Sample Booklet Mathematics — Book II Grade 6 All

Transcription

Sample Booklet
Grade 6
Mathematics — Book II
Expressions and Equations
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 6, Book II
This page may not be reproduced.
Teacher Guide
Lori Mammen
Editorial Director
ISBN: 978-1-60539-915-7
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.
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© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 6—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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Blackline Masters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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 6—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 15–45 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
4
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
What You Need to Know: 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).
Try It: Students are asked to try a guided example. Teachers can present the guided
example in a whole-class or small-group setting. Teachers should engage students in
“math-talk” during these examples (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).
4
5
Descriptions of TestSMART® Complexity Levels
The following descriptions provide an overview of the three complexity levels used to
align the TestSMART® Common Core Student Work Text items to the Common Core State
Standards (CCSS) for Mathematics (NGA/CCSSO, 2010b). Each explanation details the
kinds of activities that occur within each level. However, they do not represent all of the
possible thought processes for each level.
Low Complexity
Low Complexity (L)
Low-complexity items align with
What did you say?
the CCSS at Level 1 of the Webb
Directions: Match each numerical form on the left with the correct written form on the right.
Use the box at the bottom of the page to show your work, if necessary.
(2002a) model. Activities and
a. two to the sixth power
_____ 1. 6
problems at this level require
_____ 2. 3 x 3 x 3
b. three squared
routine, single-step methods.
c. six squared
_____ 3. 5
_____ 4. 3
d. five to the seventh power
An item may ask students to
e. seven to the first power
_____ 5. 7 x 7 x 7
recognize or restate a fact,
_____ 6. 2 x 2 x 2 x 2 x 2 x 2
f. three to the sixth power
definition, or term. For example,
g. six cubed
_____ 7. 5
students may need to identify
8. 6
h. three cubed
_
9. 7
i. seven cubed
_
the attributes of a geometric figure.
3
j. seven squared
_
Items of this complexity may require students
to
follow
a
basic
procedure
with clearly
3x3x3x3x3x3
k. three to the seventh power
_
defined steps. At this cognitive level, students
may
need
to
apply
a formula or perform
l. five cubed
7
_
a simple algorithm. Some major concepts represented at this level include arithmetic
facts, perimeter, and converting units of measure. A low-complexity item may ask
students to identify, recognize, use, or measure information and concepts.
Standard 6.EE.1 (L)
2
7
7
ced.
3
3
2
T
2
1
Moderate Complexity
Moderate Complexity (M)
Moderate-complexity items
Finding Area With the Distributive Property
align with the CCSS at Level 2
Directions: Write an expression to represent the area of each rectangle below. Then, rewrite
the expression using the distributive property. Remember to simplify both expressions. The
of the Webb model. Items of
first one is completed for you.
moderate complexity involve
1.
x
12
both comprehension and the
7
subsequent processing of
7(12 + x)
l x w: ___________________________
information. Activities at this
7x + 84
Distributive Property: ______________________
level demand more than one
2.
x+4
4x
step in the reasoning process.
Students are asked to determine
how to best solve the problem. An item may ask students to generate a table of paired
numbers based on a real-life situation. Items may
involve using a model to solve a
3.
6
problem. At this cognitive level, students will need to
visualize for tasks such as
extending patterns and determining nonexamples. Items may involve interpreting
l
information from a simple graph, table, or diagram. Some major
concepts represented
at this level include classifying geometric figures
and
using
strategies
to estimate. Items
4.
x
of this complexity may ask students to classify, organize, observe, collect and display
data, or compare data. Some items also require students to apply low-complexity skills
l
and concepts.
Standard 6.EE.3 (M)
T
This p
T
6
7
High Complexity
High Complexity (H)
High-complexity items align
What’s the problem?
with the CCSS at Level 3 and/or
Directions: Read each item below. Then, write a problem situation to describe each
equation. You DO NOT have to solve the problem. The first one is completed for you.
*
4 of the Webb model. Items of
1. 2g + 4 = 12
g represents the number of grandchildren
high complexity require students
Mrs.
Chang
made
12
waffles at breakfast this morning. She
________________________________________________________________________
to use strategic, multi-step thinking;
made 4 waffles for her husband and 2 waffles for each of her
________________________________________________________________________
develop a deeper understanding
grandchildren ( g ). How many grandchildren does Mrs. Chang
________________________________________________________________________
of the information; and extend
have?
________________________________________________________________________
2. 160 – c = 33.75
c represents the cost of an item
thinking. The problems at this level
________________________________________________________________________
are non-routine and more abstract.
