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7.NS.2
2012
Domain: The Number System
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide
rational numbers.
Standards: 7.NS.2 Apply and extend previous understanding of multiplication and division and of fractions to multiply and
divide rational numbers.
a. understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to
satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the
rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, the –(p/q) = (-p)/q = p/(-q). Interpret quotients of rational
numbers by describing real-world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division,; know that the decimal form of a rational number terminates
in 0s or eventually repeats.
Essential Questions
Enduring Understandings
Activities, Investigation, and Student Experiences
 What models and
relationships help you
make sense of
multiplying and dividing
positive and negative
numbers?
 Understand that multiplication
and division of whole
numbers is extended to
integers by requiring that
operations continue to satisfy
properties of operations.
 How does knowing how
to add positive and
negative integers help
you multiply positive
and negative integers?
 There are similarities and
differences between
multiplying rational numbers
and multiplying fractions and
decimals.
Activity:
 Create a real-world situation that will help students to
discover the procedures for multiplying rational
numbers. For example, “You have a bank account that
you forgot about and it currently has a balance of $0.
The bank charges a service fee of $3 every month. What
would the balance be after 5 months?” Hold a brief
discussion about what the equation used to solve this
problem would look like (5(-3).
 Which number is negative? Why?
 How do properties of
addition and
multiplication help you
multiply positive and
 The rules for multiplying
rational numbers.
 Have students work with a partner to solve and share out
with the class. How did they know if the answer was
negative or positive? Challenge the students by telling
them the bank wants to refund the money. Have each
pair of students determine a new equation (-5)(-3), solve
and share out with the class. Again, how did they know
7.NS.2
negative integers?
 How does the
relationship between
multiplication and
division help you divide
integers and other
rational numbers?
 What models and
relationships help you
make sense of
multiplying and dividing
positive and negative
rational numbers?
 Ability to explore and justify
the result of division by 0.
 Ability to apply and extend
knowledge of addition and
subtraction of integers (i.e.
two color counters, arrows on
a number line) to extend to
division.
 Ability to use patterns and
concrete models to devise a
general rule for dividing.
 Deciding the order in which to
carry out the operations when
 Why do rational
a problem consists of more
numbers in decimal form
than one operation.
either terminate or
repeat?
 Ability to identify and apply
the Distributive, Associative,
 How are properties
Commutative, and Identity
useful in solving a
Properties.
variety of problems?
 Ability to recognize that when
rational numbers in fractional
Content Statements
2012
if the answer was positive or negative? Present students
with several other problems 3(-2), (-3)(-2), (-3)2.
Encourage students to look for patterns. Have students
create rules for multiplying with negative numbers. In
what situations is the answer negative? Positive?
Activity:
 Brainstorm a list of real-world scenarios in which
students will need to multiply negative numbers.
Activity:
 Use a spinner, dice OR number cards to build numbers to
multiply. Give students parameters to follow such as
how many digits each number must include. Student
will spin to create their number, and then spin to
determine whether it is negative or positive. (Odds can
represent a negative, evens can represent a positive).
Students partner with a classmate to play. The winner is
the student who has the largest/smallest product.
Activity:
 Have students play “war” with a deck of cards. The red
cards are negatives, black cards are positive. Students
deal out the cards, flip two cards each, multiply, and the
student with the larger product receives the cards.
7.NS.2
 Apply the rules for
multiplying integers.
 Review the
Commutative Property
of Multiplication.
 Multiply rational
numbers.
 Divide integers,
recognize when a
quotient is undefined.
 Express a quotient of
integers as a rational
number.
 Write a rational number
as a quotient of integers.
 Apply previous
understandings of
division of factions to
write division by a
rational number as
multiplication by the
reciprocal.
 Apply properties of
operations as strategies
to multiply and divide
rational numbers.
form are converted to
decimals they either terminate
or repeat.
2012
Assign number values to the Ace, King, Queen, and
Jack.
Activity:
 Pair up students. Present –(8/2). Ask for answer, (-4).
Present (-8)/4. Again ask for volunteer to answer, (-4).
Ask how they know the answer is negative. Present 8/(2). Ask for answer and how they know. Have a brief
discussion about the three problems. What do the three
problems have in common? What conclusions can you
draw about dividing negative numbers? Work in pairs to
solve several sets of problems similar to…
Ex. –(12/3), -(12)/3, 12/(-3), -12/-3
Generate a class discussion about the problems. Have students
make generalizations about dividing negative numbers and
create a set of rules detailing how to divide negative numbers.
Activity:
 Play a game where students use a spinner (or number
cards). Students spin to select numbers and create a
division problem. Challenge them to manipulate the
negative sign to create as many different problems as
they can that will all have the same answer.
Investigation:
 Present additional problems to students that include
decimals and fractions and challenge students to discover
whether the rule they established still applies.
Activity:
 Create four sets of problems with 5 problems in each set.
Ex. 3(5+4), 4 ( ½ + ¼), 1/3 (6+3), -5(-3 + 2), -3/4 (1/2 –
7.NS.2
 Simplify complex
fractions involving
rational numbers.
 Apply the distributive
property to multiplying
rational numbers.
 Solve real-world
problems involving
more than one operation
with rational numbers.
 Convert fractions to
decimals and vice versa.
 Determine whether a
decimal is terminating or
repeating.
Assessments

