Download Mathematics, Grade 7 - ECS Learning Systems

Sample Booklet
Grade 7
Mathematics — Books I–III
Published by:
ECS Learning Systems, Inc.
We make teaching easier!SM
testsmart.com
A Research-Based Program for the CCSS
As educators, we take developing new content seriously. As publishers, we have
delivered quality and rigor in standards-based instructional, learning, and
assessment materials for more than two decades. Based on thorough research and
development, we present a Common Core series that meets the cognitive
demands of the new standards and the needs of your students in the classroom.
• Based on the Common Core State Standards (CCSS)
• Original content and strategies for instruction, learning, and assessment
• Focus on open-ended and extended-response items
Sample pages from Book I ......................................................................................................2–25
Sample pages from Book II ..................................................................................................26–49
Sample pages from Book III ..................................................................................................50–75
Selected pages from
Student Work Text
Mathematics
Grade 7, Book I
This page may not be reproduced.
Ratios and Proportional Relationships
The Number System
Expressions and Equations
Lori Mammen
Editorial Director
ISBN: 978-1-60539-908-9
Copyright infringement is a violation of Federal Law.
©2015 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.
Photocopying of student worksheets by a classroom teacher at a non-profit school who has purchased this publication for
his/her own class is permissible. 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 7—Book I
Table of Contents
Section I
Ratios and Proportional Relationships .................................................................................3
Section II
The Number System..................................................................................................................63
Section III
Expressions and Equations ..................................................................................................107
Reference Materials .............................................................................................................149
This page may not be reproduced.
Mathematics Vocabulary...................................................................................................153
Teacher Guide (with Comprehensive Answer Key) ..............................................155
The Teacher Guide section contains a “How to Use the Student Work Text” section, an
explanation of the Common Core State Standards, a mathematics vocabulary section, a
master skills list, and much more. See page 155 for a complete list.
ECS Learning Systems, Inc.
P. O. Box 440, Bulverde, TX 78163-0440
ecslearningsystems.com • 1.800.688.3224 (t)
2
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
3
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
Section I—Ratios and Proportional Relationships
Section I—Ratios and Proportional Relationships
7.RP—Analyze proportional relationships, and use them to solve real-world and
mathematical problems
1. Compute unit rates associated with ratios of fractions, including ratios of lengths,
areas, and other quantities measured in like or different units.
2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by
testing for equivalent ratios in a table or graphing on a coordinate plane and
observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations,
diagrams, and verbal descriptions of proportional relationships.
This page may not be reproduced.
c. Represent proportional relationships by equations.
d. Explain what a point (x, y) on the graph of a proportional relationship means
in terms of the situation, with special attention to the points (0, 0) and (1, r)
where r is the unit rate.
3. Use proportional relationships to solve multi-step ratio and percent problems.
Note: The Common Core State Standards (CCSS) identify developing understanding
of and applying proportional relationships as one of four critical areas of instruction
for Grade 7.
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
4
© ECS Learning Systems, Inc.
3
TestSMART® Common Core Sample Booklet
Section I—Ratios and Proportional Relationships
Mathematics, Grade 7—Book I
Standard 7.RP.1 (M)
Finding Unit Rates
Directions: Find the unit rate in each problem below. Show all of your work.
1 of a wall in —
1 hour. At
1. Albert paints —
2
4
that rate, how many walls can he paint
1 mile in —
1 hour. How far
4. Ellen walks —
3
8
can she walk in 1 hour?
in 1 hour?
Answer: _________________________
4
3 gallon of water to fill —
1
5. Greg needs —
8
6
1 of a fence. How much paint did she
—
5
of an aquarium. How much water does
need to cover the whole fence?
he need to fill the entire aquarium?
Answer: _________________________
Answer: _________________________
3 teaspoon of vanilla to
3. Juan uses —
4
1 dozen cookies. How much
make —
2
This page may not be reproduced.
3 gallon of paint to cover
2. Emma used —
Answer: _________________________
2 mile in —
1 hour. At that
6. Casey can run —
3
4
rate, how far can she run in 1 hour?
vanilla would he use to make 1 dozen
cookies?
Answer: _________________________
6
TestSMART® Common Core Student Work Text
Answer: _________________________
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
5
TestSMART® Common Core Sample Booklet
Section I—Ratios and Proportional Relationships
Mathematics, Grade 7—Book I
Standard 7.RP.2 (L–M)
Writing Proportional Equations
You can write equations to represent proportional relationships. You know that proportional
relationships use the form y = kx, where k is the constant of proportionality (unit rate) and x
and y are two quantities.
Read the word problem below.
The cost of a pizza varies directly with the number of slices it has. Each slice
of pizza costs $1.15. What equation can be used to determine the total cost
of a pizza?
This page may not be reproduced.
When we write an equation, we can choose variables more closely matched to the problem
(instead of x and y). For this problem, we will replace x with s for the number of slices in a
pizza and replace y with c for the total cost of a pizza.
The total cost of the pizza will be k times the number of slices in the pizza. We already know
that each slice of pizza costs $1.15, so the value of k is 1.15.
k = 1.15
The total cost of the pizza will be 1.15 times the number of slices in the pizza.
c = ks
If a pizza has 6 slices, the equation looks like this.
c = 1.15 x 6
c = 6.9
The total cost of a pizza with 6 slices is $6.90.
continue to next page
20
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
6
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
Section I—Ratios and Proportional Relationships
Try It: Read the word problem below, and think about how you would solve it. Then,
complete the items that follow.
Mr. Miller traveled 186 miles in 3 hours. If he drove at a constant rate, what
equation can be used to represent how far Mr. Miller drove?
a. How is this problem different from the word problem on page 20?
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
This page may not be reproduced.
b. Write a variable to represent the amount of time Mr. Miller drove. _______________
Write a variable to represent the distance Mr. Miller drove. _______________
c. How can you determine the constant of proportionality?
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
d. What is the constant of proportionality for this problem? _______________
e. Write an equation to represent this problem. ______________________________
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
21
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
7
TestSMART® Common Core Sample Booklet
Section I—Ratios and Proportional Relationships
Mathematics, Grade 7—Book I
Standard 7.RP.3 (M)
Percent Error
Percent error problems represent another type of proportional relationship. Percent error
is an expression of error (or deviation) between estimated and exact values, expressed as a
percent of the exact value. The formula to find percent error is shown below.
% error =
| estimated value – actual value |
x 100%
actual value
The following steps explain how to solve percent error problems.
1. Subtract the actual value from the estimated value.
2. Find the absolute value of the difference from Step 1. (Remember: Absolute value is a
number’s distance from zero on a number line.)
This page may not be reproduced.
3. Divide the result from Step 2 by the actual value.
4. Multiply the quotient from Step 3 by 100 to find the percent.
Try It: Read the problem below, and think about how you would solve it. Then, complete the
items that follow. Show all of your work.
Kurt estimated the length of a bedroom wall as 6 feet. The actual length of
the wall was 4.5 feet. What was Kurt’s percent error?
1. Subtract the actual value from the estimated value. ____________________
2. Find the absolute value of the difference from Step 1. ____________________
3. Divide the result from Step 2 by the actual value. ____________________
4. Multiply the quotient from Step 3 by 100%. ____________________
Did you notice that finding percent error is similar to finding percent increase and percent
decrease?
42
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
8
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
Section II—The Number System
Section II—The Number System
7.NS—Apply and extend previous understandings of operations with fractions
to add, subtract, multiply, and divide rational numbers
1. Apply and extend previous understandings of addition and subtraction to
add and subtract rational numbers. Represent addition and subtraction on
a horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0.
b. Understand p + q as the number located a distance |q| from p, in the
positive or negative direction depending on whether q is positive or
negative. Show that a number and its opposite have a sum of 0 (are
additive inverses). Interpret sums of rational numbers by describing
real-world contexts.
This page may not be reproduced.
c. Understand subtraction of rational numbers as adding the additive
inverse, p – q = p + (-q). Show that the distance between two rational
numbers on the number line is the absolute value of their difference,
and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational
numbers.
2. Apply and extend previous understandings of multiplication and division
and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational
numbers by requiring that operations continue to satisfy the properties of
operations, particularly the distributive property, leading to products such
as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret
products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not
zero, and every quotient of integers (with non-zero divisor) is a rational
number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret
quotients of rational numbers by describing real-world contexts.
c. Apply properties of operations as strategies to multiply and divide rational
numbers.
d. Convert a rational number to a decimal using long division. Know that the
decimal form of a rational number terminates in 0s or eventually repeats.
3. Solve real-world and mathematical problems involving the four operations
with rational numbers.
Note: The Common Core State Standards (CCSS) identify developing understanding
of operations with rational numbers as one of four critical areas of instruction for Grade 7.
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
63
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
9
TestSMART® Common Core Sample Booklet
Section II—The Number System
Mathematics, Grade 7—Book I
Standard 7.NS.1 (L)
Is that rational?
In the coming lessons, you will add, subtract, multiply, and divide rational numbers, but let’s
review what you know about rational numbers first.
Think About It: What do you remember about rational numbers?
What You Need to Know: A rational number is any number that can be written as a ratio,
a,-—
a ), as long as b ≠ 0. Rational numbers can be represented as a point on a
or fraction (—
b
b
number line. Rational numbers include fractions, decimals, and integers, which are whole
numbers and their opposites (…-3, -2, -1, 0, 1, 2, 3, …).
This page may not be reproduced.
Using formal mathematical language, we can say that a rational number is any number
p where p and q are integers and q is not equal to zero.
that can be written as —
q
The following list includes rational numbers.
1
—
4
0.5
7
1.89
-10
You can express each number in the list as a ratio.
