products of whole numbers, e.g., interpret 5 × 7 as the total
number of objects in 5 groups of 7 objects each. For example,
describe a context in which a total number of objects can be expressed
as 5 × 7.
whole-number quotients of whole numbers, e.g., interpret 56 ÷
8 as the number of objects in each share when 56 objects are partitioned
equally into 8 shares, or as a number of shares when 56 objects are
partitioned into equal shares of 8 objects each. For example,
describe a context in which a number of shares or a number of groups
can be expressed as 56 ÷ 8.
multiplication and division within 100 to solve word problems in situations
involving equal groups, arrays, and measurement quantities, e.g.,
by using drawings and equations with a symbol for the unknown number
to represent the problem.
the unknown whole number in a multiplication or division equation
relating three whole numbers. For example, determine the unknown number
that makes the equation true in each of the equations 8 × ?
= 48 , 5 = ? ÷ 3 , 6 × 6 = ?.
properties of operations as strategies to multiply and divide. Examples:
If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known.
(Commutative property of multiplication.) 3 × 5 × 2 can
be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 ×
2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8
× 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40
+ 16 = 56. (Distributive property.)
multiply and divide within 100, using strategies such as the relationship
between multiplication and division (e.g., knowing that 8× 5
= 40, one knows 40 ÷ 5 = 8) or properties of operations. By
the end of Grade 3, know from memory all products of two one-digit
two-step word problems using the four operations. Represent these
problems using equations with a letter standing for the unknown quantity.
Assess the reasonableness of answers using mental computation and
estimation strategies including rounding.
arithmetic patterns (including patterns in the addition table
or multiplication table), and explain them using properties of
operations. For example, observe that 4 times a number is
always even, and explain why 4 times a number can be decomposed
into two equal addends.
and Operations in Base Ten (3.NBT)
place value understanding and properties of operations to perform
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.
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.
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
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
and Data (3.MD)
problems involving measurement and estimation of intervals of time,
liquid volumes, and masses of objects.
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.
and estimate liquid volumes and masses of objects using standard
units of grams (g), kilograms (kg), and liters (l).6 Add, subtract,
multiply, or divide to solve one-step word problems involving
masses or volumes that are given in the same units, e.g., by using
drawings (such as a beaker with a measurement scale) to represent
a scaled picture graph and a scaled bar graph to represent a data
set with several categories. Solve one- and two-step “how many
more” and “how many less” problems using information
presented in scaled bar graphs. For example, draw a bar graph
in which each square in the bar graph might represent 5 pets.
measurement data by measuring lengths using rulers marked with halves
and fourths of an inch. Show the data by making a line plot, where
the horizontal scale is marked off in appropriate units— whole
numbers, halves, or quarters.
measurement: understand concepts of area and relate area to multiplication
and to addition.
side lengths to find areas of rectangles with whole number side lengths
in the context of solving real world and mathematical problems, and
represent whole-number products as rectangular areas in mathematical
tiling to show in a concrete case that the area of a rectangle with
whole-number side lengths a and b + c is the sum of a × b and
a × c. Use area models to represent the distributive property
in mathematical reasoning.
area as additive. Find areas of rectilinear figures by decomposing
them into non-overlapping rectangles and adding the areas of the
non-overlapping parts, applying this technique to solve real world
measurement: recognize perimeter as an attribute of plane figures
and distinguish between linear and area measures.
real world and mathematical problems involving perimeters of polygons,
including finding the perimeter given the side lengths, finding an
unknown side length, and exhibiting rectangles with the same perimeter
and different areas or with the same area and different perimeters.
with shapes and their attributes.
that shapes in different categories (e.g., rhombuses, rectangles,
and others) may share attributes (e.g., having four sides), and that
the shared attributes can define a larger category (e.g., quadrilaterals).
Recognize rhombuses, rectangles, and squares as examples of quadrilaterals,
and draw examples of quadrilaterals that do not belong to any of these
shapes into parts with equal areas. Express the area of each part
as a unit fraction of the whole. For example, partition a shape into
4 parts with equal area, and describe the area of each part as 1/4
of the area of the shape.