Lesson Title: 7.NS.A.2b Dividing Integers
Date: _____________ Teacher(s): ____________________
Course: Mathematics 7
Start/end times: _________________________
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which Mathematical
Practices do you expect students to engage in during the lesson?
7.NS.A.2b Understand that integers can be divided, provided that the divisor is not zero, and
every quotient of integers (with non-zero divisor) is a rational number. If p and q
are integers, then −(p / q) = (− p) / q = p / (−q) . Interpret quotients of rational numbers by describing real-world
contexts.
MP1:
MP3:
MP4:
MP6:
MP7:
Make sense of problems and persevere in solving them.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Attend to precision.
Look for and make use of structure.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Describe how you multiply two integers. What
strategies do you use?
(Have students participate in a “carousel brainstorm” to
discuss strategies. Have students small groups to create
a quick idea web of what they know about multiplying
integers. Encourage them to use vocabulary and
examples to describe what they know about the
concept. After about 3-5 minutes, have students walk
around the room and look at other groups’ idea webs in
order to generate more ideas of the prior knowledge.)
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
1. What connection can you make between multiplying
and dividing integers?
2. What are some similarities and difference between
dividing and subtracting integers?
3. What other connections can we make between
integers and our own life? (Look for evidence of
MP7.)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
1. Review multiplication of integers and explore dividing integers by having students complete the matching
game to start their own thinking about dividing integers:
Source - http://www.quia.com/mc/1669.html
(Additional game boards found at this website)
2. Once the matching activity has been completed, ask the questions:
“What similarities do you see between dividing integers and multiplying integers?”
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: 7.NS.A.2b Dividing Integers
Course: Mathematics 7
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
“What did you notice about the signs with the division problems?”
“Fill in the blanks: If multiplication is repeated addition, then division is repeated _____________.”
(subtraction)
“How could we rewrite the division number sentence 12 ¸ 3 using subtraction? (Answer: 12-3 = 9, 9-3 = 6,
6-3 = 3, 3-3 = 0, therefore, we subtracted 3 (added -3) 4 times to equal 0. This is connected to
multiplication because -3(4) = -12.
*Absolute Value should also enter into the conversation when conducting division/repeated subtraction. (Look
for evidence of MP7.)
3. Next, the students will engage in a dialogue about “why” these sign rules make sense for multiplication.
Provide them with a number line, and ask them to make a visual representation of what -3(4)=-12 looks like
on the number line. Encourage them to think about what the problem means and how that may translate to a
picture. (Look for evidence of MP4.)
4. From this, have the students think about the sign rules for dividing integers in their small groups, and add
them to their idea webs. They can then walk around the room again to compare other groups’ ideas.
+/+ = +
+/- = -
-/- = +
-/+ = -
Real World Discussion Point – In real life, you can connect a negative integer with “losing money” and a
positive integer with “gaining or earning money.”
5. Use the following website as a resource to further illustrate and explain this concept:
http://www.homeschoolmath.net/teaching/integers.php. The blue “Animation” links are very helpful, as
they show the visual representations on a number line.
6. After this, engage the students in a discussion about what division of
integers may look like on a number line, in order to establish meaning.
Number Line Source www.mathamnesia.com/download.cfm?file_id=139
Additional Resources:
-Van de Walle: Elementary and Middle School Mathematics (page 484).
-www.grade8mathlinks.files.wordpress.com/.../8_4_dividing_integers.pdf
-Hands-On Standards (Grades 5-6), pages 120-121
7. Now give the students the following problem: Stephanie owes Kourtney $12. She is required to pay
Kourtney back in three equal payments. How much money is Stephanie sending each time she pays
Kourtney? Have a class discussion about why the solution would be -$4 (because Stephanie’s account is
losing $4 with each payment that she is making).
Now show the following division sentence on the board and have them use the counter ships to model the
solution: -12 ¸ 3 = -4. Encourage them to think about what this sentence means, as well as what it looks
like. Also, make sure to emphasize correct use of vocabulary, as they discuss the math processes.