The temperature in Bar Harbor, Maine, was -3 °F. It
then dropped during the night to be four times as
cold. What was the temperature then?

You are in a submarine at -15 feet below sea level.
You descend three times your current location. How
far below sea level are you now?

A teacher has 5 cups of M&M’s to be shared equally
with 6 students. How many cups would each student
receive?
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¼). Have students complete one set of problems with a
partner. Once students are finished, put students into
groups of 8 so that all of the problem sets are
represented. Ask each pair to present their problems to
the larger group and share their strategies and methods.
Hold a class discussion to summarize the properties they
applied and how they used their prior knowledge to solve
the more complex ones. How did they treat fractions?
How did they decide which answers were negative and
which were positive?
Activity:
 Partner students up. Give each pair of students a baggie.
Students will match the fractions to their decimal
equivalents. Once students have found the ten matched
pairs, have a class discussion about strategies students
used to find the answers. Make sure students are able to
articulate what process and operation they used. What
did they do when there weren’t any more digits in the
dividend but needed to avoid a remainder? What did
they do when they came across a repeating decimal?
Have students summarize the process for converting fractions to
decimals, using their journals.
***Sample fractions: ½, 3/8, ¼, 2/5, 5/8, 1/3, 2/3, ¾, 5/6, 4/5.
7.NS.2

-5/6 = ___/-6 Find the missing value.

A recipe calls for 2 ¾ cups of whole wheat flour. If
Ms. Ambrose wanted to triple the recipe, how many
cups of flour would she need?

Mr. Mischel wants to divide 3 ½ containers of
fertilizer equally amongst 8 flower beds. How many
containers will he need each day if he plans on filling
2 flower beds per day?

Provide students with one terminating decimal, one
repeating decimal in fraction form, and a number line.
Have students graph the given rational numbers on
the number line, determine one repeating and one
terminating decimal between the given rational
numbers, identify those that are terminating and
repeating and explain why.

The temperature at 6:00 a.m. was -5 0 F. At 2:00 p.m.
the temperature had increased to 140 F. What was the
temperature at 2:00 p.m.?
Equipment Needed:

Deck of cards

Real world scenarios

Spinners, Dice or number cards

Paper

Pencils
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Teacher Resources:


http://illuminations.nctm.org/
https://thinkfinity.org
SmartBoard Lessons on Fractions:
 http://exchange.smarttech.com/search.html?q=%20multi
plying%20and%20dividing%20fractions...
7.NS.2

Division Problems

Problem sets

Baggie of 10 fractions and corresponding decimals.
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Various Fraction Activities:
 http://www.khanacademy.org/#Algebra
 http://thinkfinity.org/search.jspa?peopleEnabled=true&u
serID=&containerType=&container=&q=fraction
Interactive Website on multiplying fractions
 http://www.visualfractions.com/multiply.htm


YouTube video on multiplying fractions using
rectangular models:
http://www.youtube.com/watch?v=6YLi5_U5Fk0
Reinforcing Integer Concepts with a Card Game
http://illuminations.nctm.org/LessonDetail.aspx?id=L81
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