Download Number and Operations in Base Ten Measurement and Data

Operations and Algebraic
Thinking
Write and interpret numerical
expressions.
Number and Operations in Base Ten
Understand the place value system.
Use equivalent fractions as a strategy
5.1. Recognize that in a multi-digit
to add and subtract fractions.
number, a digit in one place represents 10
times as much as it represents in the place 5.1. Add and subtract fractions with
5.1. Use parentheses, brackets, or to its right and 1/10 of what it represents unlike denominators (including mixed
braces in numerical expressions, in the place to its left.
numbers) by replacing given fractions
and evaluate expressions with
with equivalent fractions in such a way
these symbols.
5.2. Explain patterns in the number of
as to produce an equivalent sum or
zeros of the product when multiplying a difference of fractions with like
5.2. Write simple expressions
number by powers of 10, and explain
denominators. For example, 2/3 + 5/4
that record calculations with
patterns in the placement of the decimal = 8/12 + 15/12 = 23/12. (In general,
numbers, and interpret numerical point when a decimal is multiplied or
a/b + c/d = (ad + bc)/bd.)
expressions without evaluating divided by a power of 10. Use wholethem. For example, express the number exponents to denote powers of 10. 5.2. Solve word problems involving
calculation “add 8 and 7, then
addition and subtraction of fractions
multiply by 2” as 2 × (8 + 7).
referring to the same whole, including
Recognize that 3 × (18932 +
5.3. Read, write, and compare decimals to cases of unlike denominators, e.g., by
921) is three times as large as
thousandths.
using visual fraction models or
18932 + 921, without having to
a. Read and write decimals to
equations to represent the problem. Use
calculate the indicated sum or
thousandths using base-ten
benchmark fractions and number sense
product.
numerals,number names, and expanded of fractions to estimate mentally and
form, e.g., 347.392 = 3 × 100 + 4 ×10 + assess the reasonableness of answers.
Analyze patterns and
7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × For example, recognize an incorrect
relationships.
(1/1000).
result 2/5 + 1/2 = 3/7, by observing that
b. Compare two decimals to
3/7 < 1/2.
thousandths based on meanings of the
5.3. Generate two numerical
digits in each place, using >, =, and <
patterns using two given rules.
symbols to record the results of
Identify apparent relationships
comparisons.
between corresponding terms.
Form ordered pairs consisting of 5.4. Use place value understanding to
corresponding terms from the
round decimals to any place.
two patterns, and graph the
ordered pairs on a coordinate
Perform operations with multi-digit whole
plane. For example, given the
numbers and withdecimals to hundredths.
rule “Add 3” and the starting
number 0, and given the rule
5.5. Fluently multiply multi-digit whole
“Add 6” and the starting number numbers using the standard algorithm.
0, generate terms in the resulting
sequences, and observe that the 5.6. Find whole-number quotients of
terms in one sequence are twice whole numbers with up to four-digit
the corresponding terms in the
dividends and two-digit divisors, using
other sequence. Explain
strategies based on place value, the
informally why this is so.
properties of operations, and/or the
relationship between multiplication and
division. Illustrate and explain the
calculation by using equations, rectangular
5.7. Add, 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 used.
Number and Operations—Fractions
Apply and extend previous understandings of multiplication and
division to multiply and divide fractions.
5.3. 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?
5.6. Solve real world problems
involving multiplication of fractions
and mixed numbers, e.g., by using
visual fraction models or equations to
represent the problem.
Measurement and Data
Convert like measurement units within a
given measurement system.
5.1. Convert among different-sized
standard measurement units within a
given measurement system (e.g., convert
5 cm to 0.05 m), and use
these conversions in solving multi-step,
real world problems.
Geometry
Graph points on the coordinate
plane to solve real-world and
mathematical problems.
