Revisions 2014
Using Benchmark Fractions Plus
Operation Sense with Fractions
LESSON SAMPLER
EMPower Plus was created by the Adult Numeracy Center at TERC.
To learn more about our work with EMPower and other projects, please visit
http://adultnumeracy.terc.edu.
Authors
Donna Curry, Mary Jane Schmitt, Myriam Steinback, and Martha Merson
Contributing Authors
Tricia Donovan, Marlene Kliman, and Pam Meader
Technical Team
Production and Design Team: Valerie Martin and Sherry Soares
Copyeditor: Beverly Cory
© 2013 TERC. All rights reserved.
TERC
2067 Massachusetts Avenue
Cambridge, Massachusetts 02140
The original EMPower™ series was developed at TERC in Cambridge, Massachusetts. This material is
based upon work supported by the National Science Foundation under award number ESI-9911410
and by the Education Research Collaborative at TERC. Any opinions, findings, and conclusions or
recommendations expressed in this publication are those of the authors and do not necessarily reflect the
views of the National Science Foundation.
TERC is a not-for-profit education research and development organization dedicated to improving
mathematics, science, and technology teaching and learning.
All other registered trademarks and trademarks in this book are the property of their respective holders.
Reproduction Permission
The contents of this sampler may be distributed and photocopied for use in classrooms or for other
instructional purposes. However, please contact empower@terc.edu if you wish to use any of the
contents in a published work.
Using Benchmark Fractions Plus: Operation Sense with Fractions
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EMPower™ © 2013
Acknowledgments
Many friends and family members supported teachers’ and EMPower’s efforts. Thank you, Michelle Allman,
Kate Butler, Denise Deagan, Judith Diamond, Cara DiMattia, Ellen McDevitt, Sandy Strunk, and the board of
the Adult Numeracy Network. Roberta Froelich, Jill Gorneau, Brad Hamilton, Michael Hanish, David Hayes,
Judy Hikes, Alisa Izumi, Esther D. Leonelli, Myra Love, Pam Meader, Lambrina Mileva, Luz Rivas, Connie
Rivera, Johanna Schmitt, Rachael Stark, Jonathan Steinback, and Sean Sutherland all made unique and timely
contributions to the project. EMPower’s advisors and evaluators with expertise in adult and math education
made substantive contributions to the first edition. We appreciate the encouragement and advice from John
(Spud) Bradley, EMPower’s program officer, and Gerhard Salinger of the National Science Foundation.
The EMPower Plus revisions were made possible with funding from the MetLife Foundation and support
from District 79 Alternative Schools and Programs and the Office of Adult and Continuing Education in
conjunction with the American Council on Education and MDRC. A special thanks to Laura Feijoo, RoseMarie Mills, Robert Zweig, Zully Tejada, Lisa Hertzog, Lavern Nelson, Ira Yankwitt, Damion Frye, Patrick
Cravillion, Rhonda Naidich and the Office of Adult and Continuing Education’s Instructional Facilitators, and
Michelle Ballard for their efforts. In addition, we thank the Fund for Public Schools for its support.
We are indebted to every adult student who participated in the piloting of EMPower. EMPower authors thank
the teachers who were part of original pilot testing in AZ, ME, MA, NJ, NY, PA, RI, and TN in 2001-2004, as
well as NYC teachers who participated in the pilot of EMPower Plus in 2012-2013. The honest feedback and
suggestions for what worked and what did not work were invaluable.
EMPower NYC Teacher Participants and Instructional Facilitators,
2012–2013
Betty Aderman
Robert Andruskiewicz
Teresa Bell
Victoria Capeci
Les Chassagne
Charles Doughlin
Kathleen Downey
Robert Evans
Amity Gottschalk
Billy Green
David Gutmann
Maria Guzman
Betsy Hill
Kathleen Huggard
Sylvester Jaward
EMPower™ © 2013
Tanaquil Jones
Martha Kimball
Mara Komoska
George Lamptey
Molly Litvintchouk
Sharyn Marsh
Rhonda Naidich
Stephanie Nails
Katie Naplatarski
Jolan Ostane
Matthew Pickert
Rodolfo Rabadad
Diana Raissis
Shanita Rapatalo
Jay Rasin-Waters
Maribel Rivera
Mildred Rodriguez
Carolyn Rudder
Victoria Rush
Kenneth Schebandach
Doreen Sloan
Vanesia Smith
Paul Tardy
Kim Walker
Adam Weiss
Jordan Yarwood
Muriel Zwick
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EMPower™ © 2013
Contents
Get started with EMPower Plus
Unit Introduction
Facilitating Lesson 5a: A Look at One-Eighth
Facilitating Lesson 6a: Equal Measures
Facilitating Lesson 7a: Visualizing and Estimating Sums
and Differences
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Get started with EMPower Plus
Overview
The lessons presented here are excerpted from a set of seven lessons expanding EMPower’s Using Benchmarks:
Fractions, Decimals, and Percents to introduce operations with benchmark fractions.
Brief Overview of the Supplemental Lessons and Published Materials
For a sequence focusing on fractions, start with the published book, Using Benchmarks: Fractions, Decimals,
and Percents. The lessons in this sampler are supplementary materials.
The new lessons start with a grid showing:
• Lesson objectives, Common Core Standards for Mathematical Practice (for easy reference, the list of the
eight math practices is included)
• GED Assessment Targets from Quantitative Problem Solving, Strands 1 and 2
• List of Activities, Math Inspections, Practices, and Extensions
In addition to the grid, the lessons include:
Common Core Standards for Mathematical Practices
• New teacher book chapter (with objectives,
materials list, opening discussion and activities)
• New facilitation notes for teachers
1. Make sense of problems and persevere in solving them. *
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically. *
6. Attend to precision. *
FACILITATING
5a
A Look at
One-Eighth
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning. *
• New activities for teachers and students
• New Math Inspections for students
Objectives
•
Determine1/8ofagivenamount
•
Buildoneighthstodiscover
sixteenths
GED Mathematics Assessment Targets
Anticipated
Q1.a Order fractions and decimals, including on a
number line.
Q2.a Solve single-step or multistep real-world
arithmetic problems involving the four
operations with rational numbers, including
those involving scientific notation.
• New Practice sheets for students
• New Test Practice page for students
• New Blackline Masters
• Answer Keys for all new student pages
The grid references the activities in the published
version of Using Benchmarks, Split It Up, and
Operation Sense that will be used during and
outside of class.
Section
NEW
Activity 1: Fractions of a Yard
NEW: Students measure to the nearest eighth of a yard.
Activity 2: Finding One-Eighth on a
Number Line
NEW: Students find one-eighth on a number line, and then find
eighths of different amounts.
Math Inspection: A Look at OneSixteenth
NEW: Students find a pattern for moving from halves to quarters
to eighths to sixteenths.
Practice: One-Eighth
NEW: Students review the steps to find one-eighth and threeeighths
Practice: Pound It Out
NEW: Students use ounces and pounds to explore sixteenths
Practice: Looking at Both Sides of 0
NEW: Students find integers on the number line
Practice: A Furlong Long
NEW: Students use the notion of furlong as 1/8 mile to answer
questions.
Extension: Fat Quarters and Fat
Eighths
NEW: Students explore the difference between an eighth of a
yard and a fat eighth of a yard.
Test Practice
NEW
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Adaptation
Opening Discussion
EMPower™ © 2013
Solely for use by NYC Department of Education
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Unit Introduction
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Unit Goals
By the end of this unit, students should be able to:
• Describe part-whole situations with fractions;
• Use objects, diagrams, number line segments, and arrays to represent part-whole situations;
• Determine whether a wide variety of fractions is more than, less than, or equal to the benchmark
fractions 1/2, 1/4, and 3/4; • Visually represent the operations of addition and subtraction with fractions;
• Use benchmark fractions to estimate the reasonableness of answers.
See the Materials List on p. xii.
These lessons extend students’ understanding and ability to work with benchmark fractions, decimals, and
percents in a way that makes sense to them. Teachers tell us they love this unit because for the first time,
their students are able to retain what they learn about fractions. By taking the time to establish the concepts
of part-whole, a portion of an amount, and benchmark fractions, students develop a solid foundation for
understanding rational numbers that will serve them well as they reason about data, work with percents, and
continue to make sense of numbers in their everyday lives.
Prerequisites
Don’t hesitate to use these supplementary lessons even if some students do not know their multiplication
facts. This unit could help students become more proficient with some of the multiplication facts, as the unit
offers extensive practice in halving, doubling, quartering, and quadrupling. Students who can quickly recall
the multiplication and division facts for four and eight may move more quickly through the activities and
practices, but students whose multiplication facts for four and eight are shaky can easily halve and halve again.
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Get a Sense of the Sampler
This sampler contains
• Facilitation notes
• 3 lessons
In Lesson 5a, students build on what they know to explore eighths. In Lesson 6a, the emphasis shifts to
equivalent fractions. From there, students are well-positioned to generalize rules for adding and substracting
fractions in Lessons 7a.
Each lesson has one or more activities or investigations. Allowing time for opening and summary discussions
(including time for student reflections) and assuming a thoughtful pace, will take two hours or more for most
lessons.
In Lesson 5a, students
• Identify the part and the whole in various cases;
• Compare amounts to more or less than or equal to a benchmark fraction;
• State the fraction that represents the whole;
• Relate less familiar fractions (such as eighths and sixteenths) to halves and quarters;
• Find fraction amounts using various strategies;
• Determine the whole when a part is known
In Lesson 6a, students
• Use visual tools to reason about fraction equivalencies;
• Continue to build comfort with the meaning and value of any rational number written in the form p/q,
extending their bank of benchmark fractions to include thirds and sixths.
It is in this lesson that EMPower offers an extension so students can combine positive and negative fractions.
In Lessons 7a, students
• Use visual models (e.g., fraction strips, rulers, pattern blocks) as tools to reason about combining
fractions, finding differences;
• Use understanding of the meanings of the four operations with whole numbers to reason about the four
operations with fractions;
• Decide upon workable and reasonable procedures for operating with fractions;
• Understand and apply the properties (e.g., commutative, associative, and distributive properties) to
fractions;
• Apply what they’ve learned to solve design problems.
Using Benchmark Fractions Plus: Operation Sense with Fractions
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Using Benchmark Fractions Plus Materials List
Lesson
Recommended materials to have on hand
5a
Colored markers or pencils
Clear tape
Masking tape
Scissors
Yard sticks (without marked fractions of the yard)
6a
Colored markers (non-permanent)
Blocks or Pattern Blocks
Tape
Blackline Master 6a
Plastic sheet protectors or transparency, 1 per student
12” rulers marked to the nearest 1/16” (or Blackline Master 5a)
Strips of paper (1” x 11”), in different colors if possible (5 strips per student)
7a
1/16” ruler (or Blackline Master 5a)
Plastic sheet protectors, 1 per students
Clear tape
Blackline Master 7a
Sheet protectors
EMPower™ © 2013
Rulers (marked in eighths)
Roll of ribbon or string
Paper strips
Blackline Master 8 (from SIU book)
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EMPower™ © 2013
Common Core Standards for Mathematical Practices
*1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
FACILITATING
3. Construct viable arguments and critique the reasoning of
others.
5a
A Look at
One-Eighth
4. Model with mathematics.
*5. Use appropriate tools strategically.
*6. Attend to precision.
7. Look for and make use of structure.
*8. Look for and express regularity in repeated reasoning.
Objectives
• Determine 1/8 of a given amount
GED Mathematics Assessment Targets
Anticipated
• Build on eighths to discover
sixteenths
Q1.a Order fractions and decimals, including on a
number line.
Q2.a Solve single-step or multistep real-world
arithmetic problems involving the four
operations with rational numbers, including
those involving scientific notation.
Section
Adaptation
Opening Discussion
NEW
Activity 1: Fractions of a Yard
NEW: Students measure to the nearest eighth of a yard.
Activity 2: Finding One-Eighth on a
Number Line
NEW: Students find one-eighth on a number line, and then find
eighths of different amounts.
Math Inspection: A Look at OneSixteenth
NEW: Students find a pattern for moving from halves to quarters
to eighths to sixteenths.
Practice: One-Eighth
NEW: Students review the steps to find one-eighth and threeeighths
Practice: Pound It Out
NEW: Students use ounces and pounds to explore sixteenths
Practice: Looking at Both Sides of 0
NEW: Students find integers on the number line
Practice: Fractions of a Mile
NEW: Students use the notion of furlong as 1/8 mile to answer
questions.
