Lesson 9: Applying the Properties of Operations to Add and...
Transcript of Lesson 9: Applying the Properties of Operations to Add and...
Lesson 9: Applying the Properties of Operations to Add and Subtract Rational Numbers
Date: 10/27/14
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
Lesson 9: Applying the Properties of Operations to Add and
Subtract Rational Numbers
Student Outcomes
Students use properties of operations to add and subtract rational numbers without the use of a calculator.
Students recognize that any problem involving addition and subtraction of rational numbers can be written as
a problem using addition and subtraction of positive numbers only.
Students use the commutative and associative properties of addition to rewrite numerical expressions in
different forms. They know that the opposite of a sum is the sum of the opposites; e.g., β (3 β 4) = β3 + 4.
Classwork
Fluency Exercise (6 minutes): Integer Subtraction
Photocopy the attached 2-page fluency-building exercises so that each student receives a copy. Time the students,
allowing one minute to complete Side A. Before students begin, inform them that they may not skip over questions and
that they must move in order. After one minute, discuss the answers. Before administering Side B, elicit strategies from
those students who were able to accurately complete many problems on Side A. Administer Side B in the same fashion,
and review the answers. Refer to the Sprints and Sprint Delivery Script sections in the Module Overview for directions to
administer a Sprint.
Exercise 1 (6 minutes)
Students are given scrambled steps to one possible solution to the following problem1. They work independently to
arrange the expressions in an order that leads to a solution and record their solutions in the student materials.
Exercise 1
Unscramble the cards, and show the steps in the correct order to arrive at the solution to πππ
β (π. π + πππ
).
1 The scrambled steps may also be displayed on an interactive whiteboard, and students can come up one at a time to slide a step into
the correct position.
πππ
+ (βπ. π + (β πππ
))
βπ. π
πππ
+ (β πππ
+ (βπ. π))
π + (βπ. π) (πππ
+ (β πππ
)) + (βπ. π)
MP.8
Lesson 9: Applying the Properties of Operations to Add and Subtract Rational Numbers
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
ππ
π+ (βπ. π + (βπ
π
π)) The opposite of a sum is the sum of its opposites.
ππ
π+ (βπ
π
π+ (βπ. π)) Apply the commutative property of addition.
(ππ
π+ (βπ
π
π)) + (βπ. π) Apply the associative property of addition.
π + (βπ. π) A number plus its opposite equals zero.
βπ. π Apply the additive identity property.
After 2 minutes, students share the correct sequence of steps with the class.
What allows us to represent operations in another form and rearrange the order
of terms?
The properties of operations.
Specifically which properties of operations were used in this example?
Students recall the additive inverse property and commutative property of addition. (Students are
reminded to focus on all of the properties that justify their steps today.)
Why did we use the properties of operations?
Students recognize that using the properties allows us to efficiently (more easily) calculate the answer
to the problem.
Examples 1β2 (7 minutes)
Students record the following examples. Students assist in volunteering verbal explanations for each step during the
whole-group discussion. Today, studentsβ focus is not on memorizing the names of each property but rather knowing
that each representation is justifiable through the properties of operations.
Examples 1β2
Represent each of the following expressions as one rational number. Show and explain your
steps.
1. πππ
β (πππ
β ππ)
= πππ
β (πππ
+ (βππ)) Subtracting a number is the same as adding its inverse.
= πππ
+ (βπππ
+ ππ) The opposite of a sum is the sum of its opposites.
= (πππ
+ (βπππ
)) + ππ The associative property of addition.
= π + ππ A number plus its opposite equals zero.
= ππ
First, predict the answer. Explain your prediction.
Answers may vary.
Scaffolding:
Provide a list of the properties of operations as a reference sheet for students.
Scaffolding:
Adapt for struggling learners by having fewer steps to rearrange or by including only integers.
Adapt for proficient students by requiring that they state the property of operations to justify each step.
Lesson 9: Applying the Properties of Operations to Add and Subtract Rational Numbers
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
The answer will be between 0 and 1
2 because 5 + (β5) = 0 and β4
4
7 is close to β5, but 5 has a larger absolute
value than β44
7. To add 5 + (β4
4
7), we subtract their absolute values. Since β4
4
7 is close to β4
1
2, the answer
will be about 5 β 41
2=
1
2.
