Fifth Grade Math Alignment Document
Numbers and Operations
Number of Problems: 15
1. Analyze relationships among numbers and the four basic operations, compute fluently, and make reasonable estimates.
Objective
a. Compare
and order
integers,
decimals to
the nearest
thousandths,
like and
unlike
fractions,
and mixed
numbers
using >, <,
and =. (DOK
1)
Envision or
Investigation
6–8, 12–13,
230–231
Unit 4: 61–62,
64–65, 66–67,
68–69, 73–75,
77–78, 82–84,
119
Unit 6: 39–40,
41–42, 44, 45–
46, 47–48, 50–
51, 52–55, 57,
77, 80
PLD Level
Proficient:
Compare integers,
decimals, like and
unlike fractions, and
mixed numbers.
Basic: Order
integers, decimals,
like and unlike
fractions, and mixed
numbers.
Assessment Item
1. We measured the height of the following students in inches and
placed the results of our
measurements in the chart below.
Name Height
Robert 52.7
John 51.5

Andrew 52.9
Meagan 50.4
Latecia 47.2
Of the following individuals, which one is the shortest?
A. Andrew
B. Latecia *
C. Meagan
D. Robert
Depth of Knowledge Level: 1 Performance Level: Proficient
2. Which of the following set of numbers is ordered from greatest
to least?
A. 1,321.101 1,321.11 1,232.110 1,231.111
B. 1,321.11 1,321.101 1,232.110 1,231.111 *
C. 1,321.101 1,321.11 1,231.111 1,232.110
D. 1, 231.111 1,232.110 1,321.101 1,321.11
Depth of Knowledge Level: 1 Performance Level: Basic
Suggested Teaching Strategy
Using a string about 3 feet long, mark 0, 1, (or 0,
1, 2, and 3). Give students post-it notes with
integers such as –1 and 4; fractions such as
7
1 5
3
, 2 , 1 , and 2 ; and decimals such as 0.3,
8
3 9
4
1.45, and 0.78. Have them place the post-it
notes on the string in the order and appropriate
place that they think it should be found. Have
other students explain why they agree or
disagree with the placements.
b. Compose
and
decompose
seven-digit
numbers
and
decimals
through
4–5, 10–11, 72–
73
Unit 6: 24, 26–
27, 28–29, 31,
33–34, 35–36,
37, 86, 91, 92,
97, 102, 107,
112, 115, 117
Proficient:
Compose and
decompose sevendigit numbers and
decimals through
thousandths.
1. What number represents four hundred thousand, four and four
tenths in standard form?
A. 404,000.00
B. 400,004.04
C. 400,000.4
D. 400,004.4 *
Depth of Knowledge Level: 1 Performance Level: Proficient
Give students 2 seven-digit numbers, like
1,110,223 and 1,112,323. Students may notice
that the millions, hundred thousands, ten
thousands, tens and ones digits are the same,
but the digits in the thousands and hundreds
places are greater in 1,112,323. Therefore,
1,112,323 would be greater. This comparison
could be represented as 1,110,223 < 1,112,323
or 1,112,323 > 1,110,223. It is important that
students recognize that both statements are true.
thousandths
in word,
standard,
and
expanded
forms. (DOK
1)
Not only can the greater than (>) and less than
(<) signs be used but students should also
realize that if 1,110,223 < 1,112,323, then
1,110,223 ≠ 1,112,323.
Unit 8: 97, 102
Select one of the two quantities, say 1,110,223.
What two numbers can be added together to
create this sum? Students should create as
many pairs of numbers as they can. The
numbers can be recorded in equation form:
1,110,223 = 110,000 + 1,000,223
1,110,223 = 1,000,000 + 110,223
1,110,223 = 1,100,000 + 10,223 and so on.
The sum is the whole and each addend is a part.
The way in which the equation is written
indicates decomposition.
c. Identify
factors and
multiples of
102–104, 270–
271
Unit 1: 29–33,
Basic: Identify
factors and multiples
of whole numbers.
1. Which set of numbers contains factors of 8?
A. 1, 2, 4, 8 *
B. 2, 4, 6, 8
Assign pairs of students a composite number,
such as 12, 16, 18, 24, 30, 36, and 40. Ask each
pair to find all the factors of their assigned
number. Have the pairs share the numbers
whole
numbers.