________________________________________________________________________
Students are asked to demonstrate
________________________________________________________________________
more flexible thinking, apply prior knowledge, ________________________________________________________________________
make and test conjectures, and support
their responses. High-complexity items may require students to make generalizations
________________________________________________________________________
from patterns. Items may involve interpreting information
from a complex graph, table,
________________________________________________________________________
or diagram. At this cognitive level, students must
justify the reasonableness of a
________________________________________________________________________
solution process when more than one solution________________________________________________________________________
exists. Students will use concepts to
solve and explain problems, such as how changes in dimensions affect the volume of
________________________________________________________________________
a figure. A high-complexity item may ask students
to plan, reason, explain, compare,
________________________________________________________________________
differentiate, draw conclusions, cite evidence, analyze,
synthesize, apply, or prove.
________________________________________________________________________
Some items also require students to apply low-________________________________________________________________________
and/or moderate-complexity skills
and concepts.
Standard 6.EE.6 (H)
1
—
4
This p
c
T
* Note: Although the CCSS or state standards may include expectations that require extended thinking,
many large-scale assessment activities are not classified as Level 4. Performance and open-ended
assessment may require activities at Level 4.
8
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-three 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
8
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)
Answer Key
10
Thi
pp. 4–5
Try It: 2. 5 x 5 x 5 x 5; 625 3. 6 x 6 x 6; 216 4. 3 x 3 x 3 x 3 x 3 x
3; 729 5. 1 x 1; 1 6. 2; 2 Working Together: 1. four to the fifth
power 2. five to the fourth power 3. six to the third power
OR six cubed 4. three to the sixth power 5. one to the
second power OR one squared 6. two to the first power
p. 6
2. 62; 2; 36 3. 7; seven cubed OR seven to the third power; 7 x
7 x 7 4. 4; 4; 256 5. 25; 2; two to the fifth power 6. 9; 5; 9 x 9 x
9 x 9 x 9; 59,049 7. 83; 3; 512 8. 12; 4; twelve to the fourth
power; 12 x 12 x 12 x 12; 20,736 9. 11; eleven squared OR
eleven to the second power; 11 x 11 10. 94; 9; 4; nine to the
fourth power; 6,561
p. 7
A. 1. 3 2. 4 3. 6 4. 2 5. 3 6. 2 7. 4 8. 1 9. 3 10. 8 B. 11. 3 12. 8
13. 5 14. 2 15. 12 16. 9 17. 7 18. 6 19. 4 20. 11
p. 8
1. c 2. h 3. d 4. k 5. i 6. a 7. l 8. g 9. j 10. b 11. f 12. e
0
pp. 14–15
Talk About It: Answers may vary. Order of operations is
a set of rules that describes the correct sequence to use
when evaluating expressions and solving equations. If you
do not perform the operations in the correct order, you will
not find the correct answer. Try It: 2. a. 33 b. (27 –10) c. (2 x
7) d. 142 e. 17 x 196; Answer: 3,332 3. a. 32 b. (9 x 9) c. [85 –
81] d. {7 + 4} e. 23 f. 10 x 11 g. 110 + 8; Answer: 118 4. a. 122
b. (144 ÷ 48) c. 62 d. [3 + 36] e. {4 x 39} f. 2 x 156 g. 312 ÷ 8
h. 75 – 39; Answer: 36
p. 16
1. 17 2. 335 3. 13 4. 516 5. 39 6. 131 7. 150 8. 168 9. 61 10. 51
11. 205 12. 1 13. 47 14. 6 15. 55/66 OR 5/6
p. 17
Working Together: Answers will vary. Examples: five times
the difference of some number and seven OR five multiplied
by some number minus seven
p. 18
1. (n + 2) + 2 2. n/3 – 5 3. n/6 + 5 4. 8(n – 8) 5. 6 + 3n 6. 2(n +
3) 7. 8/n – 5 8. 4 – 2n 9. 3 + 1/2n3 10. 8 – 2n 11. n/3 – 6 12. n2
TH A
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.
