Download Go On - Triumph Learning

Contents
Domain 1 Operations and Algebraic Thinking. . . . . . . . . 4
Common Core
State Standards
Lesson 1
Evaluating Numerical Expressions . . . . . . . . . . . . . . . . . . . . 6
5.OA.1
Lesson 2
Writing and Interpreting Numerical Expressions. . . . . . . . 10
5.OA.2
Lesson 3
Analyzing and Generating Numerical Patterns . . . . . . . . . 14
5.OA.3
Domain 1 Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Domain 2 Number and Operations in Base Ten. . . . . . 24
Lesson 4
Multiplying and Dividing by Powers of Ten. . . . . . . . . . . . . 26
5.NBT.1, 5.NBT.2
Lesson 5
Using Place Value to Read and Write Decimals . . . . . . . . . 32
5.NBT.3.a
Lesson 6
Comparing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.NBT.3.b
Lesson 7
Rounding Decimals Using Place Value. . . . . . . . . . . . . 42
5.NBT.4
Lesson 8
Multiplying Whole Numbers . . . . . . . . . . . . . . . . . 48
5.NBT.5
Lesson 9
Dividing Whole Numbers. . . . . . . . . . . . . . . . . . . . . . . 54
5.NBT.6
Lesson 10
Adding and Subtracting Decimals. . . . . . . . . . . . . . . . . . . . 62
5.NBT.7
Lesson 11
Multiplying Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.NBT.7
Lesson 12
Dividing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.NBT.7
Domain 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Lesson 13
Adding and Subtracting Fractions
and Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.NF.1
Problem Solving: Adding and Subtracting
Fractions and Mixed Numbers . . . . . . . . . . . . . . . . . . 100
5.NF.2
Problem Solving: Interpreting Fractions
as Division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.NF.3
Lesson 14
Lesson 15
Lesson 16
Multiplying Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.NF.4.a, 5.NF.4.b
Lesson 17
Interpreting Multiplication of Fractions. . . . . . . . . . . . . . . 120
5.NF.5.a, 5.NF.5.b
Lesson 18
Problem Solving: Multiplying Fractions and
Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2 CC12_MTH_G5_SE_FM_Final.indd 2
Problem
Solving
Fluency
Lesson
5.NF.6
Duplicating any part of this book is prohibited by law.
Domain 3 Number and Operations—Fractions . . . . . . . 90
Performance
Task
15/06/12 9:50 AM
Common Core
State Standards
Lesson 19
Dividing with Unit Fractions and Whole Numbers. . . . . . 132
Lesson 20
Problem Solving: Dividing with Unit Fractions . . . . . 138
5.NF.7.a, 5.NF.7.b
5.NF.7.c
Domain 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Domain 4 Measurement and Data. . . . . . . . . . . . . . . . . . . 146
Converting Units of Measure to Solve Problems . . . . . . . 148
5.MD.1
Lesson 22 Line Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.MD.2
Lesson 23 Understanding and Measuring Volume . . . . . . . . . . . . . . 160
5.MD.3.a, 5.MD.3.b, 5.MD.4
Lesson 21
Lesson 24
Lesson 25
Finding Volume of Rectangular
Prisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.MD.5.a, 5.MD.5.b
Recognizing Volume as Additive . . . . . . . . . . . . . . . . 170
5.MD.5.c
Domain 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Domain 5 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Lesson 26 Graphing Points on the Coordinate Plane . . . . . . . . . . . . 180
5.G.1
The Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.G.2
Lesson 27
Lesson 28 Extending Classification of
Two-Dimensional Figures. . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.G.3, 5.G.4
Domain 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Duplicating any part of this book is prohibited by law.
Math Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
CC12_MTH_G5_SE_FM_Final.indd 3
3
15/06/12 9:50 AM
LE
SS
O
N
1
EXAMPLE A
Evaluating Numerical Expressions
Evaluate this numerical expression. (15 2 5) 1 (4 3 3)
1
Do the operation inside the first set
of parentheses ( ).
Subtract.
(15 2 5) 5 10
10 1 (4 3 3)
2
Do the operation inside the second
set of parentheses.
Multiply.
4 3 3 5 12
3
10 1 12
Evaluate the remaining expression.
Add.
10 1 12 5 22
22
CU S S
DIS
Explain how you would evaluate
(3 3 9) 2 (8 1 8). What is the value
of the expression?
6 Duplicating any part of this book is prohibited by law.
▸ The value of the expression is 22.