(DOK 1)
34–35, 49–51,
54–56, 59–60,
121–122
C. 4, 8, 12, 16
D. 8, 16, 24, 36
Depth of Knowledge Level: 1 Performance Level: Basic
(factors) they found. Ask the other pairs to verify
that they have found all of the factors and that
those listed are accurate. Then, ask students to
count the number of factors. What do they
notice? In the given set above, they should
notice that 16 and 36 have an odd number of
factors. Why is that? What other numbers would
have an odd number of factors? Why?
Using grid paper, have students create the
rectangles that would have the factors as the
lengths of the sides. [Note that you will need to
establish that the length of a side of the grid’s
squares will be one length-unit. Each square on
the grid will be one area-unit.] What do they
notice? They should notice that for the numbers
that have an odd number of factors, one of the
rectangles is a square
d. Model and
distinguish
between
prime and
composite
numbers.
(DOK 1)
106–108
Unit 1: 37–38,
55, 63
e. Model and
identify
equivalent
fractions
226–229, 234–
236,
396–397
Unit 4: 23–24,
Proficient: Model
prime and composite
numbers.
Basic: Distinguish
between prime and
composite numbers.
Proficient: Model
equivalent fractions.
Basic: Identify
equivalent fractions
1. Which set of numbers contains only prime numbers?
A. 1, 2, 3
B. 5, 17, 51
C. 7, 11, 53 *
D. 8, 12, 54
Depth of Knowledge Level: 1 Performance Level: Basic
Give pairs of students a bag of square tiles (up
to 24 tiles). Ask students to take out 2 tiles. How
many different ways can the tiles be put together
to make a rectangle?
1. What fraction is equivalent to the shaded regions in the drawing
below?
A. 7/12
B. 10/12
Models for fractions at this grade level should
focus on continuous quantities such as length
and area rather than on discrete models that use
collections of objects. As students model
equivalent fractions, they can use models. Give
Record how many different ways the tiles can be
configured for each number. For example, with 2
tiles, there is 1 way. The same is true for 3 tiles.
With 4 tiles, you can make a rectangle that is 1 x
4 or 2 x 2 so there are 2 ways. Continue through
24. What do students notice? They should notice
that for prime numbers, there is only 1 way to
make the rectangle. Composite numbers have
more than one way.
including
conversion
of improper
fractions to
mixed
numbers
and vice
versa. (DOK
1)
28–29, 30–31,
32–33, 36–39,
42–43, 44, 49–
51, 59–61, 71–
72, 73–74, 77–
78, 122–123
Unit 6: 25–26,
28–29, 34–35,
36–37, 67–68,
69–70, 74–76,
80
including conversion of
improper fractions.
C. 5/24
D. 10/24 *
Depth of Knowledge Level: 2 Performance Level: Basic
2. What is the correct representation of the mixed number 5 7/8 as
an improper fraction?
A. 35/8
B. 40/8
C. 43/8
D. 47/8 *
Depth of Knowledge Level: 1 Performance Level: Basic
3. A store has 12 slices of pizza. If they sold 3 slices, what fraction
of pizza is left?
A. 1/4
B. 3/9
C. 3/4 *
D. 9/3
Depth of Knowledge Level: 2 Performance Level: Basic

students a strip of adding machine tape. This
represents one unit of length. If it is used to
measure length, a beginning (0) point needs to
be indicated as well as the end of the unit (1).
Have students mark those points. Direct students
to measure the length of some object, such as
the width of their desk, their arm and so on. It will
probably not measure exact. Ask them how we
could get a more accurate measure. They
usually suggest folding the paper to find the
halfway mark between 0 and 1. When they fold,
they will notice that the fold creates two equal
lengths, parts of the whole unit. They continue
folding until they have found 16ths.] This is a
length model for fractions. When all the folds are
made, students will notice that, for example,
1 2 4
8
, , , and
all represent the same length.
2 4 8
16
Area models can also be used in a similar way.
