Sample Booklet Mathematics — Book II Grade 5 All

Transcription

Sample Booklet
Grade 5
Mathematics — Book II
Number and Operations—Fractions
Published by:
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Sample pages from Teacher Guide...................................................................................... 2–11
Sample pages from Student Work Text ............................................................................12–26
Selected pages from
Student Work Text
Mathematics
Grade 5, Book II
This page may not be reproduced.
Teacher Guide
Lori Mammen
Editorial Director
ISBN: 978-1-60539-882-2
Copyright infringement is a violation of Federal Law.
©2014 by ECS Learning Systems, Inc., Bulverde, Texas. All rights reserved. No part of this publication may be reproduced,
translated, stored in a retrieval system, or transmitted in any way or by any means (electronic, mechanical, photocopying,
recording, or otherwise) without prior written permission from ECS Learning Systems, Inc.
Reproduction of any part of this publication for an entire school or for a school system, by for-profit institutions and tutoring
centers, or for commercial sale is strictly prohibited.
Printed in the United States of America.
Disclaimer Statement
ECS Learning Systems, Inc. recommends that the purchaser/user of this publication preview and use his/her own judgment
when selecting lessons and activities. Please assess the appropriateness of the content and activities according to grade level
and maturity of your students. The responsibility to adhere to safety standards and best professional practices is the duty of
the teachers, students, and/or others who use the content of this publication. ECS Learning Systems is not responsible for
any damage, to property or person, that results from the performance of the activities in this publication.
TestSMART is a registered trademark of ECS Learning Systems, Inc.
2
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 5—Book II
Table of Contents
What’s Inside the Student Work Text?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
How to Use the Student Work Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Understanding Rigor and Cognitive Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Descriptions of TestSMART® Complexity Levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Fostering Mathematical Understanding and Inquiry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Definition of the Common Core State Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
The Precise Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Mathematics Manipulatives and Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Text-Marking in Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Integrating the Literacy Strands in the Mathematics Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Master Skills List. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
ECS Learning Systems, Inc. • P. O. Box 440 • Bulverde, TX 78163-0440
ecslearningsystems.com
1.800.688.3224 (t) • 1.877.688.3226 (f ) • [email protected]
2
TestSMART® Common Core Student Work Text, Teacher Guide
TestSMART® Common Core Teacher Guide—Mathematics, Grade 5—Book II
3
What’s Inside the Student Work Text?
Overview
The TestSMART ® Common Core Student Work Text addresses the Common Core State
Standards (CCSS) for Mathematics (National Governors Association Center for Best
Practices/Council of Chief State School Officers [NGA/CCSSO], 2010b) in separate books.
However, students benefit from an integrated view of mathematics (cross-domain
experiences). For instance, instead of isolating concepts, this approach groups ideas
and draws parallels. Students move beyond memorization and routine procedures to
construct mathematics using their own strategies and representations. As they grow in
understanding, they begin to generalize and transfer patterns of responding to other
mathematical and non-mathematical problems and situations.
The exercises included in the work text focus on the critical areas (major work) of the
grade as defined in the CCSS (NGA/CCSSO, 2013). The work text provides practice in
a variety of mathematical and real-world contexts. Tasks require appropriate use of
manipulatives, tools, and technology.
The TestSMART Common Core Student Work Text should supplement and support
research, planning, instruction, and both informal and formal assessment. It is
recommended that teachers introduce new math concepts through everyday problems
and situations.
How to Use the Student Work Text
Time Requirement
The time requirement depends on the activity type and topic. Activity types include
guided (whole-class and small-group), independent, and extension/homework. Most
activities will take about 15–30 minutes.
Getting Started
Teachers should implement the activities from the TestSMART Common Core Student
Work Text in sequential order. The activities logically progress within each domain,
building upon prior knowledge and personal experience. The activities also
appropriately relate thinking across domains and grades. The activities should move
students toward self-directed mathematics learning and problem solving.