Extension: Fat Quarters and Fat
Eighths
NEW: Students explore the difference between an eighth of a
yard and a fat eighth of a yard.
Test Practice
NEW
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
5a
Lesson 5a, p. 1
FACILITATING
A Look at
One-Eighth
How much is one-eighth
of a yard?
Synopsis
This lesson extends students’ bank of benchmark fractions to include eighths. By
folding, reasoning, and measuring, students use the concept of repeated halving to
find one-eighth. They then further extend repeated halving to find one-sixteenth.
1. Students fold circles to explore one-eighth wedge.
2. The whole class starts a problem about measuring yards of ribbon, and
then student pairs or small groups continue to find eighths of a yard in
inches.
3. Students look at eighths on a number line.
Objectives
Using Benchmark Fractions Plus:
Operation Sense with Fractions
•
Determine one-eighth of a given amount
•
Build on eighths to discover sixteenths
EMPower™ © 2013
Lesson 5a, p. 2
Materials/Prep
• Masking tape
• Paper strips
• Roll of ribbon or string
• Rulers (marked in eights)
• Scissors
• Yard sticks (without marked fractions of the yard)
• Colored markers or pencils
• Clear tape
Make copies of Blackline Master 8: Circle for every student for the Opening Discussion.
Cut three pieces of differently colored ribbons for Activity 1, 40.5”, 45”, and 67.5”long. Mark the
pieces “B”, “C”, and “D”, and post them around the room. Cut a fourth piece 54” long, mark it “A”
and post it on the board. For a large class, make two ribbons of each length. [Substitute masking
tape strips or string if ribbon is not available.]
Cut 36” strips of masking tape for groups to use in Activity 1. (Cut extras.)
Cut strips of blank paper approximately 1” wide, enough for students to tape together into paper
strips one yard long for Activity 1.
Opening Discussion
Start with a brief review of the fractions students have encountered.
What fractions have you learned about so far?
List students’ responses and promt for others by reminding student of lesson topics (if needed).
Today we will discuss a new fraction: one-eighth.
Distribute one copy of Blackline Master 8: Circle to each student. Direct students to cut the circle out
and then fold it into fourths. Ask:
How did you fold the circle into fourths?
Emphasize that everyone folded the paper in half and then in half again. Take some suggestions
from students for how they might make eighths. Then ask students to form eighths.
Summarize the process used to find eighths:
You created eighths in the circle by first splitting a half in half, which gave you fourths. Then you
split those fourths in half to get eighths. Another way to say how you made eighths is that you
divided by two, then divided by two again, and then divded by two again.
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 5a, p. 3
Suggest students label each wedge on the circle with “1/8” fractions so that their circles look like
this:
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
Draw four circles on the board. Name the first circle the “whole.” Then invite students to
demonstrate how they would mark halves in the second circle, fourths in the third circle, and
eighths in the fourth circle, and ask them to shade the relevant fraction in each circle. Point out the
connection between students’ actions and dividing by two, or halving.
Tell students that today’s activities will focus on one-eighth as a fraction.
Activity 1: Fractions of Yards
Ask students to get into small groups of three or four. Read aloud the directions to Problem 1 in
Fractions of Yards (Student Book, p. 58). Say:
Together we will measure and mark ribbon A (the 54” piece posted on the board).
Do you think it is more than a yard, less than a yard, or exactly one yard long?
Write down a few estimates and then ask a volunteer to use a yardstick to measure
the ribbon.
Mark off one yard and the full length of the ribbon, both in inches, resulting in a
model like this:
0
36”
54”
I yard
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 5a, p. 4
Heads Up!
Introduce the yardstick, if necessary, by distributing yardsticks for students to examine for length
(36”) and markings.
Discuss the following:
One yard is measured at the 36-inch mark.
How many more inches does the ribbon measure?
What part of a yard is that? How do you know?
0
36”
54”
I yard
When everyone agrees that 18 inches equal half a yard, ask:
How many fourths of a yard are in a half yard? How do you know?
How many eighths of a yard are in a half yard? How do you know?
Ask students to support their reasoning with another physical model. Cut and tape together strips
of paper measuring 36 inches long. Fold the strip to demonstrate the number of eighths represented
by one half yard.
36” , or 1 yard
18” , or
1
2
yard
Post a copy of the table found in Fractions of Yards, and record the data for ribbon A.
Item
Measurement in Inches
Measurement in Yards
A
54” = 36” + 18”
1 yd. + 1/2 yd.
B
C
D
Assign small groups of students to measure ribbons B, C, and D. Make available yardsticks, rulers,
colored markers, masking tape, paper strips, ribbon or string, scissors, and clear tape. Tell the class:
Measure your ribbon to the nearest half inch.
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 5a, p. 5
If some students do not know the half-inch mark on a yardstick, ask others to point it out to them
and explain how they know this mark represents one-half inch.
Students who finish early may continue with Problems 2 and 3 and then share their results with the
class after everyone discusses Problem 1.
Sharing Results for Problem 1
Choose a group with a correct measurement for the 45-inch ribbon. Ask them to share their
solution and method for finding the measurement in yards (1 1/4 or 1 2/8). Ask:
What is the length in inches of this ribbon?
Do not proceed further until you reach consensus. Then ask:
How did you determine themeasurement of the ribbon in yards?
What fraction of a yard did the nine inches represent? How do you know?
Did anyone do it differently?
Share all methods. Then post a piece of masking tape nine inches long below the half-yard piece of
tape on the board. Label the new piece of tape “9”, or 1/4 or 2/8 yd.”
Continue with the other ribbons. Repeat the process with the 40 1/2-inch ribbon, asking the same
questions of a different volunteer group and posting the one eighth yard (4 1/2” piece) of masking
tape below the one-quarter yard piece on the board. You should have a series of tape pieces that look
like this:
36” , or 1 yd
18” , or
1
2
yd
9” , or
2
8
yd
4 12 ” , or
1
8
yd
Make sure everyone sees the relationship between two-eighths and one-quarter yard. Then ask:
We said earlier that you could find an eighth by folding or splitting something in half, then in half
again, and then in half again.Where do you see a half of a half of a half as you look at the lines and
numbers?
Once students grasp the repeated halving process to find an eighth, share results for the 67 1/2-inch
line. Students should have completed the table by now.
Ask students to work on Problems 2 and 3.
Post the results to both problems. Discuss how students made their eighth marks on the yard-long
piece of tape. Did they start by finding one-half, then one-half of one-half, then half of that? If not,
ask:
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 5a, p. 6
How could finding the halfwaymark on the tape have helped?
Move to Problems 4, 5, and 6 as a class.
In Problem 4, students further apply knowledge of measurement and eighths; they are given
fractional measurements of yards and asked to provide the number of inches each represents.
In Problem 5, students speculate about sixteenths and thirty-seconds. Problem 6 presents an
opportunity to repeatedly halve a piece of yarn until 1/32 of it is shown. The piece of yarn should
look like this:
1 1
32 16
1
8
1
4
1
2
Ask:
What can you say about repeated halving?
Students may remark that each time you halve, the pieces or amounts get smaller and the fraction
denominators double.
Activity 2: Finding One-Eighth on a Number Line
Direct students to Activity 2 in their student book (p. 5). Explain that the purpose of this activity
is to get them to break different amounts into eighths. At the same time, they should begin to
informally notice the relationship among eighths, fourths, and halves.
Have students work in pairs, then debrief the activity.
What did you notice about the relationship between one-eighth and one-fourth?
What did you notice about the relationship between one-eighth and one-half?
Math Inspection: A Look at One-Sixteenth
Why This Math Inspection?
This Inspection is meant to help students understand and be able to show sixteenths on a number
line, and then make the connection between halves, fourths, eighths and sixteenths. Rulers and
measuring tapes are typically divided into sixteenths, so making the jump to this fraction makes
sense. Helping students see the relationships between the fractions they already know and sixteenths
is key. Using 1/2 of an eighth to find a sixteenth continues the logic established for the other
fractions they’ve studied thus far.
Facilitation Suggestions
Treat the Math Inspection as an opportunity for students to notice patterns, come to generalizations,
and justify their reasoning. This will provide them with a solid base for understanding important
concepts like fractions.
Ask students to explain how they know that how many sixteenths there are in an eighth; then how
they know how many are in each fourth, and then how many in each half. By now, they should be
very familiar with taking halves, so extending that to taking fourths and eighths, and ask them to
explain why and how they are doing that.
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 5a, p. 7
Summary Discussion
Ask:
How can you find an eighth of $32?
Record strategies, making sure that these three methods are included:
• Find a quarter and then halve that amount.
• Find a half, then half of that, then half of that (or divide by two, then by two again, and
then by two again).
• Divide by eight.
Tell students to turn to their Vocabulary (Student Book) to write down a definition of or
information about “one-eighth”. Collect ideas to post on the class vocabulary list.
Practice
One Eighth, p. 8
For a review of steps to find one-eighth and three-eighths.
Pound It Out, p. 9
For practice using sixteenths.
Looking at Both Sides of 0, p. 10
For practice placing integers on a number line.
Fractions of a Mile, p. 11
For practice using eighths, introduce the unit of “furlong,” still common in some contexts.
Extension
Fat Quarters and Fat Eighths, p. 12
Appropriate for students with textile experience or interest, as another exploration of quarters and
eighths.
Test Practice
Test Practice, p. 13
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 5a, p. 8
Pilot teachers told us...
A new student quickly and spontaneously joined one of the groups since she found the activities fun
and challenging. The class was engaged to the extent that they were almost missing their break.
Once students use half of half to find one fourth and half of a fourth to find an eighth, they are
hooked into using that method. Students are eager to discover what other fractions they can learn
using this strategy.
The string and measurements were puzzling to some students. One student said, “I’m lost,” but I
knew from past experience that using money was one great way to overcome difficulty. I said, “Let’s
just relate this to money. What is one WHOLE yard (equivalent) equal to?” Students knew it was 36”.
I suggested we think of $36 instead as being 1 whole (amount), and asked “If $36 is the whole, what
is half?” We talked it through a little more and then students used our school’s “money kit,” to divide
$36 into fractions. This approach made strings A, B, and C much easier to manage than they had
been before and even the most challenging string length for this activity (67.5”), was conquered by
many students. Once we had established the fact that 4.5” or $4.50 was equivalent to 1/8 (and this
fact was clear to even the previously struggling students), they were able to simply count the number
of groups of $4.50 (7) and they knew that 31.5 (or $31.50) was equivalent to 7/8. Combining that
7/8 with the other 36” (or $36.00), which of course they knew equaled 1 yard, they were able to tell
that the 67.5” string was 1 7/8 yards long.
Questions I’m glad I asked:
I’m glad I ased the students to work together to solve each problem. They came up with the the
correct answers!
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
9
4
3
Centimeters
2
1
4
3
9
9
8
8
Centimeters
2
1
5
5
5
6
6
6
7
4
7
7
7
8
8
8
9
9
9
6
7
8
3
10 11
10 11
10 11
5
6
7
9
Centimeters
2
1
12 13 14
12 13 14
12 13 14
12 13 14
4
10 11
15 16
15 16
15 16
15 16
3
9
17 18
17 18
17 18
17 18
19 20 21 22 23
19 20 21 22 23
19 20 21 22 23
19 20 21 22 23
2
5
8
1
4
6
8
Inches
3
5
7
Inches
2
4
7
1
3
6
6
2
5
5
1
4
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Inches
3
4
2
3
Inches
1
Centimeters
2
1
Blackline Master 5a: Rulers
EMPower™ © 2013
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
5a
Lesson 5a, p. 1
LESSON
A Look at
One-Eighth
How much is one-eighth
of a yard?
A half is too big. A fourth is still too much. What about half of that?
Call it half of a quarter or one-eighth. They are the same thing.
One-eighth is a sliver of a whole. When you count by eighths, you can
be more precise than with halves and quarters. For example, 87 is less
3
than a whole, but more than 4 . An eighth sounds like a little bit, but as
with all fractions, it depends on the whole amount. NYC’s population
is over 8 million. One-eight of the whole is over a million residents.
In this lesson, you will measure various items to explore one-eighth as
1
a fraction. You will also explore what happens when you divide 8 into
two equal parts.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 5a, p. 2
Activity 1: Fractions of Yards
Fabric samples and ribbons are sometimes sold in
1
1
8 -yard lengths. How long is 8 of a yard?
2
3
How long is 8 or 8 ?
1. Measure the ribbons posted around the room, and complete the following table.
Keep track of your work.
