Overview
In Unit 12, students focus on rates. The use of rates is prevalent in
the everyday world. The ability to handle rate, ratio, and proportion
problems with ease is an indication of good “number sense” and
“measurement sense”. Everyday Mathematics takes the position that
the key to understanding rates is repeated exposure to their many uses
in everyday life. Unit 12 has three main areas of focus:
◆ To introduce rates and provide practice collecting and comparing
rate data,
◆ To provide practice solving rate problems, to provide practice
comparing unit prices and identifying information needed for
comparison shopping, and
◆ To reflect on this year’s World Tour experiences and progress on
50-facts tests.
894
Unit 12 Rates
Contents
Lesson
Objective
12 ◆ 1
Introducing Rates
12 ◆ 2
Solving Rate Problems
12 ◆ 3
Converting between Rates
12 ◆ 4
Comparison Shopping: Part 1
12 ◆ 5
Comparison Shopping: Part 2
12 ◆ 6
World Tour and 50-Facts Test Wrap-Ups
12 ◆7
Progress Check 12
Page
908
To introduce rates; and to provide practice collecting and comparing
rate data.
914
To provide practice using a rate table to record rate information; and to
provide practice solving rate problems.
920
To provide practice checking the validity of data by converting the data
to more accessible rates.
926
To introduce calculating the unit price for a product; and to provide
practice comparing unit prices and identifying information needed for
comparison shopping.
931
To provide practice calculating and comparing unit prices that involve
fractions of cents.
936
To reflect on this year’s World Tour experiences and progress on
50-facts tests.
941
To assess students’ progress on mathematical content through
the end of Unit 12.
Unit Organizer
895
Learning In Perspective
896
Lesson Objectives
Links to the Past
Links to the Future
12◆1
To introduce rates; and to provide
practice collecting and comparing
rate data.
Grades 2 and 3: Collect, organize, interpret, and
display data.
Grades 5 and 6: Collect, organize, interpret, and
display data.
12◆2
To provide practice using a rate table to
record rate information; and to provide
practice solving rate problems.
Grades 2 and 3: Organize data in tables. Solve
“What’s My Rule?” problems.
Grade 5: Represent rates with formulas, tables of
values, and graphs. Solve rate and ratio number
stories. Grade 6: Estimate travel time based on
rate information. Model rate and ratio problems
with proportions; solve proportions by cross
multiplication and other methods.
12◆3
To provide practice checking the validity
of data by converting the data to more
accessible rates.
Grades 2 and 3: Collect, organize, interpret, and
display data.
Grades 5 and 6: Collect, organize, interpret, and
display data.
12◆4
To introduce calculating the unit price
for a product; and to provide practice
comparing unit prices and identifying
information needed for comparison
shopping.
Grade 2: Solve comparison number stories; use
comparison diagrams and write number models.
Grade 3: Practice multiplication in number stories.
Compare estimated costs to exact costs. Make up
and solve problems about costs of multiple items.
Grade 5: Review the meaning and uses of rates;
represent rates with formulas, tables of values,
and graphs. Solve rate and ratio number stories.
Measure heart rates. Grade 6: Model rate and
ratio problems with proportions; solve proportions
by cross multiplication and other methods.
12◆5
To provide practice calculating and
comparing unit prices that involve
fractions of cents.
Grade 2: Solve comparison number stories; use
comparison diagrams and write number models.
Grade 3: Practice multiplication in number stories.
Compare estimated costs to exact costs. Make up
and solve problems about costs of multiple items.
Grade 5: Review the meaning and uses of rates;
represent rates with formulas, tables of values,
and graphs. Solve rate and ratio number stories.
Measure heart rates. Grade 6: Model rate and
ratio problems with proportions; solve proportions
by cross multiplication and other methods.
12◆6
To reflect on this year’s World Tour
experiences and progress on
50-facts tests.
