
3-6 42 and 46
  NumberSense   4-6 92-9
  SuperSource: Color Tiles 3-4 34-37
  Fractions with Pattern Blocks   72

Math In Practice

Teaching Third-Grade Math

 3 Module 9 & 10

 

More Ideas

 

 

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Centers

These print resources can be used during independent or center time. These resources could also be used as lesson seeds.

 

 

INDEPENDENT PRACTICE/HOMEWORK/ASSESSMENT

 These resource sheets can be used for independent practice, homework, or assessment. They are intended to reinforce procedures and concepts. They should not be used as a source of direct instruction or whole-group practice.

 

 

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Learning Targets


  • Represent whole numbers as fractions.
  • Explain why equivalent fractions must describe the same-size whole.
  • Explain equivalent fractions with representations.
  • Locate equivalent fractions on a number line.
  • Generate equivalent fractions.
  • Explain that when comparing fractions one must compare same-size wholes.
  • Use benchmarks of 0, 1/2, or 1 to compare fractions.
  • Use the number of same-size parts to compare fractions (without calling them common denominators).
  • Use the same number of parts to compare fractions (without calling them common numerators).
  • Record comparisons using <, >, or =.

 

Learning targets identify what students should be able to do.  This rubric can be applied to tasks and observations for assessment and/or grading. 

 

Visit the SBIR (Standards Based Instruction and Reporting) tab in Course Essentials for more information and clarification. There you will find the measurement topic crosswalk, report card comments, links to professional learning/resources and guidance.  

 

 

 

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 This course content is offered under a CC Attribution Non-Commercial Share Alike Links to an external site. license. Content in this course can be considered under this license unless otherwise noted.