Domain 1: Operations and Algebraic Thinking
CC12_MTH_G5_SE_D1_Final.indd 6
15/06/12 9:51 AM
EXAMPLE B
Evaluate this numerical expression. 30 2 [(5 3 4) 4 2] 1 17
1
Do the operation inside the
parentheses.
Multiply.
(5 3 4) 5 20
30 2 [20 4 2] 1 17
2
Do the operation inside the
brackets [ ].
Divide.
[20 4 2] 5 10
3
30 2 10 1 17
Evaluate the remaining expression.
The subtraction comes first in the
expression.
Subtract first.
30 2 10 5 20
20 1 17
Then add.
Duplicating any part of this book is prohibited by law.
20 1 17 5 37
37
▸ The value of the expression is 37.
TRY
Insert brackets in 36 4 (12 2 9) 3 3
so that the value of the expression is 4.
Lesson 1: Evaluating Numerical Expressions CC12_MTH_G5_SE_D1_Final.indd 7
7
15/06/12 9:51 AM
Practice
Use the following expression to answer questions 1 through 3.
13 1 (8 3 3)
What operation should you do first?
H
IN
T
1.
Which expression is inside
the parentheses?
2. What operation should you do next?
3. What is the answer?
Use the order of operations to evaluate each expression.
4. [8 1 (12 2 6)] 3 5
The value of the expression is
5. 7 3 (6 2 2) 4 2
.
The value of the expression is
.
REMEMBER Start with the
operation inside the parentheses.
Then work inside the brackets.
The value of the expression is
7.
.
8. (6 1 8) 4 2 1 1
The value of the expression is
8 [8 3 (5 1 5)] 4 4
The value of the expression is
.
9. 55 4 5 2 [(3 3 2) 1 4]
.
The value of the expression is
.
Duplicating any part of this book is prohibited by law.
6. 2 3 (3 3 4) 2 (5 1 13)
Domain 1: Operations and Algebraic Thinking
CC12_MTH_G5_SE_D1_Final.indd 8
15/06/12 9:51 AM
Insert parentheses so that the value of the expression matches the value given.
10. 10 1 7 3 6 2 2
The value of the expression is 50.
12. 25 2 10 1 5 3 4
The value of the expression is 35.
11. 9 3 4 1 8 4 2
The value of the expression is 40.
13. 4 1 6 4 2 2 5
The value of the expression is 2.
Choose the best answer.
14. Which operation should you do first to
evaluate 5 1 [(3 3 4) 2 7]?
15. Which expression does not have
a value of 6?
A. addition
A. 3 1 (7 3 2) 2 11
B. subtraction
B. 2 3 [(2 1 5) 2 3]
C. multiplication
C. (8 4 4) 1 (2 3 2)
D. division
D. [4 2 (8 2 7)] 3 2
Solve.
16. Denzel bought a large pizza for $14 and 2 drinks for $3 each. He had a coupon for $1
off the total amount of his purchase. The following expression represents how much
Denzel spent.
Duplicating any part of this book is prohibited by law.
14 1 (2 3 3) 2 1
How much did Denzel spend on the pizza and the 2 drinks?
17. DECIDE Marcus says the value of 1 1 [4 3 (9 2 6)] is 31. Erica says the value of the
expression is 13. Who is correct? Explain.
Lesson 1: Evaluating Numerical Expressions CC12_MTH_G5_SE_D1_Final.indd 9
9
15/06/12 9:51 AM
Contents
Domain Assessment—Operations and Algebraic Thinking. . . . . . . . . . . . . . . . . . . . . . . . . . 4
Domain Assessment—Number and Operations in Base Ten. . . . . . . . . . . . . . . . . . . . . . . . 14
Domain Assessment—Number and Operations—Fractions. . . . . . . . . . . . . . . . . . . . . . . . 22
Domain Assessment—Measurement and Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Domain Assessment—Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Duplicating any part of this book is prohibited by law.
Summative Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3
CC12_MTH_G5_NAA_FM_Final.indd 3
19/06/12 10:14 AM
Domain Assessment • Operations and Algebraic Thinking
1.
What does the expression 75 2 5 mean?
5.
A. 75 divided by 5
A. 76 2 (6 3 2) is 76 less than (6 3 2).
B. 5 less than 75
2.
Which of the following is a true
statement?
C. 5 more than 75
B. (100 2 25) 4 3 is 3 times as great
as (100 2 25).
D. 75 times 5
C. 5 3 (5 1 10) is 5 more than (5 1 10).
D. 10 3 (43 2 3) is 10 times as great
as (43 2 3).