Item
Measurement in Inches
Measurement in Yards
A
B
C
D
8
1 2 3 4 5 6 7
2. Mark off eighths on a yard-long strip of masking tape: 8 , 8 , 8 , 8 , 8 , 8 , 8 , and 8 .
Draw a picture to represent your marked tape.
3. Determine the length in inches at each mark:
1
a. 8 yd. = _____ in.
2
b. 8 yd. = _____ in.
3
c. 8 yd. = _____ in.
4
d. 8 yd. = _____ in.
5
e. 8 yd. = _____ in.
6
f. 8 yd. = _____ in.
7
g. 8 yd. = _____ in.
8
h. 8 yd. = _____ in.
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 5a, p. 3
4. Familiar Fractions:
1
a. You bought 2 yard of silk; how long in inches is your piece of fabric?
1
b. You bought 4 yard of cotton; how long in inches is your piece of fabric?
3
c. You bought 4 yard of wool; how long in inches is your piece of fabric?
d. You bought one yard of nylon; how long in inches is your piece of fabric?
1
5. a. If you cut your 8 yard of ribbon in half, what fraction of a yard would you have? How do you know?
b. How many inches long would the ribbon piece be? How do you know?
c. What fraction of a yard would you have if you cut this new piece of ribbon in
half? How do you know?
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 5a, p. 4
1
32
1
16
1
8
1
4
1
2
6. The following line represents one piece of yarn. Show how you would cut it in half,
then cut that half in half again, then cut it in half again, then in half again. Label
each cut mark with a fraction. Note: You will have five lines and five fractions on
your piece of yarn.
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 5a, p. 5
Activity 2: Finding One-Eighth on a Number Line
1a.Complete the number line below.
1
1
a. Locate 2 between 0 and 1. Use one color pencil to show 2 of the distance from
0 to 1.
1
3
b. Use another color pencil to show 4 and 4 of the distance from 0 to 1.
c. Use another color pencil to divide the distance from 0 to 1 into four equal
parts, and one more color to divide the line into eight equal parts.
0
1
1d.Are there any overlapping fractions? If so, what are they?
2. Divide the number line below into eighths.
0
200
a. What is one-eighth of 200?
b. What is three-eighths of 200?
c. What is seven-eighths of 200?
Using Benchmark Fractions Plus:
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Lesson 5a, p. 6
3. Divide the number below into eighths.
0
1,000
a. What is one-eighth of 1,000?
b. What is five-eighths of 1,000?
c. What is six-eighths of 1,000?
d. What is three-fourths of 1,000?
4. Create your own number line below. Then divide it into eighths.
a. What did your number line begin with?
b. What did your number line end with?
c. How much did one-eighth of it represent?
d. How much did one-fourth of it represent?
e. How much did one-half of it represent?
f. How do one-eighth and one-fourth of the distance compare with one another?
g. How do one-fourth and one-half of the distance compare with one another?
h. How do one-eighth and one-half of the distance compare with one another?
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 5a, p. 7
Math Inspection: A Look at One-Sixteenth
1. The number line below is divided into 16 even parts. What is each part called?
0
1
2. Use a red color pencil to divide the number line in half. How many sixteenths are
there in each half?
3. Use a green color pencil to divide the number line into fourths. How many
sixteenths are there in each quarter?
4. Use a blue color pencil to divide the number line into eighths. How many
sixteenths are there in each eighth?
1 1 1
1
5. Describe the pattern you see with 2 , 4 , 8 , and 16
.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 5a, p. 8
Practice: One-Eighth
Fill in the blanks to make the statements true.
1. Half of a half is the same as____________________________.
2. Half of a fourth is the same as __________________________.
3. Half of an eighth is the same as _________________________.
4. To find half of any number, I can split that number into _____ groups. That is the
same as dividing the number by _____.
5. To find a fourth of a number, I can split that number into _____ groups. That is
the same as dividing the number by _____.
6. To find an eighth of a number, I can split that number into _____ groups. That is
the same as dividing the number by _____.
3
7. If I want to find 8 of a number, I can
_____________________________________________________________
_____________________________________________________________.
3
8. Another way that I can find 8 is to
____________________________________________________________
_____________________________________________________________.
EMPower™ © 2013
Solely for use by NYC Department of Education
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 5a, p. 9
Practice: Pound It Out
Pounds and Ounces
There are 16 ounces in one pound. The abbreviation for pound is lb. The abbreviation for ounce is oz.
Answer each question below. Create a number line to help you if you want.
1. Sage wants to buy a half pound of hamburger. How many ounces would she buy?
2. Tyrone orders a 12-oz. steak. What fraction of a pound is that?
3. A quarter-pounder is supposed to have how many ounces of hamburger?
4. A surf-and-turf meal includes a 6-ounce steak. What fraction of a pound is that?
5. Dale and Kim order a fourteen-oz. steak, thinking they will share it together.
What fraction of a pound is that? If they split it evenly, what fraction of a pound
would they each get?
6. Jeff ’s favorite diner offers a three-quarter pounder steak on Friday nights. How
many ounces is that?
7. After winning a game, the star running back ordered a double burger which
included two quarter-pound hamburger patties. How many ounces of hamburger
were in the hamburger?
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 5a, p. 10
Practice: Looking at Both Sides of 0
1. Label each point on the number line.
-3
-4
-5
A
B
0
-1
-2
C
D
1
E
2
3
4
F
5
G
2. Create a number line below. Label the following points on your number line:
3
a. – 4 8
b. 4
1
c. -3 8
1
d. 2 2
1
e. -1 2
3
f. - 8
g. 0
7
h. 8
7
i. 2 8
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Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 5a, p. 11
Practice: Fractions of a Mile
Furlongs
A furlong is 1/8 mile. Furlongs are still used in horse races. In cities such as Chicago and Salt Lake City,
many city blocks are approximately 1/8 mile, or one furlong, in length.
Answer each question below. Create a number line to help you if you want.
1. Adam lives in Chicago and walks from his apartment to work, which is seven
blocks away. About what fraction of a mile does Adam walk to work?
2. If Adam walks around his Chicago block once, what fraction of a mile would he walk?
3. In order for a horse to run a half-mile, how many furlongs would it run?
1
4. Many horse racing tracks are 1 8 miles long. How many furlongs is this?
5. When Milton visited Salt Lake City, he walked twelve blocks to get from his hotel to the
museum. He claimed he walked at least two miles. Is he correct? Explain your answer.
6. Danni wants to walk a mile
a day, even while she’s on
vacation in Salt Lake City.
Draw a one-mile route on the
map.
7. How does a furlong compare
to the blocks where you live?
Using Benchmark Fractions Plus:
Operation Sense with Fractions
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Lesson 5a, p. 12
Extension: Fat Quarters and Fat Eighths
In quilting, fabric is often cut into quarters by cutting a one-yard long piece of fabric in
half both by length and width.
1 yard
Instead of cutting it into fourths like this:
1 yard
yardage is cut into quarters (called fat quarters)
like this:
1. a. If a one-yard piece of cloth is 42 inches long, what would be the dimensions of a
quarter yard?
b. What would be the dimensions of a fat quarter?
1 yard
A fat eighth begins with a fat quarter and cuts
each quarter in two:
2.a. What would be the dimensions of an eighth yard of fabric cut the usual way?
b. What would be the dimensions of a fat eighth yard of fabric 43 inches long?
3. Would it matter whether you bought a fat eighth or a regular eighth yard of cloth?
Why or why not?
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 5a, p. 13
Test Practice
4. The following fractions need to be ordered from least to
greatest. What is the proper order for these fractions?
1. Which of the following does not show 81 ?
(1)
(2)
3 4
1
8
1
4
3
8
1
8
7
8
1 1 3 3 7
(1) 4 , 8 , 8 , 4 , 8
1 1 3 3 7
(2) , , , ,
(3)
4 8 4 8 8
1 1 3 3 7
(3) 8 , 4 , 8 , 4 , 8
1 1 3 3 7
(4) 8 , 4 , 4 , 8 , 8
1 1 3 7 3
(5) 8 , 4 , 8 , 8 , 4
(4) 1125
, 000
(5) 0.125
2. Jose ate 16 cookies at the holiday party. His sister,
Marcela, ate only two cookies because she is watching
her weight. Marcela ate
5. Charlie is health conscious. He reads nutrition labels
carefully. Eating an M’s Dark Chocolate serving provides
close to 81 of the daily value for which nutrient?
M‘s Dark Chocolates
(1) 1 as many cookies as Jose
Nutrition Facts
16
(2) 81 as many cookies as Jose
Serving Size 6 pieces (42g)
Servings Per Container about 7
Amount Per Serving
Calories 220
Calories from Fat 120
14
(3) 16
as many cookies as Jose
Total Fat 13g
Saturated Fat 8g
(4) 41 as many cookies as Jose
% Daily Value
20%
40%
Polyunsaturated Fat
Monosaturated Fat
Cholesterol 5mg
Sodium 0mg
Total Carbohydrate 26g
Dietary Fiber 3g
Sugars 21g
Protein 2g
(5) 21 as many cookies as Jose
3. Ellen pays her mother a fraction of the rent each month.
If Ellen pays $250 and the total rent is $2,000, what
fraction of the rent does Ellen’s mother pay?
2%
0%
9%
12%
(1) Sodium
(2) Cholesterol
(1) 81
(3) Dietary fiber
(2) 41
(3) 3
8
(4) Saturated fat
(5) Protein
(4) 5
8
(5) 7
8
Using Benchmark Fractions Plus:
Operation Sense with Fractions
6. Mara brings home about $1,000 per month. She
estimates that 81 of her monthly income goes for
transportation. How much does Mara budget monthly
for transportation?
EMPower™ © 2013
Answer Key
Lesson 5a: A Look at One-Eighth
Activity 2: Finding One-Eighth on a Number
Line (p. 5)
Activity 1: Fractions of Yards (p. 2)
1. a.-c.
1. a. 54" = 1 1/2 yd.
b. 40.5" - 1 1/8 yd.
d. ½ overlaps with 2/4 and 4/8; ¼ overlaps with
2/8, ¾ overlaps with 6/8
c. 45" = 1 1/4 yd.
d. 67.5" = 1 7/8 yd.
2.
2.
inches
0 4.5
9
0
1
8
8
yards
13.5 18 22.5 27 31.5 36
2
8
3
8
4
8
5
8
6
8
7
8
25
50 75 100 125 150 175 200
a. 25
8
8
b. 75
c. 175
3.
125 250 375 500 625 750 875 1000
3. a. 1/8 yd. = 4.5 in.
b. 2/8 yd. = 9 in.
a. 125
c. 3/8 yd. = 13.5 in.
4.
d. 4/8 yd. = 18 in.
b. 625
c. 750
d. 750
Depending on what the student chooses:
a. it will begin with 0
e. 5/8 yd. = 22.5 in.
b. it will end with whatever value the student
chooses
f. 6/8 yd. =27 in.
g. 7/8 yd. =31.5 in.
c. 1/8 will be half of a half of a half of the whole
number, or the whole number divided by 8
h. 8/8 yd. = 36 in.
d. 1/4 will be half of a half of their whole number
or that number divided by 4
4. a. 18 inches
b. 9 inches
e. 1/2 will be half of their chosen value or that
value divided by 2.
c. 27 inches
d. 36 inches
f. 1/8 will be half the distance of 1/4
5. a. 1/16
b. 2.25 inches
g. 1/4 will be half the distance of 1/2
h. 1/8 will be 1/4 the distance of 1/2
c. 1/32
Math Inspection: A Look at One-Sixteenth (p. 7)
6.
1
32
1
16
1
8
1
4
1. a sixteenth
1
2
2. 8 sixteenths in each half
3. 4 sixteenths in each fourth
4. 2 sixteenths in each eighth
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
5. Possible answers: Each denominator is a power
of 2, each fraction’s denominator is ½ of the
denominator that preceeds it. As you start with
1/16 it takes 2 sixteenths to make an eighth, 2
eighths to make a fourth, 2 fourths to make ½.
Practice : One-Eighth (p. 8)
2. Check that the students have labeled their blank
lines appropriately so that there are points
marked at each of the positions listed in exercises
a. through i.
Practice: Fractions of a Mile (p. 11)
1. 1/4
1. 7/8 mile
2. 1/8
2. 1/2 mile
3. 1/16
3. 4 furlongs
4. To find half of any number, I can split that
number into 2 groups. That is the same as
dividing the number by 2.
4. 9 furlongs
5. To find a fourth of any number, I can split
that number into 4 groups. That is the same as
dividing the number by 4.
6. The picture should show 8 blocks.
6. To find an eighth of any number, I can split
that number into 8 groups. That is the same as
dividing the number by 8.
