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CCSS Operations and Algebraic
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Thinking (OA)
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Unpacking the Standards
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Grade 4
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Math Practices: MP2, MP4
Standard: 4.OA.1 Cluster (m/s/a)
Related CA Standard
NEW
Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as
many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication
equations.
Essential Skills/Concepts
--Recognize a multiplicative comparison
problem
--Understand the order and meaning of
numbers in multiplicative comparison
problems (see examples below)
--Identify and verbalize all three
quantities of the problem OR which
quantity is being multiplied (smaller
quantity), which number tells how many
times, and which number is the product
(bigger quantity)
--Solve word problems using
multiplicative comparisons (see
examples below)
Examples:
a. Sally is five years old. Her mom is
eight times older. How old is Sally’s
Mom?
b. Sally has five times as many pencils
as Mary. If Sally has 5 pencils, how
many does Mary have?
Teaching Notes/Strategies
Manipulatives
linking cubes
two-color chips
Use comparison bar diagrams to model
problems and show the comparisons
Sally = 5 pencils
? pencils
? pencils
? pencils
? pencils
Resources
? pencils
Use manipulatives (ex: counters, linking
cubes) to visually represent objects
Have students work in pairs to solve
problems using manipulatives
Use language to show comparisons (i.e., how
many times more, how many times greater)
Draw a picture to represent the
problem/equation
Board Math
Cooperative Logic Activities from the
book Group Solutions, by Jan. M
Goodman (GEMS, Lawrence Hall of
Science).
Representing multiplicative comparison
problems at http://www.k5mathteachingresources.com/
Students create or play a matching game
using mult. comparison phrases such as 4
times as many as 2 matches 4 x 2 or 2 x 4
Academic Vocabulary:
Use information in the problem to write an
equation (i.e., 5 x ___ = 5 OR 5 x m = 5)
comparison problems
Use equation frames for students to fill in
missing information:
This shows ___ times ___
___ times as much is ___
___ times more is ___
___ times as many is ____
equation
times
times more/greater
Math Practices: MP2, MP4, MP5, MP7
Standard: 4.OA.2 Cluster (m/s/a)
Related CA Standard
4.AF.1.0
Multiply or divide to solve word problems involving multiplicative comparison, e.g, by using drawings and equations
with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from
additive comparison.
Essential Skills/Concepts
--Recognize an additive and multiplicative
comparison problem
--Know the difference between an additive
and multiplicative comparison problem
--Understand the order and meaning of
numbers in both types of comparison
problems (see examples below)
--Identify and verbalize the quantities of
the equation
--Solve word problems using additive and
multiplicative comparisons (see examples
below)
Examples:
a. A blue scarf costs $3. A red scarf
costs $6 more than the blue scarf.
How much does the blue scarf cost?
b. Lane saved $5 from his allowance.
Michael saved nine times of Lane’s
amount. How much did Michael save?
Teaching Notes/Strategies
Compare the operations of addition and
multiplication including phrases used in each
type of equation (i.e., how many times more
vs. how many more or how much less)
Cooperative Logic Activities from the
book Group Solutions, by Jan. M
Goodman (GEMS, Lawrence Hall of
Science).
Use comparison bar diagrams to model
problems and show the comparisons
Sally = 5 pencils
? pencils
? pencils
? pencils
Manipulatives:
linking cubes
two-color chips
Board Math
Directly teach the word “variable” or
“symbol” and it’s use in equations to
represent unknown quantities
? pencils
Resources
? pencils
Use manipulatives (ex: counters, linking
cubes) to visually represent objects
Have students work in pairs to solve both
additive and multiplicative comparison
problems using manipulatives
Draw a picture to represent the
problem/equation using variables/symbols
for unknown quantities
Use information in the problem to write an
equation (i.e., 5 x ___ = 5 OR 5 x m = 5)
Sample Multiplicative Comparison
problems at http://www.k5mathteachingresources.com/
Academic Vocabulary:
Equation
How many times more
How many more
How much less
Multiplicative Comparison problems
Additive Comparison problems
Variable or symbol
Standard: 4.OA.3 Cluster (m/s/a)
Math Practices: MP1, MP2, MP4,
Related CA Standard
4.NS1.4, 4.AF.1.1
MP5, MP6, MP7
Solve multistep word problems posed with whole numbers and having whole-number answers using the four
operations, including problems in which remainders must be interpreted. 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.
Essential Skills/Concepts
Fluency of basic facts in all four
operations (not essential, but desired)
Recognize/Determine the
operations/steps that are needed to
solve the problem
Be able to interpret remainders in a
problem
Use estimation to check for
reasonableness
Examples:
a. There are 146 students going on
a field trip. If each bus held 30
students, how many buses are
needed?
b. Suppose that 250 pencils were
distributed equally among 33
students for a geometry project.
