Enhanced Instructional Transition Guide
Grade 6/Mathematics
Unit 03:
Suggested Duration: 9 days
Unit 03: Numerical Operations – Addition and Subtraction of Fractions and Decimals (12 days)
Possible Lesson 01 (9 days)
Possible Lesson 02 (3 days)
POSSIBLE LESSON 01 (9 days)
This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing
with district-approved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and
districts may modify the time frame to meet students’ needs. To better understand how your district is implementing CSCOPE lessons, please contact your
child’s teacher. (For your convenience, please find linked the TEA Commissioner’s List of State Board of Education Approved Instructional Resources and
Midcycle State Adopted Instructional Materials.)
Lesson Synopsis:
Students develop a conceptual understanding with concrete models (fraction circles and number lines) to estimate and compute sums or differences of fractions and
mixed numbers. Students make connections between concrete models and algorithms for adding and subtraction fractions and mixed numbers.
TEKS:
The Texas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Texas
law. Any standard that has a strike-through (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit.
The TEKS are available on the Texas Education Agency website at http://www.tea.state.tx.us/index2.aspx?id=6148
6.2
Number, operation, and quantitative reasoning.. The student adds, subtracts, multiplies, and divides to solve problems and justify
solutions. The student is expected to:
6.2A
Model addition and subtraction situations involving fractions with objects, pictures, words, and numbers. Supporting Standard
6.2B
Use addition and subtraction to solve problems involving fractions and decimals. Readiness Standard
Underlying Processes and Mathematical Tools TEKS:
6.11
Underlying processes and mathematical tools.. The student applies Grade 6 mathematics to solve problems connected to everyday
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experiences, investigations in other disciplines, and activities in and outside of school. The student is expected to:
6.11A
Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with
other mathematical topics.
6.11B
Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating
the solution for reasonableness.
6.11D
Select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation,
and number sense to solve problems.
6.13
Underlying processes and mathematical tools.. The student uses logical reasoning to make conjectures and verify conclusions. The
student is expected to:
6.13B
Validate his/her conclusions using mathematical properties and relationships.
Performance Indicator(s):
Grade 06 Mathematics Unit 03 PI 01
Estimate and solve addition and subtraction problems involving fractions and mixed numbers in problem situations (e.g., recipes, pizza slices). Create a graphic organizer (e.g.,
flowchart, flip book, etc.) describing, in words, the process for adding and subtracting fractions and mixed numbers with like and unlike denominators. Use the organizer to
solve the problems, verifying each step of the process with the solution.
Sample Performance Indicator:
Create a flow chart describing in words the process for adding and subtracting mixed numbers with like and unlike denominators. Use the flow
chart to answer the questions in the following problem scenario, verifying each step in the solution process.
Betty is making her favorite cookies. See the recipe below:
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(1) First, she is asked to combine the sugars. How much sugar will she have in her bowl?
(2) Then, she is asked to add the flour. When measuring the flour, Betty realizes she only has
cups of flour. How much more flour does she need?
(3) The recipe said to mix the salt and baking soda together prior to adding it to the mixture. About how many teaspoons of salt and baking soda will Betty add?
Standard(s): 6.2A , 6.2B , 6.11A , 6.11B , 6.11D , 6.13B
ELPS ELPS.c.1C , ELPS.c.3J , ELPS.c.5G
Key Understanding(s):
Objects, pictures, words, and numbers may be used to model and validate conclusions of addition, subtraction, and estimation of fractions with unlike
denominators and mixed numbers.
Estimation is an effective strategy to check for reasonableness and validate conclusions when solving everyday situations requiring addition and subtraction of
fractions and mixed numbers, including problems involving ranges of numbers.
Converting mixed numbers to improper fractions and generating equivalent fractions using a common denominator are effective strategies when solving problem
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situations involving addition and subtractions of mixed numbers.
Misconception(s):
Some students may think that they should add the numerators and denominators when adding fractions instead of getting a common denominator, adding
numerators, and keeping the common denominator.
Vocabulary of Instruction:
benchmark
common denominator
equivalent fraction
fraction
improper fraction
least common denominator
mixed number
proper fraction
whole number
Materials List:
Fraction Circle Models (optional): (1 set per 4 students)(previously created in Unit 01 Lesson 01 Elaborate 1)
fraction circles (1 set per 4 students)
fraction circles (2 sets per 4 students)
fraction circles (3 sets per 4 students)
fraction circles (4 sets per 4 students)
Local Resource(s)
map pencil (2 different colors) (1 set per student)
math journal (1 per student)
Attachments:
All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments
that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website.
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Who Ate More? KEY
Who Ate More?
One-Half and Number Line Relations KEY
One-Half and Number Line Relations
Reason It Out Sums and Differences KEY
Reason It Out Sums and Differences
Is This Wrong? KEY
Is This Wrong?
Pie Sums and Differences KEY
Pie Sums and Differences
Equivalent Fractions KEY
Equivalent Fractions
What’s Your Name? KEY
What’s Your Name?
Estimated Sums and Differences KEY
Estimated Sums and Differences
Pie Sums and Differences: Mixed Numbers KEY
Pie Sums and Differences: Mixed Numbers
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Pie Eating Contest KEY
Pie Eating Contest
Fraction Applications KEY
Fraction Applications
GETTING READY FOR INSTRUCTION
Teachers are encouraged to supplement and substitute resources, materials, and activities to meet the needs of learners. These lessons are one approach to
teaching the TEKS/Specificity as well as addressing the Performance Indicators associated with each unit. District personnel may create original lessons using
the Content Creator in the Tools Tab. All originally authored lessons can be saved in the “My CSCOPE” Tab within the “My Content” area. Suggested
Day
1
Suggested Instructional Procedures
Notes for Teacher
Topics:
Spiraling Review
Compare parts of a whole
Engage 1
Students use logic and reasoning skills to compare parts of a whole.
ATTACHMENTS
Teacher Resource: Who Ate More?
KEY (1 per teacher)
Instructional Procedures:
1. Place students in pairs. Display teacher resource: Who Ate More?. Instruct student pairs to read
Teacher Resource: Who Ate More? (1
per teacher)
and complete the displayed problem. Allow time for students to complete the activity. Monitor and
assess pairs to check for understanding. Facilitate a class discussion about comparing parts of a
whole.
Ask:
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Notes for Teacher
What fraction represents one slice of Juan’s pie? Sam’s pie? Explain. (One slice of Juan’s
pie = one­eighth pie cut into 8 equal size slices. Fractional parts are eighths. One slice of Sam’s
pie = one-fourth pie cut into 4 equal size slices. Fractional parts are fourths.)
What fraction represents two slices of Juan’s pie? Sam’s pie? Explain. (One slice Juan’s
pie = one-eighth. 1 slice + 1 slice = 2 slices: Count 2 slices, one-eighth, two-eighths. One slice of
Sam’s pie = one­fourth. 1 slice + 1 slice = 2 slices: Count 2 slices, one­fourth, two­fourths.)
What fraction represents three slices of Juan’s pie? Sam’s pie? Explain. (One slice Juan’s
pie = one-eighth. 1 slice + 1 slice + 1 slice = 3 slices: Count 3 slices, one-eighth, two-eighths,
three­ eighths. One slice Sam’s pie = 1 fourth. 1 slice + 1 slice + 1 slice = 3 slices: Count 3
slices, one-fourth, two-fourths, three-fourths.)
What fraction represents four slices of Juan’s pie? Sam’s pie? Explain. (One slice Juan’s
pie = 1 eighth. 1 slice + 1 slice + 1 slice = 4 slices: Count 4 slices, one-eighth, two-eighths,
three­eighths, four­eighths. One slice Sam’s pie = one­fourth. 1 slice + 1 slice + 1 slice + 1 slice
= 4 slices: Count 4 slices, one-fourth, two-fourths, three-fourths, four-fourths: four-fourths is a
whole pie.)
How does one slice of Juan’s pie compare to one slice of Sam’s pie? Explain. (Juan’s pie
and Sam’s pie are the same size. One slice of Juan’s pie is smaller because Juan’s whole pie
was cut into more equal size pieces. So each piece of Juan’s pie is smaller than each piece of
Sam’s pie. Sam’s pie is cut into fewer equal size pieces, so each piece of Sam’s pie is larger than
each piece of Juan’s pie.)
How many slices of Juan’s pie would equal one slice of Sam’s pie? Explain. (Juan’s pie
and Sam’s pie are the same size. 2 slices of Juan’s pie would equal one slice of Sam’s pie
because 2 slices of Juan’s pie cover 1 slice of Sam’s pie. Every 2 slices in Juan’s pie = 1 slice in
Sam’s pie
. One-eighth + one-eighth = two-eighths; two-eighths = one fourth.)
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What fraction of the pie does Juan have left to eat? Sam? Explain. (Juan has four-eighths
of the pie left to eat because he ate four-eighths. four-eighths + four-eighths = eight-eighths and
eight-eighths is a whole pie.) (Sam has two-fourths of the pie left to eat because he ate twofourths and two-fourths + two-fourths = four-fourths and four-fourths is a whole pie.)
Who, Juan or Sam, ate more pie? Explain. (Both pies are the same size. Juan’s pie was cut
into eight pieces and he ate 4 pieces or four­eighths of his pie. Sam’s pie was cut into four pieces
and he ate 2 of them or two-fourths of his pie. Four-eighths is equivalent to two-fourths
.
Therefore Juan and Sam both ate the same amount of pie. Every 2 slices of Juan’s pie is
equivalent to 1 slice of Sam’s pie.)
What misconceptions did Juan have? Sam? Answers may vary. Juan felt he ate more pie
because he ate more slices than Sam, Juan did not consider the size of each slice in relation to
the whole pie, Juan’s slices were smaller since his pie was cut into more slices, Sam felt he ate
more pie because he looked at the size of one slice of his pie in comparison to the size of one
slice from Juan’s pie, Sam did not consider the relation of number of slices to the whole pie, Sam
only looked at one slice for his comparison, etc.
What is another fraction name Juan and Sam could use to describe the amount of pie
each one ate? Explain. Answers may vary. One-half, three-sixths, five-tenths, etc. These are all
equivalent fractions.
Topics:
ATTACHMENTS
Rounding fractions to benchmarks
Teacher Resource: One-Half and
Fractions on a number line
Number Line Relations KEY (1 per
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Notes for Teacher
teacher)
Explore/Explain 1
Handout: One-Half and Number Line
Students use pictorial models in order to determine if a fraction is less than, equal to, or greater than
Relations (1 per student)
one-half. Students use a number line to explore the relationship between one-half and a given fraction.
Teacher Resource: One-Half and
Number Line Relations (1 per
Instructional Procedures:
teacher)
1. Place students in groups of 4. Distribute a set of fraction circles to each group and handout: OneHalf and Number Line Relations to each student.
MATERIALS
2. Display teacher resource: One-Half and Number Line Relations. Instruct student groups to
complete problems 1 – 4 on handout: One-Half and Number Line Relations using fraction circles
to model the given fraction, one-half, and equivalent fractions for each fraction with a common
denominator. Allow time for students to complete the activity. Monitor and assess students to check
for understanding. Facilitate a class discussion about the relationship between one-half and the given
fraction circles (1 set per 4 students)
Fraction Circle Models (optional): (1 set
per 4 students)(previously created in
Unit 01 Lesson 01 Elaborate 1)
fractions.
Ask:
What model did you use for
? Explain. (Hold up the fraction piece for
parts that cover the whole and 1 of the 2 equal parts represents
What model did you use for
.)
? Explain. (Hold up the fraction piece for
parts that cover the whole and 1 of the 4 equal parts represents
. There are 2 equal
. There are 4 equal
.)
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What common fraction pieces can you use to model
four is the LCD for the denominators of 2 and 4.
How is
related to
? Explain. (
Notes for Teacher
and
? Explain. (Fourths because
is equivalent to
is less than
because
and I already have
is less than
.)
which equals
.)
According to your models,
is greater than
. How can you use symbolic notation to
compare these two fractions?
What model did you use for
? Explain. (Hold up the fraction piece for
parts that cover the whole and 2 of the 3 equal parts represents
What common fraction pieces can you use to model
6 is the LCD for the denominators of 2 and 3.
How is
larger than
related to
? Explain. (
, which is equal to
According to your models,
and
is equivalent to
is more than
because
. There are 3 equal
.)
? Explain. (Sixths because
and
is equivalent to
, which is equal to
.)
, is
.)
is greater than
. How can you use symbolic notation to
compare these two fractions?
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What model did you use for
? Explain. (Hold up the fraction piece for
parts that cover the whole and 3 of the 8 equal parts represents
What common fraction pieces can you use to model
8 is the LCD for the denominators of 2 and 8.
How is
related to
equal to
.)
? Explain. (
According to your models,
Notes for Teacher
is less than
is less than
.)
and
is equivalent to
. There are 8 equal
? Explain. (Eighths because
and I already have
because
is less than
.)
, which is
. How can you use symbolic notation to
compare these two fractions?
What model did you use for
? Explain. (Hold up the fraction piece for
parts that cover the whole and 5 of the 6 equal parts represents
What common fraction pieces can you use to model
is the LCD for the denominators of 2 and 6.
and
is equivalent to
How is five-sixths related to one-half? Explain. (
. There are 6 equal
.)
? Explain. (Sixths because 6
and I already have
is more than
because
.)
is larger than
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which is equal to
Grade 6/Mathematics
Unit 03:
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Notes for Teacher