1 =—
1
—
4
4
1
0.5 = —
2
7
7= —
1
1.89 = 189
—
100
-10 = -10
—
1
Talk About It: As a class, list some examples of rational and irrational numbers.
Try It: Complete the chart below by using a checkmark to identify each number as either
rational or irrational.
Number
Rational
Irrational
1. -0.85
6.
2. √3
7. 0.121133122133
3. 0.33
27
8. - —
4. -0.175
9. 4.26
5. √16
64
Number
TestSMART® Common Core Student Work Text
Rational
Irrational
3
9
10. —
2
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
10
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Section II—The Number System
Mathematics, Grade 7—Book I
Standard 7.NS.1 (L–M)
Adding Positive & Negative Numbers II
So far, you’ve used a number line to solve addition problems where the first addend was a
negative number and the second addend was a positive number. How would you use a
number line to solve the following problem?
4 + (-3) = ?
Again, you begin by locating the first addend (4) on the number line. This time, however, you
“add” the other number by moving to the left on the number line because you are “adding” a
negative amount.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
1
2
3
4
5
6
7
8
9 10
This page may not be reproduced.
3 2 1
You move three places to the left, and you find the answer.
4 + (-3) = 1
Working Together: Working with a partner, solve each problem below. You may use a
number line to help you.
1. 7 + (-5) = ____________
2. 4 + (-6) = ____________
3. 8 + (-2) = ____________
4. 2 + (-7) = ____________
5. 3 + (-3) = ____________
Think About It
• How does adding a negative number to a positive number differ from adding a positive
number to a negative number?
• Why are some of the solutions negative numbers while others are positive numbers?
• How does the last problem differ from the others?
68
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
11
TestSMART® Common Core Sample Booklet
Section II—The Number System
Mathematics, Grade 7—Book I
Standard 7.NS.1 (L)
Problem Solving IX
Directions: Solve each problem below. You may use the number line to help you.
This page may not be reproduced.
-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
1. 9 + (-6) =
____________
2. 4 + (-8) =
____________
3. 6 + (-10) =
____________
4. 1 + (-8) =
____________
5. 3 + (-2) =
____________
6. 4 + (-3) =
____________
7. 7 + (-7) =
____________
8. 5 + (-9) =
____________
9. 2 + (-5) =
____________
0
1
2
3
4
5
6
7
8
9
10 11 12
10. At 6 p.m., the outside temperature was 13 °F. By midnight, the outside temperature had
fallen by 5 °F. What was the temperature at midnight? ____________
11. Nathan scored 96 points on a math test, but then the teacher deducted 7 points because
he did not show his work. What was Nathan’s final score on the math test?
____________
12. An ill child has a fever of 103.8 °F. After taking medicine, the child’s temperature
decreases by 0.5 °F each hour. What is the child’s temperature after 3 hours?
____________
72
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
12
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
Section III—Expressions and Equations
Section III—Expressions and Equations
7.EE—Use properties of operations to generate equivalent expressions
1. Apply properties of operations as strategies to add, subtract, factor, and
expand linear expressions with rational coefficients.
2. Understand that rewriting an expression in different forms in a problem
context can shed light on the problem and how the quantities in it are
related.
7.EE—Solve real-life and mathematical problems using numerical and algebraic
expressions and equations
This page may not be reproduced.
3. Solve multi-step real-life and mathematical problems posed with positive
and negative rational numbers in any form (whole numbers, fractions, and
decimals), using tools strategically. Apply properties of operations to calculate
with numbers in any form, convert between forms as appropriate, and assess
the reasonableness of answers using mental computation and estimation
strategies.
4. Use variables to represent quantities in a real-world or mathematical problem,
and construct simple equations and inequalities to solve problems by
reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and
p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations
of these forms fluently. Compare an algebraic solution to an arithmetic
solution, identifying the sequence of the operations used in each
approach.
b. Solve word problems leading to inequalities of the form px + q > r or
px + q < r, where p, q, and r are specific rational numbers. Graph the
solution set of the inequality, and interpret it in the context of the
problem.
Note: The Common Core State Standards (CCSS) identify working with expressions
and linear equations as one of four critical areas of instruction for Grade 7.
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
107
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
13
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
Section III—Expressions and Equations
This page may not be reproduced.
On Your Own: Use the properties of operations in the table on page 108 to write an
equivalent expression for each item below. List each property you used to create the
equivalent expression. The first one is completed for you.
1. 4(9 + 6)
_________________________
(4 x 9) + (4 x 6)
_________________________
2. (16 x 8) x 6
_________________________
_________________________
3. 15 x 42
_________________________
_________________________
4. 12 + (28 + 26) _________________________
_________________________
5. 412 + 960
_________________________
_________________________
6. 7(10 + 15)
_________________________
_________________________
7. 3(3y + 2x)
_________________________
_________________________
8. (5y + 8) + 16
_________________________
_________________________
9. 12x + 44 + 2x
_________________________
_________________________
10. 8y + 20
_________________________
_________________________
11. 3 x 4y x 6
_________________________
_________________________
12. 12x + 16y + 8
_________________________
_________________________
© ECS Learning Systems, Inc.
Distributive Property
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
14
© ECS Learning Systems, Inc.
109
TestSMART® Common Core Sample Booklet
Section III—Expressions and Equations
Mathematics, Grade 7—Book I
Standard 7.EE.3 (M–H)
Judging Reasonable Answers
After solving a problem, you should always check that your answer is reasonable. A
reasonable answer makes sense. Why is checking for reasonableness an important step in
problem-solving? The ability to recognize reasonable answers shows that you understand a
problem and have used correct reasoning to solve it.
Talk About It–1: How do you check the reasonableness of your answers? Which method(s)
do you find most helpful? Why?
Talk About It–2: When you estimate an answer, you find an answer close to the exact
answer. How could you use estimation to determine the reasonableness of an answer?
This page may not be reproduced.
Working Together–1: Working with a classmate, list different ways to estimate the solution
to a math problem. You will share your list during a class discussion.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Estimation is a useful way to judge the reasonableness of a solution to a math problem. You
can estimate before you solve a problem and check your final answer against your original
estimation. You can also estimate after you solve a problem to judge the correctness of your
answer.
What You Need to Know: Estimation should never take the place of finding exact answers.
Estimation strategies simply provide useful ways to judge the reasonableness of answers.
Working Together–2: Working with a classmate, explain each estimation strategy listed
below. Provide an example of each strategy.
1. Rounding whole numbers
_________________________________________________________________________
_________________________________________________________________________
Example: _________________________________________________________________
continue to next page
128
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
15
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
Section III—Expressions and Equations
Standard 7.EE.4 (M–H)
Problem Solving With Inequalities
Directions: Read each problem below. Choose a variable, and briefly explain the unknown
quantity it represents. Write an inequality to represent the problem situation, and solve the
inequality. Show all of your work. The first one is started for you.
1. Patti earns $2.70 more per hour than Gary. Patti worked 25 hours last week and
earned more than $313. How much does Gary earn per hour?
p
Gary’s hourly pay
Variable: ____________ represents ________________________________________
This page may not be reproduced.
Inequality: __________________________ Answer: ____________________
2. Scott’s weekly expenses include buying lunch 5 days each week and paying $20.75 for
gasoline. He wants to limit his weekly expenses to $60 or less. If Scott always pays the
same amount for lunch, what is the maximum amount he can spend on lunch each
day?
Variable: ____________ represents ________________________________________
Inequality: __________________________ Answer: ____________________
3. A landscaper has 328 pounds of gravel to use for a job. He will use 160 pounds of the
gravel in one large flowerbed. He will put the remaining gravel in several small flower
boxes, using 7 pounds for each box. If the landscaper uses gravel in the greatest
possible number of flower boxes, into how many boxes will he pour gravel?
Variable: ____________ represents ________________________________________
Inequality: __________________________ Answer: ____________________
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
16
© ECS Learning Systems, Inc.
145
TestSMART® Common Core Sample Booklet
Teacher Guide
103
= ÷15
711 49
+
x
What’s Inside the Student Work Text? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
How to Use the Student Work Text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
This page may not be reproduced.
Understanding Rigor and Cognitive Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Descriptions of TestSMART® Complexity Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Fostering Mathematical Understanding and Inquiry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Definition of the Common Core State Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
The Precise Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Mathematics Manipulatives and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Text-Marking in Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Master Skills List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
155
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
17
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
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.
This page may not be reproduced.
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 minutes to 1 hour.
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—
•
•
•
•
•
156
analytical thinking
evaluative thinking
reflective thinking
metacognitive thinking
communication
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
18
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
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 section 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 169).
This page may not be reproduced.
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 166–167).
Question: Students are asked open-ended questions that focus on the underlying
structures and logic of mathematics.
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 166–167).
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
169–171).
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 169–171).
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
157
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
19
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
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 (L)
Low Complexity
This page may not be reproduced.
Low-complexity items align with
the CCSS at Level 1 of the Webb
(2002a) model. Activities and
problems at this level require
routine, single-step methods.
An item may ask students to
recognize or restate a fact,
definition, or term. For example,
students may need to identify
the attributes of a geometric figure.
Items of this complexity may require students to follow a basic procedure with clearly
defined steps. At this cognitive level, students may need to apply a formula or perform
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.
Moderate Complexity (M)
Moderate Complexity
Moderate-complexity items
align with the CCSS at Level 2
of the Webb model. Items of
moderate complexity involve
both comprehension and the
subsequent processing of
information. Activities at this
level demand more than one
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
problem. At this cognitive level, students will need to visualize for tasks such as
extending patterns and determining nonexamples. Items may involve interpreting
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
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
and concepts.
160
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
20
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
High Complexity (H)
High Complexity
This page may not be reproduced.
High-complexity items align
with the CCSS at Level 3 and/or
4 of the Webb model.* Items of
high complexity require students
to use strategic, multi-step thinking;
develop a deeper understanding
of the information; and extend
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.