7. Answers will vary. Sample answer: If I want to
find 3/8 of a number, I can divide the number by
8, then multiply that result by 3.
8. Answers will vary. Sample answer: Another way
to find 3/8 of a number is to find 1/2 and 1/8,
then subtract 1/8 from 1/2.
5. No, he walked 12/8 or 1 1/2 miles
7. Answers will vary. In Manhattan, for example,
north-south blocks are 1/20 of a mile, or 2 1/2
furlongs.
Extension: Fat Quarters and Fat Eighths (p. 12)
1. a. 9 inches by 42 inches
b. 18 inches by 21 inches
2. a. 4 1/2 inches by length of fabric
b. 9 inches by 21 1/2 inches
c. no the area of the cloth is the same in both.
Therefore cutting it into 8 equal pieces will yield
the same area for each eighth.
Practice: Pound It Out (p. 9)
1. 8 oz.
2. ¾ lb.
3. 4 oz.
4. 3/8 pound
5. 7/8 pound, they will each get 7/16 pound
6. 12 ounces
7. 8 ounces
Practice: Looking at Both Sides of 0 (p. 10)
1. a. label should be at -4 1/8
Test Practice (p. 13)
1. (3)
2. (2)
3. (5)
4. (3)
5. (3)
6. $125
b. label should be at -3 1/4
c. label should be at -1 3/8
d. label should be at 1/4
e. label should be at 5/8
f. label should be at 2 3/4
g. label should be at 4 7/8
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Using Benchmark Fractions Plus:
Operation Sense with Fractions
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Common Core Standards for Mathematical Practices
*1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
*5. Use appropriate tools strategically.
FACILITATING
6a
Equal Measures
*6. Attend to precision.
*7. Look for and make use of structure.
*8. Look for and express regularity in repeated reasoning.
Objectives
• Continue to build comfort with the
meaning and value of any rational
number written in the form p/q.
• Use visual tools (e.g., fraction
strips, rulers, pattern blocks) to
reason about fraction equivalencies
and comparisons.
GED Mathematics Assessment Targets
Anticipated
Q1.a Order fractions and decimals, including on a
number line.
Q1.b Perform addition, subtraction, multiplication,
and division on rational numbers.
Q1.c Apply number properties involving multiples
and factors, such as using the least common
multiple, greatest common factor, or distributive
property to rewrite numeric expressions.
Q2.a Solve single-step or multistep real-world
arithmetic problems involving the four
operations with rational numbers.
Section
Opening Discussion
Adaptation
NEW: Launch from students’ familiarity with 1/2s, 1/4s, and 1/8s.
Activity 1: Fraction Strips and Rulers NEW: Students create fraction strips for 1, 1/2 , 1/4. 1/8 and 1/16s.
They examine equivalencies. They see these same fractions on a ruler.
Activity 2: Pattern Blocks
NEW: Students use pattern blocks to explore 1, 1/2, 1/3, 1/6
equivalencies, along with different ways to make a whole.
Activity 3: Fraction Strips with
Thirds and Sixths
NEW: Students add thirds and sixths to their set of fraction strips.
Activity 4: About Ones and Zeroes
NEW: Students review the multiplicative identity and property of
zero. They generate equivalent fractions and conjecture about a math
algorithm that creates equivalent fractions.
Math Inspection: One in the
Denominator
NEW: Students become familiar with the convention of writing whole
numbers with one in the denominator
Practice: Equivalent Fractions
NEW: Students use their fraction strips to find equivalent fractions.
Practice: Between 1/3 and 1/2
NEW: Students estimate the size of fractions in relation to 1/3 and 1/2.
Practice: Ratcheting Up (or Down)
a Notch
NEW: Students practice finding larger and smaller fractions while using
the concept of equivalent fractions.
Practice: Where to Place It?
NEW: Students estimate with fractions on the number line.
Practice: 2/3 and 3/4
NEW: Students plot benchmark fractions on a line.
Mental Math Practice: Using
Properties
NEW: Review of mixed numbers, positive and negative, and order of
operations
Test Practice
NEW
Using Benchmark Fractions Plus:
Operation Sense with Fractions
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Using Benchmark Fractions Plus:
Operation Sense with Fractions
6a
Lesson 6a, p. 1
FACILITATING
Equal Measures
What do fashion design,
carpentry, gourmet cooking,
and computer graphics have
in common?
Synopsis
Students draw upon their prior knowledge of benchmark fractions to reason
about fractional equivalents. They use a variety of tools to visualize fractions
and equivalencies: fraction strips, inch-rulers, and Pattern Blocks. They reason
and solve problems with the visual tools. Once their visual understanding is well
grounded, students examine the more formal mathematics of equivalent fractions
and the role the multiplicative identity (multiplication by 1) plays in generating
equivalent fractions. They connect their visual models with the mathematical
procedures, or algorithms.
1. Students reflect upon fraction equivalencies with which they have become
familiar (1/2s, 1/4s, 1/8s), using a chosen visual model.
2. They create a set of fraction strips, add 1/16s to their repertoire and make
connections with the inch-ruler.
3. They explore 1/2, 1/3, 1/6 equivalencies with Pattern Blocks and then
fraction strips.
4. They use the multiplicative identity to generate equivalent fractions and
connect the formal math to their visual models.
Objectives
Using Benchmark Fractions Plus:
Operation Sense with Fractions
•
Continue to build comfort with the meaning and value of any rational
number written in the form p/q.
•
Use visual tools (e.g., fraction strips, rulers, pattern blocks) to reason
about fraction equivalencies and comparisons.
EMPower™ © 2013
Lesson 6a, p. 2
Materials/Prep
For Activity 1:
• Prepare paper strips (1“ by 11“), in different colors if possible. Make 5 unmarked strips
per student.
• A plastic sheet protector, 1 per student
• Tape
• Colored markers (non-permanent)
• 12-inch rulers marked to the nearest 1/16“ (or Blackline Master 5a)
For Activity 2:
• Pattern Blocks. Use only yellow, blue, red, and green (or Blackline Master 6a)
For Activity 3:
• Prepare paper strips (1“ by 11“), in different colors if possible. Each student gets 2
unmarked strips, which they will add to their fraction strip kit.
Heads Up!
Lesson 6a grounds student reasoning in the informal and visual before dealing with formal mathematical procedures. Even if students remember procedures, ask them to solve problems visually first,
then connect to the math procedures. We are guided by the writing of well-respected mathematics
teacher educators, and believe their insights hold true for adults as well as younger learners:
“It is important to give students ample opportunity to develop fraction number sense prior to and
during instruction about common denominators and other procedures for computation. Even
in grade 7 or 8 it makes sense to delay computation and work on concepts if students are not
conceptually ready.
Premature attention to rules for fraction computation has a number of serious drawbacks. None of
the algorithms helps students think about the operations and what they mean. When students follow
a procedure they do not understand, they have no means of assessing their results to see if they make
sense. Second, mastery of the poorly understood algorithm in the short term is quickly lost. When
mixed together, the differing procedures for each operation soon become a meaningless jumble.
Students ask, “Do I need a common denominator, or do I just add or multiply the bottom numbers?”
“Which one do you invert, the first or the second number?” When the numbers in a problem are
altered slightly, for example, a mixed number appears, students think the algorithm does not apply.”
— Van deWalle, Karp, & Bay-Williams (2012)
Opening Discussion
Offer some examples to remind students that another way to write 1, as in one whole, is with the
same number as the numerator and denominator.
Help me complete the story:
I had a total of $20 in my wallet and I lost it all. Not half, but all, so $20 out of $20. I can write the
whole amount as ... [20/20].
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 3
My niece used up an entire 16 oz bottle of shampoo. Not half, but all, so 16 oz out of 16 oz. I can
write the whole amount as ... [16/16].
I threw out every one of my 30 plastic bags in the house. Not 1/2, but all. So I can write that whole
amount as ... [30/30].
There’s more than one way to write 1 when it means ALL, the “whole.” Fractions are flexible. It’s
useful because you have to change them up when adding, subtracting, and simplifying answers
from multiplying and dividing. But it also means they are tricky, slippery. This whole lesson gets at
equivalent fractions, because there is more than one way to name the same relationship.
Write the following 17 fractions on the board in random order:
0/8, 1/2, 3/4, 3/8, 2/2, 7/8, 4/4, 2/4, 5/8, 0/4, 1/4, 6/8, 8/8, 0/2, 1/8, 2/8, 4/8
Say:
I just wrote 17 benchmark fractions you have been working with on the board.
a. Put them in order, from smallest to largest values.
b. How many different amounts are there really?
Ask pairs to work together on creating a context - a visual to show how someone could “see” the
relative values and those that are equal. Use all 17 fractions. Be creative! The visual model could be
anything they want – pizzas, cakes or other food, money, a number line, a group of objects, so long
as you are clear about what One Whole (1) represents.
There should be clear pictures that others could understand.
When pairs have completed their diagrams, ask them to exchange with another pair, answering the
following questions:
• How clear is the other pair’s drawing to you? Do you have suggestions to make it clearer?
• What is the same about the other pair’s and your drawing?
• What is different?
• Did you both come to the same conclusion?
• What is that conclusion?
Listen carefully to students’ understandings.
Make sure there is sound understanding of the correct order and equivalencies, and that there are 9
different values represented by the 17 fractions
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 6a, p. 4
0/8= 0/4 = 0/2
1/8
2/8 = 1/4
3/8
4/8 = 2/4 = 1/2
5/8
6/8 = 3/4
7/8
8/8 = 4/4= 2/2
Today we are going to focus attention on fractional equivalents - equal fractions. It’s important
that you and “see” the equivalence and that you then think about some mathematical rules for
equivalent fractions.
Activity 1: Fraction Strips with Rulers: Tools to Think With
For each student, have 5 strips of different colored paper, each 1“ by 11“, tape, clear plastic sheet
protector, colored markers). They will keep this fraction kit for the rest of the lessons.
Pass out 5 different colored strips to each student. Hold up the first piece and say:
Let’s call this 1, or one whole. Mark it with a big “1”.
1
I would like to fold this strip in half. How could I do this?
Students will probably recall from prior lessons to how to do this. Make sure they fold width to
width not length to length. Have them fold and open the strip up and say:
How do you know this is 1/2? What does 1/2 mean?
Students should say 2 equal parts, one part out of two is 1/2. Direct them to label each 1/2 piece as
1/2 and mark the crease line with markers.
1/
2
1/
2
Take another strip of a different color and say:
I would like to fold this strip into fourths. How could I do this?
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 5
Students should recall half of a half, or fold it 2 times. Open it up and say:
How do you know this is 1/4? What does 1/4 mean?
Students should recall 4 equal parts, one part of the four is 1/4. Have them label every part with ¼
and mark each crease line with markers.
1/
4
1/
4
1/
4
1/
4
Take a third strip and say:
I wonder what would happen if I fold this strip 3 times? How many parts will there be?
Listen for answers. Some students might say 6 thinking the segments are increasing by 2. Fold the
strip 3 times (half of a half of a half) and open. There will be 8 sections. Ask:
What fractional parts are represented here? How do you know?
Like before have them label each part as 1/8 and mark each crease line.
1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/
8 8 8 8 8 8 8 8
Pass out the last strip.
Do you see any patterns unfolding? How many times do you think we will fold this strip? How
many equal parts do you think we’ll find?
Listen for answers. Most should say 16 parts. Again, unfold and label each part as 1/16.
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16
Do you think we could continue to do this? Fold it 5, 6, or 7 times?
Most students will think it is impossible because of the thickness of the paper but if that didn’t
restrict us could we continue to do this? Yes. This is called density of fractions and there are an
infinite number of fractions between 0 and 1.
Have students take the 4 strips and tape together, one under the other with sixteenths on top,
eighths next, quarters next , and halves last. Put this inside a sheet protector with the sixteenths at
the top.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 6a, p. 6
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
166 166 16 16 16 16 16 16 16 16 166 16 16 16 16
1 16
1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/
8 8 8 8 8 8 8 8
1/
4
1/
4
1/
4
1/
2
1/
4
1/
2
Using dry erase markers, tell students we are going to find fraction equivalencies and draw lines
down the ends of ones that match. Start by drawing a line along the top of the sixteenths
Say:
Look at 1/16. Does it match (line up) with any other fractions? (No.)
Look at 2/16. Does it match up with any other fractions? (Yes, 1/8.)
