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Grade 3 
Grade 4 
Grade 5 
3.NF.1
Develop understanding of fractions as numbers.
1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a
fraction a/b as the quantity formed by a parts of size 1/b.

4.NF.1
1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how
the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to
recognize and generate equivalent fractions.

5.NF.1
Use equivalent fractions as a strategy to add and subtract fractions.
1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent
fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example,
2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

3.NF.2
2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning
it into b equal parts.
b. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the
number 1/b on the number line.

4.NF.2
2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or
numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two
fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions,
e.g., by using a visual fraction model.

5.NF.2
2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike
denominators, e.g., by using visual fraction models or equations to represent the problem.
Use benchmark fractions and
number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an
incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

3.NF.3
3. 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 comparisos with the
symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

4.NF.3
3. 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.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.3
3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).
Solve word problems involving division
of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or
equations to represent the problem.
For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied
by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people
want to share a 50pound sack of rice equally by weight, how many pounds of rice should each person get? Between what
two whole numbers does your answer lie?


4.NF.4
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5. (In general,
n x (a/b) = (n x a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and
equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and
there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers
does your answer lie?

5.NF.4
4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence
of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context
for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side
lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side
lengths to nd areas of rectangles, and represent fraction products as rectangular areas.


4.NF.5
5. Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit numbers, using
strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.


5.NF.5
5. Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing
the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given
number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a
given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of
fraction equivalence a/b = (n × a)/(n b) to the effect of multiplying a/b by 1.



Understand decimal notation for fractions, and compare decimal fractions.
6.NF.6
6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as
0.62 meters; locate 0.62 on a number line diagram.

5.NF.6
6. Solve realworld problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or
equations to represent the problem.


4.NF.7
7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the
conclusions, e.g., by using the number line or another visual model. CA

5.NF.7
7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit
fractions.1
a. Interpret division of a unit fraction by a nonzero whole number, and compute such quotients. For example, create
a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between
multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context
for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and
division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve realworld problems involving division of unit fractions by nonzero whole numbers and division of whole numbers
by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much
chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3cup servings are in 2 cups
of raisins?