Within each activity are opportunities for students to question, think about, and talk
about their learning. In addition to the specific mathematic expectations involved,
these moments during activities help students develop the following types of skills—
•
•
•
•
•
analytical thinking
evaluative thinking
reflective thinking
metacognitive thinking
communication
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3
For instance, students may need to connect information with prior knowledge or
personal experience, make predictions, infer, determine importance, visualize,
synthesize, or monitor comprehension. The Teacher Guide provides specific guidance
for supporting students throughout the learning process.
Lesson Features
Remember/What You Need to Know: Sometimes, students are prompted to recall
prior learning. Occasionally, students are given key background information to activate
or support their subject-area knowledge. Some students will not have prior knowledge
about the concept or skill. Others may have developed misconceptions.
Think About It: Students are asked to think about math-related questions and
situations and to think about their thinking. Students can think independently, or
teachers can guide “think-aloud” sessions in small or large groups (see Box 4
“Scaffolding through ‘Think Aloud,’” page 16).
Talk About It: Students are asked to talk about math concepts and situations and
to talk about their thinking. This includes examining problem situations, making
observations, explaining their problem-solving processes, and discussing math
terminology and concepts (see “Math-Talk,” pages 13–14).
Working Together: Students are asked to work together, or collaborate, in guided
settings (pairs, small-group, whole-class). Teachers can support students with openended questions (see Box 5 “Scaffolding through Open-Ended Questions,” pages 16–18).
On Your Own: Students are asked to independently explore a concept or skill, as well as
their own ways of problem solving. Teachers can support students with open-ended
questions (see Box 5 “Scaffolding through Open-Ended Questions,” pages 16–18).
Did You Know…?: Students are given tidbits and trivia about their world and how it
works. These relate to the math domains, math vocabulary, the history of mathematics,
or real-life applications. The following are suggestions for using these tidbits and trivia:
• Challenge students to find other interesting facts related to the topic. This
provides an opportunity for students to learn effective research techniques.
• Create a “Did You Know…?” display where students can post the facts they learn.
• Use the tidbits and trivia as prompts for a class discussion. Talk about what
students already know and what they would like to learn more about. Have
students generate questions for further research or discussion.
• Have students respond to the tidbits and trivia in their math journals. Provide
time for students to share their journal entries with classmates. (Students may
benefit from a guiding question related to the tidbit or trivia.)
• Have students represent the idea(s) from the tidbits and trivia in a new way.
Provide time for students to share their representations with classmates.
4
5
Understanding Rigor and Cognitive Complexity
Increased Rigor
The Common Core State Standards (CCSS) (NGA/CCSSO, 2012) hope to create more
rigorous instruction and learning across the country. But what does academic rigor
mean? Academic rigor is a measure of cognitive demand. In a rigorous system,
standards, curriculum, instruction, and assessment tightly align with congruent
measures of cognitive complexity. Students must demonstrate a deep mastery of
processes, skills, and understandings through rich, complex tasks. The TestSMART®
Common Core Student Work Text provides items written at varying levels of complexity
to accommodate the demands of the expectations in the CCSS for Mathematics
(NGA/CCSSO, 2010b). (Refer to the “Depth of Knowledge” section on this page and
the “Descriptions of TestSMART® Complexity Levels” section on pages 7–8 for more
information about the complexity levels of practice items.)
In addition, the work text provides a range of opportunities to develop mathematical
practices. The content was designed to support student progress to algebra by—
focusing on the critical areas of the grade (as defined in the CCSS)
linking major topics within each grade
thinking across grades
providing balanced attention to all aspects of rigor (NGA/CCSSO, 2013)
•
•
•
•
Depth of Knowledge
Norman Webb’s (2002a) “depth-of-knowledge” model is currently one of the most
influential alignment models in the field of education. “Depth of knowledge” describes
the degree of complexity of knowledge required for a curricular item. Webb identifies
four levels of depth of knowledge: recall (Level 1), skill or concept (Level 2), strategic
thinking (Level 3), and extended thinking (Level 4). Distinct cognitive demands occur
during each activity, or thinking process, level.
The items in the TestSMART Common Core Student Work Text were aligned to the CCSS
for Mathematics using a modified version of the “depth-of-knowledge” model (see
“Descriptions of TestSMART® Complexity Levels,” pages 7–8). During the alignment
process, the complexity level of each item (designated “Low,” “Moderate,” or “High”)
was determined. The level can be found in the skill tag of each practice item.
