simple expressions that record calculations with numbers, and interpret
numerical expressions without evaluating them. For example, express
the calculation “add 8 and 7, then multiply by 2” as 2
× (8 + 7). Recognize that 3 × (18932 + 921) is three times
as large as 18932 + 921,without having to calculate the indicated
sum or product.
two numerical patterns using two given rules. Identify
apparent relationships between corresponding terms. Form ordered pairs
consisting of corresponding terms from the two patterns, and graph
the ordered pairs on a coordinate plane. For example, given the
rule “Add 3” and the starting number 0, and given the
rule “Add 6” and the starting number 0, generate terms
in the resulting sequences, and observe that the terms in one sequence
are twice the corresponding terms in the other sequence. Explain informally
why this is so.
patterns in the number of zeros of the product when multiplying
a number by powers of 10, and explain patterns in the placement
of the decimal point when a decimal is multiplied or divided by
a power of 10. Use whole-number exponents to denote powers of
whole-number quotients of whole numbers with up to four-digit dividends
and two-digit divisors, using strategies based on place value, the
properties of operations, and/or the relationship between multiplication
and division. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
subtract, multiply, and divide decimals to hundredths, using concrete
models or drawings and strategies based on place value, properties
of operations, and/or the relationship between addition and subtraction;
relate the strategy to a written method and explain the reasoning
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.)
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.
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?
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.)
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.
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.
division of a unit fraction by a non-zero 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.
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.
real world problems involving division of unit fractions by non-zero
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 1/3-cup servings are
in 2 cups of raisins?
and Data (5.MD)
like measurement units within a given measurement system.
a line plot to display a data set of measurements in fractions of
a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade
to solve problems involving information presented in line plots. For
example, given different measurements of liquid in identical beakers,
find the amount of liquid each beaker would contain if the total amount
in all the beakers were redistributed equally.
measurement: understand concepts of volume and relate volume to
multiplication and to addition.
the volume of a right rectangular prism with whole-number side lengths
by packing it with unit cubes, and show that the volume is the same
as would be found by multiplying the edge lengths, equivalently by
multiplying the height by the area of the base. Represent threefold
whole-number products as volumes,
e.g., to represent the associative property of multiplication.
the formulas V = l × w × h and V = b × h for rectangular
prisms to find volumes of right rectangular prisms with whole number
edge lengths in the context of solving real world and mathematical
volume as additive. Find volumes of solid figures composed of two
non-overlapping right rectangular prisms by adding the volumes of
the non-overlapping parts, applying this technique to solve real world
points on the coordinate plane to solve real-world and mathematical
a pair of perpendicular number lines, called axes, to define a coordinate
system, with the intersection of the lines (the origin) arranged to
coincide with the 0 on each line and a given point in the plane located
by using an ordered pair of numbers, called its coordinates. Understand
that the first number indicates how far to travel from the origin
in the direction of one axis, and the second number indicates how
far to travel in the direction of the second axis, with the convention
that the names of the two axes and the coordinates correspond (e.g.,
x-axis and x-coordinate, y-axis and y-coordinate).
that attributes belonging to a category of two-dimensional figures
also belong to all subcategories of that category. For example, all
rectangles have four right angles and squares are rectangles, so all
squares have four right angles.