Draw a line down the edge of 2/16 to the edge of 1/8 (see example)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16 16 1166 16 16 16
16 16 16 16 16 16
1 16 16 16 166 16
1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/
8 8 8 8 8 8 8 8
1/
4
1/
4
1/
2
1/
4
1/
4
1/
2
Does 3/16 line up with any other fractions? (No.)
Does 4/16 line up with any other fractions?
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 7
This time it will match both 2/8 and 1/4. Draw a line down the edge of 4/16, 2/8, and 1/4. See
example:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16 16 1166 16 16 16
16 16 16 16 16 16
1 16 16 16 166 16
1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/
8 8 8 8 8 8 8 8
1/
4
1/
4
1/
2
1/
1/
4
4
1/
2
Continue with 5/16 (no matches), 6/16 (will line up with 3/8), 7/16 (no matches), and then 8/16
which will have the most matches. At this point it should look like this,
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16 16 1166 16 16 16
16 16 16 16 16 16
1 16 16 16 166 16
1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/
8 8 8 8 8 8 8 8
1/
4
1/
4
1/
2
Using Benchmark Fractions Plus:
Operation Sense with Fractions
1/
4
1/
4
1/
2
EMPower™ © 2013
Lesson 6a, p. 8
Continue this process all the way across the strip. When done, remove the fractions strips from the
sheet protector and what remains are the markings of a ruler, where the 1 whole could be thought
of as a “blown up” inch on the inch-ruler.
Ask individuals to work with a partner on Activity 1: Fraction Strips and Rulers—Tools to Think With
in the Student Book, p. 2.
Look at the fraction strips you created and the markings of the ruler to record as many
equivalencies you notice. Then create some of your own. Be prepared to justify your thinking with
a picture.
Activity 2: Pattern Blocks: Another Tool to Think With
Form pairs. Distribute some Pattern Blocks to each pair, asking folks to take a few yellow, red, blue,
and green shapes. If actual Pattern Blocks are not available, make copies of Blackline Master 6a:
Pattern Blocks with colored markers.
Say:
In this activity, you are going to explore the fraction equivalents and values you “see” in these
shapes.
Hold up the yellow hexagon and say:
For the purposes of this activity, the yellow hexagon is equal to 1 (one whole).
Direct students to Activity 2, p 3 in the Student Book and ask them to work together to answer the
questions.
If some pairs finish sooner than others, challenge them with the question:
What are all the values if, instead of the hexagon being the whole, the red trapezoid is the whole
(equal to 1)?
Activity 3: Thirds and Sixth Strips
Distribute three more 1“ by 11“ strips and ask them to add thirds and sixths to their kit of fraction
strips. Have them first focus on the strip representing thirds.
Say:
First, predict which benchmark fractions (1/2, 1/4, 3/4) you think one-third is between, then see
if your prediction is correct. Based on your strips, which of the two benchmark fractions is 1/3
closer to? Why do you think that is so?
Now, look at all your benchmark fraction strips, including eighths and sixteenths compared to
1/3. One-third is between which two fractions?
Now, predict which set of benchmark fraction 1/6 falls between, then test out your prediction.
What reasoning did you use to make your prediction?
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Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 9
Ask students to think back about how they created their earlier strips, beginning with halves, then
fourths, etc.
Predict what size fraction you would get when you divide sixths in half. Then create a new strip.
If some pairs finish sooner than others, challenge them with the following:
Show a fraction between 1/3 and 1/2.
Activity 4: About Ones and Zeroes
Ask pairs to think about the following four statements (that you have written on the board).
Are they true or false?
Any number multiplied by 1 gives you that number (a x 1 = a)
Any number multiplied by 0 is 0 (a x 0 = 0)
Any number added to 0 is zero (a + 0 = 0)
Any number added to 1 is that number (a + 1 = a)
After pairs have made decisions, ask for a vote, and facilitate a discussion. Finally, direct students’
attention to their Student Book, Activity 4, p. 7. The first section will be a place to record their notes
of the discussion of the four statements. And the second section, they are asked to put those ideas
into play.
Ask students to say what they are noticing in their own words. Write their words on the board
verbatim. The take home points from this investigation might be:
• 1 can be written in many ways. In fact, any number can be written in many ways.
• No matter the form, when one multiplies a number by 1, the product is that number.
(Focus on this as the important idea that underlies the math of equivalent fractions.)
• If you start with a fraction, you can always get a fraction equal to it by multiplying by 1.
• 2/2, 3/3, 4/4..., etc. are forms of 1 that are useful for getting equivalent fractions.
Math Inspection: One in the Denominator
Why This Math Inspection?
This Inspection is meant to help students understand the meaning of a fraction with a 1 in the
denominator. It will further allow them to differentiate between a 1 in the denominator and a 1 in
the numerator.
Facilitation Suggestions
This Inspection could be developed using visual models. Students often have a hard time
understanding fractions greater than one. You might encourage students to focus on the size of
the piece that makes the whole. For example, if the size of the piece is 1 and you have 5 of these
then you have 5/1. This builds on students’ understanding that if they have a whole that is divided
into quarters, they have 4/4, noting that the 4 in the denominator refers to the size of the slice. This
might help students understand why we sometimes put 1 in the denominator.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 6a, p. 10
Ask students to explain each of their answers (True or False); not just the False ones. Problem 3 is
one that often confuses students, and one that will allow you to assess their understanding of the
meaning of a fraction – and the difference between 1/5 and 5/1. Question 7 is a generalization about
the equivalence between a denominator of 1 and the number in the numerator. You may want to ask
them to restate that using their own words. You will have further evidence of their understanding of
this concept in the examples they provide through the last question.
Summary Discussion
Ask each person to write down two or three statements about equivalent fractions. Then ask them
to pass those statements to a partner to discuss each statement’s accuracy. Finally, ask each pair to
offer one statement they feel confident about, and any that they might be uncertain about.
Be sure to see that points such as the following are mentioned:
Any fraction can be turned into an equivalent fraction by splitting each piece in the whole into two
equal parts. The denominator for the new fraction will be twice as large as the original fraction
because there a twice as many pieces in the whole. The numerator for the new fraction will be twice
as large because there are twice as many pieces being considered. Example: In 3/4 = 6/8, 4 pieces in
the whole turn into 8 pieces and 3 pieces being considered turn into 6. There are more pieces, but
they are smaller.
Practice
Equivalent Fractions, p. 10
For more practice on finding equivalent fractions.
Between 1/3 and 1/2, p. 11
For practice using the benchmark fractions 1/3 and1/2.
Ratcheting Up (or Down) a Notch, p. 12
For practice with sixteenths and thirty-secondths.
Where to Place It?, p. 13
For practice placing fractions on a number line.
2/3 and 3/4, p. 14
For practice placing fractions on a number line.
Mental Math Practice
Using Properties, p. 15
Test Practice
Test Practice, p. 16
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 11
Pilot teachers told us...
Students responded positively to the algebraic extension (Activity 4: About Ones and Zeroes). The use
of variables “elevated” the activity for many learners to adult level.
It was interesting because the students knew the rule, but panicked when we used variables to write
a rule (Activity 4: About Ones and Zeroes).
I found the hexagon pattern blocks very helpful.
The pattern blocks helped students understand fraction equivalencies.
Students liked the fraction strips. Students initially struggled with Activity 3, but the resulting
conversation was great.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 6a, p. 12
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Blackline Master 6a: Pattern Blocks
Red
Red
Red
Red
Yellow
Blue
Blue
Blue
Yellow
Green
Blue
Green
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Green
Blue
Green
Green
Green
Green
Green
Green
Green
Blue
Green
Green
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Using Benchmark Fractions Plus:
Operation Sense with Fractions
6a
Lesson 6a, p. 1
LESSON
Equal Measures
What do fashion design,
carpentry, gourmet cooking,
and computer graphics have
in common?
Every number can be written in many forms. How many ways do you
1
know to write a number that means the same as 2 2 ? How are you sure
3
that a number is equal to 4 ? Numbers may look different, but have the
1
same value; they are equivalent (like 2 and 50%). Sometimes numbers
look almost alike, but do not have the same value (like $250 and $2.50);
they are not equivalent.
People working in professions where measurement is important know
this and keep alert, especially when it comes to fractions. Errors can
be expensive! Carpenters have a saying: Measure twice, cut once. Some
recipes are unforgiving when you have doubled the amount of sugar
by mistake. In this lesson, you will need to keep your eyes sharp and
pay close attention! Examine the value of the quantities carefully. Give
yourself time to think about the meaning of the numbers you encounter.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 6a, p. 2
Activity 1: Fraction Strips and Rulers: Tools to Think With
Inches
1/2
1/2
1
2
3
1
1. Think with the fraction strips and the inch ruler!
a. Look at the fraction strips you created and the markings of the ruler. With
your partner, write all of the fraction equivalencies you see. Be prepared to
show your thinking.
1
8
=2
Example: 16
2. With a partner, create five examples for each exercise below. Be prepared to show
your thinking with a picture.
3
a. Five fractions that are equivalent to 4
b. Five fractions that are equivalent to 1 21
4
c. Five fractions that are equivalent to 16
d. Describe the strategy or strategies you used to create an equal fraction.
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Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 3
Activity 2: Pattern Blocks—Another Tool to Think With
You will need a pile of pattern blocks for this activity.
Green
Yellow
Blue
Red
1. Using the yellow hexagon as the whole (1), find the fractional value of
a. The green triangle.
b. The red trapezoid.
c. The blue parallelogram.
2. Examine the Pattern Blocks to answer each of the following questions. Show the
equivalence in 3 ways: draw a picture, write a math equation, and say the math in
words.
a. How many green triangles equal 1 yellow hexagon?
The picture
The math equation
The math in words
b. How many blue parallelograms equal 1 yellow hexagon?
The picture
The math equation
The math in words
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EMPower™ © 2013
Lesson 6a, p. 4
c. How many red trapezoids equal 1 yellow hexagon?
The picture
The math equation
The math in words
d. How many green triangles equal 1 red trapezoid?
The picture
The math equation
The math in words
e. How many green triangles equal 1 blue parallelogram?
The picture
The math equation
The math in words
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 5
1
f. How many green triangles equal 2 2 yellow hexagons?
The picture
The math equation
The math in words
g. How many blue parallelograms equal 1 red trapezoid?
The picture
The math equation
The math in words
h. How many blue parallelograms equal 2 red trapezoids?
The picture
The math equation
The math in words
3. Show two more equivalent statements.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 6a, p. 6
Activity 3: Fraction Strips with Thirds and Sixths
1. Line up the following unit fractions in order from smallest to largest. Then
describe the pattern you see.
1
12
1
2
1
4
1
8
1
16
1
6
1
3
2. Now look at the fraction strips you created, including the three new ones. With
your partner write all the fraction equivalencies you see. Be prepared to show your
thinking.
2
1
Example: 4 = 2
1
3. When you divided 2 into two equal smaller parts, you got a set of fourths. When
you divided each of the fourths into two smaller parts, you got eighths.
a. Describe the rule for breaking a fraction into two equal parts.
b. Create some new fractions based on your rule.
1
2
Example: 16 = 32
1
4. Use your strips to explain why 31 is closer to 41 than 2 .
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 7
Activity 4: About Ones and Zeroes
1. True or false? Explain your reasoning with examples. Rewrite any false statements
to make them true.
a. Any number multiplied by 1 gives you that number. In other words, a × 1 = a.
True or false?
Examples:
b. Any number multiplied by 0 is 0. In other words, a × 0 = 0. True or false?
Examples:
c. Any number added to 0 is zero. In other words, a + 0 = 0. True or false?
Examples:
d. Any number added to 1 is that number. In other words, a + 1 = a. True or
false?
Examples:
2. Use the true statements about 0 and 1 to make math easy.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
a. 3 • 0 =
4
3 (4 – 3) =
e. 4
b. 5 • 0 =
16
1
f. 2 (100 – 99) =
5 +0=
c. 16
1 1
1
g. 2 ( 2 + 2 ) =
d. 5 + 1 =
16
1
1
h. 5 (2 4 – 1 4 ) =
16
EMPower™ © 2013
Lesson 6a, p. 8
3. Create some of your own examples using the rules about 0 and 1.
2
4
4. You already know that 2 = 1, that 4 = 1, and that 8
8 = 1. Use that
understanding to make the math below easy.
1
2
1
4
a. 2 × 2 =
b. 2 × 4 =
1
8
1
16
c. 2 × 8
d. 2 × 16 =
Do you agree or disagree with this statement about equation (a)?
1
2
1
1
2
2
“There are at least two correct answers: 2 × 2 = 2 and 2 × 2 = 4 .”