Grade 2: Introduce the Multiplication/Division Facts
Table; introduce multiplication and division fact
families; use Fact Triangles to begin memorizing the
basic multiplication facts. Grade 3: Use Fact
Triangles and the Facts Table to explore the
relationship between multiplication and division;
play games that promote recall of multiplication and
division facts.
Grades 5 and 6: Applications and maintenance,
including continuous practice of extended
multiplication and division facts and
identifying factors.
Unit 12 Rates
Key Concepts and Skills
Key Concepts and Skills
Grade 4 Goals*
Describe examples of rates.
Collect and organize data to create a table.
Find the median and mean of a data set.
Use data to draw conclusions and make predictions.
Write a number sentence with parentheses.
Operations and Computation Goal 7
Data and Chance Goal 1
Data and Chance Goal 2
Data and Chance Goal 2
Patterns, Functions, and Algebra Goal 3
Find multiples.
Solve multiplication and division facts.
Multiply and divide decimals by whole numbers.
Use repeated addition and scaling to model multiplication problems.
Use patterns and rules to solve rate problems.
Number and Numeration Goal 3
Operations and Computation Goal 3
Operations and Computation Goal 4
Operations and Computation Goal 7
Patterns, Functions, and Algebra Goal 1
Find multiples.
Divide whole numbers.
Round decimals and whole numbers.
Convert between rates.
Analyze and interpret data.
Number and Numeration Goal 3
Operations and Computation Goal 4
Operations and Computation Goal 6
Operations and Computation Goal 7
Data and Chance Goal 2
Find multiples.
Multiply and divide decimals by whole numbers.
Use repeated addition and scaling to model multiplication problems.
Analyze and interpret data.
Use patterns and rules to solve rate problems.
Number and Numeration Goal 3
Operations and Computation Goal 4
Operations and Computation Goal 7
Data and Chance Goal 2
Patterns, Functions, and Algebra Goal 1
12◆5
Compare decimals.
Divide decimals by whole numbers.
Round decimals.
Use repeated addition and scaling to solve rate problems.
Number and Numeration Goal 6
Operations and Computation Goal 4
Operations and Computation Goal 6
Operations and Computation Goal 7
12◆6
Demonstrate automaticity with multiplication facts.
Solve problems involving division of multidigit whole numbers; interpret the remainder.
Use data to create a line graph.
Use the median, the mean, and a line graph to draw conclusions about a data set.
Operations and Computation Goal 3
Operations and Computation Goal 4
Data and Chance Goal 1
Data and Chance Goal 2
12◆1
12◆2
12◆3
12◆4
* See the Appendix for a complete list of Grade 4 Goals.
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Ongoing Learning and Practice
Math Boxes
Math Boxes are paired across lessons as shown in the brackets below.
This makes them useful as assessment tools.
Mixed practice
[12◆ 1, 12◆ 3], [12◆ 2, 12◆ 4, 12◆ 6], [12◆ 5, 12◆ 7]
Mixed practice with multiple choice
12 ◆ 1, 12 ◆ 4, 12 ◆ 6
Mixed practice with writing/reasoning opportunity
12 ◆ 2, 12 ◆ 3, 12 ◆ 5
Practice through Games
1 2
4 3
Games are an essential component of practice in the Everyday Mathematics
program. Games offer skills practice and promote strategic thinking.
Lesson
Game
Skill Practiced
12 ◆ 2
Credits/Debits Game
(Advanced Version)
Adding and subtracting positive and
negative integers
Operations and Computation Goal 2
12 ◆ 4
Name That Number
Representing numbers in different ways
Number and Numeration Goal 4
12 ◆ 5
Fraction Top-It
Comparing and ordering fractions
Number and Numeration Goal 6
See the Differentiation Handbook for ways to adapt games to meet students’ needs.
Home Communication
▲
Study Links provide homework and home communication.
Home Connection Handbook provides more ideas to communicate
effectively with parents.
Unit 12 Family Letter provides families with an overview, Do-Anytime
Activities, Building Skills Through Games, and a list of vocabulary.