What is the largest number of
pencils each student can receive?
Academic Vocabulary:
Operation
Equation
Variable
Teaching Notes/Strategies
Resources
Use manipulatives (ex: counters, linking
cubes, base ten blocks) to visually
represent objects
Manipulatives
linking cubes
two-color chips
Have students work in pairs to solve
multistep word problems using
manipulatives
Multistep word problems
Interpreting remainders
Both at http://www.k5mathteachingresources.com/
Draw a picture/diagram to represent
the problem/equation using
variables/symbols for unknown
quantities
Use the PUSD Universal Problem Solving
Strategy to solve word problems
Use error-analysis to determine what
was done incorrectly when solving a
problem
Problem solving strategy anchor chart
Board Math
Standard: 4.OA.4 Cluster (m/s/a)
Math Practices: MP2, MP7
Related CA Standard
4.NS4.1, 4.NS.2
Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of
its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number.
Determine whether a given whole number in the range 1-100 is prime or composite.
Essential Skills/Concepts
Identify (find) the factor pairs for
numbers 1 through 100.
Recognize that a number is a multiple of
each of its factors (Ex: 21 is a multiple
of 3 and a multiple of 7 because 3x7=21)
Determine multiples for the numbers 1
through 100.
Identify numbers 1 through 100 as
prime or composite.
Know that 1 is neither prime nor
composite because prime numbers have
exactly 2 factors (itself and 1).
Academic Vocabulary:
Multiples
Products
Factors and factor pairs
Prime
Composite
Teaching Notes/Strategies
Search systematically for factor pairs by
checking each number (2, 3, 4, etc.) until
“reversals” in the pairs are found
Create Factor trees for prime factorization
Alternate Model for prime factorization for
the number 180.
180
2 90
5 45
180 = 2 x 3 x 3 x 3 x 5
3 9
3 3
1
Using a 100 chart, create the “Sieve of
Eratosthones” to find primes (see Teacher
Share for this resource/directions.)
Use a multiplication chart to find multiples and
factors
Draw arrays for factor pairs
Write out factor pairs in a “T” table
Use skip counting (by 3’s, 5’s, etc.) to find
multiples
Resources
Manipulatives:
Two-color chips
100 charts
Anchor charts of the factor
tree and the alternate model.
Multiplication charts
Finding Multiples
Prime Number Hunt
Common Multiples
Least Common Multiple
Find the Factor are available
at
http://www.k5mathteachingresources.com
Math Practices: MP2, MP4, MP5, MP7
Standard: 4.OA.5 Cluster (m/s/a)
Related CA Standard
Partial 7.AF.1.1
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were
not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the
resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain
informally why the numbers will continue to alternate this way.
Essential Skills/Concepts
Understand patterns and explaining the
reasoning behind a given pattern (i.e.,
examine a dot pattern in which each
design has 4 more dots than the
previous one and then reason about how
the dots are organized to determine the
total number of dots in the 100th design)
Generate and analyze number and shape
patterns following a given rule
Analyze the results of a pattern using a
given rule and explain why the pattern
of the results will continue.
Sample Problems:
a. There are 4 beans in a jar. Each day
3 beans are added. How many beans
are in the jar for each for the first
5 days (Make a table)
b. Complete the table using the rule
g=m + 14
c. Fill in the missing number in this
sequence 1, 2, 4, __, 16, 32
Teaching Notes/Strategies
Resources
Examine varying types of patterns and explaining
the reasoning behind a given pattern (Ex:
patterns using dots, numbers, real objects,
colors, shapes which include “growing” and multioperational patterns, etc. See Resource column
)
Manipulatives: Two-color chips
Have students extend a given pattern and explain
reasoning (i.e., give the 20th term in the pattern)
by using an equation
“Patterns that Grow” available at
http://illuminations.nctm.org/
LessonDetail.aspx?ID=U103
Use the “Bridge Map” (see Thinking Maps) to
break down and analyze patterns.
Anchor charts for tables with patterns
and Bridge Map analyzing a pattern
2 +2 4+2 6
+2
8+2 10 Rule = +2
Have students generate patterns and write a rule
for the pattern
Use a table and function input-output tables to
map out patterns and find or follow the rule with
and without algebraic notations (f, x, y)
Use patterns in real-life situations (i.e. house
numbers, growing earnings, etc.)
“Pattern a Day” or Number Talks problem to
analyze/solve.
Examples of problems:
“Double Plus One” available at
http://illustrative
mathematics.org/standards/k8#
Board Math
Functions Foldable Chart (Toni Torres)
Square Numbers and Triangular
Numbers available at http://www.k5mathteachingresources.com
Academic Vocabulary:
Pattern
Sequence
Term
Rule (mathematical definition)