Explain your reasoning.
5. What strategies can you use for creating equivalent fractions?
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 9
Math Inspection: One in the Denominator
We’ve looked at many fractions and equivalent fractions. Some have had a 1 in the
numerator, but what about a 1 in the denominator?
Mark each of the following True or False. Rewrite any false statements to make them
true.
1. A fraction with a 1 in the denominator is equivalent to 1. True or false?
1
2. If a fraction has a 1 in the denominator, it is equal to 2 . True or false?
1
5
3. 5 = 1 . True or false?
12
4. 1 = 12. True or false?
24
5. 1 = 2. True or false?
4
6. 1 = 4. True or false?
7. A fraction with a denominator of 1 is equivalent to the number in the numerator.
True or false?
Give a few examples of your own of fractions with a 1 in the denominator and an
equivalent number.
a. ___________ = ____________
b. ___________ = ____________
c. ___________ = ____________
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 6a, p. 10
Practice: Equivalent Fractions
1
1
2
1
2
1
4
1
8
1
4
1
4
1
8
1
8
1
8
1
8
1
4
1
8
1
8
1
8
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16
1. Use the chart above to fill in the numerators to make equivalent fractions.
1 = 2 = 4 = 8 = 16 then
1=
3
=
5
=
6
=
7
=
9
=
15
=
18
...
2. Complete the following equivalent fractions using the chart above.
EMPower™ © 2013
a.
1
2 = 4
h.
1
2 = 8
o.
4
16 = 4
b.
1
4 = 8
i.
4
4 = 8
p.
4
16 = 8
c.
1
8 = 16
j.
3
4 = 8
q.
6
16 = 8
d.
1
4 = 16
k.
3
4 = 16
r.
8
16 = 4
e.
1
2 = 16
l.
3
8 = 16
s.
8
16 = 2
f.
2
4 = 16
4
m. 8 = 16
t.
1 = 16
g.
1 = 4
n.
u.
1= 2
1 = 8
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 11
Practice: Between 1/3 and 1/2
1
Which of the fractions below are between 31 and 2 ? Show with pictures or words.
11
a. 20
9
b. 16
5
c. 12
6
d. 16
7
e. 14
9
f. 20
12
g. 15
2
h. 15
4
i. 9
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 6a, p. 12
Practice: Ratcheting Up (or Down) a Notch
1“
A ratchet set consists of sockets that range from 32
to
31 “
1“
32 . Each socket is 32 larger than the one before. Use
this information to answer the questions below.
1. Sam and Dale are trying to tighten the bolts on their picnic table. Sam grabs a
5
ratchet with a 16 “ socket. He realizes that is not quite big enough. Which size
socket should he try next? Why?
17 “
2. Kim is trying to tighten a bolt on a metal frame. She grabs a 32
socket which is
just a tad bit too large. Which size should she try next? Why?
5
3. Jerry is trying to loosen a bolt on his tractor. He tries to use a 8 “ socket, which is
too small. Which size should he try next? Why?
7“
4. Danni is trying to tighten a bolt on her treadmill. She tries to use a 16
socket,
which is too big. Which size should she try next? Why?
7
5. John is trying to loosen a bolt on his grill. He tries a 8 “ socket, which is too small.
Which size should he try next? Why?
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Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 13
Practice: Where to Place It?
For each problem, first mark the given fraction on the line. Then circle the correct
answer for whether the fraction is less than (<), equal to (=), or greater than (>) the
given fraction.
35
1. 60 is less than (<)
equal to (=) 1
or greater than (>) 3 .
60
60
0
60
11
2. 15
is less than (<) equal to (=) 2
or greater than (>) 3 .
15
15
0
15
16
3. 45 is less than (<) equal to (=) 1
or greater than (>) 3 .
45
45
0
45
17
4. 21
is less than (<) equal to (=) 2
or greater than (>) 3 .
21
21
0
21
13
5. 30
is less than (<) equal to (=) 30
30
0
30
Using Benchmark Fractions Plus:
Operation Sense with Fractions
1
or greater than (>) 3 .
EMPower™ © 2013
Lesson 6a, p. 14
3
2
and 4
3
Practice:
2
3
1. Which is larger 3 or 4 ? Show how you know.
2. Name the points on the number line.
0
1
A
B
2
3
E
CD
F
3. Show each point on the number line.
2
3
0
1
2
3
a. 2 1 3
5
b. 1 12
5
c. 2 6 1
d. 2 6
7
e. 16
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Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a, p. 15
Mental Math Practice: Using Properties
How quickly can you mentally solve each of the problems below?
1. 43 ( 2 – 4 )
2 4
2. 12 ( 21 + 21 )
3. 4 ( 21 ) + 6 ( 21 – 21 )
4. 8 ( 6 – 85 )
8
5. -4 + 3 ( 21 + 21 )
6. 9 + (-3)( 42 – 42 )
7. 50 ( 43 – 43 )
8. -10 + 5( 21 )(2)
9. 40 + (12)( 1 )(4)
4
10. -2 + 7( 21 )(4)
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Lesson 6a, p. 16
Test Practice
1. Tom works 12-hour days, four days a week. Because
they are long days, he thinks about how much work
he has already finished in a day. So far today, he has
worked 7 hours. What fraction of the day has Tom
worked?
(1) 74
5
(2) 12
7
(3) 16
(4) 7
12
(5) 1
4
4. Which of the following is not equivalent to 2?
3
(1) 64
6
(2) 9
8
(3) 12
9
(4) 12
10
(5) 15
5. During a recent hurricane, about one-third of the
population was without power. If the population is
about 150,000, about how many are without power?
(1) 10,000
(2) 50,000
2. Sherry timed herself as she walked around the track.
It took 20 minutes. What part of an hour does this
represent?
(3) 75,000
(4) 100,000
(5) 450,000
(1) 20
1
20
(2) 40
(3) 31
(4) 2
3
6. A small store tracked the payment type chosen by its
customers during one day. According to the chart, about
what fraction of purchases were made with credit cards?
ATM Debit
Credit
Cash
1
(5) 20
3. Nate is trying to save $3,000 for a used car. So far he has
saved about $1,400. About what fraction of the total has
he saved?
(1) almost 1
4
(2) almost 31
(3) almost 21
(4) almost 3
4
(5) almost 2
3
EMPower™ © 2013
(1) 1
4
(2) 31
(3) 4
10
(4) 21
(5) 2
3
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 6a: Equal Measures
Answer Key
2a.
Activity 1: Fraction Strips and Rulers (p. 2)
1/ 1/ 1/
6+ 6+ 6+
1/ 1/ 1/
6+ 6+ 6 =1
1a. There will be a variety of answers such as 1/4 =
2/8 = 4/16; 2/4 = 1/2= 4/8=8/16; 3/4=6/8=12/16
etc.
2a. Some possibilities: 3/4 = 6/8 = 9/12 = 12/16 =
15/20 (last example beyond their strips) possible
picture to support their thinking:
1/
4
1/
4
1/
4
6 one sixths equal one
b.
1/ 1/ 1/
3+ 3+ 3 =1
b. Some possibilities: 1 1/2 = 1 2/4 = 1 4/8 = 1 8/16
= 3/2 = 6/4 =12/8; possible picture to support
their thinking:
1/
2
1
1/
2
1/
2
1/
2
c. Some possibilities: 14/16 = 7/8 = 21/24 = 28/32 =
35/40
d. Some possible answers: they see that the number
you multiply by to change a denominator is really
the amount of pieces (or slices) the denominator
is changed into. For example 1/4 = 2/8 because 4
x 2 = 8 and “2” is the number of pieces (or 1/8’s)
that 1/4 is divided into. Another possibility is
students will see patterns in the equivalency. For
example, 1/8 = 2/16 = 3/24, etc. students would
see the numerator increasing by one while the
denominator is increasing by increments of 8.
Activity 2: Pattern Blocks (p. 3)
Note: Responses may vary slightly for these questions.
1a.1/6
b. 1/2
c. 1/3
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Possible answer: One
third plus one third plus
one third equals 1.
c.
1/ 1/
2+ 2 =1
One half plus one half
equals one.
d.
1/ 1/ 1/
6+ 6+ 6
1
3
= /6 = /2
Three sixths equal one
half or one sixth added 3
times is one half.
EMPower™ © 2013
e.
1/ 1/
6+ 6 =
2/ 1/
6= 3
The sum of two sixths
equals one third.
3. Possible answers: 4 red trapezoids = 2 hexagons
1/2 + 1/2 + 1/2 + 1/2 = 2; four halves equal two
Activity 3: Fraction Strips with Thirds and
Sixths (p. 6)
1. 1/16, 1/12, 1/8, 1/6, 1/4, 1/3, 1/2. Possible answer:
the denominators decrease in size as the fractions
increase in size.
2. Many possible answers: 2/3 = 4/6, 1/3 = 2/6, 3/6
= 1/2, 2/4 = 1/2, etc.
f.
3
1
6/ 6/ 3/ 15/
6 + 6 + 6 = 6 = 2 /6 or 2 /2 or
1/ 1/ 1/ 1/ 1/ 1/ 1/
6+ 6+ 6+ 6+ 6+ 6+ 6+
1/ 1/ 1/ 1/ 1/ 1/ 1/
6+ 6+ 6+ 6+ 6 + 6+ 6
3
1
1/ 15/
+ 6 = 6 = 2 /6 or 2 /2
Two and a half is the same as six
sixths added to six sixths added to
three sixths.
3a. Every time you break a fraction into 2 identical
pieces, the denominator and the numerator for
the new fraction doubles.
b. Possible answers: 1/3 = 2/6; 3/8 = 6/16; 3/4 = 6/8
etc.
Activity 4: About Ones and Zeroes (p. 7)
1a. True, some possible examples: 5 × 1 = 5, 1/3 × 1
= 1/3, 4 × 1 = 4
b. True, some possible examples: 3 × 0 = 0, 1/2 × 0
= 0, 7 × 0 =0
c. False, some possible examples: 5 + 0 = 5 not 0,
3/5 + 0 = 3/5 not 0
d. False, some possible examples: 3 + 1 = 4 not 3,
3/4 + 1 = 1 3/4 not 3/4
g.
1/ 1/ 1/
3+ 6 = 2
One third added to one
sixth is one half.
2a. 0, b. 0, c. 5/16, d. 1 5/16, e. 3/4(1) = 3/4,
f. 1/2 (1) = 1/2, g. 1/2(1) = 1/2, h. 5/16 (1) =
5/16
3. Some possible answers: 3/5(1/2 – 1/2) = 0 ; 1/2
(3 – 2) = 1/2
4a. 1/2 or 2/4
b. 1/2 or 4/8
c. 1/2 or 8/16
h.
d. 1/2 or 16/32
1/ 1/ 1/ 1/
3+ 6 + 3+ 6 =
2/
2=1
Possible answers: two sets
of two thirds added to two
sets of half of two thirds is
one.
EMPower™ © 2013
Students should agree based on the
multiplication property of 1 above. They should
also see that 1/2 and 2/4 are equivalent.
5. Since multiplying by one produces the same
value, hopefully students will see that multiplying
by fractional forms of one (2/2, 3/3, 4/4, etc.) will
generate equivalent fractions.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Math Inspection: One in the Denominator (p. 9)
26. 8/16 = 1/2
1. F
27. 1 = 16/16
2. F
28. 1 = 2/2
3. F
4. T
5. F
6. T
7. T
8. Answers will vary, e.g., 9/1, 16/1, 1,000/1
Practice: Equivalent Fractions (p. 10)
Practice: Between 1/3 and 1/2 (p. 11)
a. Not between since 10/20 is 1/2, 11/20 would be
greater than 1/2
b. Not between since 8/16 is 1/2 , 9/16 would be
greater than 1/2
c. Is between, since 6/12 is 1/2, 5/12 is less than 1/2
but greater than 1/3
1. 1/2 = 2/4
d. Is between, since 6/16 is 3/8, 3/8 is less than 1/2
but greater than 1/3
2. 1/4 = 2/8
e. Not between since 7/14 equals 1/2
3. 1/8 = 2/16
f.
4. 1/4 = 4/16
5. 1/2 = 8/16
6. 2/4 = 8/16
Is between since 10/20 is 1/2 then 9/20 is less than
1/2 but greater than 1/3
g. Not between since 7.5/15 is 1/2, 12/15 is greater
than 1/2
7. 1 = 4/4
h. Not between since 3/15 is 1/3 then 2/15 is less
than 1/3
8. 1/2 = 4/8,
i.