.)
According to our models,
is greater than
. How can you use symbolic notation to
compare these two fractions?
3. Instruct student groups to complete the remainder of handout: One-Half and Number Line
Relations using the number lines provided. Allow time for students to complete the activity. Monitor
and assess student groups to check for understanding. Facilitate a class discussion about the
relationship between one-half and the given fractions using a number line.
Ask:
How can you verify the location for
on the number line? Explain. (This is the correct
location because the distance between 0 and 1 was divided into 2 equal parts and one-half was
placed at the first mark after 0 to mark 1 of the 2 equal parts between 0 and 1.)
How can you locate
on the number line? Explain. (Divide the distance between 0 and 1
into 4 equal parts. Begin at 0 and count each equal part,
represent
How is
line than
. The first mark after 0 will
.)
related to
.
is
? Explain. (
less than
is less than
because
because
is equal to
is closer to 0 on a number
.)
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According to your number line,
is less than
Grade 6/Mathematics
Unit 03:
Suggested Duration: 9 days
Notes for Teacher
. How can you use symbolic notation to
compare these two fractions?
How can you locate
on the number line? Explain. (Divide the distance between 0 and 1
into 3 equal parts. Begin at 0 and count each equal part,
represent
How is
. The second mark after 0 will
.)
related to
number line than
? Explain. (
is more than
because
is further from 0 on a
.)
According to our number line,
is greater than
. How can you use symbolic notation
to compare these two fractions?
How can you locate
on the number line? Explain. (Divide the distance between 0 and 1
into 8 equal parts. Begin at 0 and count each equal part,
after 0 will represent
How is
related to
. The third mark
.)
? Explain. (
is less than
because
is closer to 0 on a number
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line than
Notes for Teacher
.)
According to your number line,
is less than
. How can you use symbolic notation to
compare these two fractions?
How can you locate
on the number line? Explain. (Divide the distance between 0 and 1
into 6 equal parts. Begin at 0 and count each equal part,
will represent
How is
. The fifth mark after 0
.)
related to
number line than
? Explain. (
is more than
because
is further from 0 on a
.)
According to your number line,
is greater than
. How can you use symbolic notation
to compare these two fractions?
What part of the number line represents the denominator in each fraction? (The number of
equal parts between the whole numbers 0 and 1.)
What part of the number line represents the numerator in each fraction? (The number of
spaces after 0 and before the mark.)
2
Topics:
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Notes for Teacher
Spiraling Review
Estimate the sum of fractions using benchmarks
Estimate the difference of fractions using benchmarks
ATTACHMENTS
Explore/Explain 2
Teacher Resource: Reason It Out
Students use logic and reasoning skills to estimate the sums and differences of fractions without the
Sums and Differences KEY (1 per
models. Students validate their estimations with an explanation and a pictorial model.
teacher)
Handout: Reason It Out Sums and
Differences (1 per student)
Instructional Procedures:
1. Place students in groups of 3 – 4. Distribute handout: Reason It Out Sums and Differences to each
student.
Teacher Resource: Reason It Out
Sums and Differences (1 per teacher)
2. Display teacher resource: Reason It Out Sums and Differences. Instruct student groups to
complete problem 1 from their handout: Reason It Out Sums and Differences. Allow time for
students to complete the activity. Monitor and assess student groups to check for understanding.
Facilitate a class discussion to debrief student solutions.
Ask:
How does the addend
relate to
?(
is less than
.)
How does the addend
relate to
? ( is less than
.)
What is the sum of
and
? (The sum of
How does the sum of the two addends,
and
and
is 1 whole.)
, relate to
? One? Explain. (The sum
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of
and
sum of
is greater than
and
is less than
because
is greater than
and the sum of
is
less than
.) (The sum of
and
fraction less than
and
, the sum of
is 1. Since
and
How can you use the diagram for
and
is greater than
is less than 1 because each addend
and
? Explain. (Since each addend is
and
is less than
is 1.
and
is
less than 1
is being added to a
is less than 1.)
and
to support your estimation for the sum of
? (From the diagram, I can draw in a line to show a benchmark for
can show each addend is less than
, so the
and
, the sum is less than one because the sum of
whole because the sum of
and
is 1.)
What is a reasonable estimate for the sum of
less than
Notes for Teacher
since each addend is less than
. In the diagram, I
and I need 2 halves to
make 1 whole, the diagrams support my estimation that the sum is greater than
and less than
1 whole.)
3. Instruct student groups to complete problems 2 – 6 on their handout: Reason It Out Sums and
Differences. Allow time for students to complete the activity. Monitor and assess student groups to
check for understanding. Facilitate a class discussion to debrief student solutions.
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4. Instruct student groups to complete problem 7 on their handout: Reason It Out Sums and
Differences. Allow time for students to complete the activity. Monitor and assess student groups to
check for understanding. Facilitate a class discussion to debrief student solutions.
Ask:
How does the fraction
relate to
?(
is
How does the fraction
relate to
?(
is less than
How does the difference of the fractions,
of
and
is less than
because
subtracting a fraction greater than
greater than
and
)
, relate to
is greater than
from
.
equivalent to
and
.
difference less than
is less than
is greater than
.
is
? Explain. (The difference
more than
, the difference is less than
What is a reasonable estimate for the difference of
difference for
)
is
more than
and
. Since I am
.)
? Explain. (A reasonable
and
, so I am subtracting more than
minus
from
is
, which is
which is a
.)
How can you use the diagram for
subtract
to support your estimation for the
difference of three-fourths and one-third? (From the diagram, I have shaded
. Since I am
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subtracting
than
, I need to cross out a little more than
. This supports my estimation that
minus
Notes for Teacher
which leaves me with a difference less
is less than
.)
5. Instruct student groups to complete the remainder of handout: Reason It Out Sums and
Differences. Allow time for students to complete the activity. Monitor and assess student groups to
check for understanding. Facilitate a class discussion to debrief student solutions.
3
Topics:
Spiraling Review
Add fractions
Subtract fractions
ATTACHMENTS
Engage 2
Teacher Resource: Is This Wrong?
Students use experience and reasoning skills to add and subtract fractions with unlike denominators
KEY (1 per teacher)
using fraction circles and pictorial models.
Teacher Resource: Is This Wrong? (1
per teacher)
Instructional Procedures:
1. Place students in pairs and display teacher resource: Is This Wrong?. Instruct student pairs to draw
MATERIALS
a diagram model of each problem in their math journal. Allow time for students to complete their
models. Monitor and assess student pairs to check for understanding. Facilitate a class discussion
math journal (1 per student)
about the models created and how Sam and Jennifer were incorrect.
Ask:
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Can you draw a diagram to represent
of a pizza?
Notes for Teacher
State Resources
of a pizza?
MTR 6 – 8: Is it Really News?
What expression will represent this situation?
What is a common denominator for these fractions? (fourths)
How can you rename
so that the denominator is fourths? (
How can you change the diagram for
to represent
)
? (Divide each of the half sections
into 2 equal parts, so there will be 4 sections representing fourths.)
How much pizza do you have when you combine the shading for
and
of a pizza?
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Why is Sam’s representation incorrect? (It is not a reasonable answer. According to Sam’s
representation, there is less than
of a pizza when
reasonable because the problem began with
Can you draw a diagram to represent
and
were added. This is not
of a pizza and added more to the
.)
of a pizza?
What expression will represent this situation?
What is a common denominator for these fractions? (fourths)
How can you rename one-half so that the denominator is fourths? (
How many fourths do you need to remove from
)
the to represent removing
? (Cross
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out
Notes for Teacher
)
How much pizza do you have when you remove
from the
?
Why is Jennifer’s representation incorrect? (It is not a reasonable answer. According to
Jennifer’s representation, there is 1 whole pizza when
was removed from
. This is not
reasonable because the problem begins with less than 1 whole pizza, removes
and ends up
with 1 whole pizza.)
Topics:
ATTACHMENTS
Add fractions
Teacher Resource: Pie Sums and
Subtract fractions
Differences KEY (1 per teacher)
Handout: Pie Sums and Differences
Explore/Explain 3
(1 per student)
Students use fraction circles and pictorial models to make conclusions about adding and subtracting
Teacher Resource: Pie Sums and
fractions, and connect their findings to the algorithms for adding and subtracting fractions.
Differences (1 per teacher)
Teacher Resource (optional):
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Instructional Procedures:
Equivalent Fractions KEY (1 per
1. Place students into groups of 4. Distribute handout: Pie Sums and Differences and 2 different color
map pencils to each student and 2 sets of fraction circles to each group.
teacher)
Handout (optional): Equivalent
Fractions (1 per student)
2. Instruct student groups to use the fraction circles to model problem 1 from handout: Pie Sums and
Differences and then create a sketch of the model on their handout using a different color map pencil
for each addend. Allow time for students to complete problem 1. Monitor and assess student groups
to check for understanding. Facilitate a class discussion about the solution process.
MATERIALS
fraction circles (2 sets per 4 students)
Ask:
map pencil (2 different colors) (1 set per
How can you represent one-fourth using symbolic notation and in the diagram? ( : shade
student)
1 of the 4 fractional parts.)
How can you represent one-third using symbolic notation and in the diagram? (
: shade
TEACHER NOTE
Students investigated LCM and GCF in Unit
02 Lesson 01.
1 of 3 fractional parts.)
How do you subdivide the fraction circle to create an equivalent fraction model?
(Subdivide each fraction circle model into the same type of fractional parts. This denominator is a
TEACHER NOTE
common multiple for the given denominators.)
Handout (optional): Equivalent Fractions can
With what fractional parts can you rename
and
with so these fractions have a
be used if students need extra practice finding
the GCF, LCM, and equivalent fractions.
common denominator? (twelfths)
Why did you choose twelfths as the common denominator? (It is a common multiple for 4
and 3.)
How do you record this action using symbolic notation?
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Suggested Instructional Procedures
How can you represent
in the diagram?
Notes for Teacher
? (Shade 3 of the 12 fractional parts. Shade
4 of the 12 fractional parts.)
What is the sum of
and
?(
. Count the shaded twelfths
.)
How can you record this action using symbolic notation?
How does this sum compare to the estimate from problem 1 on Reason It Out Sums and
Differences? (The computed sum is reasonably close to the estimated sum. The computed sum
is
and the estimated sum was greater than
.
is greater than
which is equivalent to
.)
3. Instruct student groups to use the fraction circles to model problem 2 from handout: Pie Sums and
Differences, and then create a sketch of the model on their handout using a map pencil to shade the
beginning fraction and use “x” to cross out the amount being subtracted. Allow time for students to
complete problem 2. Monitor and assess student groups to check for understanding. Facilitate a
class discussion about the solution process.
Ask:
How can you represent
using symbolic notation in the diagram? (
: shade 3 of the 4
fractional parts.)
How can you represent
using symbolic notation? (
: No shading is done in a separate
page 23 of 117 Grade 6/Mathematics
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Suggested Duration: 9 days
Enhanced Instructional Transition Guide
Suggested
Day
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diagram since I will be removing this fraction amount from
Notes for Teacher
.)
How do you subdivide the fraction circle to create an equivalent fraction model?
(Subdivide each fraction circle model into the same type of fractional parts. This denominator is a
common multiple for the given denominators.)
With what fractional parts can you rename
and
with so these fractions have a
common denominator? (twelfths)
Why did you choose twelfths as the common denominator? (It is a common multiple for 4
and 3.)
How do you record this action using symbolic notation?
How can you represent
in the diagram? (Shade 9 of the 12 fractional parts.)
What is the difference of
and
shaded twelfths:
?(
. Count and cross out 4 shaded twelfths from the 9
.)
How can you record this action using symbolic notation?
How does this difference compare to the estimate from problem 5 on Reason It Out Sums
and Differences? (The computed difference is reasonably close to the estimated difference. The
computed difference is
, which is equivalent to
and the estimated difference was less than
, and
is less than
.)
4. Display teacher resource: Pie Sums and Differences. Facilitate a class discussion to summarize
page 24 of 117 Enhanced Instructional Transition Guide
Suggested
Day
Grade 6/Mathematics
Unit 03:
Suggested Duration: 9 days
Suggested Instructional Procedures
Notes for Teacher
the symbolic actions for problems 1 – 2. Demonstrate recording the symbolic notation for each
student in the problems, allowing students to connect the diagram to the symbolic notation.
5. Instruct student groups to complete the remainder of handout: Pie Sums and Differences. Allow
students to use the fraction circles when needed. Monitor and assess student groups to check for
understanding.
6. Using the displayed teacher resource: Pie Sums and Differences. Facilitate a class discussion to
summarize the symbolic actions for problems 3 – 11. Demonstrate recording the symbolic notation
for each student in the problems, allowing students to connect the diagram to the symbolic notation.
4 – 5
Topics:
Spiraling Review