* 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.
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
161
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
21
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
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.
Forty-three states and the District of Columbia have adopted the standards. During
the 2014–2015 academic year, adopting states began formal CCSS assessments.
Assessments include the following types of items:
This page may not be reproduced.
•
•
•
•
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—
•
•
•
•
•
•
•
•
•
162
critical/evaluative thinking
creative/innovative thinking
elaborative thinking
problem solving
decision making
researching
collaboration
communication
organizing and connecting ideas
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
22
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
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...supports
a focus on teaching
for meaning and
understanding.”
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
This page may not be reproduced.
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 159 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—
•
•
•
•
•
•
•
•
•
•
•
•
•
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
© ECS Learning Systems, Inc.
“
Rich mathematics
experiences provide
students with
opportunities to see
structure, patterns,
and relationships in
many different
contexts.”
TestSMART® Common Core Student Work Text
163
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
23
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
• 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
This page may not be reproduced.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
24
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book I
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.
This page may not be reproduced.
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). Applying Webb’s depth-of-knowledge and NAEP levels of complexity in mathematics.
Retrieved from National Center for the Improvement of Educational Assessment (NCIEA) Web site:
http://www.nciea.org/publications/DOKmath_KH08.pdf
186
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book I
© ECS Learning Systems, Inc.
25
Selected pages from
Student Work Text
Mathematics
Grade 7, Book II
This page may not be reproduced.
Geometry
Lori Mammen
Editorial Director
ISBN: 978-1-60539-909-6
Copyright infringement is a violation of Federal Law.
©2015 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.
Photocopying of student worksheets by a classroom teacher at a non-profit school who has purchased this publication for
his/her own class is permissible. 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.
26
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Table of Contents
Geometry .......................................................................................................................................3
Mathematics Vocabulary...................................................................................................114
Teacher Guide (with Comprehensive Answer Key) ...............................................115
The Teacher Guide section contains a “How to Use the Student Work Text” section, an
explanation of the Common Core State Standards, a mathematics vocabulary section, a
master skills list, and much more. See page 115 for a complete list.
This page may not be reproduced.
ECS Learning Systems, Inc.
P. O. Box 440, Bulverde, TX 78163-0440
ecslearningsystems.com • 1.800.688.3224 (t)
2
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
27
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Geometry
Geometry
7.G—Draw, construct, and describe geometrical figures, and describe the
relationships between them
1. Solve problems involving scale drawings of geometric figures, including
computing actual lengths and areas from a scale drawing and reproducing
a scale drawing at a different scale.
2. Draw (freehand, with ruler and protractor, and with technology) geometric
shapes with given conditions. Focus on constructing triangles from three
measures of angles or sides, noticing when the conditions determine a
unique triangle, more than one triangle, or no triangle.
This page may not be reproduced.
3. Describe the two-dimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right
rectangular pyramids.
7.G—Solve real-life and mathematical problems involving angle measure, area,
surface area, and volume
4. Know the formulas for the area and circumference of a circle and use them to
solve problems. Give an informal derivation of the relationship between the
circumference and area of a circle.
5. Use facts about supplementary, complementary, vertical, and adjacent angles
in a multi-step problem to write and solve simple equations for an unknown
angle in a figure.
6. Solve real-world and mathematical problems involving area, volume, and
surface area of two- and three-dimensional objects composed of triangles,
quadrilaterals, polygons, cubes, and right prisms.
Note: The Common Core State Standards (CCSS) identify solving problems involving
scale drawings and informal geometric constructions and working with two- and
three-dimensional shapes to solve problems involving area, surface area, and volume
as one of four critical areas of instruction for Grade 7.
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
28
© ECS Learning Systems, Inc.
3
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Geometry
Talk About It–2
• Why do you think we use scale drawings?
• What are some real-world examples of scale drawings?
Working Together: Let’s make a scale drawing of the rectangle shown below. We will use a
scale factor of 2.
This page may not be reproduced.
• First, find the dimensions of the rectangle above (in centimeters). Dimensions are the
lengths of the sides of a two-dimensional figure. For rectangles, the dimensions are
length and width. Use a ruler to measure the dimensions of the rectangle above. Label
the sides of the rectangle above with the correct dimensions.
• To find the dimensions of the scale drawing, multiply each dimension of the rectangle
above by the scale factor of 2.
__________ x 2 = _________
__________ x 2 = _________
In the scale drawing, the rectangle will be ______ centimeters long and _______
centimeters wide.
• Using a ruler and the correct dimensions, draw the scale drawing of the original
rectangle on a separate sheet of paper.
Think About It: In the example above, you made a scale drawing using a scale factor greater
than 1. The resulting drawing was greater in size than the original.
• What will happen if you create a scale drawing with a scale factor less than 1? Explain
your reasoning.
• What will happen if you create a scale drawing with a scale factor of 1? Explain your
reasoning.
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
5
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
29
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Geometry
Standard 7.G.1 (H)
Challenger
Directions: Read and solve each problem below. Show all of your work.
1. The diagram below shows the floor plan for an apartment. What are the dimensions
(in feet) of each room in the apartment?
10 cm
10 cm
This page may not be reproduced.
4 cm
6 cm
Kitchen
10 cm
Dining
Room
10 cm
Bathroom
Bedroom
8 cm
Living
Room
10 cm
12 cm
Scale: 2 cm = 3 ft
Kitchen: ____________________
Dining Room: ____________________
Living Room: ____________________
Bedroom: ____________________
Bathroom: ____________________
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
30
© ECS Learning Systems, Inc.
29
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Geometry
Standard 7.G.2 (M)
Review: Triangle Theorems
When you studied triangles in previous grades, you learned some important theorems, or
rules, that apply to all triangles. A theorem is a mathematical statement proven to be true
based on known facts. In geometry, two theorems apply when you draw triangles. You will
use both theorems in future lessons. Let’s review them now.
The Triangle Angle Sum Theorem states that the sum of the interior (inner) angles of any
triangle is equal to 180°. Look at the triangles below.
60°
45°
30°
45°
60°
60°
75°
This page may not be reproduced.
90°
75°
Talk About It–1
• Do the triangles above follow the Triangle Angle Sum Theorem? Why or why not?
• Can a triangle have three angles with the following measurements: 30°, 50°, 60°? Why or
why not?
The Triangle Inequality Theorem states that any side of a triangle is always shorter than the
sum of the other two sides. Look at the triangles below.
6
4
8
8
8
3
5.5
10
8
Talk About It–2
• Do the triangles above follow the Triangle Inequality Theorem? Why or why not?
• Can a triangle have side lengths of 3 centimeters, 5 centimeters, and 9 centimeters?
Why or why not?
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
41
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
31
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Geometry
Standard 7.G.2 (M–H)
More About Unique Triangles
You already have two shortcuts to help with your study of triangles. Let’s review the two
theorems you have learned.
The Triangle Angle Sum Theorem states that the sum of the interior angles of any
triangle is equal to 180°.
The Triangle Inequality Theorem states that any side of a triangle is always shorter than
the sum of the other two sides.
This page may not be reproduced.
Given angle measurements or side lengths, you can decide whether the measurements form
a triangle or no triangle at all.
Talk About It–1
• Can you create a triangle with the following three angle measurements: 80°, 80°, 80°?
Why or why not?
• Can you create a triangle with the following three angle measurements: 10°, 40°, 130°?
Why or why not?
• Can you create a triangle with the following side lengths: 3 cm, 6 cm, 8 cm? Why or why
not?
• Can you create a triangle with the following side lengths: 2 cm, 4 cm, 7 cm? Why or why
not?
Given a set of conditions (three angle measurements or three side lengths), you know if you
can create a triangle. In a similar way, there are sets of conditions that allow you to determine
whether a triangle is unique or not. Example #2 of the previous lesson (p. 47) included the
directions below.
Draw a triangle with side lengths of 2 centimeters, 2 centimeters, and
3 centimeters.
Talk About It–2
• What did you and your classmates conclude after studying Example #2?
• Based on your conclusions, what set of conditions determines a unique triangle?
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
32
© ECS Learning Systems, Inc.
49
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Geometry
Standard 7.G.3 (M–H)
Plane Sections
A plane is a flat surface that extends forever in each direction. When a plane intersects (cuts
through) a three-dimensional (solid) shape, it creates a plane section. A plane section is a
two-dimensional “slice” of the solid shape. Cross section is another name for a plane section.
The diagram below shows a plane intersecting a rectangular prism and creating a plane
section.
plane section
plane
This page may not be reproduced.
A prism is a three-dimensional object with two identical bases and flat sides. A right
rectangular prism, like the one above, is a prism with right angles between the base and
the sides.
A pyramid is a three-dimensional object with a polygon base and triangular sides that share
a common vertex. A right rectangular pyramid has a rectangular base and four congruent
triangular sides that meet at a vertex above the base. A right rectangular prism and a right
rectangular pyramid are shown below.
right rectangular prism
right rectangular pyramid
Let’s explore the plane sections of a right rectangular prism and a right rectangular pyramid.
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
55
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
33
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Geometry
Angled Plane
In the diagram to the right, an angled plane
intersects all four lateral faces of a right rectangular
prism.
When an angled plane intersects all four lateral faces
of a right rectangular prism, the resulting plane
section is a parallelogram. In this example, the plane
section is a rectangle, which is also a parallelogram.
Talk About It–3: What other plane sections could result when an angled plane intersects all
four lateral faces of a right rectangular prism? Explain your reasoning.
This page may not be reproduced.
Plane Sections of Right Rectangular Pyramids
Horizontal Plane
In the diagram to the right, a horizontal plane
intersects a right rectangular pyramid. A horizontal
plane is parallel to the pyramid’s base.