36
11
Selected pages from
Student Work Text
Mathematics
Grade 6, Book II
Lori Mammen
Editorial Director
ISBN: 978-1-60539-906-5
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
Expressions and Equations ...................................................................................................3
Reference Materials .............................................................................................................124
Mathematics Vocabulary...................................................................................................127
Scratch Paper ..........................................................................................................................128
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 6—Book II
13
6.EE—Apply and extend previous understandings of arithmetic to algebraic expressions
1. Write and evaluate numerical expressions involving whole-number exponents.
2. Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing
for numbers.
b. Identify parts of an expression using mathematical terms (sum, term, product,
factor, quotient, coefficient). View one or more parts of an expression as a single
entity.
c. Evaluate expressions at specific values of their variables. Include expressions that
arise from formulas used in real-world problems. Perform arithmetic operations,
including those involving whole-number exponents, in the conventional order
when there are no parentheses to specify a particular order (Order of Operations).
3. Apply the properties of operations to generate equivalent expressions.
4. Identify when two expressions are equivalent (i.e., when the two expressions name
the same number regardless of which value is substituted into them).
6.EE—Reason about and solve one-variable equations and inequalities
5. Understand solving an equation or inequality as a process of answering a question:
Which values from a specified set, if any, make the equation or inequality true? Use
substitution to determine whether a given number in a specified set makes an
equation or inequality true.
6. Use variables to represent numbers, and write expressions when solving a real-world
or mathematical problem. Understand that a variable can represent an unknown
number, or, depending on the purpose at hand, any number in a specified set.
7. Solve real-world and mathematical problems by writing and solving equations of the
form x + p = q and px = q for cases in which p, q, and x are all non-negative rational
numbers.
8. Write an inequality of the form x > c or x < c to represent a constraint or condition in
a real-world or mathematical problem. Recognize that inequalities of the form x > c
or x < c have infinitely many solutions. Represent solutions of such inequalities on
number line diagrams.
6.EE—Represent and analyze quantitative relationships between dependent and
independent variables
9. Use variables to represent two quantities in a real-world problem that change in
relationship to one another. Write an equation to express one quantity, thought
of as the dependent variable, in terms of the other quantity, thought of as the
independent variable. Analyze the relationship between the dependent and
independent variables using graphs and tables, and relate these to the equation.
Note: The Common Core State Standards (CCSS) identify writing, interpreting, and using
expressions and equations as one of four critical areas of instruction for Grade 6.
14
3
Standard 6.EE.1 (L–M)
The Exponent’s Base
So far, your work with exponents has included only bases that are whole numbers. However,
bases can also be fractions and decimals.
In the expression below, the base is a fraction. The fraction is placed in parentheses to show
that the entire number is squared.
2 )2
(—
3
2 )2 is read as “two-thirds squared” or “two-thirds to the second power.”
The expression (—
3
2 x—
2.
It means —
3 3
2x2
4
=—
9
3x3
2 )2 = —
2 x—
2 =
(—
3
3
3
2
2 )2 with —
2 or —
2.
Do not confuse (—
2
3
3
3
3
3
3
2
2 )2, —
2 , and —
2 are not the same. Show all of your work.
Try It: In the box below, prove that (—
2
In the expression below, the base is a decimal.
0.43
The expression 0.43 is read as “four-tenths cubed” or “four-tenths to the third power.” It means
0.4 x 0.4 x 0.4.
0.43 = 0.4 x 0.4 x 0.4 = 0.064
continue to next page
9
15
Standard 6.EE.1 (M)
Exponents & Order of Operations
When trying to solve an expression, it is important to remember order of operations.
Talk About It: Define order of operations, and explain why it is important.
You learned about order of operations in fifth grade, but you did not work with exponents.
The list below shows where exponents occur in order of operations.
Order of Operations
1—any operation inside parentheses
2—any operation inside brackets
3—any operation inside braces
4—any exponents from left to right
5—multiplication and division worked left to right
6—addition and subtraction worked left to right
What You Need to Know: In fifth grade, you learned that parentheses, brackets, and
braces are punctuation symbols used to group things. Parentheses look like this: ( ). Brackets
look like this: [ ]. Braces look like this: { }.
Look at the example below. It shows the correct order in which to perform the operations.
2 + (32 x 4) = n
First, compute the exponent inside the parentheses.
2 + (32 x 4) = n
32 = 9
Second, multiply inside the parentheses.
2 + (9 x 4) = n
9 x 4 = 36
Finally, add.
2 + 36 = n
n = 38
2 + (32 x 4) = 38
14
16
Standard 6.EE.2 (M)
Writing Expressions II
A. Directions: Write each algebraic expression below in word form. Be sure to include the
correct use of sum, difference, product, or quotient.