898
Unit 12 Rates
Problem Solving
Encourage students to use a variety of strategies to solve problems and to
explain those strategies. Strategies that students might use in this unit:
◆ Acting out the problem
◆ Using computation
◆ Making a table
◆ Using estimation
Lesson
Activity
12 ◆ 1
Collect data and compare eye-blinking rates for when a person is reading
and at rest.
12 ◆ 2
Solve rate problems.
12 ◆ 3
Use rates to determine if data makes sense.
12 ◆ 4
Calculate and compare unit prices.
12 ◆ 5
Solve problems involving unit prices.
Lesson
s
teach t that
h
problem rough
s
not jus olving,
t about
problem
solving
See Chapter 18 in the Teacher’s Reference Manual for more information about problem solving.
Planning Tips
Pacing
Pacing depends on a number of factors, such as students’ individual needs
and how long your school has been using Everyday Mathematics. At the
beginning of Unit 12, review your Content by Strand Poster to help you
set a monthly pace.
MOST CLASSROOMS
APRIL
MAY
JUNE
NCTM Standards
Unit 12
Lessons
NCTM
Standards
12 ◆1
12 ◆ 2
12 ◆ 3
12 ◆ 4
12 ◆ 5
12 ◆ 6
12 ◆ 7
2, 5,
6–9
1, 2,
6–9
1, 2, 5,
6–9
2,
6–10
1, 2,
6–10
1, 2,
4–10
6–10
Content Standards: 1 Number and Operations, 2 Algebra, 3 Geometry, 4 Measurement, 5 Data Analysis and Probability
Process Standards: 6 Problem Solving, 7 Reasoning and Proof, 8 Communication, 9 Connections, 10 Representation
Unit Organizer
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Balanced Assessment
Ongoing Assessment
Recognizing Student Achievement
Opportunities to assess students’ progress toward Grade 4 Goals:
Lesson
Content Assessed
12 ◆ 1
Write number sentences comparing two numbers between 100 and –100.
[Number and Numeration Goal 6]
12 ◆ 2
Describe the rule for a pattern and use that rule to solve problems.
[Patterns, Functions, and Algebra Goal 1]
12 ◆ 3
Use given data to create a line graph.
[Data and Chance Goal 1]
12 ◆ 4
Insert parentheses in number sentences to make them true.
[Patterns, Functions, and Algebra Goal 3]
12 ◆ 5
Use scaling to model multiplication and division.
[Operations and Computation Goal 7]
12 ◆ 6
Demonstrate automaticity with multiplication
facts through 10 ⴱ 10.
[Operations and Computation Goal 3]
Use the Assessment
Management System
to collect and analyze data
about students’ progress
throughout the year.
Informing Instruction
To anticipate common student errors and to highlight
problem-solving strategies:
Lesson 12 ◆2 Note the similarities between a rate table and a “What’s
My Rule?” table
Lesson 12 ◆ 3 Note that a problem may be missing information or contain
unnecessary information
Lesson 12 ◆ 4 Be careful when assuming that the lower-cost package is
the better buy
900
Unit 12 Rates
Periodic Assessment
12 ◆7 Progress Check 12
ASSESSMENT ITEMS
CONTENT ASSESSED
Self
Oral/Slate
Written
Find whole-number factors of numbers.
[Number and Numeration Goal 3]
✔
✔
Compare and order fractions.
[Number and Numeration Goal 6]
✔
✔
Compare and order integers.
[Number and Numeration Goal 6]
Add and subtract signed numbers.
[Operations and Computation Goal 2]
Open Response
✔
✔
✔
Add and subtract decimals.
[Operations and Computation Goal 2]
✔
Solve problems involving division.
[Operations and Computation Goal 4]
✔
Use scaling to model rate situations.
[Operations and Computation Goal 7]
✔
✔
✔
Analyze and interpret data.
[Data and Chance Goal 2]
✔
Find the volume of rectangular prisms.