Rounding mixed numbers using benchmarks
Explore/Explain 4
ATTACHMENTS
Students use fraction circles and pictorial models to round mixed numbers. Students explore the
Teacher Resource: What’s Your
connections between the relationships of rounding mixed numbers to placing mixed numbers on a
Name? KEY (1 per teacher)
number line.
Handout: What’s Your Name? (1 per
student)
Instructional Procedures:
Teacher Resource: What’s Your
Name? (1 per teacher)
1. Place students into groups of 4. Distribute 3 sets of fraction circles to each group and handout:
What’s Your Name? to each student.
2. Display teacher resource: What’s Your Name?. Instruct students to use the fraction circles to model
problem 1 from handout: What’s Your Name?, and then create a sketch of the model on their
MATERIALS
fraction circles (3 sets per 4 students)
handout. Allow time for students to complete problem 1. Monitor and assess student groups to check
page 25 of 117 Grade 6/Mathematics
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Enhanced Instructional Transition Guide
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Notes for Teacher
for understanding. Facilitate a class discussion about the solution process.
Ask:
TEACHER NOTE
How many fully shaded whole circles models do you have? Why? (2. Because the 2 in
The purpose of showing
written as
represents 2 wholes.)
How many equal size parts are in the whole circle that are not fully shaded? Why? (8.
may also be
, is to prepare for subtraction
of mixed numbers.
Because the denominator, 8, represents how many equal parts the whole needs to be divided
into.)
How many of these parts did you shade? Why? (3. Because the numerator, 3, represents how
many parts need to be shaded.)
What kind of number is represented by the model? (a mixed number)
How do you write this number symbolically? (
What two whole numbers is
another whole, so
)
between? Explain. (2 and 3. There are 2 wholes and part of
is greater than 2. Since three-eighths is less than 1, then
is less than
3.)
Is
less than halfway, halfway, or more than halfway between 2 and 3? Explain. (Less
than halfway. Halfway between 2 and 3 would be
than
then
s is less than
. Another name for
is
. Since
is less
.)
How did you divide the whole models in Diagram B? Why? (Into eighths. Since the
fractional circle was already divided into eighths, the whole circles could also be divided into
eighths.)
page 26 of 117 Grade 6/Mathematics
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Suggested Duration: 9 days
Enhanced Instructional Transition Guide
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Notes for Teacher
How many eighths are there altogether? (19)
How can you write the total number of eighths as a fraction symbolically? (
What kind of fraction is
)
? (improper fraction)
What is the relationship between
and
? (They are equivalent.)
What equivalent fractional model did you represent in Diagram C? (
)
What is the relationship between all three diagrams? (They are equivalent.)
Where did you mark
on Number Line A? (Between the whole numbers 2 and 3. Divide the
part of the number line between the 2 and 3 into 8 equal sections. Since
placed at the halfway mark.
is
less than
, so
is
is the mark before
then
is
.)
3. Using the displayed teacher resource: What’s Your Name?, model problem 1 by demonstrating the
process of partitioning and representing a mixed number with a fraction circle and a number line. Use
Diagram A and Number Line A to represent
. Use Diagram B and Number Line B to represent
. Use Diagram C and Number Line C to represent
. Point to the appropriate parts of the diagram
for the fraction circle models and the number line models. Facilitate a class discussion using formal
mathematical language to connect models to symbolic notation, and address any misconceptions.
Ask:
page 27 of 117 Grade 6/Mathematics
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Enhanced Instructional Transition Guide
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Where did you mark
Notes for Teacher
on Number Line B? (Divide the number line between each whole
number into 8 equal sections. Label each mark as after 0 as
mark as
, etc.
is on the same
on Number Line A.)
Where did you mark
on Number Line C? (Divide the part of number line between 2 and 3
into 8 equal sections. Label the sections starting at 2 as
the same mark as
on Number Line B and
, then
, etc.
is on
on Number Line A.)
4. Instruct student groups to complete the remainder of handout: What’s Your Name?. Allow students
to use the fraction circles if needed. Monitor and assess student groups to check for understanding.
Facilitate a class discussion to summarize the symbolic actions for problems 2 – 6. Demonstrate
recording the symbolic notation for each student in the problems, allowing students to connect the
diagram to the symbolic notation.
Ask
How can you use the relationship between the numerator and denominator of the
fraction to determine if this fraction is less than halfway, halfway, or more than halfway
between 2 whole numbers? Explain. (Halfway is represented by the fraction
that are equivalent to
. In fractions
, the numerator is half of the denominator or the denominator is double
the amount of the numerator. When the numerator is more than
of the denominator, then the
page 28 of 117 Grade 6/Mathematics
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Suggested Duration: 9 days
Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
fraction is greater than
and is more than halfway, and when the numerator is less than
Notes for Teacher
of
the denominator, then the fraction is less than halfway.)
Topics:
ATTACHMENTS
Estimate the sum of mixed numbers using benchmarks
Teacher Resource: Estimated Sums
Estimate the difference of mixed numbers using benchmarks
and Differences KEY (1 per teacher)
Handout: Estimated Sums and
Explore/Explain 5
Differences (1 per student)
Students use fraction circles and pictorial models to make conclusions about estimating the sums and
Teacher Resource: Estimated Sums
differences of mixed numbers, and connect their findings to estimate the sums and differences of mixed
and Differences (1 per teacher)
numbers without models.
Instructional Procedures:
1. Place students into groups of 4. Distribute 3 sets of fraction circles to each group and handout:
MATERIALS
fraction circles (3 sets per 4 students)
Estimated Sums and Differences to each student.
2. Display teacher resource: Estimated Sums and Differences. Instruct student groups to use the
fraction circles to model problem 1 handout: Estimated Sums and Differences, and then create a
sketch of the model on their handout. Allow time for students to complete problem 1. Monitor and
assess student groups to check for understanding. Facilitate a class discussion about the solution
process.
Ask:
page 29 of 117 Grade 6/Mathematics
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Suggested Duration: 9 days
Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
In problem 1, what fractions are you given? (
and
Notes for Teacher
.)
What do you need to do with these two fraction values? Explain. (Find the sum since I am
given the amount eaten during the first minute and the second minute and the question asks how
much pie was eaten in 2 minutes.)
How are the addends,
and
related to
How is the sum of
and
the addends,
are both less than
and
?(
to
. When I add
are both less than
related to 1 whole? Explain. (Since
and
and
.)
is 1 whole and
, the sum is less than 1 whole.)
In problem 1, how can you estimate the sum for
is equivalent to
and
and
? Explain.
, the sum is more than
is
.
because I added more than
, and the sum is less than 1 whole because both addends were less than
reasonable estimate of the sum is greater than
less than
.A
and less than 1 whole.)
3. Instruct students groups to complete the remainder of handout: Estimated Sums and Differences.
Allow students to use the fraction circles if needed. Monitor and assess student groups to check for
understanding. Facilitate a class discussion about the solution processes used to estimate each
solution.
4. Collect handout: Estimated Sums and Differences to use for further instruction.
page 30 of 117 Enhanced Instructional Transition Guide
Suggested
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6-7
Grade 6/Mathematics
Unit 03:
Suggested Duration: 9 days
Suggested Instructional Procedures
Notes for Teacher
Topics:
Spiraling Review
Adding mixed numbers
Subtracting mixed numbers
ATTACHMENTS
Explore/Explain 6
Teacher Resource: Pie Sums and
Students use fraction circles to find the sums and differences for mixed numbers, and connect their
Differences: Mixed Numbers KEY (1
findings to the algorithms for adding and subtracting mixed numbers.
per teacher)
Handout: Pie Sums and Differences:
Instructional Procedures:
Mixed Numbers (1 per student)
1. Place students in groups of 4. Distribute 4 sets of fraction circles to each group and handout: Pie
Sums and Differences: Mixed Numbers to each student. Instruct student groups to complete the
MATERIALS
practice problems. Allow time for students to complete the handout.
fraction circles (4 sets per 4 students)
2. To facilitate student understanding of finding sums and differences of mixed numbers, use Local
Local Resource(s)
Resource(s) that provide everyday situations for students to use fraction circles to model, record their
models in pictorial form, and make connections to the algorithms.
Clarifications and/or Considerations:
Local Resource(s) should be used to meet the specificity and rigor of the Instructional Focus
Document for this unit.
Topics:
Spiraling Review
Adding fractions and mixed numbers
page 31 of 117 Grade 6/Mathematics
Unit 03:
Suggested Duration: 9 days
Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Notes for Teacher
Subtracting fractions and mixed Numbers
ATTACHMENTS
Elaborate 1
Teacher Resource: Pie Eating Contest
Students formalize and apply the algorithms for adding and subtracting fractions and mixed numbers.
KEY (1 per teacher)
Handout: Pie Eating Contest (1 per
Instructional Procedures:
student)
Teacher Resource: Pie Eating Contest
1. Place students in groups of 4. Distribute handout: Pie Eating Contest to each student. Instruct
student groups to complete the handout. Allow time for students to complete the activity. Monitor and
(1 per teacher)
assess students to check for understanding. Facilitate individual group discussions, as needed.
Ask:
What are the two fractions you are using for this problem? Answers may vary.
and
,
and
and
,
, etc.
Do you need to rename any of the fractions? Explain. Answer may vary. Rename
so it will
have the same denominator as three-eighths. 8 is a common multiple for 8 and 4, Rename
and
to have a common denominator of 12; etc.
How do you use symbolic notation when you rename one-fourth?
If Juan ate three-eighths of a pie and two-eighths of a pie, how much pie has he eaten? (
)
page 32 of 117 Grade 6/Mathematics
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Suggested Duration: 9 days
Enhanced Instructional Transition Guide
Suggested
Day
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Notes for Teacher
How do you show this using symbolic notation?
What did you do with the numerators of the two fractions? (Added the numerators.)
What did you do with the denominators of the two fractions? (If there is a common
denominator between the two fractions, it is left alone because the denominator tells us what
fractional parts I am using.)
Why don’t you add the denominators? (It would not give us a reasonable answer. I am adding
two fractional parts of a pie. Both fractional parts are less than
add
to the
, I am adding more than
numerators and the denominators,
, so more than
.
is
less than
. When I
of a pie will be eaten. If I added the
the fraction sum would be less than
, which is
not a reasonable response. The denominator tells us the type of fractional part. Also if I change
the denominator without changing the numerator proportionally, I change the value of the fraction,
and therefore, change the problem.)
How can you summarize the process you used to record adding these two fractions?
(Rename each fraction so there is a common denominator. The common denominator is a
multiple of the denominators. After renaming the fractions so the fractions have common
denominators, I add the numerators only.)
2. Distribute and the previously collected handout: Estimated Sums and Difference to each student.
3. Instruct students to check the reasonableness of their solutions from handout: Pie Eating Contest
by comparing their corresponding estimated solutions on handout: Estimated Sums and
Differences.
page 33 of 117 Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Grade 6/Mathematics
Unit 03:
Suggested Duration: 9 days
Notes for Teacher
4. Display teacher resource: Pie Eating Contest. Facilitate a class discussion about the processes
students used to solve the problems.
Ask:
What was one action you did before adding or subtracting any fractions? (Renamed each
fraction so the fractions had a common denominator. The common denominator was a multiple of
the denominators.)
How do you rename mixed numbers? (Write each whole number in fraction form with a
common denominator. Rename the fraction part of the mixed number so the fraction also has a
common denominator. Add all the parts together from the whole numbers and fraction.)
After renaming the fractions, what action did you do with the numerators? (I added the
numerators or subtracted the numerators only. The common denominators were not added or
subtracted because this would not give us a reasonable answer, and the purpose of the
denominator identifies the type of fractional part.)
8
Topics:
Spiraling Review
Applications of adding fractions and mixed numbers
Applications of subtracting fractions and mixed numbers
ATTACHMENTS
Elaborate 2
Teacher Resource: Fraction
Students solve real-life problems involving adding and subtracting fractions and mixed numbers.
Applications KEY (1 per teacher)
Handout: Fraction Applications (1 per
Instructional Procedures:
student)
1. Place students in pairs. Distribute handout: Fraction Applications to each student. Instruct pairs to
page 34 of 117 Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Grade 6/Mathematics
Unit 03:
Suggested Duration: 9 days
Notes for Teacher
complete the fraction practice problems. Allow time for students to complete the handout. Monitor
and assess student pairs to check for understanding. Facilitate a class discussion to debrief student
solutions.
9
Evaluate 1
Instructional Procedures:
1. Assess student understanding of related concepts and processes by using the Performance
Indicator(s) aligned to this lesson.
Performance Indicator(s):
Grade 06 Mathematics Unit 03 PI 01
Estimate and solve addition and subtraction problems involving fractions and mixed numbers in problem
situations (e.g., recipes, pizza slices). Create a graphic organizer (e.g., flowchart, flip book, etc.) describing,