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5
Overall, rigorous mathematics instruction focuses on the critical areas of the grade (as
defined in the CCSS) while providing balanced attention to the following processes
and proficiencies, which are considered mutually reinforcing (e.g., Grouws, 2004;
Kilpatrick, Swafford, & Findell, 2001; NAGB, 2012; NMAP, 2008; NCTM, 2000;
NGA/CCSSO, 2010b).
NCTM Process Standards
•
•
•
•
•
problem solving
reasoning and proof
communication
representation
connections
NRC’s Proficiency Strands
•
•
•
•
•
adaptive reasoning
strategic competence
conceptual understanding
procedural fluency
productive disposition
(Kilpatrick et al., 2001, Adding It Up)
The “processes and proficiencies” are the foundation for the Standards for
Mathematical Practice (NGA/CCSSO, 2010b).
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.
An effective mathematics curriculum integrates the mathematics domains of the
grade (Grouws, 2004).
“
It is the
relationship between
the knowledge types
that gives one’s
knowledge the power
of application in a
wide variety of
settings”
(p. 183, Silver, 1987).
6
•
•
•
•
•
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
Box 1: Balance in Rigorous Mathematics Instruction
7
Fostering Mathematical Understanding and Inquiry
Common Core State Standards*
The Common Core State Standards (CCSS) (NGA/CCSSO, 2012) is a standards-based
U.S. education reform initiative sponsored by the National Governors Association
(NGA) and the Council of Chief State School Officers (CCSSO). The initiative seeks to
provide a set of national curriculum standards to create more rigorous, consistent
instruction and learning across the country. These standards were developed based
on models from various states and countries, as well as recommendations from K–12
educators and students. The expectations, aimed at college and career readiness,
focus on core concepts and processes at deep and complex levels. The curriculum
standards for ELA/literacy and mathematics were released in 2010. Science and
history standards are in development.
•
•
•
•
Forty-five states and the District of Columbia have adopted the standards, but Alaska,
Minnesota, Nebraska, Texas, and Virginia have yet to adopt them. During the
2014–2015 academic year, adopting states should begin formal CCSS assessments.
Assessments will include the following types of items:
selected-response items (multiple-choice items)
constructed-response items
technology-enhanced items/tasks
performance tasks
For more information about the CCSS initiative, please visit
http://www.corestandards.org.
*
This information was current at time of publication.
Box 2: Definition of the Common Core State Standards
Mathematics Instruction and Learning
Mathematics is a study of patterns, relationships, measurement, and properties in
numbers, quantity, magnitude, shape, space, and symbols. Effective mathematics
instruction requires students to mindfully attend to elements of structure and
content—including patterns and language choice. This disciplined study involves trying
and retrying during problem solving to better understand how structure and content
work together in systems of meaning (Paul & Elder, 2008). The ability to recognize,
analyze, and use patterns and relationships is essential to problem solving.
Mathematical thinking skills are closely tied to skills that are essential for success in
school, career/work, and life, such as—
•
•
•
•
•
•
•
•
•
critical/evaluative thinking
creative/innovative thinking
elaborative thinking
problem solving
decision making
researching
collaboration
communication
organizing and connecting ideas
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9
“
Research...supports
a focus on teaching
for meaning and
understanding.”
These skills are essential to achieving learning goals in the areas of information and
communication technology (ICT) literacy and science. As students develop in
mathematics, they should also see connections in reading, language arts, social studies,
history, art, music, physical education and sports, and other areas of the curriculum.
Research (e.g., Fennema & Romberg, 1999; Hiebert et al., 1997; Simon, 2006; Skemp,
1976) supports a focus on teaching for meaning and understanding. Fluency with
computational procedures and basic facts allows students to expend less cognitive
energy when problem solving. However, drilling on isolated skills can become
meaningless (e.g., Grouws, 2004; Schoenfeld, 1988). In addition, these rote activities
sometimes involve the use of mnemonic devices. These types of “tricks” are not
suggested strategies for achieving long-term understanding and flexible use of skills.
