Topic: Number and Operations-Fractions |
Common Core Mathematics 3-5 |
| Cluster Develop understanding of fractions as numbers. |
| | Grade 3 |
| | | 3.NBT.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. |
| | | 3.NBT.2. | Understand a fraction as a number on the number line; represent
fractions on a number line diagram. |
| | | 3.NBT.2.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. 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. |
| | | 3.NBT.2.b. | Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number
line. |
| | | 3.NF.3. | Explain equivalence of fractions in special cases, and compare
fractions by reasoning about their size. |
| | | 3.NF.3.a. | Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. |
| | | 3.NF.3.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. |
| | | 3.NF.3.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. |
| | | 3.NF.3.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. |
| Cluster Solve problems involving measurement and estimation of intervals
of time, liquid volumes, and masses of objects. |
| | Grade 3 |
| | | 3.MD.1. | Tell and write time to the nearest minute and measure time intervals
in minutes. Solve word problems involving addition and subtraction
of time intervals in minutes, e.g., by representing the problem on a
number line diagram. |
| Cluster Extend understanding of fraction equivalence and ordering. |
| | Grade 4 |
| | | 4.NF.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. |
| | | 4.NF.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. |
| Cluster Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers. |
| | Grade 4 |
| | | 4.NF.3. | Understand a fraction a/b with a > 1 as a sum of fractions 1/b. |
| | | 4.NF.3.a. | Understand addition and subtraction of fractions as joining and
separating parts referring to the same whole. |
| | | 4.NF.3.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. |
| | | 4.NF.3.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. |
| | | 4.NF.3.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. |
| | | 4.NF.4. | Apply and extend previous understandings of multiplication to
multiply a fraction by a whole number. |
| | | 4.NF.4.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 × (1/4),
recording the conclusion by the equation 5/4 = 5 × (1/4). |
| | | 4.NF.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 × (2/5) as 6 × (1/5),
recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) |
| | | 4.NF.4.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? |
| Cluster Understand decimal notation for fractions, and compare decimal
fractions. |
| | Grade 4 |
| | | 4.NF.5. | Express a fraction with denominator 10 as an equivalent fraction with
denominator 100, and use this technique to add two fractions with
respective denominators 10 and 100.[Students who can generate equivalent fractions can develop strategies for adding
fractions with unlike denominators in general. But addition and subtraction with unlike
denominators in general is not a requirement at this grade]. For example, express 3/10 as
30/100, and add 3/10 + 4/100 = 34/100. |
| | | 4.NF.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. |
| | | 4.NF.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 a visual
model. |
| Cluster Use equivalent fractions as a strategy to add and subtract fractions. |
| | Grade 5 |
| | | 5.NF.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.) |
| | | 5.NF.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. |
| Cluster Apply and extend previous understandings of multiplication and
division to multiply and divide fractions. |
| | Grade 5 |
| | | 5.NF.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 50-pound sack of rice equally by weight, how many
pounds of rice should each person get? Between what two whole numbers
does your answer lie? |
| | | 5.NF.4. | Apply and extend previous understandings of multiplication to
multiply a fraction or whole number by a fraction. |
| | | 5.NF.4.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.) |
| | | 5.NF.4.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 find areas of
rectangles, and represent fraction products as rectangular areas. |
| | | 5.NF.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. |
| | | 5.NF.6. | Solve real world problems involving multiplication of fractions and
mixed numbers, e.g., by using visual fraction models or equations to
represent the problem. |
| | | 5.NF.7. | Apply and extend previous understandings of division to divide unit
fractions by whole numbers and whole numbers by unit fractions.[Apply and extend previous understandings of division to divide unit
fractions by whole numbers and whole numbers by unit fractions] |