When a horizontal plane intersects a right
rectangular pyramid, the plane section has the same
shape as the pyramid’s base. In this example, the
plane section is a square.
Talk About It–4: What other plane sections could result when a horizontal plane intersects a
right rectangular pyramid? Explain your reasoning.
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
34
© ECS Learning Systems, Inc.
57
TestSMART® Common Core Sample Booklet
Geometry
Mathematics, Grade 7—Book II
Standard 7.G.4 (L–M)
Diameter & Circumference
Remember, perimeter is the distance around a two-dimensional figure. The perimeter of a
circle is called its circumference. The variable C often represents circumference.
C
Three circles appear below. Each circle’s circumference and diameter is included.
C = 37.68 in.
d = 12 in.
B
C
C = 26.69 in.
d = 8.5 in.
C = 31.4 in.
d = 10 in.
This page may not be reproduced.
A
Talk About It–1
C , what would you
• If you used each circle’s circumference and diameter in the ratio —
d
expect for an answer? Why?
• Find the ratio of circumference to diameter for all three circles.
Circle A
C
d
37.68 in. = ______
12 in.
Circle B
26.69 in. = ______
8.5 in.
Circle C
31.4 in. = ______
10 in.
C?
• What special name do we give to the ratio —
d
continue to next page
68
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
35
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Geometry
Standard 7.G.4 (M)
Riddle Me This
This page may not be reproduced.
A. Directions: Determine the area of each item below. Round your answers to the nearest
hundredth. Then, match each answer with a letter from the code chart that follows to answer
the riddle. (Note: Use 3.14 for π.)
1. A semicircular dinner mat has a diameter of 8 inches.
A = _______________
2. The radius of a personal pizza is 3.5 inches.
A = _______________
3. A flying disc has a diameter of 13 inches.
A = _______________
4. The radius of a clock face is 9 inches.
A = _______________
5. A silver dollar has a radius of 2.5 centimeters.
A = _______________
What has a head but never weeps, has a bed but never sleeps, can run but never walks,
and has a bank but no money?
Code Chart
A
E
I
R
V
254.34 25.12 132.67 38.47
____
4
19.63
____ ____ ____ ____ ____
2
3
5
1
2
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
36
© ECS Learning Systems, Inc.
75
TestSMART® Common Core Sample Booklet
Geometry
Mathematics, Grade 7—Book II
Standard 7.G.5 (M)
Angle Partners
Directions: Write and solve an equation to find x, the measure of the missing angle in each
diagram below.
1.
4.
R
E
x
122°
x
D
C
T
74°
F
S
Equation: ____________________
U
Equation: ____________________
2.
This page may not be reproduced.
x = _______________
x = _______________
5.
O
L
29°
M
x
N
P
101°
J
Equation: ____________________
K
M
Equation: ____________________
x = _______________
3.
x
x = _______________
6.
W
E
Y
68°
x
x
18°
X
Z
Equation: ____________________
x = _______________
C
D
F
Equation: ____________________
x = _______________
continue to next page
90
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
37
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
Geometry
Standard 7.G.5 (M)
Crossing Paths
Directions: Identify pairs of adjacent and vertical angles in each diagram below. Then, find
the value of x.
1.
3.
C
R
131°
35°
x
A
E
D
x
S
V
T
U
This page may not be reproduced.
B
Adjacent angles: ___________________
Adjacent angles: ___________________
_________________________________
_________________________________
Vertical angles: ____________________
Vertical angles: ____________________
_________________________________
_________________________________
x = _______________
2.
x = _______________
4.
P
E
C
112°
126°
D
x
M
G
x
N
O
F
Adjacent angles: ___________________
Adjacent angles: ___________________
_________________________________
_________________________________
Vertical angles: ____________________
Vertical angles: ____________________
_________________________________
_________________________________
x = _______________
© ECS Learning Systems, Inc.
x = _______________
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
38
© ECS Learning Systems, Inc.
95
TestSMART® Common Core Sample Booklet
Geometry
Mathematics, Grade 7—Book II
Standard 7.G.6 (L)
Area, Surface Area, & Volume
Area is the measure of the space inside a two-dimensional figure. Area is measured in
square units.
Surface area is the total area of the faces (including the bases) of a solid figure. A face is a
flat side of a three-dimensional figure. A base is a special type of face. A rectangular prism
has two bases that are congruent and parallel.
Volume is the amount of space inside a three-dimensional object. Volume is measured in
cubic units.
A pyramid is a solid figure with a base and triangular faces that share a vertex. A right
rectangular pyramid is a three-dimensional figure with a rectangular base and four
congruent triangular sides that meet in a point above the base.
Think About It
This page may not be reproduced.
• How could you find the area of the rectangle and triangle below?
• How could you find the area of the trapezoid and parallelogram below?
continue to next page
100
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
39
TestSMART® Common Core Sample Booklet
Geometry
Mathematics, Grade 7—Book II
Standard 7.G.6 (M)
Problem Solving VI
Directions: Read and solve each problem below. Show all of your work.
1. Stan mailed the package below to his grandfather. He covered the box in brown
mailing paper. The paper cost $0.10 per square foot. How much money did it cost Stan
to cover the package?
12 in.
15 in.
25 in.
This page may not be reproduced.
Answer: _______________
2. Katelyn worked on a sewing project for one of her classes. She used the pattern shown
below. What was the area of the pattern Katelyn used?
12 in.
6 in.
1.5 in.
4.5 in.
3 in. 4.5 in.
Answer: _______________
3. Marcus built a fort out of leftover cardboard for his younger brother. The fort was
shaped like a triangular prism, as shown below. The volume of the fort was 120 cubic
feet. What was the height of the triangular base of the fort?
7 ft
7 ft
h
Answer: _______________
8 ft
5 ft
continue to next page
108
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
40
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Teacher Guide
103
= ÷15
711 49
+
x
What’s Inside the Student Work Text? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
How to Use the Student Work Text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
This page may not be reproduced.
Understanding Rigor and Cognitive Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Descriptions of TestSMART® Complexity Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Fostering Mathematical Understanding and Inquiry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Definition of the Common Core State Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
The Precise Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Mathematics Manipulatives and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Text-Marking in Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Master Skills List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text–Teacher Guide Section
115
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
41
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
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.
This page may not be reproduced.
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 minutes to 1 hour.
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—
•
•
•
•
•
116
analytical thinking
evaluative thinking
reflective thinking
metacognitive thinking
communication
TestSMART® Common Core Student Work Text–Teacher Guide Section
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
42
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
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 section 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 129).
This page may not be reproduced.
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 126–127).
Question: Students are asked open-ended questions that focus on the underlying
structures and logic of mathematics.
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 126–127).
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
129–131).
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 129–131).
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text–Teacher Guide Section
117
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
43
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
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)
Mathematics, Grade 7—Book II
Geometry
Low-complexity items align with
Triangle Task
the CCSS at Level 1 of the Webb
Directions: Use a ruler and a protractor to draw a triangle that matches the given angle
measurements in each item below. Then, write the name of the triangle on the answer line.
(2002a) model. Activities and
1. 110° 40° 30°
3. 80° 50° 50°
problems at this level require
routine, single-step methods.
An item may ask students to
recognize or restate a fact,
definition, or term. For example,
students may need to identify
the attributes of a geometric figure.
Items of this complexity may require students to follow a basic procedure with clearly
defined steps. At this cognitive level, students may need to apply a formula or perform
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.
This page may not be reproduced.
Standard 7.G.2 (L–M)
Moderate Complexity
Moderate Complexity (M)
Mathematics, Grade 7—Book II
Geometry
Moderate-complexity items
Finding Scale Factors
align with the CCSS at Level 2
Directions: In each item below, Figure A and Figure B are similar. Use the dimensions of both
figures to determine the scale factor used for Figure B (the scale drawing). Show all of your
of the Webb model. Items of
work. The first one is completed for you.
moderate complexity involve
1.
A 1.5 in.
6.9 in. = 3 OR 4.5 in. = 3
both comprehension and the
2.3 in.
2.3 in.
1.5 in.
4.5 in.
B
subsequent processing of
6.9 in.
3
Scale factor: _______________
information. Activities at this
2.
level demand more than one
2.5 cm
A
B
3.75 cm
step in the reasoning process.
3.4 cm
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
problem. At this cognitive level, students will need to visualize for tasks such as
extending patterns and determining nonexamples. Items may involve interpreting
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
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
and concepts.
Standard 7.G.1 (M)
120
TestSMART® Common Core Student Work Text–Teacher Guide Section
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
44
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
High Complexity
High Complexity (H)
Mathematics, Grade 7—Book II
Geometry
High-complexity items align
Challenger
with the CCSS at Level 3 and/or
Directions: Read and solve each problem below. Show all of your work.
*
1. The diagram below shows the floor plan for an apartment. What are the dimensions
4 of the Webb model. Items of
(in feet) of each room in the apartment?
high complexity require students
10 cm
10 cm
to use strategic, multi-step thinking;
develop a deeper understanding
Dining
10 cm
10 cm
Room
Kitchen
of the information; and extend
thinking. The problems at this level
4 cm
Bathroom
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 7.G.1 (H)
This page may not be reproduced.
* 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.
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text–Teacher Guide Section
121
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
45
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
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.
Forty-three states and the District of Columbia have adopted the standards. During
the 2014–2015 academic year, adopting states began formal CCSS assessments.
Assessments include the following types of items:
This page may not be reproduced.
•
•
•
•
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—
•
•
•
•
•
•
•
•
•
122
critical/evaluative thinking
creative/innovative thinking
elaborative thinking
problem solving
decision making
researching
collaboration
communication
organizing and connecting ideas
TestSMART® Common Core Student Work Text–Teacher Guide Section
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
46
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
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...supports
a focus on teaching
for meaning and
understanding.”