9. 9. 4/4 = 8/8,
10. 3/4 = 6/8
11. 3/4 = 12/ 16,
Is between since 4.5/9 is 1/2 then 4/9 is less than
1/2 but greater than 1/3
Practice: Racheting Up (or Down) a Notch (p. 12)
12. 3/8 = 6/16
1. 11/32 because 5/16 = 10/32 and one more 1/32
makes 11/32
13. 13. 4/8 = 8/16
2. 1/2 because 17/32 less than 1/32 is 16/32 or 1/2
14. 1= 8/8
15. 5/8 = 10/16
16. 6/8 = 3/4,
17. 6/8 = 12/16
18. 7/8 = 14/16
19. 8/8= 16/16
20. 4/8 = 2/4
3. 21/32 because 5/8 is 20/32 and one more 1/32 is
21/32
4. 13/32 because 7/16 is 14/32 and one less 1/32 is
13/32
5. 29/32 because 7/8 is 28/32 and one more 1/32 is
29/32
Practice: Where to Place It? (p. 13)
21. 2/16=1/8,
1. > 1/3
22. 4/16 = 1/4
2. > 2/3
23. 4/16 = 2/8
3. > 1/3
24. 6/16 = 3/8
4. 2/3
25. 8/16 = 2/4
5. > 1/3
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Practice: 2/3 and 3/4 (p. 14)
1. 3/4 Possible answer:
2a. 3/8
b. 15/16
c. 1 3/8
d. 1 7/16
e. 2 1/8
f. 2 7/16
Mental Math Practice: Using Properties (p. 15)
1. 0
2. 12
3. 2
4. 1
5. -1
6. 9
7. 0
8. -5
9. 52
10. 12
Test Practice (p. 17)
1. 4
2. 3
3. 3
4. 4
5. 2
6. 5
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Common Core Standards for Mathematical Practices
*1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
*4. Model with mathematics.
*5. Use appropriate tools strategically.
*6. Attend to precision.
*7. Look for and make use of structure.
*8. Look for and express regularity in repeated reasoning.
Objectives
•
•
Use understanding of the meanings
of addition (combining) and
subtraction (take away, difference,
and missing part) to reason about
fraction operations.
Use visual models (e.g., fraction
strips, rulers, pattern blocks) as
tools to reason about combining
fractions and finding differences.
FACILITATING
7a
Visualizing and
Estimating Sums
and Differences
GED Mathematics Assessment Targets
Anticipated
Q1.a Order fractions and decimals, including on a
number line.
Q1.b Perform addition, subtraction, multiplication,
and division on rational numbers.
Q1.c Apply number properties involving multiples
and factors, such as using the least common
multiple, greatest common factor, or distributive
property to rewrite numeric expressions.
Q2.a Solve single-step or multistep real-world
arithmetic problems involving the four
operations with rational numbers, including
those involving scientific notation.
Section
Adaptation
Opening Discussion
NEW
Activity 1: Adding and Subtracting on
the Ruler
NEW: Students use their fraction strips for 1, 1/2 , 1/4. 1/8 and
1/16s to show addition and subtraction ideas. They apply the
knowledge to the context of the 1/16’’ ruler.
Activity 2: Adding and Subtracting
with Pattern Blocks
NEW: Pattern blocks introduced to explore addition and
subtraction
(1, 1/2, 1/3, 1/6s).
Practice: Time in Transit
NEW: Students find totals given fractions of hours
Mental Math Practice: Estimating with NEW: Students predict the size of the sum or difference
Fractions
Practice: Build Up
NEW: Students calculate height of assembled scrapwood and
bricks
Extension: Nothing Fancy Bird House
NEW: Students fill in missing information on the construction of
a birdhouse
Test Practice
NEW
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
7a
Lesson 7a, p. 1
FACILITATING
Visualizing and
Estimating Sums and
Differences
Can you combine, take away,
or find the difference between
two fractional amounts?
Synopsis
Students draw upon their prior knowledge about addition and subtraction of
whole numbers as they find sums or differences using fractions. This lesson
continues to ground students’ visual reasoning with benchmark fractions. At
this point, students must be challenged to arrive at solutions NOT through a
mathematical algorithm, but rather, by thinking with visuals and estimating.
1. The class decides on the reasonableness of a statement about fraction
addition.
2. Student pairs carefully examine addition and subtraction on the 1/16’s
inch ruler and write mathematical statements to match.
3. Students examine addition and subtraction of 1/2s, 1/3s and 1/6s with
with Pattern Blocks.
4. Students work in pairs on a design project, using rulers and estimation.
5. The class summarizes what has become clear about working with
fractions, and what is still unclear.
Objectives
Using Benchmark Fractions Plus:
Operation Sense with Fractions
•
Use understanding of the meanings of addition (combining) and
subtraction (take away, difference, and missing part) to reason about
fraction operations.
•
Use visual models (e.g., fraction strips, rulers, pattern blocks) as tools to
reason about combining fractions and finding differences.
EMPower™ © 2013
Lesson 7a, p. 2
Materials/Prep
• 1/16 inch ruler or Blackline Master 5a: 1/16 inch ruler
• Plastic sheet protector
• 1/2 inch grid paper
• Scotch tape
Heads Up!
This lesson assumes that students have understandings of the meanings of addition as combining and
subtraction as take-away, comparison, and missing addend with whole numbers. It also assumes that
students’ understanding of several benchmark fractions is solid, and the material in Using Benchmarks,
Lessons 1 through 6a has been mastered.
Opening Discussion
Start the discussion by asking students to draw and estimate to see what makes sense. Use the
expression, “Shave off a fraction,” to start a discussion with students.
Is anyone trying to shave off time or cut back on expenses in some way? What fraction are you
cutting back by?”
Listen for examples. Then ask students to think about how they handled whole numbers when they
wanted to take away, cut back, or find a difference. Subtraction becomes handy, but some people
add up, some visualize with a number line, depending on the numbers, you might do a combination
of things.
Refer students to the scenario in the introductory text on p. 1 of Lesson 7a (the ice cream diet) and
ask students to take a minute to think about where they see addition and/or subtraction in the
example.
With whom do you agree?
Talk with someone about this exchange and prepare to share or explain your reasoning.
When students share, record what they say on the board or on an overhead transparency. Expect
some students to say that 1/2 + 1/3 is, indeed, 2/5. Invite them to share drawings with the class. The
idea that 2/5 is smaller than 1/2 is important. The exact answer is not as important as having a good
sense that 2/5 is an unreasonable answer for 1/2 + 1/3. If any students use a correct algorithm for
finding common denominators, ask them to support their answer with a picture as well.
Heads Up!
Don’t focus here on the procedure for adding fractions by finding common denominators. An
estimate—for example,”almost one, but not quite”— is good enough. The important point is that
when combining these two amounts, the sum cannot end up being smaller than one (or both) of
the amounts. For the activities in this lesson, ask students to rely upon visuals and estimation to solve
problems.
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 7a, p. 3
Activity 1: Adding and Subtracting on the Ruler
Distribute to each student a copy of Blackline Master 5a: 1/16 inch Ruler with a plastic sheet
protector and an erasable marker (so that students can draw and erase line segments).
Using an overhead projector, or an enlarged ruler for all to see, ask for a volunteer to review ruler
markings for 1/2, 1/4, 1/8, and 1/16. Also ask for volunteers to explain:
What other ways could we read the ½ inch mark? [2/4 in , 4/8 in , 8/16 in]
What other ways could we read the 1/4 inch mark? [2/8 in , 4/16 in]
What other ways could we read the 1/8 inch mark? [2/16 in]
Direct students’ attention to Activity 1: Adding and Subtracting on the Ruler in the Student Book, p.
2, where they will be asked to draw a line segment of a certain length, draw the action of adding or
subtracting from it to find a sum or difference, and record the actions in math symbols.
Encourage students to practice on the plastic-covered ruler, and then to record their work on the
Student Book Activity sheet. Ask pairs to work together on the first problem:
Draw a line segment 1/8 of an inch long. Draw a line segment 3/8 inches long
attached to it. How long is the entire line segment? What other fraction expresses
the answer?
When students are done, ask for a volunteer to show the action on the ruler and how to write
what was done in math symbols (an equation), and to show where the drawing connects to the
symbolic math.
Ask for comments and questions, and if there is another correct way to express the answer.
For example, if the first student writes:
1/8 in + 3/8 in = 4/8in
Others might offer:
1/8 in + 3/8in = 2/4in
1/8 in + 3/8 in = 1/2 in
Have student pairs work together on the rest of the problems, and then ask various pairs to present
their work publicly, first showing their drawing on the ruler, and then the record of that with math
symbols and the various ways to express the answer.
As you facilitate the sharing, emphasizing the connections between the drawings and the symbols,
there are opportunities to introduce or bring up additional ideas.
For example:
In Problems 1 - 4, use the phrase “simplest terms” to show that the preferred final answer is usually
the one that uses the smallest possible integers to express the fractions. 1/2 is preferred to 4/8, even
though 4/8 is a correct answer.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 7a, p. 4
In Problems 5, 6, 8, and 9, with amounts greater than 1 inch, use the phrases “mixed number” and
“improper fraction.” 3/4 + 3/4 = 1 1/2 = 1 3/4 = 6/4 = 3/2. To get at the improper fraction, ask,
“How many fourths altogether?” Stress also that this is 6/4 and show how that becomes 1 1/2. 6/4 =
4/4 + 2/4. Since 4/4 = 1 change that to 1 + 2/4 or 1 2/4. Then 2/4 simplifies to 1/2.
It’s easier to think when you are using the same size pieces. When combining or taking away lengths,
did anyone change one length to an equivalent length to help with the measuring? “Common
denominators” will help them in lesson 8 when we will more formally add and subtract fractions
with the algorithm.
Extension: If time allows, ask pairs to see if they can solve all the problems with their fraction strips.
Activity 2: Adding and Subtracting with Pattern Blocks
Use Pattern Blocks to show the addition or subtraction and to arrive at the answer.
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 7a, p. 5
Summary Discussion
Take a few minutes to reflect about what is clear and what is not clear about fractions.
Ask students to rate their strength, on a scale from 1 - 5, as to their ability to
• Use visuals to perform fraction addition and subtraction
• Estimate the answer to addition or subtraction of fraction problems.
• Use a ruler accurately
About fractions: What is very clear and what could be clearer?
Practice
Time in Transit, p. 6
For practice adding and subtracting on a number line.
Build Up, p. 8
For practice using fractions in a real-life context.
Mental Math Practice
Estimating with Fractions, p. 7
Test Practice
Test Practice, p. 9
Pilot teachers told us...
Activity 1: Adding and Subtracting on the Ruler was one of the most shocking experiences in my almost
30 years teaching adults. I use EMPower in two classes. I found the same thing in both classes — no
one knew how to use a ruler! I did not just hand out rulers and tell them to do the activity. I drew
a ruler on the board and went over what all the divisions are. From previous EMPower lessons they
could tell me half of a half, and half of a quarter, and half of an eighth. So, we went over the ruler
before I gave them rulers to do the activity. They were totally lost. I understand what you had in
mind — that one way to add and subtract fractions is to follow the numbers on a ruler and it eliminates the “finding a common denominator” step. It seemed like a simple activity because I thought
reading or using a ruler is an elemental skill. After a few minutes of confusion, I asked the class if they
ever use a ruler or tape measure at home to measure things. I said, “Don’t you measure something to
see if it will fit?” as an example. They said no, they just “eyeball it.” I was shocked. I thought using a
ruler was a pretty basic thing. My colleague suggested students spend some time measuring things
around the classroom. I plan to do that with them.
If students have minimal experience using rulers and tape measures, give them 5-10 minutes and a
challenge to measure objects in the room; for example: Find an item that measures 8 1/2”. Find two
items that measure 13.5”. Find an item that measures less than 3/4” .
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 7a, p. 6
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Blackline Master 7a: 1/2 Inch Grid
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
7a
Lesson 7a, p. 1
LESSON
Visualizing and
Estimating Sums
and Differences
What happens when you
combine, take away, or find
the difference between two
fractional amounts?
It seems everyone is feeling short on time or cash. Workers want to cut
their time commuting. Owners want to cut costs so they look for ways
to cut waste and use supplies well. It makes sense to compare, to find the
difference and see the amount of savings.
Cutting corners may save you a fraction of the amount of time or
money you planned to spend. Cutting back on portion size is one way
to cut calories. Is it worth it? Use math to figure it out. Think about
the situation: cutting back, finding the difference, taking part from a
whole amount — all call for subtraction. Subtraction can be tricky with
fractions. What if a dieter said:
1
“I cut back. I ate 2 of the container of ice cream last week instead of a
1
whole container. This week I ate an additional 3 of the container, so I’ve
2
eaten only 5 of the container of ice cream. My diet is going well.”