Which expression means the same as
“9 more than 18 divided by 3”?
A. (18 1 3) 4 9
C. (18 4 3) 1 9
How would you use numbers and
symbols to express “28 divided by
1
the product of ​ __
2  ​and 8”?
D. (9 4 3) 1 18
A. 28 4 ​ __
​ 8 ​ 3 8  ​
6.
B. (9 1 3) 4 18
B.
3.
( 1 )
1
28 4 ​( __
​ 2 ​ 3 8 )​
1
28 2 ​( __
​ 2 ​ 3 8 )​
What is the value of 21 1 4 3 8 2 2?
C.
A. 51
1
D. __
​ 2 ​ 3 8 3 28
B. 98
C. 150
7.
Use the order of operations to evaluate
1
__
​ 4  ​3 (40 1 16).
A. 14
4.
Which word phrase is equivalent
to 35 1 28 2 10?
A. 35 more than the quotient of
28 divided by 10
B. 26
C. 44
D. 176
B. 10 less than the sum of 35 and 28
C. 10 more than the sum of 35 and 28
D. the sum of 35 and 28 and 10
Go On
Duplicating any part of this book is prohibited by law.
D. 198
4
CC12_MTH_G5_NAA_Final.indd 4
19/06/12 10:15 AM
Look at the patterns below.
Pattern 1: 2, 5, 8, 11, 14, …
10. Khalia’s teacher wrote two rules on
the board.
Operations and
Algebraic Thinking
8.
Pattern 2: 2, 14, 26, 38, 50, …
What are the rules for the patterns?
A. Pattern 1: add 3; Pattern 2: add 6
Start at 3, add 5
Start at 6, add 4
B. Pattern 1: add 3; Pattern 2: add 12
C. Pattern 1: add 3; Pattern 2: add 14
D. Pattern 1: add 3; Pattern 2: multiply
by 7
9.
Use the order of operations to evaluate
5 3 [36 4 (4 3 3)].
A. 15
B. 29
C. 90
D. 135
If Khalia were to write out the sequences
based on these two rules, what would
they look like?
A. 3, 6, 9, 12, 15, 18 and 6, 10, 14,
18, 22, 26
B. 3, 8, 11, 14, 17, 20 and 6, 12, 18,
24, 30, 36
C. 3, 8, 11, 14, 17, 20 and 6, 12, 18,
24, 30, 36
Duplicating any part of this book is prohibited by law.
D. 3, 8, 13, 18, 23, 28 and 6, 10, 14,
18, 22, 26
Go On
5
CC12_MTH_G5_NAA_Final.indd 5
19/06/12 10:15 AM
11. Jamie and Marcus created two different patterns. The rule for Jamie’s pattern was start at 2 and
add 2. The rule for Marcus’s pattern was start at 2 and multiply by 2.
A.
Jamie’s
Pattern
2
4
6
8
10
Marcus’s
Pattern
2
4
8
16
32
Ordered Pair
(2, 2)
(4, 4)
(6, 8)
(8, 16)
(10, 32)
C.
Jamie’s
Pattern
2
2
6
8
8
Marcus’s
Pattern
4
4
8
16
10
Ordered
Pair
(2, 4)
(2, 4)
(6, 8)
(8, 16)
(8, 10)
B.
Jamie’s
Pattern
2
6
10
4
16
Marcus’s
Pattern
4
8
2
8
32
Ordered Pair
(2, 4)
(6, 8)
(10, 2)
(4, 8)
(16, 32)
D.
Jamie’s
Pattern
2
4
6
8
8
Marcus’s
Pattern
4
2
8
6
16
Ordered
Pair
(2, 4)
(4, 2)
(6, 8)
(8, 6)
(8, 16)
Go On
Duplicating any part of this book is prohibited by law.
Which of the following tables shows the ordered pairs made from the corresponding terms of
the sequences for Jamie and Marcus’s patterns?
6
CC12_MTH_G5_NAA_Final.indd 6
19/06/12 10:15 AM
14. Which expression has a value of 33?
Operations and
Algebraic Thinking
12. Which of the following shows how
to find the value of the expression
7 3 (18 2 9)?
A. 5 1 (10 3 2) 1 8 4 4 1 1
A. Find the quotient of 18 divided by 9,
then multiply by 7.
B. 5 1 10 3 2 1 (8 4 4) 1 1
B. Find the difference of 18 and 9, then
multiply by 7.