To extend
this objective, ask students if there is

ever a time when
1
4
and
do not represent the
2
8
same quantity. Students should note that they
are only equivalent when they represent a
quantity made
from

 the same unit. If the original
whole, or unit, is not the same, then their
equivalency cannot be assumed.
f. Add,
subtract,
multiply, and
divide (with
and without
remainders)
using nonnegative
rational
numbers.
(DOK 1)
24–26, 30–32,
38–
48, 60–71, 84–
87,
90–100, 110–
112,
122–139, 158–
163,
170–190, 256–
258,
262–269, 278–
287,
Proficient: Add,
subtract, multiply, and
divide using nonnegative rational
numbers.
1. Using the recipe below, how much more sugar than flour was
used?
A. 1 ¼ *
B. 2 ¼
C. 1 ¾
D. 2 ¾
Depth of Knowledge Level: 2 Performance Level: Proficient
Give students a non-negative rational number
3
8
such as 2 . Ask the students to find three pairs
3
8
of numbers whose sum is 2 . [Note, you can
use
 any operation and any non-negative rational
number as the sum, difference, product or
quotient in a similar
 way.] Allow students to work
for about 3 minutes. Record their pairs of
numbers. Note the numbers they find. For
example, if they did not find a pair that uses two
improper fractions, you can ask them to find
another pair with that condition. This is an
opportunity to push their level of thinking by
288–289, 422–
423
Unit 1: 29–31,
41, 45–47, 56,
57, 61, 71–74,
83–84, 86–87,
94–95, 98, 121–
122, 123–124,
127–128, 137–
139, 143–144,
148–149
Unit 3: 91–92
Unit 4: 95–96,
98–99, 102–
103,
104–105, 108,
109–110
Unit 6: 24, 37,
86, 87–88, 89,
92, 97, 98, 102,
103, 106, 107,
108–109, 110–
111, 114, 117
Unit 8: 97, 102
making conditions on the numbers in the pairs.
Using this reversibility-type question, the DOK
level raises from DOK 1 to at least DOK 2.
g. Estimate
sums,
differences,
products,
and
quotients of
non-negative
rational
numbers to
include
strategies
such as
front-end
rounding,
benchmark
numbers,
compatible
numbers,
and
rounding.
(DOK 2)
30–32, 62–63,
65,
68, 70, 86–87,
124–
125, 136–137,
174–
177, 184–185,
284–
285
Unit 2: 41, 42–
43, 47, 48, 64,
71, 83, 105, 110
Unit 3: 44–45,
51, 58, 64, 69,
97
Unit 6: 90–91,
105–106, 113
Unit 7: 54–56,
62, 68, 75, 93,
96, 110, 114
Unit 9: 35, 41,
68, 72, 93, 98,
104, 109
Advanced: Justify
estimations of sums,
differences, products,
and quotients of nonnegative rational
numbers.
Proficient:
Estimate sums,
differences, products,
and quotients of nonnegative rational
numbers.
1. John went to the store and bought some snacks for $4.25, a water
gun for $7.87, and some
school supplies for $8.49. About how much money did he spend?
A. $19
B. $20 *
C. $22
D. $23
Depth of Knowledge Level: 2 Performance Level: Proficient
At this grade level, students should see and use
a variety of strategies, including front-end,
benchmark numbers, rounding, and compatible
numbers. Present a contextual situation such as:
I need to go shopping after school today and I
forgot to bring my checkbook. I have only $24 in
cash. I need to buy the following items. Without
adding the costs, decide if I have enough money.
Be able to explain how you decided and what
method you used.
Item
Cost
Steak
$6.53
Lettuce
$1.39
Ice Cream
$3.69
Milk
$1.89
Cereal
$3.29
Apples
$2.79
Yogurt
$2.99
Juice
$1.74
Show this list on the overhead (or ELMO) so that
you can control the amount of time that students
have to look at it. Because they are estimating,
you want to restrict the amount of time so that
they are forced to estimate rather than compute.
After a minute or so, ask students to share their
ideas. What types of strategies did they use?
Record their strategies and ask students to
justify why that strategy would be appropriate for
this context. They should note that because you
will also have to compensate for the tax that will
be paid, they should over-estimate so that you
will have enough money for the food and the tax.
There is no right or wrong estimate—an estimate
with computations is related to the context in
which it is used.