Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency Specification Sheet
3rd Six Weeks: 25 Days
Discuss Numerical Fluency Problems with your students by sharing their multiple
correct strategies. This sharing will help students become flexible, efficient and
accurate with numerical reasoning while learning the TEKS deeply.
SPECIAL NOTE: Pages 3-4 are Student Model Sheets for use on Numerical Fluency
this six weeks. Please have them available for students’ use as they need them.
NF
Problem
Matrix
#
TEKS
#
TAKS
Obj #
3-1
117
6.1B
1
3-2
117
6.1B
1
3-3
117
6.1B
1
Answer
Answers will vary.
60
6
3
1 1
or 1
or 1 or 1.6
100
10
5
2
20
1
2 1
or 1
or 1 or 1.2
10
100
5
B
D
1
3-4
117
6.1B
1
6
1
3-5
117
6.1B
1
2
1
2
3-6
117
6.1B
1
3
4
5
1
2
3-7
117
6.1B
1
3
4
Austin ISD Secondary Mathematics Department
9
15
18
12
6
2
3
4
5
3
5
6
4
2
24
27
30
36
21
7
8
9
10
8
9
10
12
7
2
3
4 6 8
 
6 9 12
3, 6, 9, 12, 15, 18, 21, 24, 27, 30 and
5, 10, 15, 20, 25, 30, 35, 40, 45, 50
6
10
3 5 15
 
5 5 25
3 3 9
 
5 3 15
When a number is multiplied by 1, the value of the original
number does not change in value. Therefore, the product is
equivalent to the original number.
5, 10, 15, 20, 25, 30, 35, 40, 45, 50
8, 16, 24, 32, 40, 48, 56, 64, 72, 80
5 6 30
 
8 6 48
5 10 50


8 10 80
5 7 35
 
8 7 56
3rd Six Weeks 2009-2010
Page 1 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
NF
Problem
Matrix
#
TEKS
#
TAKS
Obj #
Answer
1
of the trip is more than
4
1
1 1
1
Marta’s distance of of the trip.
>
because is a larger
5
4 5
4
1
portion of one whole amount than of the whole amount. Fewer
5
parts of whole make larger portions of the whole.
1
2 Neither Marta or John has reach Pleasanton yet since
of the
3
1
1
distance is further than either
or a of the distance. Also,
5
4
20 15 12
1 20 1 15
1 12