D. (5 1 10) 3 2 1 8 4 4 1 1
C. 5 1 10 3 (2 1 8) 4 4 1 1
C. Find the sum of 18 and 9, then
multiply by 7.
D. Multiply 7 and 18, then subtract 9.
13. Use the order of operations to evaluate
5
1
1
__
__
2 ​ __
8 ​1 ​ 4 ​ 2 1 ​ 2 ​ 
1
A. 1 ​ __
4 ​ 
3
B. 1 ​ __
8 ​
C. 2
3
Duplicating any part of this book is prohibited by law.
D. 2 ​ __
8 ​
Go On
7
CC12_MTH_G5_NAA_Final.indd 7
19/06/12 10:15 AM
15. Zack created a table of ordered pairs from the sequences for two patterns. The rules for the
two patterns are shown below.
Pattern A: Start at 0 and add 4
Pattern B: Start at 0 and add 6
Pattern A gives the x-value and Pattern B gives the y-value. Zack graphed the ordered pairs,
as shown below.
y
32
28
24
20
16
12
8
4
0
2
4
6
8 10 12 14 16 18
x
Which of these tables best reflects the patterns displayed in Zack’s graph?
Pattern A Pattern B
0
4
8
12
18
B.
1
12
18
22
36
Pattern A Pattern B
1
4
8
12
16
0
6
12
18
24
Ordered Pair
(0, 1)
(4, 12)
(8, 18)
(12, 22)
(18, 36)
C.
Ordered Pair
(1, 0)
(4, 6)
(8, 12)
(12, 18)
(16, 24)
D.
Pattern A Pattern B
0
4
8
12
16
0
6
12
18
24
Pattern A Pattern B
0
4
8
12
16
12
24
36
48
60
Ordered
Pair
(0, 0)
(4, 6)
(8, 12)
(12, 18)
(16, 24)
Ordered
Pair
(0, 12)
(4, 24)
(8, 36)
(12, 48)
(16, 60)
Go On
Duplicating any part of this book is prohibited by law.
A.
8
CC12_MTH_G5_NAA_Final.indd 8
19/06/12 10:15 AM
Operations and
Algebraic Thinking
16. What is the value of the expression [(7 3 8) 4 2] 2 11? Show your work.
Duplicating any part of this book is prohibited by law.
17. Xenia told her teacher that she had found a mistake in her math textbook. In one of the
lessons, it said that (222 1 11) 4 3 is three times as great as the sum of 222 and 11. What
mistake did Xenia find?
Go On
9
CC12_MTH_G5_NAA_Final.indd 9
19/06/12 10:15 AM
18. Ms. Solis wrote the following equation on the blackboard for her fifth-grade class.
1
61 - 7 ÷ 7 × 2 + 17 = 47
Go On
Duplicating any part of this book is prohibited by law.
Where in the equation do you need to add parentheses and brackets so that the equation
is correct? Explain your answer using an example of what would happen if parentheses and
brackets are added incorrectly.
10
CC12_MTH_G5_NAA_Final.indd 10
19/06/12 10:15 AM
macarons, $2
Operations and
Algebraic Thinking
19. A fancy pastry shop charges $2 for each macaron and $6 for each petit four.
petit fours, $6
A. Write the pattern that shows the cost of 0 to 6 macarons. Write the pattern that shows
the cost of 0 to 6 petit fours.
B. Use the corresponding terms from the patterns you created in Part A to complete the
table below. Then describe the relationship between corresponding terms.
Duplicating any part of this book is prohibited by law.
Macarons Pattern
(in $)
Petit Fours Pattern
(in $)
Ordered Pair
Go On
11
CC12_MTH_G5_NAA_Final.indd 11
19/06/12 10:15 AM
20. Will chose 2 cards from a pile. The cards are shown below.
Card A
Card B
Start with 0
and
add 2.
Start with 0
and
add 3.
Will used the rules on the cards to write a sequence of numbers for each pattern. He then
created a table of ordered pairs using the corresponding terms from the patterns.
A. Use the blank table below to show how Will completed his table of ordered pairs.
Card B Pattern
Ordered Pair
Go On
Duplicating any part of this book is prohibited by law.