[Measurement and Reference Frames Goal 2]
✔
Convert among U.S. customary units of capacity.
[Measurement and Reference Frames Goal 3]
✔
✔
Solve open sentences.
[Patterns, Functions, and Algebra Goal 2]
✔
✔
✔
Assessment Handbook
Unit 12 Assessment Support
◆ Grade 4 Goals, pp. 37–50
◆ Unit 12 Assessment Overview, pp. 142–151
◆ Unit 12 Open Response
• Detailed rubric, p. 146
• Sample student responses, pp. 147–149
Unit 12 Assessment Masters
◆ Unit 12 Self Assessment, p. 211
◆ End-of-Year Assessment, pp. 234–241
◆ Unit 12 Written Assessment, pp. 212–214
◆ Unit 12 Individual Profile of Progress,
pp. 290, 291, and 302
◆ Unit 12 Open Response, p. 215
◆ Unit 12 Class Checklist, pp. 292, 293,
and 303
◆ Quarterly Checklist: Quarter 4, pp. 300
and 301
◆ Exit Slip, p. 311
◆ Math Logs, pp. 306–308
◆ Other Student Assessment Forms, pp. 304,
305, 309, and 310
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Differentiated Instruction
Daily Lesson Support
ENGLISH LANGUAGE LEARNERS
12◆ 1
12◆ 3
EXTRA PRACTICE
12◆ 2 Using rate tables to solve rate
problems
Creating a Rates All Around Museum
Analyzing life expectancy or average
lifetime data
READINESS
5-Minute Math 12◆ 5 Solving rate
problems; 12◆ 6 Solving problems involving
the mean and median of a data set
ENRICHMENT
12◆ 1 Analyzing the median and mean of a
data set
12◆ 2 Describing and illustrating situations
involving rates
12◆ 4 Using a concrete model to determine
unit prices
12◆ 5 Exploring comparison shopping
problems
12◆ 6 Solving division problems and
interpreting remainders
12◆ 1
12◆ 2
12◆ 3
12◆ 4
Creating a side-by-side bar graph
Representing rates with line graphs
Calculating mammal speeds
Analyzing and interpreting data with
Consumer Reports for Kids Online
12◆ 5 Exploring how barometric pressure can
be used to determine elevetion
Adjusting the Activity
12◆ 1 Finding the mean of a data set and
writing a number model
12◆ 1 Finding the middle value of a data
group ELL
A U D I T O R Y
䉬
12◆ 2 Using variables to state a rule as a
formula
12◆ 5 Rounding decimals more precisely
12◆ 6 Finding the median distance traveled
by students
K I N E S T H E T I C
䉬
T A C T I L E
䉬
V I S U A L
Cross-Curricular Links
Social Studies
Lesson
12 ◆ 1
Students record rates from
Consumer
Lesson 12 ◆ 4 Students learn what a unit
the World Tour section of the Student
price is.
Reference Book.
Lesson 12 ◆ 5 Students share ad information
Lesson 12 ◆ 6 Students discuss and record
they used to find unit prices of items.
their impressions of World Tour experiences.
Science
Literature
Lesson
12 ◆ 2
Students read Each Orange
Had Eight Slices and use rate language
Lesson 12 ◆ 5 Students use rates in terms of
barometric pressure to determine elevation.
▲
to describe each situation.
Differentiation Handbook
See the Differentiation Handbook for materials on Unit 12.
902
Unit 12 Rates
Language Support
Everyday Mathematics provides lesson-specific suggestions to help all
students, including non-native English speakers, to acquire, process,
and express mathematical ideas.
Connecting Math and Literacy
If Dogs Were Dinosaurs, by David M. Schwartz, Scholastic Press, 2005
If You Hopped Like a Frog, by David M. Schwartz, Scholastic Press, 1999
The Grapes of Math, by Gregory Tang, Scholastic Paperbacks, 2004
Unit 12 Vocabulary
comparison shopping
consumer
per
products
rate
rate table
services
unit price
unit rate
Student Reference Book
pp. 239, 247, 254, 271, 297, 299, and 301
Multiage Classroom ◆ Companion Lessons
Companion Lessons from Grades 3 and 5 can help you meet instructional
needs of a multiage classroom. The full Scope and Sequence can be found
in the Appendix.