in words, the process for adding and subtracting fractions and mixed numbers with like and unlike
denominators. Use the organizer to solve the problems, verifying each step of the process with the solution.
Sample Performance Indicator:
Create a flow chart describing in words the process for adding and subtracting mixed
numbers with like and unlike denominators. Use the flow chart to answer the questions
in the following problem scenario, verifying each step in the solution process.
page 35 of 117 Grade 6/Mathematics
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Suggested Duration: 9 days
Enhanced Instructional Transition Guide
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Notes for Teacher
Betty is making her favorite cookies. See the recipe below:
(1) First, she is asked to combine the sugars. How much sugar will she have in her bowl?
(2) Then, she is asked to add the flour. When measuring the flour, Betty realizes she only has
cups of
flour. How much more flour does she need?
(3) The recipe said to mix the salt and baking soda together prior to adding it to the mixture. About how many
teaspoons of salt and baking soda will Betty add?
Standard(s): 6.2A , 6.2B , 6.11A , 6.11B , 6.11D , 6.13B
ELPS ELPS.c.1C , ELPS.c.3J , ELPS.c.5G
page 36 of 117 Enhanced Instructional Transition Guide
Suggested
Day
Suggested Instructional Procedures
Grade 6/Mathematics
Unit 03:
Suggested Duration: 9 days
Notes for Teacher
04/01/13
page 37 of 117 Grade 6
Mathematics
Unit: 03 Lesson: 01
Who Ate More? KEY
Juan and Sam each ordered a lemon meringue pie from a local café. Each pie was
the same size. Juan cut his pie into 8 equal size slices. Sam cut his pie into 4
equal size slices. Juan said he ate more pie because he ate 4 slices of his pie. Sam
said he ate more pie because his pieces were larger, and he ate 2 slices of his pie.
Who is correct? Explain.
Since both pies are the same size, neither Juan nor Sam ate more pie. Both ate the same amount of pie. Refer to diagram
below:
4
Juan ate of the pie.
8
4
Shade 1 for every 2 parts =
8
Create 2 groups from the eighths:
halves
4
1
1 of the 2 groups are shaded:
=
8
2
4 ÷ 4
1
=
8 ÷ 4
2
Sam ate
2
of the pie.
4
Shade 1 for every 2 parts =
2
4
Create 2 groups from the fourths.
2
1
1 of the 2 groups are shaded:
=
4
2
2 ÷ 2
1
=
4 ÷ 2
2
4
which is equivalent to
8
2
For every 2 pieces in Sam’s pie shade 1: Shade
which is equivalent to
4
For every 2 pieces in Juan’s pie shade 1: Shade
©2012, TESCCC
05/17/12
1
.
2
1
.
2
page 1 of 1
Grade 6
Mathematics
Unit: 03 Lesson: 01
Who Ate More?
Juan and Sam each ordered a lemon meringue pie from a local café. Each pie was
the same size. Juan cut his pie into 8 equal size slices. Sam cut his pie into 4
equal size slices. Juan said he ate more pie because he ate 4 slices of his pie. Sam
said he ate more pie because his pieces were larger, and he ate 2 slices of his pie.
Who is correct? Explain.
©2012, TESCCC
08/13/12
page 1 of 1
Grade 6
Mathematics
Unit: 03 Lesson: 01
One-Half and Number Line Relations KEY
1. Model the given fraction and one-half using fraction pieces, then shade each representation for
the given fraction and one-half using the models below to match the fraction pieces.
2. Create equivalent fractions using a common denominator for the given fraction and one-half using
fraction pieces, then shade the equivalent fraction for each model below the original model to
match the fraction pieces.
3. Circle the correct symbol, explain how the given fraction is related to one-half, and write a
comparison statement for the relationship.
Problem 1: three- fourths
Problem 3: one-third
<
=
<
=
>
<
=
>
<
=
>
>
Explain
Explain
3
1
is greater than
. (Compare pieces)
4
2
1
1
is less than
. (Compare pieces)
3
2
1
2
is equal to
.
2
4
1
2
1
3
is equal to
and
is equal to
3
6
2
6
3
1
1 2
is
greater than
( ).
4
4
2 4
1 2
1
1 3
( ) is
less than
( ).
3 6
6
2 6
3 1
>
4 2
1 1
<
3 2
Problem 2: two-fifths
Problem 4: five-eighths
<
<
=
=
>
<
>
<
=
=
>
©2012, TESCCC
>
Explain
Explain
2
1
is less than
. (Compare pieces)
5
2
5
1
is greater than
. (Compare pieces)
8
2
2
4
1
5
is equal to
and
is equal to
.
5
10
2
10
1
4
is equal to
.
2
8
2
4
1
1
5
(
is
less than
(
)
5 10
10
2 10
5
1
1 4
is
greater than
( ).
8
8
2 8
2 1
<
5 2
5 1
>
8 2
08/13/12
page 1 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
One Half and Number Line Relations KEY
Use the number lines below to show the relationship between one-half and the given fractions.
1. Plot the given fraction on the number line.
2. Divide and label the number line to show equivalent fractions for the given fraction and onehalf.
3. Describe how the given fraction and one-half are related and write a comparison statement.
Problem 5: one-fourth
0
1
4
1 2
=
2 4
3
4
1=
4
4
Explain
1 1
2 1
1
Divided the number line into 4 equal parts – fourths – to plot .
is equivalent to .
is less than .
4 2
4 4
2
1
1
1 2 1 1
is
less than
( ). < .
4
4
2 4 4 2
Problem 6: two-thirds
0
1
6
2
6
1
3
=
2
6
2
4
=
3
6
5
6
1=
6 3
=
6 3
Explain
Divided the number line into 6 equal parts – sixths – to plot
2 4 1
3
( ). is equivalent to .
3 6 2
6
2 4
1 3 2 4
1
1 3
2 1
( ) is greater than
( ). ( ) is
greater than
( ).
> .
3 6
2 6 3 6
6
2 6
3 2
Problem 7: three-eighths
0
1
8
2
8
3
8
1
4
=
2
8
5
8
6
8
7
8
1=
8
8
1=
6
6
Explain
3 1
4
Divided the number line into 8 equal parts – eighths – to plot .
is equivalent to .
8 2
8
3
1 4 3
1
1 4 3 1
is less than
( ). is
less than
( ). < .
8
2 8 8
8
2 8 8 2
Problem 8: five-sixths
0
1
6
2
6
1
3
=
2
6
4
6
5
6
Explain
5 1
3
Divided the number line into 6 equal parts – sixths – to plot .
is equivalent to .
6 2
6
5
1 3
5
2
1 3
5 1
is greater than
( ).
is
greater than
( ). > .
6
2 6
6
6
2 6
6 2
©2012, TESCCC
08/13/12
page 2 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
One-Half and Number Line Relations
1. Model the given fraction and one-half using fraction pieces, then shade each representation for
the given fraction and one-half using the models below to match the fraction pieces.
2. Create equivalent fractions using a common denominator for the given fraction and one-half using
fraction pieces, then shade the equivalent fraction for each model below the original model to
match the fraction pieces.
3. Circle the correct symbol, explain how the given fraction is related to one-half, and write a
comparison statement for the relationship.
Problem 1: three-fourths
Problem 3: one-third
<
<
=
=
>
<
>
<
=
=
>
>
Explain
Explain
Problem 2: two-fifths
Problem 4: five-eighths
<
<
=
=
>
<
>
<
=
=
>
>
Explain
©2012, TESCCC
Explain
08/13/12
page 1 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
One-Half and Number Line Relations
Use the number lines below to show the relationship between one-half and the given fractions.
1. Plot the given fraction on the number line.
2. Divide and label the number line to show equivalent fractions for the given fraction and onehalf.
3. Describe how the given fraction and one-half are related and write a comparison statement.
Problem 5: one-fourth
0
1
2
1
Explain
Problem 6: two-thirds
0
1
2
1
Explain
Problem 7: three-eighths
0
1
2
1
Explain
Problem 8: five-sixths
0
1
2
1
Explain
©2012, TESCCC
08/13/12
page 2 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences KEY
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 1
Juan ate one-fourth of a pie and Sam ate one-third of a pie. How much pie have the two boys eaten?
Estimate and Reasoning:
1
1
1 1
1
and , are both less than .
plus
equals 1.
4
3
2 2
2
1
Since each addend is less than , the sum will be less than 1.
2
1
1
1
1
1
1
1
1
is
less than . Since
is greater than , the sum of
and
is greater than .
4
4
2
3
4
4
3
2
Each of the addends,
Reasonable Estimate:
1
< sum < 1
2
Problem 2
Juan ate one-eighth of a pie and Sam ate one-fourth of a pie. How much pie have the two boys
eaten?
Estimate and Reasoning:
1
1
1 1
1
and , are both less than .
plus
equals 1.
8
4
2 2
2
1
Since each addend is less than , the sum will be less than 1.
2
1
3
1
1
2
1
1
1
is
less than . Since
is equivalent to , the sum of
and
is less than .
8
8
2
4
8
8
4
2
Each of the addends,
Reasonable Estimate:
sum <
©2012, TESCCC
08/13/12
1
2
page 1 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences KEY
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 3
Juan ate one-fourth of a pie and Sam ate two-thirds of a pie. How much pie have the two boys eaten?
Estimate and Reasoning:
1
1
2
1
is less than . The addend
is greater than .
4
2
3
2
1
1
Since one addend is less than
and the other addend is greater than , the sum is greater
2
2
1
1
than
and we need to determine if the sum is less than 1 or greater than 1 because
plus
2
2
1
2
1
2
1
equals 1.
is
less than 1 ( plus
= 1).
2
3
3
3
3
1
1
2
1
Since
is less than , the sum of
and
is less than 1.
4
3
3
4
The addend,
Reasonable Estimate:
1
< sum < 1
2
Problem 4
Juan ate five-sixths of a pie and Sam ate one-fourth of a pie. How much pie have the two boys
eaten?
Estimate and Reasoning:
1
1
5
1
is less than . The addend
is greater than .
4
2
6
2
1
1
Since one addend is less than
and the other addend is greater than , we know the sum is
2
2
1
greater than
and we need to determine if the sum is less than 1 or greater than 1 because
2
1
1
5
1
5
1
plus
equals 1.
is
less than 1 ( plus
= 1).
2
2
6
6
6
6
1
1
5
1
Since
is greater than , the sum of
and
is greater than 1.
4
6
6
4
The addend,
Reasonable Estimate:
1 < sum
©2012, TESCCC
08/13/12
page 2 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences KEY
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 5
Juan ate one-third of a pie and Sam ate three-fourths of a pie. How much pie have the two boys
eaten?
Estimate and Reasoning:
1
1
3
1
is less than . The addend
is greater than .
3
2
4
2
1
1
and the other addend is greater than , we know the sum is
Since one addend is less than
2
2
1
greater than
and we need to determine if the sum is less than 1 or greater than 1 because
2
1
1
3
1
3
1
plus
equals 1.
is
less than 1 ( plus
= 1).
2
2
4
4
4
4
1
1
3
1
Since
is greater than , the sum of and
is greater than 1.
3
4
4
3
The addend,
Reasonable Estimate:
1 < sum
Problem 6
Juan ate five-eighths of a pie and Sam ate three-fourths of a pie. How much pie have the two boys
eaten?
Estimate and Reasoning:
5
3
1
and , are greater than
and less than 1.
8
4
2
1
1
Since both addends are greater than , we know the sum is greater than 1 because
plus
2
2
1
equals 1.
2
We also know the sum is less than 2 because 1 whole plus 1 whole = 2.
Both addends,
Reasonable Estimate:
1 < sum < 2
©2012, TESCCC
08/13/12
page 3 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences KEY
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 7
Juan had three-fourths of a pie. He ate one-third of the pie. How much pie does Juan have left to eat?
Estimate and Reasoning:
3
1 3
1
1
is greater than . is
greater than .
4
2 4
4
2
3
1
2
1 1
1
minus
=
(equivalent to ).
is greater than .
4
4
4
2 3
4
1
3
1
Since we are subtracting more than
when we subtract
minus , the
4
4
3
1
3
1
1
difference will be less than because minus
= .
2
4
4
2
Reasonable Estimate:
difference <
1
2
Problem 8
Sam had five-sixths of a pie. He ate two-thirds of the pie. How much pie does Sam have left to eat?
Estimate and Reasoning:
5
1 5
2
1
is greater than .
is
greater than .
6
2 6
6
2
5
2
3
1 2
2
minus
=
(equivalent to ).
is greater than .
6
6
6
2 3
6
2
5
2
Since we are subtracting more than
when we subtract
– , the
6
6
3
1
5
2
1
difference will be less than
because minus
= .
2
6
6
2
Reasonable Estimate:
difference <
©2012, TESCCC
08/13/12
1
2
page 4 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences KEY
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 9
Juan had seven-eighths of a pie. He ate one-fourth of the pie. How much pie does Juan have left to
eat?
Estimate and Reasoning:
7
1 7
3
1
is greater than .
is
greater than .
8
2 8
8
2
7
3
4
1 1
3
minus
=
(equivalent to ).
is less than .
8
8
8
2 4
8
3
7
1
Since we are subtracting less than
when we subtract
minus , the
8
8
4
1
7
3
1
difference will be greater than
because
minus
= .
2
8
8
2
Reasonable Estimate:
1
< difference < 1
2
Problem 10
Juan had five-eighths of a pie. He ate one-half of a pie. How much pie does Juan have left to eat?
Estimate and Reasoning:
5
1 5
1
4
1
is greater than .
minus
=
(equivalent to ).
8
2 8
8
8
2
1
5
1
Since we are subtracting
when we subtract
minus , the difference
2
8
2
1
5
1
1
will be less than
because
minus
= .
2
8
8
2
Reasonable Estimate:
difference <
©2012, TESCCC
08/13/12
1
2
page 5 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences KEY
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 11
Sam three-fourths of a pie. He ate one-eighth of a pie. How much pie does Sam have left to eat?
Estimate and Reasoning:
3
1 1
2
is greater than .
is equivalent to .
4
2 4
8
1
1
1
Since we are subtracting less than when we subtract
minus , the
4
4
8
1
2
1
1 1
difference will be greater than
because we have more than and
plus = .
2
4
4
4 2
Reasonable Estimate:
1
< difference < 1
2
©2012, TESCCC
08/13/12
page 6 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 1
Juan ate one-fourth of a pie and Sam ate one-third of a pie. How much pie have the two boys eaten?
Estimate and Reasoning:
Reasonable Estimate:
Problem 2
Juan ate one-eighth of a pie and Sam ate one-fourth of a pie. How much pie have the two boys
eaten?
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 1 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 3
Juan ate one-fourth of a pie and Sam ate one-thirds of a pie. How much pie have the two boys
eaten?
Estimate and Reasoning:
Reasonable Estimate:
Problem 4
Juan ate five-sixths of a pie and Sam ate one-fourth of a pie. How much pie have the two boys
eaten?
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 2 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 5
Juan ate one-third of a pie and Sam ate three-fourths of a pie. How much pie have the two boys
eaten?
Estimate and Reasoning:
Reasonable Estimate:
Problem 6
Juan ate five-eighths of a pie and Sam ate three-fourths of a pie. How much pie have the two boys
eaten?
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 3 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 7
Juan had three-fourths of a pie. He ate one-third of the pie. How much pie does Juan have left to eat?
Estimate and Reasoning:
Reasonable Estimate:
Problem 8
Sam had five-sixths of a pie. He ate two-thirds of the pie. How much pie does Sam have left to eat?
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 4 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 9
Juan had seven-eighths of a pie. He ate one-fourth of the pie. How much pie does Juan have left to
eat?
Estimate and Reasoning:
Reasonable Estimate:
Problem 10
Juan had five-eighths of a pie. He ate one-half of a pie. How much pie does Juan have left to eat?
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 5 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Reason It Out Sums and Differences
1. Discuss how the numbers used in the problem are related to one-half.