Card A Pattern
12
CC12_MTH_G5_NAA_Final.indd 12
19/06/12 10:15 AM
Operations and
Algebraic Thinking
B. Graph the ordered pairs on the graph below. Be sure to label each axis and scale.
Then describe the relationship between the terms that make up the ordered pairs.
y
Duplicating any part of this book is prohibited by law.
x
STOP
13
CC12_MTH_G5_NAA_Final.indd 13
19/06/12 10:15 AM
Contents
Instructional Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Common Core State Standards Correlation Chart . . . . . . . . . . . . . . . . . 12
Domain 1 Operations and Algebraic Thinking. . . . . . . . . . . 16
Common Core
State Standards
Lesson 1
Evaluating Numerical Expressions . . . . . . . . . . . . . . . . . . . . . 18
5.OA.1
Lesson 2
Writing and Interpreting Numerical Expressions. . . . . . . . . . 20
5.OA.2
Lesson 3
Analyzing and Generating Numerical Patterns . . . . . . . . . . . 22
5.OA.3
Domain 2 Number and Operations in Base Ten. . . . . . . . . 24
Lesson 4
Multiplying and Dividing by Powers of Ten. . . . . . . . . . . . . . . 26
5.NBT.1, 5.NBT.2
Lesson 5
Using Place Value to Read and Write Decimals . . . . . . . . . . . 28
5.NBT.3.a
Lesson 6
Comparing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.NBT.3.b
Lesson 7
Rounding Decimals Using Place Value. . . . . . . . . . . . . . . 32
5.NBT.4
Lesson 8
Multiplying Whole Numbers . . . . . . . . . . . . . . . . . . . 34
5.NBT.5
Lesson 9
Dividing Whole Numbers. . . . . . . . . . . . . . . . . . . . . . . . . 36
5.NBT.6
Lesson 10
Adding and Subtracting Decimals. . . . . . . . . . . . . . . . . . . . . . 38
5.NBT.7
Lesson 11
Multiplying Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.NBT.7
Lesson 12
Dividing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.NBT.7
Adding and Subtracting Fractions and Mixed Numbers. . . . 46
5.NF.1
Problem Solving: Adding and Subtracting Fractions
and Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.NF.2
Lesson 15
Problem Solving: Interpreting Fractions as Division. . . . 50
5.NF.3
Lesson 16
Multiplying Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.NF.4.a, 5.NF.4.b
Lesson 17
Interpreting Multiplication of Fractions. . . . . . . . . . . . . . . . . . 54
5.NF.5.a, 5.NF.5.b
Lesson 18
Problem Solving: Multiplying Fractions and
Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Lesson 13
Lesson 14
Lesson 19
Dividing with Unit Fractions and Whole Numbers . . . . . . . . 58
Lesson 20
Problem Solving: Dividing with Unit Fractions . . . . . . . . 60
Problem
Solving
Fluency
Lesson
5.NF.6
5.NF.7.a, 5.NF.7.b
5.NF.7.c
Duplicating any part of this book is prohibited by law.
Domain 3 Number and Operations—Fractions . . . . . . . . . . 44
Performance
Task
2
CC12_MTH_G5_TM_Final.indd 2
23/07/12 3:26 PM
Common Core
State Standards
Domain 4 Measurement and Data. . . . . . . . . . . . . . . . . . . . . . . 62
Converting Units of Measure to Solve Problems . . . . . . . . . . 64
5.MD.1
Lesson 22 Line Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.MD.2
Lesson 23 Understanding and Measuring Volume . . . . . . . . . . . . . . . . . 68
5.MD.3.a, 5.MD.3.b,
Lesson 21
5.MD.4
Lesson 24
Finding Volume of Rectangular Prisms. . . . . . . . . . . . . . . 70
5.MD.5.a, 5.MD.5.b
Lesson 25
Recognizing Volume as Additive . . . . . . . . . . . . . . . . . . . 72
5.MD.5.c
Domain 5 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Lesson 26 Graphing Points on the Coordinate Plane . . . . . . . . . . . . . . . 76
5.G.1
The Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.G.2
Lesson 27
Lesson 28 Extending Classification of Two-Dimensional Figures. . . . . . 80
5.G.3, 5.G.4
Answer Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Math Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Appendix A: Fluency Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A
Duplicating any part of this book is prohibited by law.
Appendix B: Standards for Mathematical Practice. . . . . . . . . . . . . . . . . . B
3
CC12_MTH_G5_TM_Final.indd 3
23/07/12 3:26 PM
LE
SS
O
N
1
Evaluating Numerical Expressions
Learning Objective
• Students will evaluate numerical expressions using the correct
order of operations.