Students understand more when they actively construct meaning during rich, complex
tasks (e.g., Fosnot, 1996; Fosnot, 2005; Noddings, 1990).
Appropriate Tasks
The CCSS emphasize the need for understanding and its impact on carrying out
effective mathematical practices and true mastery of mathematical content
(NGA/CCSSO, 2010b). (Refer to Box 1 “Balance in Rigorous Mathematics Instruction” on
page 6 for a list of the Standards for Mathematical Practice.) Rich mathematics tasks
often involve persistent problem solving and, therefore, can require time. Rich tasks
allow all students—even struggling learners—the opportunity to adequately explore
and discuss complex problems, situations, and ideas. Rich mathematics experiences
provide students with opportunities to see structure, patterns, and relationships in
many different contexts.
Rich, complex mathematics tasks—
•
•
•
•
•
“
Rich mathematics
experiences provide
students with
opportunities to see
structure, patterns,
and relationships in
many different
contexts.”
10
•
•
•
•
•
•
•
•
begin with a clear, explicit, reasonable, actionable learning goal
incorporate the use of sound number sense and basic computational skills
rely on the integrated development of mathematical skills and understandings
build on prior knowledge and personal experience
utilize a variety of settings in which to explore and share mathematical ideas with
others (i.e., paired, small-group, whole-class)
encourage risk-taking to further the learning process
encourage students to work and think mathematically
invite all students to participate in constructive math inquiries and discussions
promote complex thinking and transfer of understanding by focusing on the “big
ideas” and “essential questions”
apply mathematical ideas to a broad range of real-life and imagined situations
help students learn to use the precise language of mathematics for specific
purposes
require students to make conjectures, hypothesize, test and retest ideas, justify
thinking, represent findings in meaningful ways, and reflect
require students to look for and utilize the underlying order and logic of
mathematics when problem solving
9
• allow for diversity in thinking and offer many valid entry points to mathematical
challenges for all students (e.g., multiple solution paths, multiple representations)
• explore and reinforce concepts through hands-on activities involving the use of
technology, manipulatives, tools, and play
• allow students to generalize and transfer patterns of responding to other
mathematical and non-mathematical problems and situations
• require extended engagement (e.g., Hiebert et al., 1997; National Council of
Teachers of Mathematics [NCTM], 2000)
A
Answer Key
©
10
Thi
pp. 4–5
Think About It: Answers may vary. Simplified fractions
are easier to write and work with because they have
smaller numbers and fewer digits, e.g., 1/4 vs. 125/500;
Answers may vary. It may be useful not to simplify
fractions if you need to perform more operations that
require common denominators or if a problem requires
an answer in a certain form.
pp. 6–7
A. 1. divide models into fifteenths; 10/15 + 3/15 = 13/15
2. divide models into twelfths; 3/12 + 4/12 = 7/12
3. divide models into sixths; 4/6 + 2/6 = 6/6 = 1 4. divide
models into eighths; 4/8 + 2/8 = 6/8 = 3/4 5. divide
models into thirty-fifths; 7/35 + 10/35 = 17/35 B. 6. 11/12
7. 28/30 = 14/15 8. 25/28 9. 18/40 = 9/20 10. 13/14
pp. 8–9
Talk About It: A mixed number consists of a whole
n b
d
A
shaded whole and 2 shaded sections 4. 1 1/8; shade
1 whole and 5 sections and place Xs over 4 shaded
sections 5. 2 1/15; shade 4 wholes and 10 sections and
place Xs over 2 shaded wholes and 9 shaded sections
B. 6. 2 1/6 7. 2 2/20 = 2 1/10 8. 3/20 9. 3 17/28 10. 2 2/8
= 2 1/4
pp. 22–24
A. 1. 2 3/6 = 2 1/2; shade 3 wholes and 4 sections and
place Xs over 1 shaded whole and 1 shaded section
2. divide models into tenths; 1 5/10 + 1 4/10 = 2 9/10
3. 2 7/20; shade 2 wholes and 15 sections and place Xs
over 8 shaded sections 4. 2 3/30 = 2 1/10; shade 3 wholes
and 15 sections and place Xs over 1 shaded whole and
12 shaded sections 5. divide models into fortieths; 15/40
+ 8/40 = 23/40 6. divide models into twentieths; 2 5/20 +
1 8/20 = 3 13/20 7. 1 1/12; shade 1 whole and 4 sections
and place Xs over 3 shaded sections 8. 1 7/12; shade 4
wholes and 9 sections and place Xs over 3 shaded
wholes and 2 shaded sections B. 9. 1 4/30 = 1 2/15
2
2
References
* All Web sites listed were active at time of publication.