1. 6 + 4y
_______________________________________________________
2. 3(12 – f )
_______________________________________________________
3. 7s – 3
_______________________________________________________
4.
x+5
100
_______________________________________________________
6. (5 + b) ÷ 30
_______________________________________________________
7. (12 + h) – (2 + a)
_______________________________________________________
_______________________________________________________
5. 10(m + 5)
B. Directions: Write each phrase below as an algebraic expression.
8. the product of five and the sum of some number and fifteen ______________________
9. the difference between forty-seven and twice some number ______________________
10. the quotient of nine more than some number and three
______________________
11. the sum of four and the product of three and some number
______________________
12. the quotient of nine more than some number and twelve
______________________
13. the difference between nine and the product of two and
some number
______________________
14. the sum of eight and the product of nine and some number ______________________
21
17
Standard 6.EE.3 (M)
Building Blocks
Directions: Draw a model to represent each expression. Draw a second model that
redistributes the blocks using the distributive property. Then, write an equivalent expression
to represent the new model. The first one is completed for you.
New model
Original model
3(x + 3)
Equivalent Expression: ________________________________
1. 3x + 9
2. 4y + 12
3. 2x + 4y + 6
32
18
Standard 6.EE.3 (M)
Finding Area With the Distributive Property
Directions: Write an expression to represent the area of each rectangle. Then, rewrite the
expression using the distributive property. Remember to simplify both expressions. The first
one is completed for you.
1.
12
x
7(12 + x)
7
7x + 84
l x w: ___________________________
4x
x+4
2.
3x
l x w: ___________________________
3.
6y
2x
4y
l x w: ___________________________
4.
x+3
y+6
x
l x w: ___________________________
45
19
From Expressions to Equations
You learned about numerical and algebraic expressions in previous lessons. Now, you
will learn how to solve an equation. An equation is a statement that two mathematical
expressions are equal. Hint: You can tell the difference between an expression and an
equation by looking for an equal sign (=). An equal sign tells you that you are dealing with
an equation instead of an expression.
Look at the example below.
Joey had 26 baseball cards in a shoebox under his bed. His best friend gave
him more cards, and now Joey has 100 baseball cards. How many baseball
cards did Joey receive from his best friend?
“had 26 baseball cards”
“friend gave him more”
“now [he] has 100”
Important Words: First, identify the most important words. These words best describe the
situation and will help you solve the problem.
Variable: Next, choose a variable to represent the unknown number in the situation. Ask
yourself: “What am I trying to find?” or “What do I want to know?”
n = number of baseball cards
Expression: Next, form an algebraic expression by placing one of the known numbers and
the variable together. Be sure to use the correct operation based on what you want to find or
know. Hint: This is where the important words come in handy. They help you decide which
of the known numbers and which operation to use. “More” and “now he has” let you know
that you will be adding.
26 + n
Equation: Finally, form the equation by adding an equal sign and the other known number
to the expression. Since the quantities on both sides of an equal sign must be the same, the
variable will now have an exact answer.
26 + n = 100
Now that you have your equation, how can you solve it? Let’s look at some of the strategies
you can use.
20
59
Standard 6.EE.5 (M)
Diagram of an Inequality
Directions: Complete the chart for each of the following items. Show all of your work when
determining possible solutions.
1. Bouncy World charges a $50 flat rate for a party rental, plus $6.50 per person.
Mrs. Douglas only has $115 to spend on her son’s birthday party. How many people
can she invite to her son’s party without exceeding her limit?
Important Words
Variable
Inequality
Possible Solutions
NOT Possible Solutions
2. Juan wants to buy a model plane from a catalog. Each model plane costs $18.99, with
a flat rate of $9.99 shipping. Juan only has $100 to spend. How many model planes can
Juan buy without exceeding his limit?
Important Words
Solution
Variable
Inequality
Possible Solutions
NOT Possible Solutions
Solution
69
21
Standard 6.EE.6 (M–H)
Expressions at the Grocery Store
Directions: Read and respond to each item. The first one is completed for you.
1. Hope has three more than twice as many apples as Harmony. Write an expression to
represent the number of apples Hope has.
a = number of apples Harmony has
2a + 3
Expression: ______________________________
Variable and Description: __________________________________________________
2. A baker charges $25 per cake that he makes. He also charges $0.32 per letter for
messages written on the cake. Write an expression to represent the total cost of a
cake with a message written on it.
Expression: ______________________________
3. The deli charges $21 per ham. If a customer wants the ham sliced, the deli charges
$0.10 per slice. Write an expression to represent the total cost of a sliced ham.
Expression: ______________________________
4. Jaime has five more than three times as many loaves of bread as Justin. Write an
expression to represent the number of loaves of bread Jaime has.