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
This page may not be reproduced.
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 119 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—
•
•
•
•
•
•
•
•
•
•
•
•
•
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
© ECS Learning Systems, Inc.
“
Rich mathematics
experiences provide
students with
opportunities to see
structure, patterns,
and relationships in
many different
contexts.”
TestSMART® Common Core Student Work Text–Teacher Guide Section
123
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
47
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
• 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)
This page may not be reproduced.
A
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
48
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book II
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.
This page may not be reproduced.
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). Applying Webb’s depth-of-knowledge and NAEP levels of complexity in mathematics. Retrieved
from National Center for the Improvement of Educational Assessment (NCIEA) Web site:
http://www.nciea.org/publications/DOKmath_KH08.pdf
Hess, K. K. (2006). Cognitive complexity: Applying Webb DOK levels to Bloom’s taxonomy. 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.
Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning
community. Journal for Research in Mathematics Education, 35(2), 81–116.
142
TestSMART® Common Core Student Work Text–Teacher Guide Section
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book II
© ECS Learning Systems, Inc.
49
Selected pages from
Student Work Text
Mathematics
Grade 7, Book III
This page may not be reproduced.
Statistics and Probability
Lori Mammen
Editorial Director
ISBN: 978-1-60539-910-2
Copyright infringement is a violation of Federal Law.
©2015 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.
Photocopying of student worksheets by a classroom teacher at a non-profit school who has purchased this publication for
his/her own class is permissible. Reproduction of any part of this publication for an entire school or for a school system, by forprofit 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.
50
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
Table of Contents
Statistics and Probability .......................................................................................................3
Mathematics Vocabulary...................................................................................................123
Teacher Guide (with Comprehensive Answer Key) ...............................................125
The Teacher Guide section contains a “How to Use the Student Work Text” section, an
explanation of the Common Core State Standards, a mathematics vocabulary section, a
master skills list, and much more. See page 125 for a complete list.
This page may not be reproduced.
ECS Learning Systems, Inc.
P. O. Box 440, Bulverde, TX 78163-0440
ecslearningsystems.com • 1.800.688.3224 (t)
2
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
51
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
Statistics and Probability
Statistics and Probability
7.SP—Use random sampling to draw inferences about a population
1. Understand that statistics can be used to gain information about a population by
examining a sample of the population; generalizations about a population from a
sample are valid only if the sample is representative of that population. Understand that
random sampling tends to produce representative samples and support valid inferences.
2. Use data from a random sample to draw inferences about a population with an unknown
characteristic of interest. Generate multiple samples (or simulated samples) of the same
size to gauge the variation in estimates or predictions.
7.SP—Draw informal comparative inferences about two populations
3. Informally assess the degree of visual overlap of two numerical data distributions with
similar variabilities, measuring the difference between the centers by expressing it as a
multiple of a measure of variability.
This page may not be reproduced.
4. Use measures of center and measures of variability for numerical data from random
samples to draw informal comparative inferences about two populations.
7.SP—Investigate chance processes and develop, use, and evaluate probability models
5. Understand that the probability of a chance event is a number between 0 and 1 that
expresses the likelihood of the event occurring. Larger numbers indicate greater
likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2
indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a
likely event.
6. Approximate the probability of a chance event by collecting data on the chance process
that produces it and observing its long-run relative frequency, and predict the
approximate relative frequency given the probability.
7. Develop a probability model, and use it to find probabilities of events. Compare
probabilities from a model to observed frequencies; if the agreement is not good,
explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes,
and use the model to determine probabilities of events.
b. Develop a probability model (which may not be uniform) by observing frequencies in
data generated from a chance process.
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and
simulation.
a. Understand that, just as with simple events, the probability of a compound event is
the fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized
lists, tables and tree diagrams. For an event described in everyday language (e.g.,
“rolling double sixes”), identify the outcomes in the sample space which compose
the event.
c. Design and use a simulation to generate frequencies for compound events.
Note: The Common Core State Standards (CCSS) identify drawing inferences about
populations based on samples as one of four critical areas of instruction for Grade 7.
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
52
© ECS Learning Systems, Inc.
3
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
Statistics and Probability
Standard 7.SP.1 (M–H)
Problem Solving I
Directions: Use the information provided to write a reasonable decision for each scenario.
Statistical
Question
Sampling Method
Results
1. How many
sodas do
Jensen
Middle
School
students
drink each
day?
The school’s
computer system
randomly selected
110 names from
a master list of
520 students.
3 or more sodas—48 students
2. How do
most Jensen
Middle
School girls
get to
school?
The vice-principal
polled 25 female
students chosen
randomly from a
comprehensive
list.
parent drop-off—15 students
3. How many
books do
male
students
at Jensen
Middle
School carry
in their
backpacks?
A school
administrator
surveyed every
4th male student
listed on a school
roster, for a total
of 130 students.
4 or more books—49 students
4. How late
do Jensen
Middle
School
teachers stay
after school?
The school
secretary polled
30 teachers
chosen randomly
from a complete
alphabetical list.
3 or more hours after
school—6 teachers
Decision
2 sodas—37 students
1 soda—19 students
no soda—6 students
This page may not be reproduced.
bus—7 students
walking—3 students
3 books—62 students
2 books—15 students
1 book—4 students
2 hours after school—
12 teachers
1 hour after school—
8 teachers
do not stay after school—
4 teachers
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
13
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
53
TestSMART® Common Core Sample Booklet
Statistics and Probability
Mathematics, Grade 7—Book III
Standard 7.SP.2 (M)
Estimating With Proportions
Directions: Use proportions to estimate a solution for each item below.
1. The 3,800 students in a school district voted on a mascot for a new school. The district
surveyed 200 students before the actual vote. The results of that survey appear in the
table below.
Survey Results
Lion
58
Wildcat
74
Eagle
20
Bull
48
Based on the survey results, how many of the district’s 3,800 students would you
expect to vote for each mascot?
a. Lion
____________
This page may not be reproduced.
b. Wildcat ____________
c. Eagle
____________
d. Bull
____________
2. A school librarian ordered new books for the library. The school’s 2,100 students voted
on their favorite genres, but the librarian also conducted a survey of 300 students’
preferences. The results of that survey appear in the table below.
Survey Results
Nonfiction
84
Graphic
Novels
54
Poetry
12
General
Fiction
118
Fantasy
& Sci-Fi
32
Based on the survey results, how many of the school’s 2,100 students would the
librarian expect to vote for each genre?
a. Nonfiction
____________
b. Graphic Novels
____________
c. Poetry
____________
d. General Fiction
____________
e. Fantasy & Sci-Fi
____________
continue to next page
16
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
54
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
Statistics and Probability
Standard 7.SP.3 (M–H)
Comparing Two Data Sets, Part 2
You already know how to compare two sets of data using the mean and MAD. Now we will
learn a different method to compare data. The following data was collected about two
professional athletic teams.
Population 1: Members of a professional football team (weights in pounds)
Data Set 1: 198, 210, 218, 200, 220, 198, 214, 252, 232, 220, 196, 228, 244, 240, 256, 248, 200,
264, 222, 238, 240
Population 2: Members of a professional basketball team (weights in pounds)
Data Set 2: 205, 201, 195, 213, 205, 203, 197, 221, 219, 203, 233, 191, 197, 225, 221, 215, 217,
223, 199
This page may not be reproduced.
We can compare the two data sets by using the median, another measure of center. The
median is the middle value after all the values have been placed in order from least to
greatest.
To find the median of a data set, we first rearrange the values in order from least to greatest,
as shown below.
Data Set 1: 196, 198, 198, 200, 200, 210, 214, 218, 220, 220, 222, 228, 232, 238, 240, 240, 244,
248, 252, 256, 264
Data Set 2: 191, 195, 197, 197, 199, 201, 203, 203, 205, 205, 213, 215, 217, 219, 221, 221, 223,
225, 233
Once the values are listed in order, we can find the median, or middle value, for each data set.
Try It–1: Find and circle the median in each data set above. Then complete each statement
below.
a. The median weight of members of the football team is ________________ pounds.
b. The median weight of members of the basketball team is ________________ pounds.
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
39
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
55
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
Statistics and Probability
Standard 7.SP.4 (M)
Talking Mean & Median
Mean and median are measures of center used to compare and draw inferences about
different data sets. However, each measure can give a very different picture of the data being
compared. Let’s consider an example in which we use mean to compare two data sets.
The daily profits for Company A and Company B appear in the tables below.
Company A
Sunday
$1,200
Monday
$1,200
Tuesday
$1,300
Wednesday Thursday
$1,100
$1,300
Friday
$1,200
Saturday
$1,000
Friday
$1,300
Saturday
$1,100
Company B
This page may not be reproduced.
Sunday
$1,000
Monday
$1,100
Tuesday
$1,300
Wednesday Thursday
$1,200
$1,200
Talk About It–1
• How would you determine the mean daily profit for each company, rounded to the
nearest dollar?
• What is the mean daily profit for each company?
Company A _____________
Company B ____________
• Based on the mean daily profit, what would you conclude about the profits of the two
companies?
Now, let’s use median to compare the same data sets.
Talk About It–2
• How would you determine the median daily profit for each company?
• What is the median daily profit for each company?
Company A _____________
Company B ____________
• Based on the median daily profit, what would you conclude about the profits of the two
companies?
• Was your comparison based on the mean daily profit the same as your comparison based
on the median daily profit? Why or why not?
• Why does comparing the data sets based on their means and medians lead to different
conclusions?
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
56
© ECS Learning Systems, Inc.
47
TestSMART® Common Core Sample Booklet
Statistics and Probability
Mathematics, Grade 7—Book III
Standard 7.SP.5 (L)
Hello, Probability!