Adams, T. (2003). Reading mathematics: More than words can say. Reading Teacher, 56, 786–795.
Aiken, L. R. (1972). Language factors in learning mathematics. Review of Education Research, 42(3),
359–385.
Allington, R. L., & Johnston, P. H. (2002). Reading to learn: Lessons from exemplary fourth-grade
classrooms. New York: Guilford.
Barnes, D. (1976/1992). From communication to curriculum. London: Penguin. (2nd ed., 1992,
Portsmouth, NH: Boynton/Cook-Heinemann.)
Block, C. C., & Parris, S. R. (Eds.). (2008). Comprehension instruction: Research-based best practices
(2nd ed.). New York: Guilford Press.
Brummett, B. (2010). Techniques of close reading. Thousand Oaks, California: SAGE Publications.
Butler, D. L., & Winnie, P. H. (1995). Feedback and self-regulated learning: A theoretical synthesis.
Review of Educational Research, 65(3), 245–281.
Chapin, S. H., O’Connor, C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to help
students learn (2nd ed.). Sausalito, CA: Math Solutions.
Fennema, E., & Romberg, T. (Eds.). (1999). Mathematics classrooms that promote understanding.
Mahwah, NJ: Lawrence Erlbaum Associates.
Fosnot, C. T. (Ed.). (1996). Constructivism: Theory, perspectives, and practice. New York: Teachers College
Press.
Fosnot, C. T. (2005). Constructivism revisited: Implications and reflections. The Constructivist, 16(1).
Fraivilig, J., Murphy, L. A., & Fuson, K. (1999). Advancing children’s mathematical thinking in everyday
mathematics classrooms. Journal for Research in Mathematics Education, 30(2), 148–170.
Grouws, D. A. (2004). Chapter 7: Mathematics. In G. Cawelti (Ed.), Handbook of research on improving
student achievement (3rd ed.). Arlington, VA: Educational Research Service.
Harmon, J., Hedrick, W., & Wood, K. (2005). Research on vocabulary instruction in the content areas:
Implications for struggling readers. Reading & Writing Quarterly, 21, 261–280.
Harvey, S., & Daniels, H. (2009). Comprehension and collaboration: Inquiry circles in action. Portsmouth,
NH: Heinemann.
Hattie, J., & Timperley, H. (2007, March). The power of feedback. Review of Educational Research, 77(1),
81–112.
Herbel-Eisenmann, B., & Cirillo, M. (Eds.). (2009). Promoting purposeful discourse. Reston, VA: NCTM.
Hess, K. K. (2006). Exploring cognitive demand in instruction and assessment. Retrieved
from National Center for the Improvement of Educational Assessment (NCIEA) Web site:
http://www.nciea.org/publications/DOK_ApplyingWebb_KH08.pdf
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., Olivier, A., & Human, P.
(1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH:
Heinemann.
34
11
Selected pages from
Student Work Text
Mathematics
Grade 5, Book II
Numbers and Operations—Fractions
Lori Mammen
Editorial Director
ISBN: 978-1-60539-876-1
Copyright infringement is a violation of Federal Law.
©2014 by ECS Learning Systems, Inc., Bulverde, Texas. All rights reserved. No part of this publication may be reproduced,
translated, stored in a retrieval system, or transmitted in any way or by any means (electronic, mechanical, photocopying,
recording, or otherwise) without prior written permission from ECS Learning Systems, Inc.
Reproduction of any part of this publication for an entire school or for a school system, by for-profit institutions and tutoring
centers, or for commercial sale is strictly prohibited.