Does it make sense? Reason and prove your answer with all the tools you
have used so far. Use number lines, rulers, fraction strips, a drawing, or
objects to work out solutions and show your thinking. Don’t forget to use
benchmarks like one half to predict (or estimate) an answer.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 7a, p. 2
Activity 1: Adding and Subtracting on the Ruler
Practice adding and subtracting on the rulers. For Problems 1–7, draw the segments and
read the answer from the marks on the ruler. Then, write a math equation to show all the
action.
1. Draw a line segment 1/8 of an inch long. To that line segment, attach a line
segment 3/8 of an inch long. How long is the entire line segment? Is there another
fraction to express the answer?
Inches
1
2
3
4
5
6
The math equation: ___________________________________________
2. Draw a line segment 3/16 of an inch long. Extend that line segment by 9/16 of an
inch more. How long is the entire segment? Is there more than one way to read the
mark of the ruler?
Inches
1
2
3
4
5
6
The math equation: ___________________________________________
3. Draw a line segment 15/16 of an inch long. Subtract 4/16 of an inch. How long is
the remaining length? Is there more than one way to read the mark of the ruler?
Inches
1
2
3
4
5
6
The math equation: ___________________________________________
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 7a, p. 3
4. Draw a line segment 7/8 of an inch long. Subtract 3/8 of an inch. What is the
answer? Is there more than one way to read the answer on the ruler?
Inches
1
2
3
4
6
5
The math equation: ___________________________________________
5. Draw a line segment 3/4 of an inch long. To that segment, attach another
segment 3/4 of an inch long. How long is the final segment? Is there more than one
way to read the answer on the ruler?
Inches
1
2
3
4
5
6
The math equation: ___________________________________________
6. Draw a line segment 1 1/8 inches long. Subtract 5/8 of an inch from it. What is
your answer? How long is the final segment? Is there more than one way to read the
answer on the ruler?
Inches
1
2
3
4
5
6
The math equation: ___________________________________________
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
Lesson 7a, p. 4
7. Draw a line segment 1 1/4 inches long and subtract a segment 2/4 of an inch
long. How long is the final segment? Is there more than one way to read the answer
on the ruler?
The drawing:
Inches
1
3
2
4
5
6
The math equation: ___________________________________________
Answer true or false. Refer to the ruler to figure out or confirm your answer.
Inches
1
3
2
4
5
6
8. True or false?
7
1
3
8 inches – 4 inches = 8 inches
The drawing:
9. True or false?
15 inches = 1 3 inches
2 21 inches – 16
8
The drawing:
10.What is the value of the missing measurement?
3
1
4 inches + ? inches = 4 4 inches
The drawing:
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 7a, p. 5
Activity 2: Adding and Subtracting with Pattern Blocks
Use Pattern Blocks to show the addition or subtraction and to arrive at the answer.
1. 5 – 1 =
2. 1 – 2 =
3
1=
3. 32 – 2
1 1
4. 2 – 6 =
1 1 1
5. 2 + 3 + 6 =
2 1
6. 3 + 6 =
3
6
Make up 3 more!
7.
Using Benchmark Fractions Plus:
Operation Sense with Fractions
–
=
8.
–
=
9.
–
=
EMPower™ © 2013
Lesson 7a, p. 6
Practice: Time in Transit
Use a number line or other drawing to figure out each of the problems below.
1. Elena and Ben drove from Chicago to St. Paul with their son, Sam. They started
3
at 8 a.m., drove for 4 hours, and stopped for lunch at a rest area for 4 of an hour.
1
Then they continued driving, and after 3 2 hours, Sam asked, “Are we there yet?”
a. First, estimate in your head how long they’ve been traveling.
b. What is the actual time that they have been traveling? ________ Explain.
3
2. Trey drove 4 of an hour from his house to the airport. He spent one and a half
hours there. His flight to Jackson, Mississippi took two hours. He then had a halfhour bus ride before he walked another 15 minutes to his grandmother’s house.
How many hours did Trey spend in transit?
a. Estimate.
b. Actual time to get from his house to his grandmother’s house? ________
Explain.
3. Steve says it takes 8 hours to drive to his favorite beach from his house. He drives
for four and one-quarter hours, then stops for lunch for three-quarters of an hour.
He then drives for an hour and a half and stops to pick up some supplies. How
many more hours of driving are ahead of him before he reaches the beach?
a. Estimate.
b. Actual time still to go? ________ Explain.
4. Choose a destination of your own. Write the details for the time it takes to get
from your home to that destination. Be sure to incluide any time you stop.
1
a. Does it take more or less than 3 2 hours to arrive? How much more or less?
1
b. How much more or less than 1 4 hours?
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 7a, p. 7
Mental Math Practice: Estimating with Fractions
For each problem below, decide which benchmark fractions it comes between.
between 0 and 2
1
between 2 and 1 b. 1/6 + 1/12 + 5/8 between 0 and 2
1
between 2 and 1 a. 1/3 + 4/9
1
between 1 and 1 2
1
between 1 and 1 2
1
between 1 and 1 2
1
between 1 and 1 2
1
between 1 and 1 2
1
between 1 and 1 2
1
between 1 and 1 2
1
between 1 and 1 2
c. 5/6 – 1/8
between 0 and 2
1
between 2 and 1 d. 5/8 + 9/16
between 0 and 2
1
between 2 and 1 e. 11/16 – 5/8
between 0 and 2
1
between 2 and 1 f. 5/12 + 11/12
between 0 and 2
1
between 2 and 1 g. 3/4 – 1/16
between 0 and 2
1
between 2 and 1 h. 5/16 + 5/12
between 0 and 2
1
between 2 and 1 Using Benchmark Fractions Plus:
Operation Sense with Fractions
1
1
1
1
1
1
1
1
EMPower™ © 2013
Lesson 7a, p. 8
Practice: Build Up
Maria has a lot of projects that she wants to do in her yard.
She already has the following materials available:
3
• 7 bricks, 1 4 ” thick
1
• 10 pieces of scrap wood, 1 2 ” thick
Project 1: Build a Marker
Can Maria construct a marker 2’ tall (24”)? Write an equation and make a sketch to
show how you know.
Project 2: Shore Up the Shed
1
Maria’s shed is sinking on one side. She needs 10 2 inches of material under the back
corner in order to make it level. Sketch a picture and label it to show how she can use
1
scrap wood and/or bricks to bring up the height to 10 2 inches.
Project 3: Plant bench
Maria also needs to replace one leg of her plant bench. The legs are each 3’ (or 36”) tall.
She has a 30” long piece of wood that she can use as a leg. How many pieces of scrap
wood does she need to put under it to raise the height to 3’? Write an equation and make
a sketch to show how you know.
If Maria wanted to complete all three projects, how much more in materials would she
have to get? Explain how you know.
EMPower™ © 2013
Using Benchmark Fractions Plus:
Operation Sense with Fractions
Lesson 7a, p. 9
Test Practice
1. Tim had an 18" board. He cut two small pieces off: one
1
6 2 " and the other 7 41 ". About how much does he have
left of his 18" board?
1
4. Sue cut three pieces of yarn, one 2 2 feet, another 1 41
feet, and the third 3 43 feet. How many feet of yarn did
she cut?
(1) a little less than 4"
1
(2) a little more than 4"
(1) 6 2 feet
(3) a little more than 5"
(2) 6 43 feet
(4) a little more than 13"
(3) 7 feet
(5) a little less than 14"
1
(4) 7 2 feet
2. Because Marcia likes pecans, she tends to use a little
more than recipes call for. She baked two different
batches of cookies that called for pecans. One recipe
1
(5) 11 2 feet
1
called for 1 2 cups and the other called for 2 43 cups. If
she used slightly more than the recipes called for, about
how many cups of pecans did she use?
1
1
5. Emily and her sister Jane were eating a pizza. Jane ate 3
1
of the pizza and Emily ate 2 . How much of the pizza did
they eat together?
1
(1) at least 2 2 cups but less than 3 2
(1) the whole pizza
(2) at least 3 cups but less than 4
1
(3) at least 3 2 cups but less than 4 41
(2) 52 of the pizza
(4) at least 4 cups but less than 5
5
(3) 6 of the pizza
(5) at least 5 cups but less than 6
1
(4) less than 2 of the pizza
3. If the hexagon pattern block is the whole,
then which arrangement of the shapes shows
2
3
1
(5) less than 3 of the pizza
+ 21 ?
1
6. Bev cut four pieces of ribbon, each 3 2 inches long.
How many inches of ribbon did she cut altogether?
(1)
(2)
(3)
(4)
(5)
Using Benchmark Fractions Plus:
Operation Sense with Fractions
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Answer Key
Lesson 7a: Visualizing and Eatimating Sums
and Differences
Activity 1: Adding and Subtracting on the
Ruler (p. 2)
3. 1/6
1. 1/8 + 3/8 = 4/8 = 1/2
2. 3/16 + 9/16 = 12/ 16 = 6/8 =3/4
3. 3/16 + 9/16 = 12/16; yes 6/8 or 3/4
4. 7/8 -3/8 = 4/8; yes 2/4 or 1/2
5. 3/4 + 3/4 = 6/4; yes 1 2/4 or 1 12
6. 1 1/8 – 5/8 = 4/8; yes; or 1/2
7. 1 1/4 – 2/4 =3/4 yes; 6/8 or 12/16
4. 2/6 = 1/3
8. True
9. False should be 1 9/16
10. 3 2/4 or 3 1/2
Activity 2: Adding and Subtracting with
Pattern Blocks (p. 5)
1. 3/6 or 1/2
5. 1
2. 1/3
6. 5/6
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013
7. - 9. possible examples: 5/6 – 1/6 = 4/6 or 2/3; 1-1/6
= 5/6, 1/6 + 1/3 =3/6 = 1/2 ; etc.
Mental Math Practice: Estimating with
Fractions (p. 7)
a. Between 1/2 and 1
b. Between 1/2 and 1
c. Between 1/2 and 1
d. Between 1 and 1 1/2
e. Between 0 and 1/2
f. Between 1 and 1 1/2
g. Between 1/2 and 1
h. Between 1/2 and 1
Practice: Build Up (p. 8)
1. Project 1: Build a Marker
Yes. Total wood is 15”.
(1 1/2 + 1 1/2 + 1 1/2 + 1 1/2 + 1 1/2 + 1 1/2 + 1
1/2 + 1 1/2 + 1 1/2 + 1 1/2 = 15)
Total bricks are 12 1/4”
(1 3/4” + 1 3/4” + 1 3/4” + 1 3/4” + 1 3/4” + 1
3/4” + 1 3/4” = 12 1/4”)
1 3/4
3
3 1/2
3
3 1/2
3
3 1/2
12 1/4
3
3
Practice: Time in Transit (p. 6)
15
1. a. Estimate: 8 hours +
b. 4 + 3/4 + 3 1/2 = 8 1/4 hours or 8 hours, 15
minutes
2. a. Estimate: less than 5 hours
b. 3/4 + 1 1/2 + 2 + 1/2 + 1/4 = 5 hours
2. Project 2: Shore Up the Shed
6 bricks is one way to do it, 7 pieces of wood is
another.
3. a. Estimate: 1 hour still to go
b. 4 1 4/ + 3/4 + 1 1/2 + _1 1/2__ = 8
4. a. Answers will vary.
b. Answers will vary.
EMPower™ © 2013
So combined only 27 1/4”.
3
W
W
W
W
W
W
W
B
B
B
B
B
10 1/2
10 1/2
3
30”
Using Benchmark Fractions Plus:
Operation Sense with Fractions
3. Project 3: Plant Bench
4 pieces of scrap wood will do it.
1 1/2” +1/2” = 3” + 1/2” + 1/2” = 6”
Wrap Up: All four projects
Tower: If using all available bricks and 8 pieces of wood
for the tower, Marie has 2 pieces of scrap wood left
over.
Shed: 10 1/2”, so 6 bricks for that or 5 pieces of scrap
wood (plus the 2 left over).
Plant bench: Needs 6” height, so an extra 4 bricks.
(1 3/4” + 1 3/4” + 1 3/4” + 1 3/4”)
Test Practice (p. 9)
1. 2
2. 4
3. 3 or 4
4. 4
5. 3
6. 14 inches
Using Benchmark Fractions Plus:
Operation Sense with Fractions
EMPower™ © 2013