2. π + (βπππ
)
= π + [β (π +ππ
)] The mixed number πππ
is equivalent to π +ππ
.
= π + (βπ + (βππ
)) The opposite of a sum is the sum of its opposites.
= (π + (βπ)) + (βππ
) Associative property of addition.
= π + (βππ
) π + (βπ) = π
=ππ
+ (βππ
) π
π = π
=π
π
Does our answer match our prediction?
Yes, we predicted a positive number close to zero.
Exercise 2 (10 minutes): Team Work!
Students work in groups of three. Each student has a different colored pencil. Each problem has at least three steps.
Students take turns writing a step to each problem, passing the paper to the next person, and rotating the student who
starts first with each new problem.
After 8 minutes, students partner up with another group of students to discuss or debate their answers. Students should
also explain their steps and the properties/rules that justify each step.
Exercise 2: Team Work!
a. βπ. π β (βπ. π) + π. π b. ππ + (βππππ
)
= βπ. π + π. π + π. π = ππ + (βππ + (βππ
))
= βπ. π + π. π + π. π = (ππ + (βππ)) + (βππ
)
= π + π. π = ππ + (βππ
)
= π. π = ππππ
MP.2 &
MP.3
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
c. πππ
+ ππ. π β (βπππ
) d. ππ
ππβ (βπ. π) β
π
π
πππ
+ ππ. π + πππ
=ππππ
+ π. π βππ
= πππ
+ πππ
+ ππ. π =ππππ
+ π. π + (βππ
)
= πππ
+ ππ. π =ππππ
+ (βππ
) + π. π
= π + ππ. π =ππππ
+ (βππππ
) + π. π
= ππ. π = π + π. π
= π. π
Exercise 3 (5 minutes)
Students work independently to answer the following question, then after 3 minutes, group members share their
responses with one another and come to a consensus.
Exercise 3
Explain step by step, how to arrive at a single rational number to represent the following expression. Show both a
written explanation and the related math work for each step.
βππ β (βπ
π) β ππ. π
Subtracting (βππ
) is the same as adding its inverse π
π: = βππ +
ππ
+ (βππ. π)
Next, I used the commutative property of addition to rewrite the expression: = βππ + (βππ. π) +ππ
Next, I added both negative numbers: = βππ. π +ππ
Next, I wrote π
π in its decimal form: = βππ. π + π. π
Lastly, I added βππ. π + π. π: = βππ
Closing (3 minutes)
How are the properties of operations helpful when finding the sums and differences of rational numbers?
The properties of operations allow us to add and subtract rational numbers more efficiently.
Do you think the properties of operations could be used in a similar way to aid in the multiplication and
division of rational numbers?
Answers will vary.
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
Exit Ticket (8 minutes)
Lesson Summary
Use the properties of operations to add and subtract rational numbers more efficiently. For instance,
βπππ
+ π. π + πππ
= (βπππ
+ πππ
) + π. π = π + π. π = π. π.
The opposite of a sum is the sum of its opposites as shown in the examples that follow:
βπππ
= βπ + (βππ
).
β(π + π) = βπ + (βπ).
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
Name Date
Lesson 9: Applying the Properties of Operations to Add and
Subtract Rational Numbers
Exit Ticket
1. Jamie was working on his math homework with his friend, Kent. Jamie looked at the following problem.
β9.5 β (β8) β 6.5
He told Kent that he did not know how to subtract negative numbers. Kent said that he knew how to solve the
problem using only addition. What did Kent mean by that? Explain. Then, show your work and represent the
answer as a single rational number.
_______________________________________________________________________________________________
_______________________________________________________________________________________________
_______________________________________________________________________________________________
Work Space:
Answer: ___________________
2. Use one rational number to represent the following expression. Show your work.
3 + (β0.2) β 1514
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
Exit Ticket Sample Solutions
1. Jamie was working on his math homework with his friend, Kent. Jamie looked at the following problem
βπ. π β (βπ) β π. π
He told Kent that he did not know how to subtract negative numbers. Kent said that he knew how to solve the
problem using only addition. What did Kent mean by that? Explain. Then, show your work and represent the
answer as a single rational number.
Kent meant that since any subtraction problem can be written as an addition problem by adding the opposite of the
number you are subtracting, Jamie can solve the problem by using only addition.