2. Determine an estimate through reasoning.
3. Draw a diagram to support your estimate.
Problem 11
Sam had three-fourths of a pie. He ate one-eighth of a pie. How much pie does Sam have left to eat?
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 6 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Is This Wrong? KEY
Example 1:
Sam ate one-half of a pie and then ate one-fourth of a pie. He said that to
show this using symbolic language, we would write:
1
1
1+1
2
+
=
=
2
4
2+4
6
Draw a diagram to show whether Sam is correct.
+
+
=
Sam is not correct. The correct amount is three-fourths.
Example 2:
Jennifer said she had three-fourths of a mini pizza to eat for a snack when
she got home from school. She said she planned to eat one-half of the mini
pizza. Jennifer said that to show this using symbolic language, we would
write:
3
1
3 − 1
2
−
=
=
=1
4
2
4 − 2
2
Draw a diagram to show whether Jennifer is correct.
–
–
=
Jennifer is not correct. The correct amount is one-fourth.
©2012, TESCCC
08/13/12
page 1 of 1
Grade 6
Mathematics
Unit: 03 Lesson: 01
Is This Wrong?
Example 1:
Sam ate one-half of a pie and then ate one-fourth of a pie. He said that to
show this using symbolic language, we would write:
1
1
1+1
2
+
=
=
2
4
2+4
6
Draw a diagram to show whether Sam is correct.
Example 2:
Jennifer said she had three-fourths of a mini pizza to eat for a snack when
she got home from school. She said she planned to eat one-half of the mini
pizza. Jennifer said that to show this using symbolic language, we would
write:
3
1
3 − 1
2
−
=
=
=1
4
2
4 − 2
2
Draw a diagram to show whether Jennifer is correct.
©2012, TESCCC
08/13/12
page 1 of 1
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
1. Model the given problem situations using fraction pieces by either displaying both addends for
addition or the beginning amount for subtraction.
2. Shade each addend or beginning amount in the given model to match the fraction pieces
displayed.
3. Write each number in fraction notation.
4. Create equivalent fractions using a common denominator for the given fractions using fraction
pieces.
5. Subdivide and shade the equivalent fractions for each addend or beginning amount in the model
below the original model to match the displayed fraction pieces.
6. Combine the shaded fraction models for addition or subtract the amount of pie eaten from the
beginning amount model.
7. Record your actions using symbolic notation.
Problem 1
Juan ate one-fourth of a pie and Sam ate one- third of a pie. How much pie have the two boys eaten?
One-fourth =
1
4
One-third =
1
3
+
Rename fraction if needed:
Rename
1
4
Rename fraction if needed:
with a denominator of
Rename
twelfths:
1
3
with a denominator of twelfths:
1x 4
4
=
3x4
12
1x 3
3
=
4x3
12
+
Equivalent Fraction Model:
3
12
Equivalent Fraction Model:
4
12
Summary
steps:
of
symbolic
1
1
+
4
3
1x 3
3
1x4
4
=
and
=
4x3
12
3x4
12
3
4
+
12
12
3+4
12
7
12
Combine and shade the numerators of the common denominator
3+4
7
=
12
12
©2012, TESCCC
08/13/12
page 1 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 2
Juan had three-fourths of a pie. He ate one-third of a pie. How much pie does Juan have left to eat?
Three-fourths =
3
4
Ate one-third =
–
Rename fraction if needed:
Rename
3
4
Rename fraction if needed:
with a denominator of
Rename
twelfths:
1
3
with a denominator of twelfths:
1x 4
4
=
3x4
12
3x3
9
=
4x3
12
–
Equivalent Fraction Model:
1
3
Ate
4
12
9
12
Summary of symbolic
steps:
3
1
−
4
3
3x3
9
1x4
4
=
and
=
4x3
12
3x4
12
9
4
−
12
12
9 − 4
12
5
12
Subtract the numerators of the common denominator
9 − 4
5
=
12
12
©2012, TESCCC
08/13/12
page 2 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 3
Juan ate one-eighth of a pie. Sam ate one fourth of a pie. How much pie have the two boys eaten?
One-eighth =
1
8
One-fourth =
1
4
+
Rename fraction if needed:
Rename fraction if needed:
1
8
Rename
1
4
with a denominator of eighths:
1x 2
2
=
4x2
8
+
Equivalent Fraction Model:
1
8
Equivalent Fraction Model:
Combine and shade the numerators of the common denominator
2
8
Summary of symbolic
steps:
1
1
+
8
4
1
1x2
2
and
=
8
4x2
8
1
2
+
8
8
1+2
8
3
8
1+2
3
=
8
8
©2012, TESCCC
08/13/12
page 3 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 4
Sam had five-sixths of a pie. He ate two-thirds of a pie. How much pie does Sam have left to eat?
Five-sixths =
5
6
Ate two-thirds =
–
Rename fraction if needed:
Rename fraction if needed:
5
6
Rename
2
3
with a denominator of sixths:
2x2
4
=
3x2
6
–
Equivalent Fraction Model:
2
3
Ate
4
6
5
6
Summary of symbolic
steps:
5
2
−
6
3
5
2x2
4
and
=
6
3x2
6
5
4
−
6
6
5 − 4
6
1
6
Subtract the numerators of the common denominator
5 − 4
1
=
6
6
©2012, TESCCC
08/13/12
page 4 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 5
Juan ate one-fourth of a pie. Sam ate two-thirds of a pie. How much pie have the two boys eaten?
One-fourth =
1
4
Two-thirds =
2
3
+
Rename fraction if needed:
Rename
1
4
Rename fraction if needed:
with a denominator of
Rename
twelfths:
2
3
with a denominator of twelfths:
2x4
8
=
3x4
12
1x 3
3
=
4x3
12
+
Equivalent Fraction Model:
3
12
Equivalent Fraction Model:
8
12
Summary of symbolic
steps:
1
2
+
4
3
1x3
3
2x4
8
=
and
=
4x3
12
3x4
12
3
8
+
12
12
3+8
12
11
12
Combine and shade the numerators of the common denominator
3+8
11
=
12
12
©2012, TESCCC
08/13/12
page 5 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 6
Juan had seven-eighths of a pie. He ate one-fourth of it. How much pie does Juan have left to eat?
seven eighths =
7
8
Ate one-fourth =
–
Rename fraction if needed:
Rename fraction if needed:
7
8
Rename
1
4
with a denominator of fourths:
1x2
2
=
4x2
8
–
Equivalent Fraction Model:
1
4
Ate
2
8
7
8
Summary of symbolic
steps:
7
1
−
8
4
7
1x2
2
and
=
8
4x2
8
7
2
−
8
8
7 − 2
8
5
8
Subtract the numerators of the common denominator
7 − 2
5
=
8
8
©2012, TESCCC
08/13/12
page 6 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 7
Juan ate one-third of a pie. Sam ate three-fourths of a pie. How much pie have the two boys eaten?
one third =
1
3
three fourths =
3
4
+
Rename fraction if needed:
Rename
1
3
Rename fraction if needed:
with a denominator of
Rename
3
4
with a denominator of
twelfths:
twelfths:
1x4
4
=
3x4
12
3x3
9
=
4x3
12
+
Equivalent Fraction Model:
4
12
Equivalent Fraction Model:
9
12
Summary of symbolic
steps:
1
3
+
3
4
1x4
4
3x3
9
=
and
=
3x4
12
4x3
12
4
9
+
12
12
4+9
12
13
1
or 1
12
12
Combine and shade the numerators of the common denominator
4+9
13
1
=
=1
12
12
12
©2012, TESCCC
08/13/12
page 7 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 8
Juan had five-eighths pies. He ate one-half of a pie. How much pie does Juan have left?
Five-eighths =
5
8
–
Ate one-half =
1
2
Rename fraction if needed:
Rename fraction if needed:
5
8
1x 4
4
=
2x4
8
–
Equivalent Fraction Model:
Ate
5
8
Subtract the numerators of the common denominator
4
8
Summary of symbolic
steps:
5
1
−
8
2
5
1x4
4
and
=
8
2x4
8
5
4
−
8
8
5 − 4
8
1
8
5 − 4
1
=
8
8
©2012, TESCCC
08/13/12
page 8 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 9
Juan ate five-sixths of a pie and Sam ate one-fourth of a pie. How much pie have the two boys
eaten?
5
1
Summary of symbolic
Five-sixths =
One-fourth =
6
4
steps:
+
Rename fraction if needed:
Rename fraction if needed:
5x2
10
=
6x2
12
1x 3
3
=
4x3
12
5
1
+
6
4
5x2
10
1x3
3
=
and
=
6x2
12
4x3
12
10
3
+
12
12
10 + 3
12
13
1
or 1
12
12
+
Equivalent Fraction Model:
10
12
Equivalent Fraction Model:
3
12
Combine and shade the numerators of the common denominator
10 + 3
13
1
=
or 1
12
12
12
©2012, TESCCC
08/13/12
page 9 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 10
Sam had three-fourth of a pie. He ate one-eighth of a pie. How much pie does Sam have left?
Three-fourth =
3
4
–
Ate one-eighth =
1
8
Rename fraction if needed:
Rename fraction if needed:
3x2
6
=
4x2
8
1
8
–
Equivalent Fraction Model:
Ate
1
8
Summary of symbolic
steps:
3
1
−
4
8
3x2
6
and
4x2
8
6
1
−
8
8
6 − 1
8
5
8
6
8
Subtract the numerators of the common denominator
6− 1
5
=
8
8
©2012, TESCCC
08/13/12
page 10 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences KEY
Problem 11
Juan ate five-eighths of a pie. Sam ate three-fourths of a pie. How much pie have the two boys
eaten?
Five-eighths =
5
8
Three-fourths =
3
4
+
Rename fraction if needed:
Rename
Rename fraction if needed:
3
4
with a denominator of
eighths:
5
8
3x2
6
=
4x2
8
+
Equivalent Fraction Model:
5
8
Equivalent Fraction Model:
6
8
Summary of symbolic
steps:
5
3
+
8
4
5
3x2
6
and
=
8
4x2
8
5
6
+
8
8
5+6
8
11
3
=1
8
8
Combine and shade the numerators of the common denominator
5+6
11
=
or
8
8
©2012, TESCCC
1
3
8
08/13/12
page 11 of 11
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences
1. Model the given problem situations using fraction pieces by either displaying both addends for
addition or the beginning amount for subtraction.
2. Shade each addend or beginning amount in the given model to match the fraction pieces
displayed.
3. Write each number in fraction notation.
4. Create equivalent fractions using a common denominator for the given fractions using fraction
pieces.
5. Subdivide and shade the equivalent fractions for each addend or beginning amount in the model
below the original model to match the displayed fraction pieces.
6. Combine the shaded fraction models for addition or subtract the amount of pie eaten from the
beginning amount model.
7. Record your actions using symbolic notation.
Problem 1
Juan ate one-fourth of a pie and Sam ate one-third of a pie. How much pie have the two boys eaten?
Summary of symbolic
steps:
+
Rename fraction if needed:
Rename fraction if needed:
+
Equivalent Fraction Model: ______
©2012, TESCCC
Equivalent Fraction Model: ______
08/13/12
page 1 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences
Problem 2
Juan had three-fourths of a pie. He ate one-third of a pie. How much pie does Juan have left to eat?
Summary of symbolic
steps:
–
Rename fraction if needed:
Ate how much?
Rename fraction if needed:
–
Ate how much?
Equivalent Fraction Model: ______
Problem 3
Juan ate one-eighth of a pie. Sam ate one-fourth of a pie. How much pie have the two boys eaten?
Summary of symbolic
steps:
+
Rename fraction if needed:
Rename fraction if needed:
+
Equivalent Fraction Model: ______
©2012, TESCCC
Equivalent Fraction Model: ______
08/13/12
page 2 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences
Problem 4
Sam had five-sixths of a pie. He ate two-thirds of a pie. How much pie does Sam have left to eat?
Summary of symbolic
steps:
Ate how much?
–
Rename fraction if needed:
Rename fraction if needed:
–
Ate how much?
Equivalent Fraction Model: ______
Problem 5
Juan ate one-fourth of a pie. Sam ate two-thirds of a pie. How much pie have the two boys eaten?
Summary of symbolic
steps:
+
Rename fraction if needed:
Rename fraction if needed:
+
Equivalent Fraction Model: ______
©2012, TESCCC
Equivalent Fraction Model: ______
08/13/12
page 3 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences
Problem 6
Juan had seven-eighths of a pie. He ate one-fourth of it. How much pie does Juan have left to eat?
Summary of symbolic
steps:
Ate how much?
–
Rename fraction if needed:
Rename fraction if needed:
–
Ate how much?
Equivalent Fraction Model: ______
Problem 7
Juan ate one-third of a pie. Sam ate three-fourths of a pie. How much pie have the two boys eaten?
Summary of symbolic
steps:
+
Rename fraction if needed:
Rename fraction if needed:
+
Equivalent Fraction Model: ______
©2012, TESCCC
Equivalent Fraction Model: ______
08/13/12
page 4 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences
Problem 8
Juan had five-eighth of a pie. He ate one-half of a pie. How much pie does Juan have left?
Summary of symbolic
steps:
Ate how much?
–
Rename fraction if needed:
Rename fraction if needed:
–
Ate how much?
Equivalent Fraction Model: ______
Problem 9
Juan ate five-sixths of a pie and Sam ate one-fourth of a pie. How much pie have the two boys
eaten?
Summary of symbolic
steps:
+
Rename fraction if needed:
Rename fraction if needed:
+
Equivalent Fraction Model: ______
©2012, TESCCC
Equivalent Fraction Model: ______
08/13/12
page 5 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences
Problem 10
Sam had three-fourths of a pie. He ate one-eighth of it. How much pie does Sam have left?
Summary of symbolic
steps:
Ate how much?
–
Rename fraction if needed:
Rename fraction if needed:
–
Ate how much?
Equivalent Fraction Model: ______
Problem 11
Juan ate five-eighths of a pie. Sam ate three-fourths of a pie. How much pie have the two boys
eaten?
Summary of symbolic
steps:
+
Rename fraction if needed:
Rename fraction if needed:
+
Equivalent Fraction Model: ______
©2012, TESCCC
Equivalent Fraction Model: ______
08/13/12
page 6 of 6
Grade 6
HS Mathematics
Unit: 03 Lesson: 01
Equivalent Fractions KEY
•
•
1.
Calculate the greatest common factor (GCF) for the numerator and denominator in each set
of fractions.
Simplify and write equivalent fractions by dividing the numerator and denominator in each
fraction by the greatest common factor (GCF) of the denominators.
Note: The student may select a different method other than the one shown to calculate the
GCF.
12
16
Numerator:
___12___
Denominator:
___16___
Greatest Common Factor of ___12___ and ___16___:
Factors of 12: 2 x 2 x 3 or 1 x 12, 2 x 6, 3 x 4 1, 2, 3, 4, 6, 12
Factors of 16: 2 x 2 x 2 x 2 or 1 x 16, 2 x 8, 4 x 4 1, 2, 4, 8
GCF is 2 x 2 or 4
12 ÷ 4
Divide numerator and denominator by the GCF:
=
16 ÷ 4
2.
3
4
7
21
Numerator:
___7___
Denominator:
___21___
Greatest Common Factor of ___7___ and ___21___:
Factors of 7: 1 x 7 or 1 x 7 1, 7
Factors of 21: 3 x 7 or 1 x 21, 3 x 7 1, 3, 7, 21
GCF is 7
3.
Divide numerator and denominator by the GCF:
7 ÷ 7
21 ÷ 7
=
1
3
9 ÷ 1
11 ÷ 1
=
9
11
9
11
Numerator:
___9___
Denominator:
___11___
Greatest Common Factor of ___9___ and ___11___:
Factors of 9: 3 x 3 or 1 x 9, 3 x 3 1, 3, 9
Factors of 11: 1 x 11 or 1 x 11 1, 11
GCF is 1
©2012, TESCCC
Divide numerator and denominator by the GCF:
08/13/12
page 1 of 2
Grade 6
HS Mathematics
Unit: 03 Lesson: 01
Equivalent Fractions KEY
Calculate the least common multiple for the denominators in each set of fractions.
Generate an equivalent fraction for each fraction from the given set of fractions so each
fraction has the LCM as its denominator.
Note: The student may select a different method other than the one shown to calculate the
LCM.
3
5
4.
and
8
12
3
___8___
Denominator for :
8
5
Denominator for
:
___12___
12
•
•
Least Common Multiple of ___8___ and ___12___:
Multiples of 8: 8 x 1 = 8, 8 x 2 = 16, 8 x 3 = 24, 8 x 4 = 32, … 8, 16, 24, 32, …
Multiples of 12: 12 x 1 = 12, 12 x 2 = 24, 12 x 3 = 36, 12 x 4 = 48 … 12, 24, 36, 48, …
LCM is 24
5.
Equivalent fraction for
3
with the LCM as the denominator:
8
3x 3
8x 3
=
9
24
Equivalent fraction for
5
5x 2
with the LCM as the denominator:
12
12 x 2
=
10
24
3
2
and
4
5
3
:
4
2
Denominator for :
5
Denominator for
___4___
___5___
Least Common Multiple of ___4___ and ___5___:
Multiples of 4: 4 x 1 = 4, 4 x 2 = 8, 4 x 3 = 12, 4 x 4 = 16, 4 x 5 = 20, … 4, 8, 12, 16, 20, …