Probability is the likelihood that an event will happen. The likelihood that a particular event
will happen is expressed as a number between 0 and 1. A probability near 0 indicates that an
event is highly unlikely to happen. A probability near 1 indicates that an event is highly likely
to happen. Probability is most often expressed as a ratio (fraction).
Suppose you reached into the jar shown below and selected one marble without looking.
Since you are selecting one marble at random, there is a chance you could choose any of
them. Selecting a marble from the jar in a probability experiment is called an event. An event
is a set of one or more outcomes in an experiment.
This page may not be reproduced.
Talk About It–1: If you select one marble at random from the jar shown above, what three
events are possible?
The jar above has three colors of marbles, but there are different numbers of each color—and
this is where probability comes in. Using probability, you can compare one random outcome
with all possible outcomes. The set of all possible outcomes is known as the sample space.
An event is a subset of the sample space.
Talk About It–2: Look at the jar of marbles again. If you reach into the jar and select one
marble, what is the sample space? In other words, what are all the possible outcomes?
continue to next page
64
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
57
TestSMART® Common Core Sample Booklet
Statistics and Probability
Mathematics, Grade 7—Book III
Standard 7.SP.6 (M)
Probability Grid
Directions: Answer questions 1–6 below. Round your answers to the nearest whole number.
Then, shade the boxes on the grid that correspond to the correct answers. Finally, answer
question 7.
1. A package of balloons contains 8 pink, 4 purple, and 6 white balloons. A balloon is
selected from the package and returned. If 500 trials are conducted, about how many
purple balloon selections should we expect?
Answer: ____________
2. A basket contains 14 white, 8 pink, and 12 red roses. A rose is selected from the
basket and returned. If 1,500 trials are conducted, about how many white rose
selections should we expect?
This page may not be reproduced.
Answer: ____________
3. A bag contains 14 black, 13 red, and 11 silver feathers. A feather is selected from the
bag and returned. If 2,500 trials are conducted, about how many black feather
selections should we expect?
Answer: ____________
4. A box contains 5 black, 10 orange, and 9 green wires. A wire is selected from the box
and returned. If 400 trials are conducted, about how many black wire selections
should we expect?
Answer: ____________
5. A basket contains 14 white, 13 yellow, and 11 green balls of yarn. A ball of yarn is
selected from the basket and returned. If 350 trials are conducted, about how many
green yarn selections should we expect?
Answer: ____________
6. A package contains 22 yellow, 25 black, and 21 brown rubber bands. A rubber band is
selected from the package and returned. If 25 trials are conducted, about how many
yellow rubber band selections should we expect?
Answer: ____________
8
83
101
921
618
82
530
3,009
96
3,750
111
123
1,304
10
7,400
12
106
20
315
1,850
19
TestSMART® Common Core Student Work Text
7. What shape do the correct answers on
the grid create?
Answer: ____________
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
58
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Statistics and Probability
Mathematics, Grade 7—Book III
Standard 7.SP.6 (H)
Problem Solving V
Directions: Complete the following activity in small groups. (Note: The teacher will provide
each group with a pair of 8-sided dice.)
1. Complete the chart below with all the possible outcomes (sums of the numbers rolled)
of rolling both dice. A few outcomes have been provided for you.
Die #1
1
3
4
5
6
7
8
9
4
9
This page may not be reproduced.
Die #2
1
2
3
4
5
6
7
8
2
12
10
2. How many outcomes are possible? ____________
3. What is the theoretical probability of rolling a sum of 9? ____________
4. Which two sums have the lowest theoretical probability of being rolled? ____________
5. Which sum has the highest theoretical probability of being rolled? ____________
6. What is the theoretical probability of rolling doubles? ____________
7. What is the theoretical probability of rolling numbers that total each sum below?
2 ____________
6 ____________
10 ____________
14 ____________
3 ____________
7 ____________
11 ____________
15 ____________
4 ____________
8 ____________
12 ____________
16 ____________
5 ____________
9 ____________
13 ____________
continue to next page
84
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
59
TestSMART® Common Core Sample Booklet
Statistics and Probability
Mathematics, Grade 7—Book III
Standard 7.SP.7 (M)
Dependent Events
An independent event is not connected to or dependent on any other event. For example,
selecting a marble from a bag and replacing it before selecting another marble does not
affect the outcome of any other event.
A dependent event, on the other hand, is connected to and does depend on the outcome of
another event. Suppose we have a bag of 10 marbles, 5 black and 5 white. We select a marble
from the bag, but do NOT return it before selecting another marble from the bag.
Talk About It: How would selecting and keeping the first marble affect the second selection
you make from the bag?
This page may not be reproduced.
Here is another example. Javier has the bag of marbles shown below. He selects a marble
from the bag and records its color. He does NOT replace the marble in the bag before making
his next selection.
In earlier examples, the selected item was always replaced before another selection was
5 chance of selecting a black
made. In earlier examples, we would have said there is a —
14
4
5
marble, a — chance of selecting a gray marble, and a — chance of selecting a white marble.
14
14
In other words, we assumed the sample space (a list of all possible outcomes) remained the
same for each trial.
continue to next page
98
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
60
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Statistics and Probability
Mathematics, Grade 7—Book III
Standard 7.SP.8 (M)
Compound Events
Simple events have exactly one outcome. For example, if you flip a coin, there can be only
one outcome: heads or tails. Compound events, on the other hand, consist of more than one
simple event. Consider the following example.
Julia plays a game with a spinner and a die. Spinning the spinner results in a single outcome.
Rolling the die results in a single outcome. However, if Julia spins the spinner and rolls the die
on each turn, she produces two outcomes for a single event.
Red
Blue
Green
This page may not be reproduced.
Talk About It–1: Brainstorm a list of other paired activities that would result in single
outcomes.
If Julia spins the spinner and rolls the die, what outcomes are possible? To determine the
outcomes that will make up the sample space, Julia can represent the data in an organized
list, a table, or a tree diagram. Let’s look at each one.
Organized List
The spinner has 3 possible outcomes: red, blue, green. The die has 6 possible outcomes: 1, 2,
3, 4, 5, 6. Each outcome in the list will consist of 1 spinner outcome and 1 die outcome. Your
organized list—your sample space—would look like this.
(red, 1)
(red, 2)
(red, 3)
(red, 4)
(red, 5)
(red, 6)
(blue, 1)
(blue, 2)
(blue, 3)
(blue, 4)
(blue, 5)
(blue, 6)
(green, 1)
(green, 2)
(green, 3)
(green, 4)
(green, 5)
(green, 6)
Listing all the events together creates a sample space with 18 possible outcomes.
continue to next page
102
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
61
TestSMART® Common Core Sample Booklet
Statistics and Probability
Mathematics, Grade 7—Book III
Standard 7.SP.8 (M)
Probability & Compound Events I
When determining the probability of a compound event, it is important to know whether you
are dealing with independent or dependent events. Remember, independent events are not
connected; they do not depend on each other. Dependent events are connected; the
outcome of one depends on the outcome of another. Let’s look at an example.
Macey has the spinner and die shown below. What is the probability that Macey’s spin will
land on green and she will roll a 3?
Red
Red
Blue
Yellow
This page may not be reproduced.
Green Green
First, determine if the two events are independent or dependent. Ask yourself: Does the
outcome of Macey’s spin affect the outcome of her roll of the die? OR Does the outcome
of Macey’s roll affect the outcome of her spin? The answer is no. Thus, the events are
independent.
To find the probability of Macey landing on green and rolling a 3, or P(green and 3), you can
use one of the three methods you have learned: organized list, table, or tree diagram. For this
example, we will use a table to find all the outcomes for the problem.
Spinner
Spinner
Possibilities
Possibilities
Red (R)
Red (R)
Green (G)
Green (G)
Blue (B)
Yellow (Y)
1
R1
R1
G1
G1
B1
Y1
2
R2
R2
G2
G2
B2
Y2
Die
DiePossibilities
Possibilities
3
4
R3
R4
R3
R4
G3
G4
G3
G4
B3
B4
Y3
Y4
5
R5
R5
G5
G5
B5
Y5
6
R6
R6
G6
G6
B6
Y6
From the chart, we see that there are 2 outcomes that include both green and 3 and a total of
2 , or —
1.
36 possible outcomes. That gives us a probability of —
36
18
continue to next page
108
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
62
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Statistics and Probability
Mathematics, Grade 7—Book III
Standard 7.SP.8 (M)
Designing & Using Simulations
In a simulation, you use a mathematical model to represent a real situation. You can use
simulations to predict the frequency of certain compound events. Read the example below,
and think about how you would solve the problem.
Mr. Alvarez regularly buys 16-ounce bottles of a certain brand of juice. The
manufacturer of the juice prints “winning codes” under the lids of 40% of the
16-ounce juice bottles. What is the probability that Mr. Alvarez must buy at
least 4 bottles of juice to find one with a “winning code”?
If Mr. Alvarez must buy at least 4 bottles of juice to get a winning code, then the first 3 bottles
he buys must not have a winning code.
L
L
W
L
W
W
L
L
W
This page may not be reproduced.
We can design a simulation to answer this question. First, we must choose a tool (e.g., die,
spinner, coin, deck of cards) to use for the simulation. We will use a spinner for this simulation.
L
Think About It
• Why is a spinner a better choice for this simulation than a die, a coin, or a deck of cards?
• Why did we divide and label the spinner the way we did?
On the spinner above, four of the 10 sections are labeled with a “W” to represent the 40% of
the juice bottles with a winning code.
We will use the spinner to determine the probability that Mr. Alvarez must buy at least 4 juice
bottles to receive a winning code. We will conduct multiple trials. Each trial will end when we
land on a “W.”
The results of Trial 1 are shown below.
