Lesson Title: 7.NS.A.2b Dividing Integers Course: Common Core Mathematics 7
Date: _____________ Teacher(s): ____________________ 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: Make sense of problems and persevere in solving them.
MP3: Construct viable arguments and critique the reasoning of others.
MP4: Model with mathematics.
MP6: Attend to precision.
MP7: 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.
What connection can you make between multiplying and dividing integers?
What are some similarities and difference between dividing and subtracting integers?
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.
Distribute the red/yellow counter chips (Algebra Tiles or Blocks can also be used) to the students. Using the counter ships, ask them to model: -3(4) = -12 (Look for evidence of MP4.)
Also, prompt them to think of -3(4) = 12 as “4 groups of -3.”
2. Have the students determine the fact family for the multiplication sentence -3(4) = -12.
-3(4) = -12 -12

-3 = 4 4(-3) = -12 -12

4 = -3
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand , S. (2009).
Accessible mathematics : 10 instructional shifts that raise student

Portsmouth , NH : Heinemann .


Lesson Title: 7.NS.A.2b Dividing Integers Course: Common Core Mathematics 7
Date: _____________ Teacher(s): ____________________ Start/end times:
3. Administer the matching game to the students to start their own thinking about dividing integers:
Source - http://www.quia.com/mc/1669.html
(Additional game boards found at this website)
4. Encourage them to find the matches and make connections between this and multiplying integers.
5. Ask the questions:
“What similarities do you see between dividing integers and multiplying integers?”
“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.

(Look for evidence of MP7.)
6.
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.”
7.
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.)
8.
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.
9.
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
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: Common Core Mathematics 7
Date: _____________ Teacher(s): ____________________ Start/end times:
10.
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.”
11.
Now, assign the following division sentences to the class for them to model, except by drawing the counters 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
12.
Ask the question, “Please model 12 why.
 

0.” Is this possible? Why or why not?”
 
13.
 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
14.
Present the students with the following problem:
   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 Deposit Withdrawal Balance
May 5
May 6
May 8
May 11
May 15
May 19
May 21
May 22
May 24
May 27
May 28
May 30
$10
$3
$6
$12
$5
$10
$15
$5
$8
$10
$7
$25
Procedures:
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: Common Core Mathematics 7
Date: _____________ Teacher(s): ____________________ Start/end times:
A.
Complete the balance column, based on the deposits and withdrawals.
B.
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:
 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.
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?
Red/Yellow Counter Chips
Algebra Tiles or Blocks
Van de Walle Elementary and Middle School
Mathematics
Homework: Exactly what follow-up homework tasks, problems, and/or exercises will be assigned upon the completion of the lesson?
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?
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand , S. (2009).
Accessible mathematics : 10 instructional shifts that raise student achievement. Portsmouth , NH : Heinemann .