3.NF.A.3

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

 

Summary:

Students should recognize equivalent fractions and be able to explain why they are equivalent.Using number lines or other models, students will recognize fractions that represent the same amount. For example, if a whole is broken into four pieces, two of these would be the same as one piece of that same whole if it had been broken into two pieces. Students will be able to see that if we divide the interval from 0 to 1 into b pieces and mark the end of the bth piece, then that is the location of the fraction b/b. But it is also the end of the interval from 0 to 1, so it is the number 1. This shows b/b is equivalent to 1. They also need to understand that 1/1 is one part when a whole is divided into one part; that is, a whole. So a/1 is a parts, each of which is a whole. This shows that a/1 = a. Students should be able to compare fraction with the same numerator or denominator.

Understanding the Standard:

  • Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
  • Recognize and generate simple equivalent fractions, (e.g., 1/2 = 2/4 and 4/6 = 2/3). Explain why the fractions are equivalent, (e.g., by using a visual fraction model).
  • Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. For example, express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point on a number line.
  • Compare two fractions with the same numeral or the same denominator by reasoning about their size. Recognize that comparisons are only valid 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).
  • The use of manipulatives and visual models is extremely helpful. Students can construct models that will represent the sizes of two different fractions as parts of a whole that two different fractions have within an object. By superimposing transparent number lines on one another, students can discover that fractions such as 1/2 and 3/6 are equivalent.
  • Students will need opportunities to compare fractions and order them using numeric representation and  visual models.
  • Students will represent whole numbers as fractions. For example, if some celery sticks (of equal length) were cut into six equal pieces, then students could see that the fraction 6/6 represents six pieces, which is the same amount as one. Similarly, 12/6 represents 12 pieces, which together make up two celery sticks; therefore, 12/6 = 2.

Questions to Focus Learning:

  • Can students identify fractions that share a common numerator or denominator by creating a visual or model?
  • Are students able to order fractions according to their size?
  • Can students choose a fraction that is equivalent to a given fraction?
  • Can students represent a give whole number as a fraction?
  • Can students recognize a fraction of the form b/b as equivalent to 1?

Skills

Prior to: Students can identify and write fractions as pictures, symbols, and words.
At Grade Level: Students are able to compare fractions to one another and to whole numbers. They are also able to think of an equivalent fraction for a given fraction like 1/2 (example 2/4).