Multiples of 5: 5 x 1 = 5, 5 x 2 = 10, 5 x 3 = 15, 5 x 4 = 20, … 5, 10, 15, 20, …
LCM is 20
Equivalent fraction for
3
with the LCM as the denominator:
4
3x 5
4x 5
=
15
20
Equivalent fraction for
2
with the LCM as the denominator:
5
2x 4
5x 4
=
8
20
©2012, TESCCC
08/13/12
page 2 of 2
Grade 6
HS Mathematics
Unit: 03 Lesson: 01
Equivalent Fractions
•
•
1.
Calculate the greatest common factor (GCF) for the numerator and denominator in each set
of fractions.
Simplify and write equivalent fractions by dividing the numerator and denominator in each
fraction by the greatest common factor (GCF) of the denominators.
12
16
Numerator:
______
Denominator:
______
Greatest Common Factor of ______ and ______:
Divide numerator and denominator by the GCF:
2.
12 ÷
16 ÷
=
7
21
Numerator:
______
Denominator:
______
Greatest Common Factor of ______ and ______:
Divide numerator and denominator by the GCF:
3.
7 ÷
21 ÷
=
9 ÷
11 ÷
=
9
11
Numerator:
______
Denominator:
______
Greatest Common Factor of ______ and ______:
Divide numerator and denominator by the GCF:
©2012, TESCCC
08/13/12
page 1 of 2
Grade 6
HS Mathematics
Unit: 03 Lesson: 01
Equivalent Fractions
•
•
4.
Calculate the least common multiple for the denominators in each set of fractions.
Generate an equivalent fraction for each fraction from the given set of fractions so each
fraction has the LCM as its denominator.
3
5
and
8
12
3
:
8
5
Denominator for
:
12
Denominator for
______
______
Least Common Multiple of ______ and ______:
5.
Equivalent fraction for
3
with the LCM as the denominator:
8
3x
8x
Equivalent fraction for
5
5x
with the LCM as the denominator:
12
12 x
=
=
3
2
and
4
5
3
:
4
2
Denominator for :
5
Denominator for
______
______
Least Common Multiple of ______ and ______:
Equivalent fraction for
3
with the LCM as the denominator:
4
3x
4x
=
Equivalent fraction for
2
with the LCM as the denominator:
5
2x
5x
=
©2012, TESCCC
08/13/12
page 2 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name? KEY
1.
2.
3.
4.
5.
6.
7.
8.
Model the given number using fraction pieces.
Subdivide and shade Diagram A to represent the displayed number.
Use the fractions pieces to create a model equivalent to the model in Diagram A.
Subdivide and shade Diagram B to represent the displayed number. Explain your process.
Use the fraction pieces to create another model equivalent to both models in Diagram A and B.
Subdivide and shade Diagram C to represent the displayed number. Explain your process.
Record your actions using symbolic notation.
Between what two whole number values is the fraction or mixed number from Diagram A? Indicate if the given
fraction or mixed number is less than halfway, halfway, or more than halfway between the two whole numbers.
Explain your response.
9. Plot the values for the models from Diagram A, Diagram B, and Diagram C on the corresponding number lines. If
necessary, divide the number lines to match the model. Explain how the fraction circle model and number line
model are similar.
Problem 1: Two and three-eighths
Fraction Diagram A
Fraction Diagram B
Two and three-eighths
Nineteen-eighths
Fraction Diagram C
One and eleven-eighths
Explain
Divided one whole into 8 equal parts
Have eighths in each of 2 wholes
Explain
Divided each whole into 8 equal parts
Have eighths in each of 3 wholes
Shaded 19 parts:
2
3
8
19
8
=
2
11
8
Shaded 1 whole and 11 parts: 1
19
8
11
1
8
=
3
is between the two whole numbers 2 and 3.
8
3
is less than halfway, halfway, or more than halfway between the two whole numbers 2 and 3.
8
3
1
4
1
3
Explain:
is less than
because
is equal to ; 2 is less than halfway between 2 and 3
8
2
8
2
8
2
Number Line A
0
1
2
2
1
2
19 eighths
8
8
16
8
3
3
8
Number Line B
0
3
24
8
Number Line C
0
1
2
3
11
1
8
Explain how the fraction circle model and number line model are similar:
Each model gives a representation for 1 whole. One whole for each model was divided into 8 equal size parts to represent eighths.
3
8
Two whole units and three of the eight parts in one of the whole units were shaded or marked to represent 2 . Divided two wholes
from the original model into 8 equal size parts for each whole (1 x 8 = 8) to create two models for eighths. Divided one whole from the
original model into 8 equal size parts (1 x 8 = 8) to create a model for eighths. The denominator identifies the type of parts and the
numerator identifies the number of parts.
©2012, TESCCC
08/13/12
page 1 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name? KEY
Problem 2: One and one-third
Fraction Diagram A
Fraction Diagram B
Fraction Diagram C
One and one-third
Four-thirds
One and two-sixths
Explain
Explain
Divided 1 whole into 3 equal parts
Have thirds in each of 2 wholes
Shaded 4 parts:
1
1
Explain:
1
3
Divided each third into 2 equal parts
3 x 2 = 6: now have sixths.
4
3
Shaded 1 whole and 2 parts: 1
=
4
3
1
1
is between the two whole numbers 1 and 2.
3
=
1
2
6
2
6
1
is less than halfway, halfway, or more than halfway between the two whole numbers 1 and 2.
3
1
1
3
1
1
2
1
is less than
because
is equal to
and
is equal to . 1 is less than halfway between 1 and 2.
3
2
6
2
3
6
3
Number Line A
0
1
1
3
2
4
3
2
2
6
2
1
Number Line B
0
1
Number Line C
0
1
1
Explain how the fraction circle model and number line model are similar:
Each model gives a representation for 1 whole. A whole for each model was divided into 3 equal size parts to represent thirds. One
1
3
whole and one of the three parts were shaded or marked to represent 1 . Divided one whole from the original model into three equal
size parts (1 x 3 = 3) to create a model for thirds. Divided each third from the original model into two equal size parts (3 x 2 = 6) to
create a model for sixths. The denominator identifies the type of parts and the numerator identifies the number of parts.
©2012, TESCCC
08/13/12
page 2 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name? KEY
Problem 3: One and one-fourth
Fraction Diagram A
Fraction Diagram B
Fraction Diagram C
One and one-fourth
Five-fourths
One and two-eighths
Explain
Explain
Divided 1 whole into 4 equal parts
Have fourths in each of 2 wholes
Shaded 5 parts:
1
1
Explain:
1
4
Divided each fourth into 2 equal parts
4 x 2 = 8: now have eighths.
5
4
2
8
Shaded 1 whole and 2 parts: 1 .
=
5
4
1
1
is between the two whole numbers 1 and 2.
4
=
1
2
8
1
is less than halfway, halfway, or more than halfway between the two whole numbers 1 and 2.
4
1
1
2
1
1
2
1
is less than because
is equal to
and
is less than to . 1 is less than halfway between 1 and 2.
4
2
4
2
4
4
4
Number Line A
0
1
1
1
4
2
5
4
2
2
8
2
Number Line B
0
1
Number Line C
0
1
1
Explain how the fraction circle model and number line model are similar:
Each model gives a representation for one whole. A whole for each model was divided into 4 equal size parts to represent fourths. One
1
4
whole and one of the four parts were shaded or marked to represent 1 . Divided one whole from the original model into 4 equal size
parts (1 x 4 = 4) to create a model for fourths. Divided each fourth from the original model into 2 equal size parts (4 x 2 = 8) to create a
model for eighths. The denominator identifies the type of parts and the numerator identifies the number of parts.
©2012, TESCCC
08/13/12
page 3 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name? KEY
Problem 4: Two and one-sixth
Fraction Diagram A
Fraction Diagram B
Explain
Explain
Divided each whole into 6 equal parts
Have sixths in each of 3 wholes
Divided one whole into 6 equal parts
Have sixths in each of 2 wholes
Shaded 13 parts:
2
1
6
2
Explain:
13
6
Shaded 1 whole and 7 parts: 1
13
6
=
2
Fraction Diagram C
1
=
7
6
7
6
1
is between the two whole numbers 2 and 3.
6
1
is less than halfway, halfway, or more than halfway between the two whole numbers 2 and 3.
6
1
3
1
1
is less than 1 half because
is equal to . 2 is less than halfway between 2 and 3
6
6
2
6
Number Line A
0
1
2
1
6
3
13
6
3
7
6
3
2
Number Line B
0
2
1
6
6
12
6
Number Line C
0
1
2
1
Explain how the fraction circle model and number line model are similar:
Each model gives a representation for one whole. One whole for each model was divided into six equal size parts to represent sixths.
Two whole units and one of the six parts in one of the whole units were shaded or marked to represent 2
1
. Divided two wholes from
6
the original model into six equal size parts for each whole (1 x 6 = 6) to create two models for sixths. Divided one whole from the
original model into six equal size parts (1 x 6 = 6) to create a model for sixths. The denominator identifies the type of parts and the
numerator identifies the number of parts.
©2012, TESCCC
08/13/12
page 4 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name? KEY
Problem 5: One and two-fifths
Fraction Diagram B
Fraction Diagram A
Explain
Fraction Diagram C
Explain
Explain
Divided 1 whole into 5 equal parts
Have fifths in each of 2 wholes
Shaded 7 parts:
1
2
5
1
Explain:
7
5
4
10
Shaded 1 whole and 4 parts: 1
7
5
=
1
Divided each fifth into 2 equal parts
5 x 2 = 10: now have tenths.
4
1
10
=
2
is between the two whole numbers 1 and 2.
5
2
is less than halfway, halfway, or more than halfway between the two whole numbers 1 and 2.
5
2
1
5
1
2
4
2
is less than because
is equal to
and
is equal to . 1 is less than halfway between 1 and 2.
5
2
10
2
5
10
5
Number Line A
0
1
1
2
5
2
Number Line B
0
1
7
5
5
5
2
10
5
Number Line C
0
1
4
1
10
2
Explain how the fraction circle model and number line model are similar:
Each model gives a representation for 1 whole. A whole for each model was divided into five equal size parts to represent fifths. One
2
5
whole and two of the five parts were shaded or marked to represent 1 . Divided one whole from the original model into five equal size
parts (1 x 5 = 5) to create a model for fifths. Divided each fifth from the original model into two equal size parts (5 x 2 = 10) to create a
model for tenths. The denominator identifies the type of parts and the numerator identifies the number of parts.
©2012, TESCCC
08/13/12
page 5 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name? KEY
Problem 6: two-thirds
Fraction Diagram A
two-thirds
Fraction Diagram B
Fraction Diagram C
four-sixths
six-ninths
Explain
Explain
Divided each third into 3 equal parts
3 x 3 = 9: now have ninths.
Divided each third into 2 equal parts
3 x 2 = 6: now have sixths.
Shaded 4 (2 x 2 = 4) parts:
2
3
4
6
Shaded 6 (2 x 3 = 6) parts:
4
6
=
=
6
9
6
9
2
is between the two whole numbers 0 and 1.
3
2
is less than halfway, halfway, or more than halfway between the two whole numbers 0 and 1.
3
2
1
3
1 2
4 2
2
is greater than
because
is equal to .
is equal to .
is more than halfway between 0 and 1.
is less than 1
3
2
6
2 3
6 3
3
3
because
is equal to 1 whole.
3
Explain:
Number Line A
0
2
3
1
4
6
1
6
9
1
Number Line B
0
Number Line C
0
Explain how the fraction circle model and number line model are similar:
Each model gives a representation for one whole. The whole for each model was divided into three equal size parts to represent thirds.
Two of the three parts were shaded or marked to represent
2
. Divided each third from the original model into two equal size parts (3 x
3
2 = 6) to create a model for sixths. Divided each third from the original model into three equal size parts (3 x 3 = 9) to create a model
for ninths. The denominator identifies the type of parts and the numerator identifies the number of parts.
©2012, TESCCC
08/13/12
page 6 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name?
Model the given number using fraction pieces.
Subdivide and shade Diagram A to represent the displayed number.
Use the fractions pieces to create a model equivalent to the model in Diagram A.
Subdivide and shade Diagram B to represent the displayed number. Explain your process.
Use the fraction pieces to create another model equivalent to both models in Diagram A and B.
Subdivide and shade Diagram C to represent the displayed number. Explain your process.
Record your actions using symbolic notation.
Between what two whole number values is the fraction or mixed number from Diagram A?
Indicate if the given fraction or mixed number is less than halfway, halfway, or more than
halfway between the two whole numbers. Explain your response.
9. Plot the values for the models from Diagram A, Diagram B, and Diagram C on the
corresponding number lines. If necessary, divide the number lines to match the model. Explain
how the fraction circle model and number line model are similar.
1.
2.
3.
4.
5.
6.
7.
8.
Problem 1: Two and three-eighths
Fraction Diagram A
______
=
Fraction Diagram B
Fraction Diagram C
Explain
Explain
______
=
______
____ is between the two whole numbers _____ and _____.
_____ is less than halfway, halfway, or more than halfway between the two whole numbers
_____ and _____.
Explain:___________________________________________________________________________
___________________________________________________________________________
Number Line A
0
1
2
3
2
3
2
3
Number Line B
0
1
Number Line C
0
1
Explain how the fraction circle model and number line model are similar:
________________________________________________________________________________
©2012, TESCCC
08/13/12
page 1 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name?
Problem 2: One and one-third
Fraction Diagram A
______
=
Fraction Diagram B
Fraction Diagram C
Explain
Explain
______
=
______
____ is between the two whole numbers _____ and _____.
_____ is less than halfway, halfway, or more than halfway between the two whole numbers
_____ and _____.
Explain: ______________________________________________________________________________
______________________________________________________________________________
Number Line A
0
1
2
Number Line B
0
1
2
Number Line C
0
1
2
Explain how the fraction circle model and number line model are similar:
________________________________________________________________________________
________________________________________________________________________________
©2012, TESCCC
08/13/12
page 2 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name?
Problem 3: One and one-fourth
Fraction Diagram A
______
=
Fraction Diagram B
Fraction Diagram C
Explain
Explain
______
=
______
____ is between the two whole numbers _____ and _____.
_____ is less than halfway, halfway, or more than halfway between the two whole numbers
_____ and _____.
Explain: ______________________________________________________________________________
______________________________________________________________________________
Number Line A
0
1
2
Number Line B
0
1
2
Number Line C
0
1
2
Explain how the fraction circle model and number line model are similar:
________________________________________________________________________________
________________________________________________________________________________
©2012, TESCCC
08/13/12
page 3 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name?
Problem 4: Two and one-sixth
Fraction Diagram A
______
=
Fraction Diagram B
Fraction Diagram C
Explain
Explain
______
=