Printed in the United States of America.
Disclaimer Statement
ECS Learning Systems, Inc. recommends that the purchaser/user of this publication preview and use his/her own judgment
when selecting lessons and activities. Please assess the appropriateness of the content and activities according to grade level
and maturity of your students. The responsibility to adhere to safety standards and best professional practices is the duty of
the teachers, students, and/or others who use the content of this publication. ECS Learning Systems is not responsible for
any damage, to property or person, that results from the performance of the activities in this publication.
12
Number and Operations–Fractions...................................................................................3
Mathematics Vocabulary...................................................................................................145
P. O. Box 440
Bulverde, TX 78163-0440
ecslearningsystems.com
1.800.688.3224 (t)
1.877.688.3226 (f )
[email protected]
2
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 5—Book II
13
Number and Operations–Fractions
5.NF—Use equivalent fractions as a strategy to add and subtract fractions
1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing
given fractions with equivalent fractions in such a way as to produce an equivalent sum or
difference of fractions with like denominators.
2. Solve word problems involving addition and subtraction of fractions referring to the same
whole, including cases of unlike denominators, e.g., by using visual fraction models or
equations to represent the problem. Use benchmark fractions and number sense of fractions
to estimate mentally and assess the reasonableness of answers.
5.NF—Apply and extend previous understandings of multiplication and division to multiply and
divide fractions
3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word
problems involving division of whole numbers leading to answers in the form of fractions or
mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
4. Apply and extend previous understandings of multiplication to multiply a fraction or whole
number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently,
as the result of a sequence of operations a × q ÷ b.
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the
appropriate unit fraction side lengths, and show that the area is the same as would be
found by multiplying the side lengths. Multiply fractional side lengths to find areas of
rectangles, and represent fraction products as rectangular areas.
5. Interpret multiplication as scaling (resizing) by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the
other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a
product greater than the given number (recognizing multiplication by whole numbers
greater than 1 as a familiar case); explaining why multiplying a given number by a fraction
less than 1 results in a product smaller than the given number; and relating the principle
of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
6. Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by
using visual fraction models or equations to represent the problem.
7. Apply and extend previous understandings of division to divide unit fractions by whole
numbers and whole numbers by unit fractions.
a. Interpret division of a unit fraction by a non-zero whole number, and compute such
quotients.
b. Interpret division of a whole number by a unit fraction, and compute such quotients.
c. Solve real-world problems involving division of unit fractions by non-zero whole
numbers and division of whole numbers by unit fractions, e.g., by using visual fraction
models and equations to represent the problem.
Note: The Common Core State Standards (CCSS) identify developing fluency with addition and
subtraction of fractions and developing understanding of the multiplication of fractions and of division
of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by
unit fractions) as one of three critical areas of instruction for Grade 5.
14
3
Standard 5.NF.1 (L–M)
Subtracting Mixed Numbers
Sometimes fractions appear as mixed numbers. It is important to understand the difference
between whole numbers and fractional numbers. Look at the example below.
2 –1—
1
3—
5
3
–
The fraction model above makes it easy to subtract the whole numbers (3 – 1 = 2). However,
the fractions do not have common denominators. We need to find a common denominator,
as shown below.
3 2x3 – 1 1x5 =
5x3
3x5
6 – 1—
5 =
3—
15
15
1
2—
15
On Your Own: Draw fraction models to represent the problem-solving steps above.
18
15
Standard 5.NF.2 (M–H)
Problem Solving III
Directions: Solve each problem below. Show all of your work.
1 feet wide. The windows
1. The windows on the bottom floor of a building are each 4 —
2
3 feet wide. How much wider are the windows on the
on the second floor are each 4 —
4
second floor than on the first floor?
Answer: _______________________
2. Mr. Mason is cutting trim to put along the door in his bedroom. The length of trim
1 feet. Mr. Mason accidentally cuts a piece of trim measuring 12 —
5 feet.
needed is 12 —
3
6
How much does Mr. Mason need to cut from the trim so it will fit along the door?