Trial 1
Spin Number
Outcome
Winning Code
(Y or N)
Number of Spins
Until “W” was Spun
1
2
3
L
L
W
N
N
Y
3
continue to next page
116
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
63
TestSMART® Common Core Sample Booklet
Statistics and Probability
Mathematics, Grade 7—Book III
Standard 7.SP.8 (M)
Designing & Using Simulations
In a simulation, you use a mathematical model to represent a real situation. You can use
simulations to predict the frequency of certain compound events. Read the example below,
and think about how you would solve the problem.
Mr. Alvarez regularly buys 16-ounce bottles of a certain brand of juice. The
manufacturer of the juice prints “winning codes” under the lids of 40% of the
16-ounce juice bottles. What is the probability that Mr. Alvarez must buy at
least 4 bottles of juice to find one with a “winning code”?
If Mr. Alvarez must buy at least 4 bottles of juice to get a winning code, then the first 3 bottles
he buys must not have a winning code.
This page may not be reproduced.
We can design a simulation to answer this question. First, we must choose a tool (e.g., die,
spinner, coin, deck of cards) to use for the simulation. We will use a spinner for this simulation.
L
L
W
L
W
W
L
L
W
L
Think About It
• Why is a spinner a better choice for this simulation than a die, a coin, or a deck of cards?
• Why did we divide and label the spinner the way we did?
On the spinner above, four of the 10 sections are labeled with a “W” to represent the 40% of
the juice bottles with a winning code.
We will use the spinner to determine the probability that Mr. Alvarez must buy at least 4 juice
bottles to receive a winning code. We will conduct multiple trials. Each trial will end when we
land on a “W.”
The results of Trial 1 are shown below.
Trial 1
Spin Number
Outcome
Winning Code
(Y or N)
Number of Spins
Until “W” was Spun
1
2
3
L
L
W
N
N
Y
3
continue to next page
116
TestSMART® Common Core Student Work Text
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
64
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
Statistics and Probability
Standard 7.SP.8 (M–H)
Problem Solving VII
Directions: Using what you know about simulations, design a simulation for each scenario
below.
1. If 30% of the students at a middle school claim their favorite subject is science, what is
the probability that you will have to survey at least 5 students before finding one whose
favorite subject is science?
a. Tool to be used: __________________________________________________________
b. In the box below, create a chart that shows at least 20 trials.
This page may not be reproduced.
c. Probability: ____________
continue to next page
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text
119
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
65
TestSMART® Common Core Sample Booklet
Teacher Guide
103
= ÷15
711 49
+
x
What’s Inside the Student Work Text? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
This page may not be reproduced.
How to Use the Student Work Text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Understanding Rigor and Cognitive Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Descriptions of TestSMART® Complexity Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Fostering Mathematical Understanding and Inquiry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Definition of the Common Core State Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
The Precise Language of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Mathematics Manipulatives and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Text-Marking in Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Master Skills List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text–Teacher Guide Section
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
66
© ECS Learning Systems, Inc.
125
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
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.
This page may not be reproduced.
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 minutes to 1 hour.
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—
•
•
•
•
•
126
analytical thinking
evaluative thinking
reflective thinking
metacognitive thinking
communication
TestSMART® Common Core Student Work Text–Teacher Guide Section
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
67
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
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 section 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 139).
This page may not be reproduced.
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 136–137).
Question: Students are asked open-ended questions that focus on the underlying
structures and logic of mathematics.
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 136–137).
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
139–141).
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 139–141).
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text–Teacher Guide Section
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
68
© ECS Learning Systems, Inc.
127
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
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)
Mathematics, Grade 7—Book III
Statistics and Probability
Low-complexity items align with
Probability POP!
the CCSS at Level 1 of the Webb
Directions: Answer each question below by expressing the probability of the event as a
fraction, decimal, and percent. Then, circle whether the event is “impossible,” “unlikely,”
(2002a) model. Activities and
“equally likely,” “likely,” or “certain.”
problems at this level require
The Balloon POP! Game
routine, single-step methods.
22
18
6
21
7
16
An item may ask students to
3
11
15
1
20
29
recognize or restate a fact,
30
9
5
10
14
4
definition, or term. For example,
12
23
26
28
students may need to identify
the attributes of a geometric figure.
Items of this complexity may require students to follow a basic procedure with clearly
defined steps. At this cognitive level, students may need to apply a formula or perform
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 7.SP.5 (L)
This page may not be reproduced.
Moderate Complexity
Moderate Complexity (M)
Mathematics, Grade 7—Book III
Statistics and Probability
Moderate-complexity items
Valid or Invalid?
align with the CCSS at Level 2
Directions: Read the information provided for each statistical question below. Then
determine whether a valid inference has been made for each question, and explain your
of the Webb model. Items of
answer.
moderate complexity involve
1. How many hours do middle-school students at Peterson Middle School sleep
each night?
both comprehension and the
Sampling Method: School administrators randomly selected 25 male students and
25 female students to participate.
subsequent processing of
Sample Size: 50 students
Results: 6 hours—11 girls, 7 boys; 7 hours—8 girls, 8 boys; 8 hours—4 girls, 8 boys;
information. Activities at this
9 hours—2 girls, 2 boys
Inference: On average, middle-school boys get more sleep than middle-school girls.
level demand more than one
Is this inference valid? ____________
Explanation: _________________________
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
problem. At this cognitive level, students will need to visualize for tasks such as
extending patterns and determining nonexamples. Items may involve interpreting
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
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
and concepts.
Standard 7.SP.1 (M)
130
TestSMART® Common Core Student Work Text–Teacher Guide Section
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
69
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
High Complexity
High Complexity (H)
Mathematics, Grade 7—Book III
Statistics and Probability
High-complexity items align
It’s All Uniform
with the CCSS at Level 3 and/or
Directions: Complete each item below. Express all probabilities in percents rounded to the
nearest tenth.
*
4 of the Webb model. Items of
1. The owner of a fruit stand sells assorted grapes. He has 24 bunches of grapes in various
colors—5 red bunches, 3 purple bunches, 9 green bunches, and 7 black bunches.
high complexity require students
a. Susan randomly selects one color of grapes from the display. What is the theoretical
probability of Susan selecting each of the following colors?
to use strategic, multi-step thinking;
red grapes ____________
green grapes ____________
develop a deeper understanding
purple grapes ____________
black grapes ____________
of the information; and extend
b. Suppose Susan conducts 10 trials, randomly selecting one color of grapes each time
and replacing it after each selection. Her trials yield the following results: 4 red,
0 purple, 3 green, 3 black. What is the experimental probability of Susan
thinking. The problems at this level
selecting each color of grapes?
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.
This page may not be reproduced.
Standard 7.SP.7 (M–H)
* 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.
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text–Teacher Guide Section
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
70
© ECS Learning Systems, Inc.
131
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
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.
Forty-three states and the District of Columbia have adopted the standards. During
the 2014–2015 academic year, adopting states began formal CCSS assessments.
Assessments include the following types of items:
selected-response items (multiple-choice items)
constructed-response items
technology-enhanced items/tasks
performance tasks
This page may not be reproduced.
•
•
•
•
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—
•
•
•
•
•
•
•
•
•
132
critical/evaluative thinking
creative/innovative thinking
elaborative thinking
problem solving
decision making
researching
collaboration
communication
organizing and connecting ideas
TestSMART® Common Core Student Work Text–Teacher Guide Section
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
71
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
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...supports
a focus on teaching
for meaning and
understanding.”
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).
This page may not be reproduced.
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 129 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—
•
•
•
•
•
•
•
•
•
•
•
•
•
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
© ECS Learning Systems, Inc.
“
Rich mathematics
experiences provide
students with
opportunities to see
structure, patterns,
and relationships in
many different
contexts.”
TestSMART® Common Core Student Work Text–Teacher Guide Section
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
72
© ECS Learning Systems, Inc.
133
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
• 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)
Adequate Processing Time
Appropriate instructional pacing allows students adequate processing time, or “think
time” (e.g., Allington & Johnston, 2002; Barnes, 1976/1992; Rowe, 1986). Some teachers
and students may find it difficult to tolerate long work periods, underdeveloped
responses, or silences during discussion. However, authentic reasoning and problem
solving require time to—
•
•
•
•
•
•
•
•
“
Processing time
involves allowing
‘space’ in the
conversation
for everyone’s
contributions.”
134
This page may not be reproduced.
•
•
•
•
•
•
•
•
•
truly interact with a problem, question, or idea
apply prior knowledge and personal experience
encounter new ideas and perspectives
gather relevant data and information
challenge assumptions and other “default thinking”
make and justify conjectures
model or represent situations and relationships
identify correspondences across situations and representations
use patterns and relationships to solve problems (e.g., recognize, describe,
replicate, extend, construct)
try and retry problems and experiments
use appropriate tools to develop understanding and solve problems
invent effective and reasonable alternatives to known routines and procedures
progress from understanding of concrete to more abstract concepts
formulate precise solutions, responses, arguments, and questions
evaluate the reasonableness of results, arguments, or methods
make generalizations based on findings
express generalizations in mathematical and non-mathematical terms
Teachers should pause for processing time whenever necessary, including after posing
a question, after a student responds to a question, and when students are problem
solving. When students are given proper time, they are more likely to use correct logic
or reasoning to link ideas and examine the relationships between concepts more
critically. Leaving this space in the conversation also encourages students to develop
more effective listening skills. Processing time involves allowing “space” in the
conversation for everyone’s contributions. To encourage students to sustain the
thought process, teachers can continually return students to the original problem or
question. Modeling the reflective thought process by “thinking aloud” is also helpful.
TestSMART® Common Core Student Work Text–Teacher Guide Section
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
73
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
This page may not be reproduced.
•
•
•
•
•
•
•
•
•
•
•
How can we show an amount of things?
How can we share things equally?