______
____ is between the two whole numbers _____ and _____.
_____ is less than halfway, halfway, or more than halfway between the two whole numbers
_____ and _____.
Explain: ______________________________________________________________________________
______________________________________________________________________________
Number Line A
0
1
2
3
2
3
2
3
Number Line B
0
1
Number Line C
0
1
Explain how the fraction circle model and number line model are similar:
________________________________________________________________________________
©2012, TESCCC
08/13/12
page 4 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name?
Problem 5: One and two-fifths
Fraction Diagram A
Fraction Diagram B
Fraction Diagram C
Explain
Explain
Explain
______
=
______
=
______
____ is between the two whole numbers _____ and _____.
_____ is less than halfway, halfway, or more than halfway between the two whole numbers
_____ and _____.
Explain: ______________________________________________________________________________
______________________________________________________________________________
Number Line A
0
1
2
Number Line B
0
1
2
Number Line C
0
1
2
Explain how the fraction circle model and number line model are similar:
________________________________________________________________________________
________________________________________________________________________________
©2012, TESCCC
08/13/12
page 5 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
What’s Your Name?
Problem 6: two-thirds
Fraction Diagram A: ____
______
Fraction Diagram B: ____
=
______
Fraction Diagram C: ____
=
______
____ is between the two whole numbers _____ and _____.
_____ is less than halfway, halfway, or more than halfway between the two whole numbers
_____ and _____.
Explain: ______________________________________________________________________________
______________________________________________________________________________
Number Line A
0
1
Number Line B
0
1
Number Line C
0
1
Explain how the fraction circle model and number line model are similar:
________________________________________________________________________________
________________________________________________________________________________
©2012, TESCCC
08/13/12
page 6 of 6
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences KEY
1. Model the given problem situation using fraction pieces by either displaying both addends for
addition or the beginning amount for subtraction.
2. Shade each addend or beginning amount in the given model to match the fraction pieces
displayed.
3. Use the fraction pieces to estimate the sum or difference of each situation.
4. Do not give an exact answer by working the problem.
5. Draw a diagram to support your estimate.
6. Explain your reasoning and write a reasonable estimate using symbols.
Problem 1
Sam needs to eat one and one-half pies in 2 minutes. At the end of 1 minute, he has eaten seveneighths of a pie. How much pie does Sam have left to eat?
Estimate and Reasoning:
7
8
is a little less than 1 whole pie. (
If Sam ate
7
8
7
8
plus
1
8
= 1 whole.)
of a pie, we will subtract 1 whole pie, since
7
8
is almost 1 whole.
We subtract a majority of the 1 whole pie, but leave a part of the 1 whole pie. Still have
1
2
a pie
plus a part from the 1 whole pie.
Difference is greater than
1
2
because we still have a little extra from the 1 whole pie we
subtracted.
Reasonable Estimate:
1
2
©2012, TESCCC
08/13/12
< difference < 1
page 1 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences KEY
Problem 2
Juan needs to eat two and one-half pies in 2 minutes. During the first minute he ate one and one-third
pies. How much pie does Juan have left to eat?
Estimate and Reasoning:
1
3
1
ate 1
3
One and
is between 1 and 2, but closer to 1 because
Juan
pies ( 1
1
3
1
3
1
2
is less than .
is close to 1) so we subtract 1 whole pie and a part of
of a pie. There is still a part of the
Still have 1 whole pie and a part of
1
2
1
2
of a pie remaining.
1
2
of a pie left.
Reasonable Estimate:
1 < difference < 1
1
2
Problem 3
Sam ate one and one-half pies in 2 minutes and seven-eighths of a pie in 1 minute. How much pie
has Sam eaten in 3 minutes?
Estimate and Reasoning:
7
8
is a little less than 1 whole pie. (
Since
7
8
7
8
plus
1
8
= 1 whole.)
is almost 1 whole pie, we will consider it as 1 whole pie.
To estimate, 1 whole pie plus
Since added a little extra to
1
2
7
to
8
of a pie plus 1 whole pie is 2
1
2
pies.
make 1 whole pie, the sum is a little less than 2
1
2
pies.
Reasonable Estimate:
2 < sum < 2
©2012, TESCCC
08/13/12
1
2
page 2 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences KEY
Problem 4
Juan has two and one-fourth pies to eat. In one minute he ate five-eighths of a pie. How much pie
does Juan have left to eat?
Estimate and Reasoning:
8
8
is equal to 1 whole pie.
Juan ate
5
8
of a pie, so we subtract
5
8
from the
8
8
and the difference is a little less than
1
2
of a
pie.
Still have 1 whole pie plus
1
4
of a pie plus a little less than
have more than 1 whole pie and less than 1
3
4
1
2
of a pie (equivalent to
2
4
) so we
of a pie.
Reasonable
Estimate:
1 < difference < 1
3
4
Problem 5
Sam ate one and one-fourth pies and five eighths of another pie. How much pie has Sam eaten?
Estimate and Reasoning:
5
8
is a little more than
1 whole plus
1
4
plus
Since only added
4
8
4
8
1
2
(equivalent to
(equivalent to
2
4
of a pie instead of
4
8
).
3
4
) equals 1 .
5
8
3
4
of a pie, the sum is a little more than 1 , but less than 2.
Reasonable Estimate:
1
©2012, TESCCC
08/13/12
3
4
< sum < 2
page 3 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences KEY
1.
2.
3.
4.
Write a problem that will fit the given diagram.
Do not work the problem and do not give an exact answer.
Estimate the sum for each situation.
Determine your estimate through reasoning.
Problem 6
Juan ate one and five-eighths of a pie during the first minute. He ate one-fourth of a pie during the
second minute. How much pie has Juan eaten during 2 minutes?
Estimate and Reasoning:
Since
5
8
plus
5
8
is
1
4
3
8
less than 1 whole (
5
8
plus
is not quite 1 whole because
When we add the 1 whole pie,
The sum is greater than
1
2
5
8
3
8
1
4
=
8
8
3
8
is less than .
of a pie, and
because
5
8
= 1 whole),
1
4
of a pie, the sum is not quite 2.
is greater than
1
2
(
4
8
).
Reasonable Estimate:
1
1.
2.
3.
4.
1
2
< sum < 2
Write a problem that will fit the given diagram.
Do not work the problem and do not give an exact answer.
Estimate the difference for each situation.
Determine your estimate through reasoning.
Problem 7
Juan has two and one-fourth pies to eat. In one minute he ate six-eighths of a pie. How much pie
does Juan have left to eat?
Estimate and Reasoning:
8
8
is equal to 1 whole pie.
Juan ate
6
8
of a pie, so we subtract
Still have 1 whole pie plus
1
4
6
8
from the
of a pie plus
2
8
8
8
and the difference is 2 eighths of a pie.
of a pie so we have 1 whole pie
plus less than another 1 whole pie.
Reasonable Estimate:
1 < difference < 2
©2012, TESCCC
08/13/12
page 4 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences KEY
Problem 8
Sam has one and one-third of a pie to eat. He ate one-fourth of a pie. How much pie does Sam have
left to eat?
Estimate and Reasoning:
1 whole is equivalent to 4 fourths.
The difference between 1
1
4
from
4
4
3
4
plus
1
3
there is
3
4
1
3
and
1
4
of a pie and
4
4
–
1
4
=
3
4
.
is more than 1 because when we subtract
1
3
of a pie left.
is more than 1 whole because
1
3
is more than
1
4
.
Reasonable Estimate:
1 < difference < 2
©2012, TESCCC
08/13/12
page 5 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences
1. Model the given problem situation using fraction pieces by either displaying both addends for
addition or the beginning amount for subtraction.
2. Shade each addend or beginning amount in the given model to match the fraction pieces
displayed.
3. Use the fraction pieces to estimate the sum or difference of each situation.
4. Do not give an exact answer by working the problem.
5. Draw a diagram to support your estimate.
6. Explain your reasoning and write a reasonable estimate using symbols.
Problem 1
Sam needs to eat one and one-half pies in 2 minutes. At the end of 1 minute, he has eaten seveneighths of a pie. How much pie does Sam have left to eat?
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 1 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences
Problem 2
Juan needs to eat two and one-half pies in 2 minutes. During the first minute he ate one and one-third
pies. How much pie does Juan have left to eat?
Estimate and Reasoning:
Reasonable Estimate:
Problem 3
Sam ate one and one-half pies in 2 minutes and seven-eighths of a pie in 1 minute. How much pie
has Sam eaten in three minutes?
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 2 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences
Problem 4
Juan has two and one-fourth pies to eat. In 1 minute he ate five-eighths of a pie. How much pie does
Juan have left to eat?
Estimate and Reasoning:
Reasonable Estimate:
Problem 5
Sam ate 1 and one-fourth pies and five-eighths of another pie. How much pie has Sam eaten?
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 3 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences
1.
2.
3.
4.
5.
Write a problem that will fit the given diagram.
Do not work the problem and do not give an exact answer.
Estimate the sum for each situation.
Determine your estimate through reasoning.
Explain your reasoning and write a reasonable estimate using symbols.
Problem 6
________________________________________________________________________________
________________________________________________________________________________
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 4 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Estimated Sums and Differences
1.
2.
3.
4.
5.
Write a problem that will fit the given diagram.
Do not work the problem and do not give an exact answer.
Estimate the difference for each situation.
Determine your estimate through reasoning.
Explain your reasoning and write a reasonable estimate using symbols.
Problem 7
________________________________________________________________________________
________________________________________________________________________________
Estimate and Reasoning:
Reasonable
Estimate:
Problem 8
________________________________________________________________________________
Estimate and Reasoning:
Reasonable Estimate:
©2012, TESCCC
08/13/12
page 5 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences: Mixed Numbers KEY
1. Model the given problem situation using fraction pieces by either displaying both addends for
addition or the beginning amount for subtraction.
2. Shade each addend or beginning amount in the given model to match the fraction pieces
displayed.
3. Write each number in fraction notation.
4. Create equivalent fractions using a common denominator for the given fractions using fraction
pieces.
5. Subdivide and shade the equivalent fractions for each addend or beginning amount in the model
below the original model to match the displayed fraction pieces.
6. Combine the shaded fraction models for addition or subtract the amount of pie eaten from the
beginning amount model.
7. Record your actions using symbolic notation.
Problem 1
Juan ate two and two-thirds of a pie. Sam ate one and three-fourths of a pie. How much pie have the
two boys eaten?
two and two-thirds = 2
2
3
one and three-fourths = 1
3
4
+
Rename fraction if needed:
Rename fraction if needed:
Rename 2
2
3
Rename 1
with a denominator of twelfths:
3
4
with a denominator
Summary of symbolic steps:
of twelfths:
2 8 8x4
32
2 = =
=
3 3 3x4
12
3 7 7x3
21
1 = =
=
4 4 4x3
12
+
Equivalent Fraction Model:
32
12
Equivalent Fraction Model:
2
3
+1
3
4
2 8 8x4
32
3 7 7x3
21
2 = =
=
and 1 = =
=
3 3 3x4
12
4 4 4x3
12
32
21
+
12
12
32 + 21
12
53
5
=4
12
12
2
21
12
Combine and shade the numerators of the common denominator
32+ 21 53
=
12
12
©2012, TESCCC
08/13/12
page 1 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences: Mixed Numbers KEY
Problem 2
Juan had one and one-third of a pie. He ate three-fourths of it. How much pie does Juan have left to
eat?
one and one-third = 1
1
3
Ate three-fourths =
–
Rename fraction if needed:
Rename 1
1
3
3
4
Rename fraction if needed:
with a denominator of
Rename
twelfths:
3
4
with a denominator of twelfths:
3x3
9
=
4x3
12
1 4 4x4
16
1 = =
=
3 3 3x4
12
Summary of symbolic
steps:
1
3
−
3
4
1 4 4x4
16
3x3
9
1 = =
=
and
=
3 3 3x4
12
4x3
12
16
9
−
12
12
16 − 9
12
7
12
1
–
Equivalent Fraction Model:
Ate
9
12
16
12
Subtract the numerators of the common denominator
16 − 9
7
=
12
12
©2012, TESCCC
08/13/12
page 2 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences: Mixed Numbers
1. Model the given problem situation using fraction pieces by either displaying both addends for
addition or the beginning amount for subtraction.
2. Shade each addend or beginning amount in the given model to match the fraction pieces
displayed.
3. Write each number in fraction notation.
4. Create equivalent fractions using a common denominator for the given fractions using fraction
pieces.
5. Subdivide and shade the equivalent fractions for each addend or beginning amount in the model
below the original model to match the displayed fraction pieces.
6. Combine the shaded fraction models for addition or subtract the amount of pie eaten from the
beginning amount model.
7. Record your actions using symbolic notation.
Problem 1
Juan ate two and two-thirds of a pie. Sam ate one and three-fourths of a pie. How much pie have the
two boys eaten?
Summary of symbolic
steps:
+
Rename fraction if needed:
Rename fraction if needed:
+
Equivalent Fraction Model: ______
©2012, TESCCC
Equivalent Fraction Model: ______
08/13/12
page 1 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Sums and Differences: Mixed Numbers
Problem 2
Juan had one and one-third of a pie. He ate three-fourths of it. How much pie does Juan have left to
eat?
Summary of symbolic
steps:
–
Rename fraction if needed:
Rename fraction if needed:
–
Equivalent
Fraction Model:
______
©2012, TESCCC
Ate how much?
Ate how much?
Equivalent
Fraction Model:
______
08/13/12
page 2 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Eating Contest KEY
Find the sum or difference for each problem. Use symbolic notation to record your
process. Compare the computed sum or difference with the estimated sum or
difference from the handout: Estimated Sums and Differences.
1. Juan at three-eighths of a pie the first minute of a pie eating contest and one-fourth
of a pie the second minute. How much pie has Juan eaten after 2 minutes?
3
1
+
8
4
Rename fraction if needed:
3
1x2
2
and
=
8
4x2
8
3
2
+
8
8
3+2
8
5
8
The computed sum of
5
8
is reasonable with the estimation of:
1
2
< sum < 1.
2. Sam needs to eat one and one-half pies in 2 minutes. At the end of 1 minute, he
has eaten seven-eighths of a pie. How much pie does Sam have left to eat?
1
7
−
2
8
Rename fraction if needed:
1
1x4
4
7
8
 8 and 2 x 4 = 8  and 8