Expression: ______________________________
5. Cheryl bought four less than five times as many cans of corn as Joaquin. Write an
expression to represent the number of cans of corn Cheryl bought.
Expression: ______________________________
78
22
Standard 6.EE.6 (M)
Problem Solving V
Directions: Read each item, and complete the chart that follows. The first one is started for
you.
1 times as much as Sarah, and Billy weighs 34 pounds more than
1. Mr. Henson weighs 3 —
2
Sarah. Write an expression to represent each person’s weight.
Sarah
Mr. Henson
Billy
Person
w
Weight
3 —21 w
Person
2. During a school fundraiser, Cybil sold 8 more boxes of chocolate than Jeff. Jeff sold
double the amount of boxes of chocolate that Greg sold. Write an expression to
represent each person’s sales.
Boxes of Chocolate Sold
3. Larry has $1 bills, $5 bills, and $20 bills in his wallet. He has 4 times as many $1 bills as
1 as many $20 bills as $5 bills. Write an expression to represent
he has $5 bills and only —
4
each type of bill in Larry’s wallet.
Type of Bill
Number of Bills
83
23
Standard 6.EE.9 (M)
Independent & Dependent Variables
You know that a variable is a symbol or letter that represents an unknown value in an
expression, equation, or inequality. Now, you will learn how to recognize the relationship
between two different variables in the same equation. The independent variable is the
variable that can be changed. The dependent variable is the variable that is affected by
the change in the independent variable. Note: To identify independent and dependent
variables, look closely at the equation. The dependent variable will almost always be isolated
on one side of the equal sign.
Look at the example below.
Gabriel earns $7 per hour mowing yards during the summer.
You can write an equation to express one quantity (the dependent variable) in terms of the
other quantity (the independent variable), as shown below.
m = 7h
In this situation, the amount of money Gabriel earns (m) depends on the number of hours he
works (h). The number of hours Gabriel works DOES NOT depend on the amount of money
he earns. Therefore, the dependent variable is money (m), and the independent variable is
hours (h).
Try It: Identify the independent and dependent variables in each scenario below.
1. the number of representatives from a state in the House of Representatives (r)
the state’s population (p)
Independent variable: _________
Dependent variable: _________
2. the number of raffle tickets purchased (t)
the chances of winning (c)
3. a toddler’s behavior (b)
the length of the toddler’s nap (n)
24
103
Standard 6.EE.9 (M–H)
Tabling the Variables II
Directions: Read each item, and write an algebraic equation to represent each. Then, create
a table to represent the values of the independent and dependent variables in each
equation. The first one is completed for you.
1. One-half of a number is equal to another number.
2
4
6
8
x
1
2
3
4
y
2
4
6
8
y
1
OR
2
3
4
1 x OR x = —
1y
y=—
2
2
x
Equation: ____________________________________
2. Five less than a number is equal to another number.
x
y
Equation: ____________________________________
3. A number is equal to two less than three times another number.
x
y
Equation: ____________________________________
110
25
Discrete vs. Continuous Data
When you graph equations, the data is not always connected with a line. Let’s examine why
this is sometimes true.
Data that contains distinct values is called discrete data. Discrete data is information that
cannot be represented with fractional parts (e.g., people, animals, cars, etc.). Data that
contains any values within a range is called continuous data. Continuous data is
information that can be represented with fractional parts (e.g., time, temperature, distance,
etc.). An easy way to think about this is to remember that discrete data is usually counted
and continuous data is usually measured.
Continuous data is graphed with a line connecting the points. For example, a graph showing
the time it takes to travel a certain distance would have a line connecting the points because
both time and distance could be represented with fractional parts. If all variables can be
represented in fractional parts, the data is continuous.
Discrete data is graphed with the coordinates only—the points are NOT connected with a
line. For example, a graph showing the cost of food per person would be graphed with
coordinates only (no line connecting them) since a fraction of a person would not be
considered. Even though the cost of food could be represented with fractional parts, people
cannot. If at least one variable must be represented in whole parts, the data is discrete.
On Your Own: Identify whether the data in each item below is discrete or continuous. Then,
circle the correct word to indicate if the graph of the data would have a connecting line.
1. the amount of time worked and the amount of money earned
Type of Data: _________________________
The graph
WOULD
WOULD NOT
have a connecting line.
2. the number of people at a campout and the number of tents needed
Type of Data: _________________________
The graph
116
WOULD
WOULD NOT
have a connecting line.
26
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