Work Space:
βπ. π β (βπ) β π. π
= βπ. π + π + (βπ. π)
= βπ. π + (βπ. π) + π
= βππ + π
= βπ
Answer: βπ
2. Use one rational number to represent the following expression. Show your work.
π + (βπ. π) β ππππ
= π + (βπ. π) + (βππ + (βπ
π))
= π + (βπ. π + (βππ) + (β π. ππ))
= π + (βππ. ππ)
= βππ. ππ
Problem Set Sample Solutions
Show all steps taken to rewrite each of the following as a single rational number.
1. ππ + (βπππ
ππ) 2. ππ + (βπ
ππ
)
= ππ + (βππ + (β πππ
)) = ππ + (βπ + (βππ
))
= (ππ + (βππ)) + (βπ
ππ) = (ππ + (βπ)) + (β
ππ
)
= ππ + (βπ
ππ) = π + (β
ππ
)
= ππππππ
= πππ
Lesson 9: Applying the Properties of Operations to Add and Subtract Rational Numbers
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
3. π
π+ ππ. π β (βπ
π
π) 4.
ππ
ππβ (βππ) β
π
π
=ππ
+ ππ. π + πππ
= ππ ππ
+ ππ + (βππ
)
=ππ
+ πππ
+ ππ. π = ππ ππ
+ (βππ
) + ππ
= πππ
+ ππ. π = ππ ππ
+ (β ππ ππ
) + ππ
= πππ
+ πππ
ππ =
π ππ
+ ππ
= ππ
ππ+ ππ
πππ
= ππ π
ππ
= ππππππ
= πππ
ππ
5. Explain, step by step, how to arrive at a single rational number to represent the following expression. Show both a
written explanation and the related math work for each step.
π βπ
π+ (βππ
π
π)
First, I rewrote the subtraction of π
π as the addition of its inverse β
ππ
: = π + (βππ
) + (βππππ
)
Next, I used the associative property of addition to regroup addend: = π + ((βππ
) + (βππππ
))
Next, I separated βππππ
into the sum of βππ and βππ
: = π + ((βππ
) + (βππ) + (βππ
))
Next, I used the commutative property of addition: = π + ((βππ
) + (βππ
) + (βππ))
Next, I found the sum of βππ
and βππ
: = π + ((βπ) + (βππ))
Next, I found the sum of βπ and βππ: = π + (βππ)
Lastly, since the absolute value of ππ is greater than the absolute value = βππ
of π, and it is a negative ππ, the answer will be a negative number.
The absolute value of ππ minus the absolute value of π equals ππ,
so the answer is βππ.
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
Integer Subtraction β Round 1
Directions: Determine the difference of the integers, and write it in the column to the right.
1. 4 β 2 23. (β6) β 5
2. 4 β 3 24. (β6) β 7
3. 4 β 4 25. (β6) β 9
4. 4 β 5 26. (β14) β 9
5. 4 β 6 27. (β25) β 9
6. 4 β 9 28. (β12) β 12
7. 4 β 10 29. (β26) β 26
8. 4 β 20 30. (β13) β 21
9. 4 β 80 31. (β25) β 75
10. 4 β 100 32. (β411) β 811
11. 4 β (β1) 33. (β234) β 543
12. 4 β (β2) 34. (β3) β (β1)
13. 4 β (β3) 35. (β3) β (β2)
14. 4 β (β7) 36. (β3) β (β3)
15. 4 β (β17) 37. (β3) β (β4)
16. 4 β (β27) 38. (β3) β (β8)
17. 4 β (β127) 39. (β30) β (β45)
18. 14 β (β6) 40. (β27) β (β13)
19. 23 β (β8) 41. (β13) β (β27)
20. 8 β (β23) 42. (β4) β (β3)
21. 51 β (β3) 43. (β3) β (β4)
22. 48 β (β5) 44. (β1,066) β (β34)
Number Correct: ______
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
Integer Subtraction β Round 1 [KEY]
Directions: Determine the difference of the integers, and write it in the column to the right.