How can we divide things equally?
How can unknowns affect problems?
How can we represent unknown information?
How can we represent variable information?
How can variable information affect problems?
How can we represent a problem situation?
How can we accomplish a task in steps?
How can we show an equal relationship?
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
74
© ECS Learning Systems, Inc.
TestSMART® Common Core Sample Booklet
Mathematics, Grade 7—Book III
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.
This page may not be reproduced.
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). Applying Webb’s depth-of-knowledge and NAEP levels of complexity in mathematics.
Retrieved from National Center for the Improvement of Educational Assessment (NCIEA) Web site:
http://www.nciea.org/publications/DOKmath_KH08.pdf
Hess, K. K. (2006). Cognitive complexity: Applying Webb DOK levels to Bloom’s taxonomy. Retrieved from
National Center for the Improvement of Educational Assessment (NCIEA) Web site:
http://www.nciea.org/publications/DOK_ApplyingWebb_KH08.pdf
© ECS Learning Systems, Inc.
TestSMART® Common Core Student Work Text–Teacher Guide Section
155
TestSMART® Common Core Student Work Text—Mathematics, Grade 7—Book III
© ECS Learning Systems, Inc.
75
It’s On The Test
From TestSMART® Student Practice Books to elementary-level skills practice,
ECS has all the test preparation materials you need.
ECS2401
ECS241X
ECS2428
ECS2436
ECS2444
ECS2452
ECS2460
ECS2479
ECS2487
ECS2495
ECS2509
ECS2517
ECS1030
ECS1057
ECS1065
ECS1049
BH88931
BH88932
BH88933
BH88934
BH88941
BH88942
BH88943
BH88944
BH88951
BH88952
BH88953
BH88954
BH88955
BH88956
BH88957
BH88958
BH88959
Math
TestSMART® Math Concepts Gr. 3
TestSMART® Math Operations &
Problem Solving Gr. 3
TestSMART® Math Concepts Gr. 4
TestSMART® Math Operations &
Problem Solving Gr. 4
TestSMART® Math Concepts Gr. 5
TestSMART® Math Operations &
Problem Solving Gr. 5
TestSMART® Math Concepts Gr. 6
TestSMART® Math Operations &
Problem Solving Gr. 6
TestSMART® Math Concepts Gr. 7
TestSMART® Math Operations &
Problem Solving Gr. 7
TestSMART® Math Concepts Gr. 8
TestSMART® Math Operations &
Problem Solving Gr. 8
Math Whiz Kids™ at the Amusement
Park Gr. 3–5
Math Whiz Kids™ at Home Gr. 3–5
Math Whiz Kids™ at the Mall Gr. 3–5
Math Whiz Kids™ at the Zoo Gr. 3–5
Dot-to-Dot 1–100+ Gr. 2–4
Math Art Gr. 1–2
Math Art Gr. 2–3
Multiplication Dot-to-Dot Gr. 3–4
Math Drill, Practice & Apply Gr. 1–2
Math Drill, Practice & Apply Gr. 2–3
Math Drill, Practice & Apply Gr. 3–4
Math Drill, Practice & Apply Gr. 4–5
First Number Skills Gr. K–1
Time & Money Skills Gr. 1–2
Number Facts to 10 Gr. 1–2
Basic Facts to 18 Gr. 2–3
Regrouping Skills Gr. 2–3
Multiplication Facts Gr. 3–4
Multiplication Skills Gr. 3–5
Place Value Gr. 1–2
Fraction Basics Gr. 2–3
BH88891
BH88892
BH88893
BH88894
BH88901
BH88902
BH88903
BH88904
BH88905
BH88911
BH88912
BH88913
BH88914
BH88915
BH88918
BH88919
BH88920
BH88961
BH88962
BH88963
BH88972
BH88973
BH88981
BH88982
BH88983
BH88984
BH88985
BH88986
BH88991
BH88992
BH88994
BH88995
BH88996
BH88997
BH88998
BH88999
NU783XRH
NU8437RH
NU5524RH
Reading
ECS2363 TestSMART® Reading Gr. 2
ECS1987 TestSMART® Reading Gr. 3
ECS1995 TestSMART® Reading Gr. 4
ECS2002 TestSMART® Reading Gr. 5
ECS2010 TestSMART® Reading Gr. 6
ECS2029 TestSMART® Reading Gr. 7
ECS2037 TestSMART® Reading Gr. 8
ECS91373 An Introduction to POWer Words™ Gr. 4
ECS8414 POWer Words™ Gr. 5–6
ECS5214 POWer Words™ Gr. 7–8
ECS5494 POWer Words™ Gr. 9–12
NU5958RH
ECS6564
ECS6571
Plurals & Possessives Gr. 2–3
Prefixes, Suffixes, Root Words Gr. 2–3
Synonyms, Antonyms, and Homonyms
Gr. 2–3
Analogies & Multiple Meanings Gr. 2–3
Alphabet Skills Gr. K–1
Consonant Sounds Gr. K–1
Vowel Sounds Gr. 1–2
Rhyming Words Gr. 1–2
Sight Words Gr. 1–2
Sight Word Stories Gr. K–2
Sight Word Rhymes Gr. K–2
Sight Words Word Search Gr. K–2
Wall Words Word Search Gr. 1–2
My First Crosswords Gr. 1–2
Sight Words in Context Gr. K–2
Rhyming Words in Context Gr. K–2
Word Endings in Context Gr. K–2
Poems & Rhymes Gr. 1–2
Fairy Tales Gr. 2–3
Fables & Tall Tales Gr. 3–4
Animals Gr. 1–2
Space, Stars, & Planets Gr. 3–4
The 5 W’s: Who? What? Where?
When? Why? Gr. 1–3
Getting the Sequence Gr. 1–3
Main Idea and Details Gr. 1–3
Fact and Opinion Gr. 1–3
Drawing Conclusions and Inferences Gr. 1–3
Context Clues Gr. 1–3
My First Sight Words Gr. K–1
Mastering Sight Words Gr. 1–2
Consonants Gr. K–1
Blends & Digraphs Gr. 1–2
Short Vowels Gr. 1–2
Long Vowels Gr. 1–2
Rhyming Words Gr. 1–2
Compounds & Contractions Gr. 1–2
Graphic Organizer Collection
Reacting to Literature: Writing
Activities for Every Book Gr. 6–8
Reacting to Literature: Writing
Activities for Every Book Gr. 9–12
Tackling Literary Terms Gr. 9–12
POWer Strategies™ for Reading
Comprehension Gr. 3–5
POWer Strategies™ for Reading
Comprehension Gr. 6–8
ECS3645
ECS3580
ECS3599
ECS3602
ECS3610
ECS3629
ECS3637
ECS9072
ECS9455
ECS9463
ECS0484
ECS9900
ECS0476
BH88925
BH88926
BH88927
ECS2371
ECS238X
ESC2398
Writing
TestSMART® Language Arts Gr. 2
TestSMART® Language Arts Gr. 3
TestSMART® Language Arts Gr. 4
TestSMART® Language Arts Gr. 5
TestSMART® Language Arts Gr. 6
TestSMART® Language Arts Gr. 7
TestSMART® Language Arts Gr. 8
Writing Warm-Ups™ Gr. K–6
Writing Warm-Ups Two™ Gr. K–6
Writing Warm-Ups Two™ Gr. 7–12
Not More Writing?! Gr. 9–12
Foundations for Writing Bk. I Gr. K–2
Foundations for Writing Bk. II Gr. 3–8
Scrambled Sentences Gr. 1–2
Writing Sentences Gr. 2–3
Writing Paragraphs Gr. 3–4
Grammar Notebook Book 1 Gr. 9–12
Grammar Notebook Book 2 Gr. 9–12
Grammar Notebook Book 3 Gr. 9–12
BH1469
BH1477
BH1493
BH1485
BH140X
BH1418
BH1426
BH1442
BH1434
Spanish-Reading
The 5 W’s: Who? What? Where?
When? Why? Gr. 1–3
Getting the Sequence Gr. 1–3
Main Idea and Details Gr. 1–3
Fact and Opinion Gr. 1–3
Drawing Conclusions and Inferences Gr. 1–3
The 5 W’s & H Gr. 4–5
Getting the Sequence Gr. 4–5
Main Idea & Details Gr. 4–5
Fact & Opinion Gr. 4–5
Drawing Conclusions & Inferences Gr. 4–5
BH1639
BH1646
BH1653
BH1660
BH1592
BH1608
BH1615
BH1622
BH1507
BH1515
BH1523
BH1530
BH1547
BH1554
BH1578
BH1585
BH1561
Spanish-Math
Dot-to-Dot 1–100+ Gr. 2–4
Math Art Gr. 1–2
Math Art Gr. 2–3
Multiplication Dot-to-Dot Gr. 3–4
Math Drill, Practice & Apply Gr. 1–2
Math Drill, Practice & Apply Gr. 2–3
Math Drill, Practice & Apply Gr. 3–4
Math Drill, Practice & Apply Gr. 4–5
First Number Skills Gr. K–1
Time & Money Skills Gr. 1–2
Number Facts to 10 Gr. 1–2
Basic Facts to 18 Gr. 2–3
Regrouping Skills Gr. 2–3
Multiplication Facts Gr. 3–4
Place Value Gr. 1–2
Fraction Basics Gr. 2–3
Multiplication Skills Gr. 3–5
BH1450
Need leveled, thematic kits?
Elementary • Middle • High School
Fiction • Nonfiction
Get Reading!!™ kits use the best of young people’s literature to emphasize common elements among three literature selections.
Ideal for RTI and leveled assessment, Get Reading!!™ helps you reinforce important skills in reading and literature at the same time.
TestSMART® books are used by thousands of teachers nationwide.
www.ecslearningsystems.com
8