Vocabulary
numerical expression a combination of numbers and
operation signs
Before the Lesson
Write a numerical expression that includes
parentheses, such as (12 1 8) 2 (3 3 5), and ask for
a volunteer to evaluate it. Then write an expression
such as the following: 18 1 [(2 3 3) 4 3] 2 2. Ask
another student to evaluate the expression. Ask the
students to explain what is different about the two
expressions, aside from the numbers. Encourage
them to notice that the first expression includes two
sets of parentheses, while the second expression
Common Core State Standard
5.OA.1 Use parentheses,
brackets, or braces in numerical
expressions, and evaluate
expressions with these symbols.
includes both parentheses and brackets. They
should notice, too, that the parentheses are inside
the brackets. Be sure students understand that they
should work from the inside to the outside, so that
they start with the innermost set of parentheses and
do the work inside. Then they work on what is inside
the brackets. After all the grouping symbols have
been removed, they use the remaining operation
symbols to finish evaluating the expression.
Examples
This example illustrates how to apply
the order of operations to a numerical expression
that includes parentheses. Emphasize that the
operations that are inside parentheses must be
performed first. Ask: Which two operations must be
performed first? As students look over their work,
point out how after each operation is performed,
the entire expression is rewritten with the result
of that operation substituted into it. Stress the
importance of rewriting the expression to help
students keep track of their work.
DISCUSS MP7 Use this feature to encourage
students to talk about the procedures involved in
evaluating a numerical expression.
First perform the operations in parentheses:
3 3 9 5 27 and 8 1 8 5 16. Then subtract:
27 2 16 5 11; the answer is 11.
EXAMPLE B This example shows how to evaluate
a numerical expression that includes two types of
grouping symbols: parentheses and brackets. Use
this example to demonstrate that students must
evaluate the entire expression inside the brackets,
(5 3 4) 4 2, before they evaluate anything else.
Ask: Which operation must be performed first?
Which must be performed second?
Point out how the entire expression is rewritten
with 10 replacing the entire expression inside the
brackets. Reinforce the importance of rewriting the
entire expression to help students keep track of
their work.
To ensure that students understand this content,
rewrite the problem on the board as [30 2 (5 3 4)] 4
2 1 17. Ask: How would the value of the expression
change if the brackets were moved like this? Use the
fact that the value is now 22 to show students how
important it is to pay attention to the positioning of
the parentheses and brackets.
Duplicating any part of this book is prohibited by law.
EXAMPLE A 18
CC12_MTH_G5_TM_Final.indd 18
23/07/12 3:27 PM
Practice
Common Errors
When evaluating numerical expressions with
parentheses inside brackets, some students
may not do the work inside the parentheses
first before evaluating the rest of the expression
inside the brackets. Remind students to treat
the entire expression inside the brackets as a
numerical expression and do what is inside the
parentheses first.
Domain 1
TRY MP1 Encourage students to perform
operations mentally or use trial and error to
determine where to place the brackets to get the
correct result. If students use trial and error, ask
students how many tries it took to find the correct
answer.
36 4 [(12 2 9) 3 3]
Duplicating any part of this book is prohibited by law.
As students are working, pay special attention to
problem 16, which provides an opportunity for
students to see how a numerical expression that
includes parentheses can be used to represent a
real-world problem.
For answers, see page 82.
19
CC12_MTH_G5_TM_Final.indd 19
23/07/12 3:27 PM
Learning Objectives
• Students will write numerical expressions to represent verbal
phrases or statements using grouping symbols to indicate which
operations are performed first.
• Students will write verbal phrases or statements to represent
numerical expressions.
Before the Lesson
Before class, get a bottle of water, with the cap screwed on, and a
cup. Tell students to imagine that you are an alien from outer space.
Tell them that the alien wants to know how to pour a cup of water for
itself. Ask: How would you tell the alien to do this, using simple steps?
Write students’ responses on the board. Using bullets, list each step
on the board or on a sheet of paper. You can list them one above the
other or next to one another. For example, students might suggest
these steps:
• Unscrew the cap of the bottle.
• Remove the cap.
• Tilt the bottle over the glass so that its top is over the glass.
Then follow the directions out of order, stating aloud which step you
are currently following. For example, tilt the bottle with the cap still
screwed on over the glass. Then unscrew the cap and remove it.
Ask: Why didn’t it work? The alien followed all your steps. Use this
to explain that sometimes, when you write directions, the order of
the steps matters and sometimes it does not. Ask students how you
could have made this clearer to the alien. They may suggest replacing
the bullets with ordinal numbers. Explain that sometimes students
will need to write a numerical expression to represent a phrase or a
statement. If the order in which the operations must be performed
matters, students should use parentheses or other grouping symbols
to indicate the correct order.