Answer: _______________________
3 gallons of paint on the inside of
3. Dylan and Joyce are painters. Today, Dylan used 8 —
8
1 gallons of paint on the outside of the house. How much
the house. Joyce used 7 —
3
more paint did Dylan use than Joyce?
Answer: _______________________
continue to next page
16
31
Problem Solving V
Directions: Read each problem below, and answer the questions. Use the benchmark
1 to explain your response. You do NOT have to compute the answer. The first
fraction of —
2
one is completed for you.
1 full of medicine A and —
3 full with medicine B. Will
1. Nurse Sandra filled a syringe —
3
7
both medicines fit into one syringe at the same time?
1 is less than half and —
3 is slightly
yes, because —
3
7
Explanation: ____________________________________________________________
1
3 <1
less than half: —+ —
3 7
_______________________________________________________________________
2 full of bandages. Later, he adds alcohol swabs to
2. Dr. Nelson fills his surgeon’s bag —
8
2 of the bag. Does Dr. Nelson have any room left in his surgeon’s bag?
fill —
5
Explanation: ____________________________________________________________
_______________________________________________________________________
3. A pharmacist wants to combine the contents of two opened bottles of cough syrup.
3 full, and the other bottle is —
2 full. Can the pharmacist combine the
One bottle is —
5
3
two amounts of cough syrup into one bottle?
Explanation: ____________________________________________________________
_______________________________________________________________________
38
17
Standard 5.NF.3 (M)
What exactly IS a fraction?
You already know that a fraction has a numerator and denominator. You also know how to
identify, write, convert, add, subtract, and multiply fractions. You know a lot about fractions!
It’s also important to know that fractions relate to division. A fraction is the division of the
numerator by the denominator, as shown below.
a = a ÷ b
—
b
1
If you divide a objects equally among b shares, each of the a objects should contribute —
b
of itself to each share. In other words, you are dividing a specific amount of objects (a) into
There are different ways to show a fraction as division. Let’s take a look at these.
equally sized portions (b).
Example #1
Melinda’s mother bought 5 papayas at the local market. She wants to give each of her
3 daughters equal amounts of the papaya. How much papaya will each daughter receive?
5 papayas divided equally among 3 daughters
Daughter 1
Daughter 2
Daughter 3
1
2
3
1
2
3
As you can see from the model above, each daughter will receive 1 whole papaya. The
2 .
remaining two papayas are divided into thirds, and each daughter will receive —
3
2 papayas.
Therefore, each daughter will receive 1 —
3
42
18
Make It Match
Directions: Write or draw the missing information in the chart below. The first row is
completed for you.
Word Problem
1.
If 4 men want
to share 3 pizzas
equally, how much
will each man get?
Explanation/
Numerical Expression
Fraction Model
When 3 wholes are
shared equally among
4 people, each person’s
3;—
3 = 3÷4
share is —
4 4
2.
3;—
3 = 3÷5
share is —
5 5
If 3 boys want to share
4 funnel cakes equally,
how much will each
boy get?
3.
4.
When 5 wholes are
When 3 wholes are
If 5 girls want to share
3 cookies equally, how
much will each girl get?
shared equally
between 2 people,
each person’s share is
5 , or 2 —
1;—
5 = 5÷2
—
2
2 2
If 4 farmers want to
share 15 bales of hay
equally, how many
bales will each farmer
get?
When 15 wholes are
5.
3;
share is 15
—, or 3 —
4
15
— = 15 ÷ 4
4
4
51
19
Standard 5.NF.4 (L)
Multiplying Fractions I
You already know how to multiply fractions by a whole number. For example, you know that
3 can be represented as 3 x —
1.
—
4
4
Think About It: How can the multiplication of a fraction by a whole number be represented
as repeated addition of that fraction?
Now, we will look at other ways to represent the multiplication of fractions by a whole
number. Look at the example below.
3 ? In other words, what is —
3 of 3 wholes?
What is 3 x —
4
Let’s begin with 3 whole crackers, and then divide each cracker into fourths.
4
3 of each cracker.
Shade —
4
Count how many fourths have been shaded to find the answer.
3 =—
9 OR 2 —
1
3x—
4 4
4
If you cannot use a model, you can still find the answer using the method below.