5.1. Use 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
Represent and interpret data.
plane located by using an ordered
5.7. Apply and extend previous
pair of numbers, called its
understandings of division to divide
5.2. Make a line plot to display a data set coordinates. Understand that the
unit fractions by whole numbers and
of measurements in fractions of
first number indicates how far to
whole numbers by unit fractions.
a unit (1/2, 1/4, 1/8). Use operations on travel from the origin in the
a. Interpret division of a unit fraction fractions for this grade to solve problems direction of one axis, and the
by a non-zero whole number,and
involving information presented in line
second number indicates how far to
compute such quotients. For
plots. For example,
travel in the direction of the second
5.4. Apply and extend previous understandings of
example, create a story context for given different measurements of liquid in axis, with the convention that the
multiplication to multiply a fraction or whole
(1/3) ÷ 4, and use a visual fraction
identical beakers, find the amount of
names of the two axes and the
number by a fraction.
model to show the quotient. Use the liquid each beaker would contain if the
coordinates correspond (e.g., x-axis
a. Interpret the product (a/b) × q as a parts of a
relationship between multiplication total amount in all the beakers were
and x-coordinate, y-axis and ypartition of q into b equal parts; equivalently, as
and division to explain that (1/3) ÷ 4 redistributed equally.
coordinate).
the result of a sequence of operations
a × = 1/12 because (1/12) × 4 = 1/3.
q ÷ b. For example, use a visual fraction model to
Geometric measurement: understand
show (2/3) × 4 = 8/3, and create a story context
concepts of volume and relate volume to 5.2. Represent real world and
for this equation. Do the same with (2/3) × (4/5)
b. Interpret division of a whole
multiplication and to addition.
mathematical problems by graphing
= 8/15.
(In general, (a/b) ×
number by a unit fraction, and
points in the first quadrant of the
(c/d) = ac/bd.)
compute such quotients. For
5.3. Recognize volume as an attribute of coordinate plane, and interpret
b. Find the area of a rectangle with fractional side example, create a story context for solid figures and understand concepts of coordinate values of points in the
lengths by tiling it with unit squares of the
4 ÷ (1/5), and use a visual fraction
volume measurement.
context of the situation.
appropriate unit fraction side lengths, and
model to show the quotient. Use the
a. A cube with side length 1 unit, called
show that the area is the same as would be found
relationship between multiplication
a “unit cube,” is said to have
Classify two-dimensional figures
by multiplying the side lengths. Multiply
and division to explain that 4 ÷ (1/5)
“one cubic unit” of volume, and can be into categories based on their
fractional side lengths to find areas of rectangles,
= 20 because 20 × (1/5) = 4.
used to measure volume.
properties.
and represent fraction products as rectangular
b. A solid figure which can be packed
areas.
without gaps or overlaps using n unit 5.3. Understand that attributes
cubes is said to have a volume of n
belonging to a category of
5.5. Interpret multiplication as scaling (resizing),
c. Solve real world problems
cubic units.
twodimensional figures also belong
by:
involving division of unit fractions
to all subcategories of that category.
a. Comparing the size of a product to the size of
by non-zero whole numbers and
5.4. Measure volumes by counting unit
For example, all rectangles have
one factor on the basis of the size of the other
division of whole numbers by unit cubes, using cubic cm, cubic in, cubic ft, four right angles and squares are
factor, without performing the
fractions, e.g., by using visual
and improvised units.
rectangles, so all squares have four
indicated multiplication.
fraction models and equations to
right angles.
b. Explaining why multiplying a given number by represent the problem. For example, 5.5. Relate volume to the operations of
a fraction greater than 1 results in a product
how much chocolate will each
multiplication and addition and solve real 5.4. Classify two-dimensional
greater than the given number (recognizing
person get if 3 people share 1/2 lb of world and mathematical problems
figures in a hierarchy based on
multiplication by whole numbers greater than 1
chocolate equally? How many 1/3- involving volume.
properties.
as a familiar case); explaining why multiplying a
cup servings are in 2 cups of raisins?
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.