1. 4 β 2 π 23. (β6) β 5 βππ
2. 4 β 3 π 24. (β6) β 7 βππ
3. 4 β 4 π 25. (β6) β 9 βππ
4. 4 β 5 βπ 26. (β14) β 9 βππ
5. 4 β 6 βπ 27. (β25) β 9 βππ
6. 4 β 9 βπ 28. (β12) β 12 βππ
7. 4 β 10 βπ 29. (β26) β 26 βππ
8. 4 β 20 βππ 30. (β13) β 21 βππ
9. 4 β 80 βππ 31. (β25) β 75 βπππ
10. 4 β 100 βππ 32. (β411) β 811 βπ, πππ
11. 4 β (β1) π 33. (β234) β 543 βπππ
12. 4 β (β2) π 34. (β3) β (β1) βπ
13. 4 β (β3) π 35. (β3) β (β2) βπ
14. 4 β (β7) ππ 36. (β3) β (β3) π
15. 4 β (β17) ππ 37. (β3) β (β4) π
16. 4 β (β27) ππ 38. (β3) β (β8) π
17. 4 β (β127) πππ 39. (β30) β (β45) ππ
18. 14 β (β6) ππ 40. (β27) β (β13) βππ
19. 23 β (β8) ππ 41. (β13) β (β27) ππ
20. 8 β (β23) ππ 42. (β4) β (β3) βπ
21. 51 β (β3) ππ 43. (β3) β (β4) π
22. 48 β (β5) ππ 44. (β1,066) β (β34) βπ, πππ
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
Integer Subtraction β Round 2
Directions: Determine the difference of the integers, and write it in the column to the right.
1. 3 β 2 23. (β8) β 5
2. 3 β 3 24. (β8) β 7
3. 3 β 4 25. (β8) β 9
4. 3 β 5 26. (β15) β 9
5. 3 β 6 27. (β35) β 9
6. 3 β 9 28. (β22) β 22
7. 3 β 10 29. (β27) β 27
8. 3 β 20 30. (β14) β 21
9. 3 β 80 31. (β22) β 72
10. 3 β 100 32. (β311) β 611
11. 3 β (β1) 33. (β345) β 654
12. 3 β (β2) 34. (β2) β (β1)
13. 3 β (β3) 35. (β2) β (β2)
14. 3 β (β7) 36. (β2) β (β3)
15. 3 β (β17) 37. (β2) β (β4)
16. 3 β (β27) 38. (β2) β (β8)
17. 3 β (β127) 39. (β20) β (β45)
18. 13 β (β6) 40. (β24) β (β13)
19. 24 β (β8) 41. (β13) β (β24)
20. 5 β (β23) 42. (β5) β (β3)
21. 61 β (β3) 43. (β3) β (β5)
22. 58 β (β5) 44. (β1,034) β (β31)
Number Correct: ______
Improvement: ______
Lesson 9: Applying the Properties of Operations to Add and Subtract Rational Numbers
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NYS COMMON CORE MATHEMATICS CURRICULUM 7β’2 Lesson 9
Integer Subtraction β Round 2 [KEY]
Directions: Determine the difference of the integers, and write it in the column to the right.
1. 3 β 2 π 23. (β8) β 5 βππ
2. 3 β 3 π 24. (β8) β 7 βππ
3. 3 β 4 βπ 25. (β8) β 9 βππ
4. 3 β 5 βπ 26. (β15) β 9 βππ
5. 3 β 6 βπ 27. (β35) β 9 βππ
6. 3 β 9 βπ 28. (β22) β 22 βππ
7. 3 β 10 βπ 29. (β27) β 27 βππ
8. 3 β 20 βππ 30. (β14) β 21 βππ
9. 3 β 80 βππ 31. (β22) β 72 βππ
10. 3 β 100 βππ 32. (β311) β 611 βπππ
11. 3 β (β1) π 33. (β345) β 654 βπππ
12. 3 β (β2) π 34. (β2) β (β1) βπ
13. 3 β (β3) π 35. (β2) β (β2) π
14. 3 β (β7) ππ 36. (β2) β (β3) π
15. 3 β (β17) ππ 37. (β2) β (β4) π
16. 3 β (β27) ππ 38. (β2) β (β8) π
17. 3 β (β127) πππ 39. (β20) β (β45) ππ
18. 13 β (β6) ππ 40. (β24) β (β13) βππ
19. 24 β (β8) ππ 41. (β13) β (β24) ππ
20. 5 β (β23) ππ 42. (β5) β (β3) βπ
21. 61 β (β3) ππ 43. (β3) β (β5) π
22. 58 β (β5) ππ 44. (β1,034) β (β31) βπ, πππ