(Note: q = whole number quantity)
Expression
Example
Improper Fraction
Mixed Number
a = qxa
qx—
b
b
3 = 3x3
3x—
4
4
9
—
4
1
2—
4
3 . How many different ways can it be written?
On Your Own: Consider this expression: 6 x —
5
List these ways on a separate sheet of paper.
56
20
Standard 5.NF.4 (M)
Fractions, Fractions Everywhere
Directions: Read each expression below, and shade the models to represent the expressions.
Then, solve each problem. The first one is completed for you.
1 x—
1
1. —
2
3
1
—
6
Answer: _________________
2 x—
3
2. —
3
4
Answer: _________________
1 x—
2
3. —
4
3
Answer: _________________
3 x—
4
4. —
4
6
Answer: _________________
1 x—
2
5. —
3
5
Answer: _________________
64
21
Cover It Up II
Directions: Write an expression to represent each model. Then, solve the expression. The first
one is completed for you.
4.
1.
1
1
1
1
2 x—
3 =—
6 OR —2
—
3
5
15
5
_______________________________
2.
_______________________________
5.
=1
=1
_______________________________
_______________________________
6.
3.
1
1
1
_______________________________
76
1
_______________________________
22
Problem Solving X
Directions: Using the information given, determine the value of the missing side (s) for each
item below. The first one is completed for you.
3 = 2 —1 x s
3—
4
2
3 square units
1. Area = 3 —
4
1
2—
2
s
3 OR 1 —1
—
2
2 __
Answer: _____________________
15 = —
5 xs
—
4
2
What number times 5 equals 15? 3
What number times 2 equals 4? 2
1 square units
2. Area = 10 —
12
2
3—
3
s
Answer: _______________________
5 square units
3. Area = 5 —
6
s
2
1—
3
Answer: _______________________
110
23
Dividing Fractions II
You have already learned about dividing a unit fraction by a whole number. Now, we will look
at dividing a whole number by a unit fraction. Read the problem below.
Thomas has 4 pounds of gumdrops. He wants to give each of his friends
1 of a pound of gumdrops. To how many friends can Thomas give the
—
3
gumdrops?
4
Look at the model below. It represents the 4 pounds of gumdrops Thomas has.
Look at the next model. It shows the gumdrops divided into thirds, which is how Thomas
wants to share them with his friends.
1
4÷—
3
Remember: One-third is a unit fraction. A unit fraction is a fraction with a numerator of 1.
It represents the basic unit of that fraction.
To solve the problem, you should count the number of thirds in the diagram. Thomas can
1 of a pound of gumdrops.
give each of 12 friends —
3
122
24
Problem Solving XII
Directions: Read and solve each problem. Show all of your work.
1 -cup servings are in 3 cups of sugar?
1. How many —
3
Answer: _______________________
1 -inch slices can be cut from a loaf of banana bread that is 12 inches long?
2. How many —
3
Answer: _______________________
1 -cup servings are in 5 cups of butter?
3. How many —
4
Answer: _______________________
1 of a bag of flour to make each batch of cookies. If he has 8 bags of
4. Gerardo uses —
3
flour, how many batches of cookies can he make?
Answer: _______________________
132
25
Problem Solving XIII
Directions: Read each problem below. Complete the model that best represents the
scenario. Then, solve each problem. Bonus: For each problem, write the multiplication
equation that proves your answer. Use the example below to help you solve these problems.
Example
1 of a fruitcake to serve to her 4 grandchildren. If each child gets an
Grandma had —
2
equal share, what part of the fruitcake will each receive?
1
1
—
2
1
—
8
1
—
8
1
—
8
1 ÷4=—
1
—
2
8
Answer: ___________________________
1
—
8
1
—
8
1
—
8
1
—
8
1
—
8
1
—
2
1 X4= —
4 =—
1
—
8
8 2
Bonus: ___________________________
1 of a box of chocolates equally among themselves. What part of the
1. Five friends divide —
3
whole box will each friend receive?
Answer: ___________________________ Bonus: ___________________________
26
135
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Sample Booklet Mathematics — Book II Grade 5 All

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