Common Core State Standard
5.OA.2 Write simple
expressions that record
calculations with numbers, and
interpret numerical expressions
without evaluating them. For
example, express the calculation
“add 8 and 7, then multiply by
2” as 2 3 (8 1 7). Recognize
that 3 3 (18,932 1 921) is three
times as large as 18,932 1 921
without having to calculate the
indicated sum or product.
Examples
EXAMPLE A This example introduces how to write
a numerical expression to represent a phrase.
When they are considering the first part of the
expression, be sure students understand that
“subtract 6 from 10” means 10 2 6, not 6 2 10.
If necessary, review the fact that subtraction is not
commutative to help students understand why it
is important to pay attention to which number is
being subtracted.
Emphasize that parentheses must be included in
the expression to clarify which operation should be
performed first. Since 6 must be subtracted from 10
and only then multiplied by 5, parentheses must be
placed around the subtraction expression 10 2 6.
Duplicating any part of this book is prohibited by law.
LE
SS
O
N
2
Writing and Interpreting Numerical
Expressions
20
CC12_MTH_G5_TM_Final.indd 20
23/07/12 3:27 PM
As students are working, pay special attention
to problems 21 and 22, which require students
to translate real-world problem situations
into numerical expressions. These are less
straightforward than other translations in this
lesson. For example, help any students who are
confused by problem 21 see that having 12 crackers
and eating 2 can be represented as the subtraction
expression 12 2 2. Then help them see why John’s
dividing the crackers equally between himself and
his friend is a division by 2.
For answers, see page 82.
Common Errors
A common error may be not understanding that
the work inside parentheses is done first, no matter
where parentheses fall in the expression. If students
do not understand this, they may write “multiply
24 by 709 and then add 54” for question 13. In
addition, they may believe that problem 19 has
no correct answer and that only the expression
(593 1 88) 3 12 is a correct way to represent
the sentence. Review the fact that multiplication
is commutative to help students see that 12 3
(593 1 88) 5 (593 1 88) 3 12, so either is the
correct answer for problem 19.
Duplicating any part of this book is prohibited by law.
EXAMPLE B This example is the reverse of
Example A. It shows students how to write a
word phrase to represent a numerical expression.
Stress that there may be more than one way
to do this. For example, after working through
Example B, ask: Can you think of another way to
use words to represent this expression? Students
may suggest “add 2 and 7, then divide by 3” or
“divide the sum of 7 and 2 by 3.”
DISCUSS MP7 MP8 Emphasize that students
should not evaluate (9 1 1) 3 6 but should instead
apply their understanding of the work they
performed for (10 2 6) 3 5 to reason the answer.
If some students are not convinced, you may need
to evaluate (9 1 1) 3 6 to show that it is six times as
large as (9 1 1).
(10 2 6) 3 5 5 20; (10 2 6) 5 4; 20 is five times as
great as 4.
Answers may vary. Possible answer: Without
calculating, you can tell that the value of (9 1 1) 3 6
is six times as great as the value of (9 1 1).
Practice
Domain 1
TRY MP2 Encourage students to include
parentheses in the numerical expressions they
write. Emphasize that including the parentheses
makes it very clear which operation is to be
performed first.
12 1 8; (6 2 5) 1 3; (18 4 6) 3 9
21
CC12_MTH_G5_TM_Final.indd 21
23/07/12 3:27 PM
LE
SS
O
N
3
Analyzing and Generating Numerical
Patterns
Learning Objective
• Students will find the rule for a numerical pattern, identify the
relationship between corresponding terms in two different
numerical patterns, write ordered pairs to represent corresponding
terms in two different numerical patterns, and plot ordered pairs
on the coordinate plane.
Vocabulary
coordinate plane a grid formed by a horizontal line, called
the x-axis, and a vertical line, called the y-axis
ordered pair two numbers that give a location on a
coordinate plane
pattern a series of numbers or figures that follows a rule
rule tells how the numbers in a pattern are related
term a number or figure in a pattern
x-axis the left-right or horizontal axis on a coordinate plane
x-coordinate the first number in an ordered pair
y-axis the up-down or vertical axis on a coordinate plane
Common
CommonCore
CoreState
StateStandard(s)
Standard
5.OA.3 Generate two
numerical patterns using two
given rules. Identify apparent
relationships between
corresponding terms. Form
ordered pairs consisting of
corresponding terms from the
two patterns, and graph the
ordered pairs on a coordinate
plane. For example, given the
rule “Add 3” and the starting
number 0, and given the rule
“Add 6” and the starting number
0, generate terms in the resulting
sequences, and observe that
the terms in one sequence are
twice the corresponding terms
in the other sequence. Explain
informally why this is so.
y-coordinate the second number in an ordered pair
Write the following on the board: 1, 3, , ….