4
7
8
8 + 8 − 8


(8 + 4) − 7
8
12 − 7
8
5
8
The computed difference of
©2012, TESCCC
5
8
is reasonable with the estimation of:
08/13/12
1
2
< difference < 1.
page 1 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Eating Contest KEY
3. Juan needs to eat two and one-half pies in 2 minutes. During the first minute he ate
one and one-third pies. How much pie does Juan have left to eat?
1
1
− 1
2
3
Rename fraction if needed:
2
6
1x3
3
1x2
2
6
6
 6 and 6 and 2 x 3 = 6  and  6 and 3 x 2 = 6 




6
6
3
6
2




6 + 6 + 6 − 6 + 6




(6 + 6 + 3) − (6 + 2)
6
15 − 8
6
7
1
=1
6
6
The computed difference of
7
6
1
2
is reasonable with the estimation of: 1 < difference < 1 .
4. Sam ate one and one-half pies in 2 minutes and seven-eighths of a pie in 1 minute.
How much pie has Sam eaten in 3 minutes?
1
7
+
2
8
Rename fraction if needed:
1
1x4
4
7
8
 8 and 2 x 4 = 8  and 8


8
4
7


8 + 8 + 8


(8 + 4) + 7
8
12 + 7
8
19
3
=2
8
8
The computed sum of
©2012, TESCCC
19
8
1
2
is reasonable with the estimation of: 2 < sum < 2 .
08/13/12
page 2 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Eating Contest KEY
5. Juan has two and one-fourth pies to eat. In one minute he ate five-eighths of a pie.
How much pie does Juan have left to eat?
1
5
−
4
8
Rename fraction if needed:
2
8
1x2
2
5
8
 8 and 8 and 4 x 2 = 8  and 8


8
8
2
5


8 + 8 + 8 − 8


(8 + 8 + 2) − 5
8
18 − 5
8
13
5
=1
8
8
The computed difference of
13
8
3
4
is reasonable with the estimation of: 1 < difference < 1 .
6. Sam ate one and one-fourth pies and five-eighths of another pie. How much pie
has Sam eaten?
1
5
+
4
8
Rename fraction if needed:
1
1x2
2
5
8
 8 and 4 x 2 = 8  and 8


2
5
8
8 + 8 + 8


(8 + 2) + 5
8
10 + 5
8
15
7
=1
8
8
The computed sum of
©2012, TESCCC
19
8
is reasonable with the estimation of: 1
08/13/12
3
4
< sum < 2.
page 3 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Eating Contest KEY
7. Juan ate one and five-eighths of a pie during the first minute. He ate one-fourth of a
pie during the second minute. How much pie has Juan eaten during 2 minutes?
5
1
+
8
4
Rename fraction if needed:
1
5
1x2
2
8
 8 and 8  and 4 x 2 = 8


8
5
2

8 + 8 + 8


(8 + 5) + 2
8
13 + 2
8
15
7
=1
8
8
The computed sum of
15
8
is reasonable with the estimation of: 1
1
2
< sum < 2.
8. Sam ate three-fourths of a pie and seven-eighths of another pie. How much pie has
Sam eaten?
3
7
+
4
8
Rename fraction if needed:
3x2
6
7
=
and
4x2
8
8
6
7
+
8
8
6+7
8
13
5
=1
8
8
The computed sum of
©2012, TESCCC
13
8
is reasonable with the estimation of: 1
08/13/12
1
2
< sum < 2.
page 4 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Eating Contest KEY
9. Juan has two and one-fourth pies to eat. In one minute he ate six-eighths of a pie.
How much pie does Juan have left to eat?
1
6
−
4
8
Rename fraction if needed:
2
8
1x2
2
6
8
 8 and 8 and 4 x 2 = 8  and 8


8
8
2
6


8 + 8 + 8 − 8


(8 + 8 + 2) − 6
8
18 − 6
8
12
4
1
=1 =1
8
8
2
The computed difference of
12
8
is reasonable with the estimation of: 1 < difference < 2
10. Sam has one and one-third of a pie to eat. He ate one-fourth of a pie. How much
pie does Sam have left to eat?
1
1
−
3
4
Rename fraction if needed:
1
1x4
4 
1x3
3
 12
 12 and 3 x 4 = 12  and 4 x 3 = 12


4 
3
 12
 12 + 12  − 12


(12 + 4) − 3
12
16 − 3
12
13
1
=1
12
12
The computed difference of
©2012, TESCCC
13
12
is reasonable with the estimation of: 1 < difference < 2
08/13/12
page 5 of 5
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Eating Contest
Find the sum or difference for each problem. Use symbolic notation to record your
process. Compare the computed sum or difference with the estimated sum or
difference from the handout: Estimated Sums and Differences.
1. Juan at three-eighths of a pie the first minute of a pie eating contest and one-fourth
of a pie the second minute. How much pie has Juan eaten after 2 minutes?
2. Sam needs to eat one and one-half pies in 2 minutes. At the end of 1 minute, he
has eaten seven-eighths of a pie. How much pie does Sam have left to eat?
3. Juan needs to eat two and one-half pies in 2 minutes. During the first minute he ate
one and one-third pies. How much pie does Juan have left to eat?
4. Sam ate one and one-half pies in 2 minutes and seven-eighths of a pie in 1 minute.
How much pie has Sam eaten in 3 minutes?
5. Juan has two and one-fourth pies to eat. In one minute he ate five-eighths of a pie.
How much pie does Juan have left to eat?
©2012, TESCCC
08/13/12
page 1 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
Pie Eating Contest
6. Sam ate one and one-fourth pies and five-eighths of another pie. How much pie
has Sam eaten?
7. Juan ate one and five-eighths of a pie during the first minute. He ate one-fourth of a
pie during the second minute. How much pie has Juan eaten during 2 minutes?
8. Sam ate three-fourths of a pie and seven-eighths of another pie. How much pie has
Sam eaten?
9. Juan has two and one-fourth pies to eat. In one minute he ate six-eighths of a pie.
How much pie does Juan have left to eat?
10. Sam has one and one-third of a pie to eat. He ate one-fourth of a pie. How much
pie does Sam have left to eat?
11. Summarize in words the process of adding and subtracting fractions.
©2012, TESCCC
08/13/12
page 2 of 2
Grade 6
Mathematics
Unit: 03 Lesson: 01
Fraction Applications KEY
Show your work for each problem.
2
1. Which of the following is not equivalent to 1 feet?
3
(A)
3
2
ft +
ft
3
3
(B)
5
2
ft +
ft
3
3
(C)
5
ft
3
2. Which number line below represents the distance Tina walked in two days if she walked
mile on day 1 and
3
of a
4
5
of a mile on day 2?
8
(A)
0
1
miles
2
(B)
0
1
miles
2
(C)
0
3
4
1
5
8
2
miles
©2012, TESCCC
08/13/12
page 1 of 3
Grade 6
Mathematics
Unit: 03 Lesson: 01
Fraction Applications KEY
3
3. Write three different expressions for 2 . Draw a diagram to support each expression.
4
1+
4
4
+
4
4
4
4
+
+
1
4
4
3
4
4
4
4
4
3
4
3
4
3
4
11
4
4. Which of the given expressions do not have the same sum?
(A)
1
2
+
2
3
(B)
8
7
+
12
14
(C)
4
8
+
2
6
Expression A and B are equivalent expressions.
8
7
1
2
is equivalent to and
is equivalent to
12
14
2
3
©2012, TESCCC
08/13/12
page 2 of 3
Grade 6
Mathematics
Unit: 03 Lesson: 01
Fraction Applications KEY
Before working any problem, give a reasonable estimate. Work each problem and show your work to
support your answer.
5. Rachel has walked three-fourths of a mile. She has one more minute to reach the seveneighths mile marker. How far must Rachel walk to reach the seven-eighths mile marker?
Estimate: less than one-fourth
7
3
−
8
4
7
6
−
8
8
1
8
6. Alex mows a large rectangular lot once a month. Alex sets aside three consecutive afternoons
1
1
to mow the lot. On day 1 he mows 18 square yards. For day 2 he mows 22 square yards.
2
3
1
On the final day he mows 15 square yards. What is the area in square yards of the
4
rectangular lot Alex mows?
Estimate: 18 + 22 + 15 = 55 square yards at least
18
1
1
1
1
+ 22 + 15 = 56
square yards
2
4
12
3
For each problem, describe why the given answer is not reasonable. Do not work the problem. Give
an estimate that would be reasonable for each problem. Justify your response.
7. Sylvia walked two-thirds of a mile on Monday and one-half of a mile on Tuesday.
How far did Sylvia walk on Monday and Tuesday?
The answer is: three-fifths of a mile
1
2
is greater than .
3
2
1
1
1
1
to , the sum is greater than 1 because
+
= 1.
2
2
2
2
1
2
2
1
A more reasonable estimate for
+
is greater than 1 because
is
less than 1 and you are adding a
3
2
3
3
1 1
2
fraction greater than
( ) to .
3 2
3
When you add a number greater than
8. Tina had a goal to walk three and one-fourth miles in two days. On the first day
she walked one and seven-eighths miles. How many miles does Tina need to
walk on the second day to reach her goal?
The answer is: two and five-eighths miles
When you find the difference: 3
1
7
7
1
– 1 , you are subtracting almost 2 whole units since
is
less than 1
4
8
8
8
whole.
A more reasonable estimate is 3
©2012, TESCCC
1
7
1
– 1 is 1 .
4
8
4
08/13/12
page 3 of 3
Grade 6
Mathematics
Unit: 03 Lesson: 01
Fraction Applications
Show your work for each problem.
2
3
1. Which of the following is not equivalent to 1 feet?
(A)
3
2
ft +
ft
3
3
(B)
5
2
ft +
ft
3
3
(C)
5
ft
3
2. Which number line below represents the distance Tina walked in two days if she walked
mile on day 1 and
3
of a
4
5
of a mile on day 2?
8
(A)
0
1
miles
2
(B)
0
1
miles
2
(C)
0
©2012, TESCCC
1
miles
2
08/13/12
page 1 of 3
Grade 6
Mathematics
Unit: 03 Lesson: 01
Fraction Applications
3
3. Write three different expressions for 2 . Draw a diagram to support each expression.
4
4. Which of the given expressions do not have the same sum?
(A)
1
2
+
2
3
(B)
8
7
+
12
14
(C)
4
8
+
2
6
Before working any problem, give a reasonable estimate. Work each problem and show your work to
support your answer.
5. Rachel has walked three-fourths of a mile. She has one more minute to reach the seven eighths
mile marker. How far must Rachel walk to reach the seven eighths mile marker?
6. Alex mows a large rectangular lot once a month. Alex sets aside three consecutive afternoons to
1
1
mow the lot. On day 1 he mows 18 square yards. For day 2 he mows 22 square yards. On
2
3
1
the final day he mows 15 square yards. What is the area in square yards of the rectangular lot
4
Alex mows?
©2012, TESCCC
08/13/12
page 2 of 3
Grade 6
Mathematics
Unit: 03 Lesson: 01
Fraction Applications
For each problem, describe why the given answer is not reasonable. Do not work the problem. Give
an estimate that would be reasonable for each problem. Justify your response.
7. Sylvia walked two-thirds of a mile on Monday and one-half of a mile on Tuesday.
How far did Sylvia walk on Monday and Tuesday?
The answer is: three-fifths of a mile
8. Tina had a goal to walk three and one-fourth miles in two days. On the first day she walked one
and seven-eighths miles. How many miles does Tina need to walk on the second day to reach her
goal?
The answer is: two and five-eighths miles.
©2012, TESCCC
08/13/12
page 3 of 3