, 
, and 
.
.
60 60 60
3 60 4 60
5 60
4
Jesse paid more for his soda because
of a dollar is $0.80 and
5
3
Andrea paid
of a dollar, which is $0.75 for her soda.
4
1
3-8
110
6.1A
1
110
6.1A
1
117
6.1B
1
110
6.1A
1
117
6.1B
1
110
6.1A
1
117
6.1B
1
110
6.1A
1
117
6.1B
1
3-9
3-10
3-11
3-12
John made more progress because
Bill finished the race before Holly because 1.25 < 1.33333…
1
1.33333… is equivalent to 1 .
3
North Austin received more rain than South Austin because 2.4 >
3
2.375. 2.375 is another name for 2 .
8
1
The investors lost more money on Monday since of a dollar which
4
2
1
equals of a dollar is greater than of a dollar.
8
8
7
9
1
2
9
4
3
9
5
1
7
2
2
7
3
3
7
See the number line below for approximations.
1
3-13
110
6.1A
1
3-14
110
6.1A
1
3-15
110
117
6.1A
6.1B
1
Austin ISD Secondary Mathematics Department
3rd Six Weeks 2009-2010
Page 2 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
NF
Problem
3-16
Matrix
#
TEKS
#
110
6.1A
117
6.1B
TAKS
Obj #
Answer
1
1
2
2
7
5
2 3
7
3 See the number line below for approximations.
See the number line below for approximations.
1 B
2 D
1
3-17
110
6.1A
1
3-18
110
6.1A
1
3-19
110
6.1A
1
3-20
217
217
6.3B
6.3B
2
2
3-21
218
6.3A
2
3-22
218
6.3A
2
3-23
132
6.1C
1
Austin ISD Secondary Mathematics Department
See number line below for approximate locations.
3 5
5 11 9
3
6 7
1 
 0.9 ; 2  2
;  1 ;  2.5
;
6 6 10
5
10 4
4 2
1
C
For NF #72, any equivalent ratio is acceptable as well, but students
should be able to demonstrate why it is an equivalent ratio.
1 9 to 15 or 9 : 15
2 6 to 15 or 6 : 15
3 Two other ways to answer #1 are 3 to 5 or 3 : 5. Two other ways
to answer #2 are 2 to 5 or 2 : 5.
4 6 to 10 or 6 : 10
5 4 to 10 or 4 : 10
6 Two other ways to answer #4 are 3 to 5 or 3 : 5. Two other ways
to answer #5 are 2 to 5 or 2: 5.
1 C
2 D
1 D
2 B
3rd Six Weeks 2009-2010
Page 3 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
1
2
3
3-24
217
6.3B
2
4
3-25
134
1
These strategies help students remember the value
of the digits. Tradition multiplication algorithms,
especially when communicated without regard to
place value, do NOT help students understand the
magnitude of the numbers involved.
80  21  2  21  1680  42  1722
20  82  1 82  1640  82  1722
40  21  40  21  2  21  820  820  42  1722
Austin ISD Secondary Mathematics Department
3rd Six Weeks 2009-2010
Page 4 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Additional answers are below.
NF 3-17
3
1
6
1
4
0
1
1
2
2
1
3
1
5
NF 3-18
0
1
1
2
1
3
4
3
0.3
NF 3-15
0
9
10
NF 316
1
0
Austin ISD Secondary Mathematics Department
1
1
5
2
1
3
4
1
2
1
4
7
4
5
6
2
3rd Six Weeks 2009-2010
2
3
5
3
Page 5 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Student Model Sheet
Choose any model or draw your own model to justify your work.
Austin ISD Secondary Mathematics Department
3rd Six Weeks 2009-2010
Page 6 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 7 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-1
1
The shaded area below represents 1 whole.
Record the amount shaded below using at least 5
different names.
2
Give at least 5 different names for the bold point on
the number line.
0
AISD Secondary Mathematics Department
1
3rd 6 Weeks 2009-2010
2
Page 8 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-2
Which statement about 2.07 is true?
A
2.07 < .207
B
2.07 > 2.007
C
.027 > 2.07
D
20.7 < 2.07
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 9 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-3
To collect data for an experiment, four classmates
measured their heights. Roberto was 1.76 meters
tall, Jorge was 1.6 meters tall, Tony was 1.67 meters
tall, and Luis was 1.7 meters tall. Which list shows
their heights in order from tallest to shortest?
A
1.6 m, 1.67 m, 1. 76 m, 1.7 m
B
1.6 m, 1.7 m, 1.67 m, 1.76 m
C
1.76 m, 1.67 m, 1.7 m, 1.6 m
D
1.76 m, 1.7 m, 1.67 m, 1.6 m
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 10 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-4
Three fraction strips are placed end to end as shown below.
Record your answer to each question as a fraction.
1
How many halves are equivalent to 3 fraction strips?
2
How many thirds are equivalent to 3 fraction strips?
3
How many fourths are equivalent to 3 fraction strips?
4
How many fifths are equivalent to 3 fraction strips?
5
How many sixths are equivalent to 3 fraction strips?
6
How many sevenths are equivalent to 3 fraction
strips?
7
How many eighths are equivalent to 3 fraction strips?
8
How many ninths are equivalent 3 fraction strips?
9
How many tenths are equivalent to 3 fraction strips?
10 How many twelfths are equivalent to 3 fraction strips?
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 11 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-5
The three large rectangles below represent cake
pans of equal size. The shaded part represents the
number of equal-size pieces of cake remaining in
each cake pan.
Cake Pan #1
Cake Pan #2
Cake Pan #3
1
What part of the cake remains in each cake pan?
2
Using the shaded part of the cake pans as a guide,
2
write three fractions equivalent to . Be prepared
3
to explain why the fractions are equivalent.
2
=
3
AISD Secondary Mathematics Department
=
=
3rd 6 Weeks 2009-2010
Page 12 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-6
1
Complete the multiplication table for the unshaded rows.
X 1
1
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
Complete each equation in #2-#4.
2
3
x
5
2
=
3
x
5
4
9
=
2
5
3
3
x
5
=
Explain why multiplying
by a form of one produces
an equivalent fraction.
5
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 13 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-7
1
Complete the multiplication table for the unshaded rows.
x 1
1
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
Complete each equation in #2-#4.
6
2
3
5
x
8
5
x
8
=
10
4
5
=
AISD Secondary Mathematics Department
5
x
8
=
56
What other fractions are
5
equivalent to ?
8
3rd 6 Weeks 2009-2010
Page 14 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-8
John and Marta are driving from Austin to Harlingen for
the holidays. John left Austin at 3:00 p.m. and has
completed
1
of the trip. Marta left at 2:45 p.m. and has
4
completed
1
of the trip. The small town of Pleasanton,
5
is about
1
of the way to Harlingen.
3
Austin
Harlingen
1
Who has made more progress in the trip? Explain