In this case, “-12” is the dividend, “3” is the divisor, and “-4” is the quotient.
“-12 broken into 3 equal groups, results in -4 in each group.”
8. Now, assign the following division sentences to the class for them to model, except by drawing the counters
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: 7.NS.A.2b Dividing Integers
Course: Mathematics 7
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
and coloring each to indicate the correct signs. Each group will receive their own sentence to (1) model, (2)
explain the meaning of, and (3) write the corresponding fact family. They can then jigsaw and have
representatives from each group travel around to explain their own sentences. (Look for evidence MP3 and
MP4.)
Sample problems: -20
¸ 5 = -4
15
¸3 = 5
-12
¸ 6 = -2
-8
¸ 4 = -2
9. Ask the question, “Please model 12 ¸ 0.” Is this possible? Why or why not?”
It is crucial that the students understand that the divisor (‘0’) cannot be a zero. Guide the students to understand
why.
10. Formative Assessment – Have the students individually match the quotients with the division sentences.
Some suggested methods for doing this are Every Pupil Response (EPR) cards or Four Squares (they walk to
a corner of the room based on what they think the correct quotient is).
Sample problems: -24
¸ 3 = -8
63
¸ -9 = -7
-56
¸ -8 = 7
100
¸ -50 = -2
11. Present the students with the following problem:
You decide to open a savings account with the Columbia Bank. For this particular account, you must maintain
an *average monthly balance of $10, or they will charge you a $15 penalty fee that is deducted from your
account. However, if you do maintain a balance of $30, then you will be rewarded with a deposit in your
account of $5. A “deposit” is the same as a positive number and a “withdrawal” is the same as a negative
number. Your first monthly bank statement is as follows:
*“Average” may be a skill that needs to be reviewed with the students.
Date
May 5
May 6
May 8
May 11
May 15
May 19
May 21
May 22
May 24
May 27
May 28
May 30
Deposit
Withdrawal
Balance
$10
$15
$3
$5
$6
$8
$12
$10
$5
$7
$10
$25
Procedures:
12. Complete the balance column, based on the deposits and withdrawals.
13. Calculate the approximate average balance for the month of May in order to determine if you will be
charged a penalty fee. Show how you calculated your work.
C. Based on whether you received a $15 penalty fee or a $5 reward in your account, calculate your new
average monthly balance. (Look for evidence of MP1 and MP6.)
Answers – Monthly balance = -$24 ¸ 12 deposits/withdrawals = average monthly balance of -$2.00. Based on
this, you will be charged a penalty fee of $15.00. Therefore, the monthly balance of -$24 plus -$15 = new
monthly balance of -$39.00. Based on this, your new average monthly balance will be -$3.00.
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: 7.NS.A.2b Dividing Integers
Date: _____________ Teacher(s): ____________________
Course: Mathematics 7
Start/end times: _________________________
The students will be able to successfully divide integers, and apply the correct sign rules. The teacher will know that
the students have mastered the concept of dividing integers as evidence of the bank activity.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Vocabulary: dividend, divisor, quotient, average, balance, checking account, deposit, withdrawal
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Red/Yellow Counter Chips
Algebra Tiles or Blocks
Van de Walle Elementary and Middle School
Mathematics
Please apply the concept of dividing integers to the
following real world problems. For each situation, write a
division sentence, using integers and the correct signs.
(“Owing money” is a negative number). (Look for
evidence of MP1.)
1. Kenny owes GameStop $35. Each of 5 friends will
help him pay off his debt. How much will each friend
pay? Write a number sentence to show your
understanding of the problem, then solve.
2. This week’s daily temperatures for Bismark, North
Dakota are as follows: -2, 5, 8, -4, -3, 8. Find the
average daily temperature of the town. Write a
number sentence to show your understanding of the
problem, then solve.
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
Were the students able to make the connection between multiplying and dividing integers?
Were the students able to make the connection between dividing and subtracting?
Were the students able to explain “why” multiplication and division of integers makes sense?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.