Ask: What number goes in the blank to continue
the pattern?
Some students may say that the next number is 5.
Others may say it is 9. Still others may correctly state
that not enough information is given to determine
the next number. Whatever answers students give,
use them to illustrate the fact that the numbers,
or terms, in a pattern follow a rule. Students who
believe the next number is 5 believe that the rule
is “add 2.” Students who believe that the next
number is 9 believe that the rule is “multiply by 3.”
Additional information is needed to determine the
rule, but if these numbers form a pattern, they must
follow a rule.
Examples
EXAMPLE A This example introduces how to find a
rule for a numerical pattern. Emphasize the fact that
since the numbers are increasing, the pattern could
involve either addition or multiplication.
TRY MP8 Explain that students should start with
the term 57 and then apply the rule “add 9” to find
the next three consecutive terms.
66, 75, 84
EXAMPLE B This example requires students to
relate the corresponding terms of two patterns. Be
sure to stress to students that even though the first
terms in both patterns are the same, 0, the other
consecutive terms are not the same, so they should
try to determine how the terms are related.
Duplicating any part of this book is prohibited by law.
Before the Lesson
22
CC12_MTH_G5_TM_Final.indd 22
23/07/12 3:27 PM
DISCUSS MP6 To benefit visual learners, show
how to plot (3, 5) and (5, 3) on the same grid.
This will help students understand that the two
ordered pairs represent different locations on the
coordinate plane.
Answers may vary. Possible answer: To graph (3, 5),
move 3 units to the right and then move 5 units up.
To graph (5, 3), move 5 units to the right and then
move 3 units up.
EXAMPLE C This example is similar to Example B,
except that the rules and first terms are given and
students must generate the first several terms in
each pattern. Explain that the ellipsis (…) is placed
after the last term found for each pattern to show
that the established pattern will continue.
Practice
Duplicating any part of this book is prohibited by law.
TRY MP7 Encourage students to find at least
five terms in each pattern and to organize the
terms in a table so it is easier for them to compare
corresponding terms.
The first pattern is 0, 2, 4, 6, 8, …. The second
pattern is 0, 6, 12, 18, 24, …. Answers may vary.
Possible answer: The terms in the second pattern
are three times the corresponding terms in the first
pattern. This is true because adding 6 to each term
in the second pattern is three times as much as
adding 2 to each term in the first pattern.
EXAMPLE D This example shows how
corresponding terms in two patterns can be
considered as ordered pairs and how those
ordered pairs can be plotted on a grid to help
students see the relationship between them.
If necessary, review how to plot points on a
coordinate plane. For example, show students
that (2, 4) can be located by starting at the origin
and moving 2 units to the right and 4 units up.
Show that the next point, (4, 8), can be found by
starting at (2, 4) and moving 2 units to the right and
4 units up. Emphasize that for each unit you move
to the right, you move twice as many units up. This
shows that each term in Pattern 2 is twice that of its
corresponding term in Pattern 1.
Domain 1
TRY MP7 Encourage students to use the rule
for each pattern to find its next term. Emphasize
that if they do this correctly, the relationship
between the two terms they find will be the same
as the relationship between Pattern 1 and Pattern 2
corresponding terms in Example B.
25; 75; Answers may vary. Possible answer:
The next term in Pattern 2 is three times the
corresponding term in Pattern 1.
As students are working, pay special attention
to problems 6 through 8, which provide an
opportunity for students to generate patterns and
then explore the relationship between them by
writing and plotting ordered pairs. While reviewing
the answers, ask: What is the relationship between
the corresponding terms in the pattern? After
establishing that each term in the second pattern is
3 times its corresponding term in the first pattern,
have students describe all the different ways they
could determine this—by comparing the rules for
the patterns, by comparing the x- and y-coordinates
of the ordered pairs, and by examining the graph.
For answers, see pages 82 and 83.
Common Errors
One error that students may make when determining
the rule for a pattern is not considering all of the
terms. If students consider only the first and second
terms, 1 and 4, for problem 2 on page 18, they may
mistakenly believe that the rule is “add 3.” Reinforce
the importance of considering more than three terms
in a pattern to be sure the correct rule has been
determined.
23
CC12_MTH_G5_TM_Final.indd 23
23/07/12 3:27 PM