your reasoning.
2
Has either Marta or John reached Pleasonton yet?
Explain your reasoning.
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 15 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-9
Jesse paid
4
of a dollar for a soda. Andrea paid $0.75
5
for her soda. Who spent the most for their soda?
Explain your reasoning.
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 16 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-10
Holly ran the Capitol 10K race in 1
1
hours. Her
3
husband, Bill ran the race in 1.25 hours. Who finished
the race in the least amount of time? Explain your
reasoning.
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 17 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-11
After a stormy night, a family from north Austin
reported they had received 2.4 inches of rain. A rain
gauge in south Austin received 2
3
inches of rain.
8
According to these reports, what part of Austin
received the most rain? Explain your reasoning.
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 18 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-12
On Friday, Dell stock dropped
1
of a dollar. The
8
following Monday, the stock dropped another
1
of a
4
dollar. On which day did the investors lose more
money? Explain your reasoning.
Extension: About how much money did one share of
Dell stock lose over the 2 days?
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 19 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-13
4
1
7
9
Choose 2 of the digits above to find the “Mystery
Fraction” for each description below.
1
Mystery Fraction:
Description: the number closest to one whole
2
Mystery Fraction:
Description: the number closest to zero
3
Mystery Fraction:
Description: the number closest to one-half
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 20 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-14
3
5
7
2
Choose 2 of the digits above to find the “Mystery
Fraction” for each description below.
1
Mystery Fraction:
Description: the number closest to one whole
2
Mystery Fraction:
Description: the number closest to zero
3
Mystery Fraction:
Description: the number closest to one-half
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 21 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-15
Use the number line below to locate the following
numbers. Be prepared to explain your reasoning.
0
1
4
.3
0
3
4
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
1
Page 22 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-16
1
Using the number line below, find and label the
following:
9
5
3 7 5
1 ,
, 2 ,
,
6 10
5 4 2
1
2
2
3
For each of the numbers listed above, give at least
one more number that describes an equivalent
amount.
1
5
=
6
9
=
10
2
3
=
5
7
=
4
5
=
2
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 23 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-17
A
0
1
B
2
3
4
1
What number is indicated by point A on the
number line above?
2
What number is indicated by point B on the
number line above?
3
On the number line below, locate
and
1 1 1 1
, , , ,
6 5 4 3
1
.
2
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 24 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-18
Use the number line below to locate the following
numbers. Be prepared to explain your reasoning.
1
2
1
0
AISD Secondary Mathematics Department
4
3
1
3
3rd 6 Weeks 2009-2010
Page 25 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-19
1
2
By 12:30 p.m., 75% of the Burnet Middle School
students had eaten lunch. What fractional part of
the Burnet Middle School students had NOT yet
eaten lunch?
A
1
25
B
1
4
C
7
5
D
3
4
1
What statement about the mixed number 1 is
5
true?
A
B
C
D
1
1
1 >1
5
4
1
1
1
>1
10
5
1
2<1
5
1
1
1
<1
10
5
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 26 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-20
Each parallelogram below is divided into sections of
2
equal size. Which parallelogram has 66 % of its total
3
area shaded?
A
B
C
D
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 27 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-21
1
What is the ratio of the shaded part of the
rectangle to the total rectangle below?
2
What is the ratio of the unshaded part of the
rectangle below to the total rectangle below?
3
What is another way to answer #1 and #2?
4
What is the ratio of the number of stars to the
total number of objects in the set below?
5
What is the ratio of the number of moons to the
total number of objects in the set below?
6
What is another way to answer #4 and #5?
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 28 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-22
1
2
A used car lot has 24 cars and 16 trucks for sale.
What is the ratio of cars to trucks?
A
2 to 3
B
3 to 5
C
3 to 2
D
5 to 3
If the ratio of tulips to roses is 2 to 5, which of these
shows possible numbers of tulips and roses?
A
12 tulips, 35 roses
B
25 tulips, 10 roses
C
15 tulips, 6 roses
D
16 tulips, 40 roses
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 29 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-23
1
2
The high temperature on Monday was 12 degrees
below zero. On Tuesday, the high temperature was
3 degrees above zero. What integer represents the
high temperature on Monday?
A
3
B
–3
C
12
D
–12
On Wednesday, a video store gave away a free
video when the total DVD rental fees were $30 or
more. There were 22 DVD rental fees that were $30
or higher that day. What integer represents the
change in the number of free videos the store had
in stock by the end of the day on Wednesday?
A
–30
B
–22
C
22
D
30
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 30 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-24
Three out of every five 6th graders use the Internet to
check their email. Represent this ratio in each of the
following grid models. Be prepared to justify your
reasoning.
1
2
3
4
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 31 of 32
Grade 6 Mathematics-----------------------------------------Numerical Fluency Problems
Numerical Fluency 3-25
A theatre has 82 rows of seats. There are 21 seats
in each row.
How many seats are in the theatre?
Solve this problem using two different strategies.
Be prepared to explain your strategies.
AISD Secondary Mathematics Department
3rd 6 Weeks 2009-2010
Page 32 of 32