Grade 3
2 ◆6
Grade 4
12 ◆1
Grade 5
2 ◆6
12 ◆ 2
12 ◆ 3
12 ◆ 5
12 ◆ 6,
12 ◆ 8
7 ◆ 7,
9 ◆5
12 ◆ 4
7 ◆ 7,
9 ◆5
12 ◆ 5
12 ◆ 6
8 ◆ 9,
12 ◆ 4
Professional Development
Teacher’s Reference Manual Links
Lesson
Topic
Section
Lesson
Topic
12 ◆ 1
Collecting and Recording Data
12.2.2
12 ◆ 3
See 12 ◆ 1.
Data Analysis
12.2.4
12 ◆ 4
See 12 ◆ 1.
Rates, Ratios, and Proportions
9.3.3
12 ◆ 5
Calculators
Museums
1.2.6
Rates, Ratios, and Proportions
9.3.3
"What's My Rule?"/Function
Machines
1.3.7
12 ◆ 2
Section
3.1.1
See also 12 ◆ 2.
12 ◆ 6
Basic Facts and Fact Power
16.3.2
Fact Practice
16.3.3
Unit Organizer
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Materials
Lesson
12◆ 1
12◆ 2
12◆ 3
12◆ 4
12◆ 5
12◆ 6
12◆ 7
Masters
Manipulative Kit Items
Other Items
Teaching Aid Masters, pp. 403
and 423
Study Link Master, p. 339
Teaching Master, p. 340
calculator*
slate
timer or clock with a second hand;
index cards; chart paper;
colored pencils or crayons
Study Link 12◆1
Teaching Aid Masters, pp. 388,
389, 443, and 454
transparency of Math Masters,
p. 454*
Study Link Master, p. 341
Game Master, p. 469
slate
deck of number cards
penny; Each Orange Had Eight Slices:
A Counting Book; colored pencils or
crayons; straightedge
Study Link 12◆2
Study Link Master, p. 342
Teaching Masters, pp. 343 and 344
calculator
slate
straightedge
Study Link 12◆3
Teaching Aid Masters, pp. 428
and 454*
Study Link Master, p. 345
Game Master, p. 489
Teaching Master, p. 346
calculator
slate
coins
computer with Internet access
Study Link 12◆4
Study Link Master, p. 347
Game Master, p. 506
Teaching Masters, pp. 348–351
Teaching Aid Master, p. 428
calculator
slate
coins
overhead calculator* ; supermarket ads;
Fraction Cards
Study Link 12◆5
Teaching Aid Masters, pp. 389,
412, and 414–417
Study Link Master, p. 352
Teaching Master, p. 353
calculator
slate
pen or colored pencil; ruler
Study Link 12◆ 6
Assessment Masters, pp. 211–215
and pp. 234–241*
Study Link Masters, pp. 354–357
slate
calculator
* Denotes optional materials
904
Unit 12 Rates
Technology
Assessment Management System, Unit 12
iTLG, Unit 12
Mathematical Background
The discussion below highlights the major content areas presented
in Unit 12 and helps establish instructional priorities.
Solving Rate Problems
(Lesson 12◆ 2)
After students have discussed examples of rates in Lesson 12-1, they
begin to solve rate problems in Lesson 12-2. Rate tables, which some
students may recognize as a special kind of “What’s My Rule?” table, are
also introduced as an aid to problem solving. By completing such tables,
students develop a sense that rate problems usually involve a search for
equivalent rates, leading to the solution of problems. As you and your
students make up and solve problems, three basic types of rate problems
should emerge. These types are illustrated by the following examples:
1. Bill’s new car can travel 35 miles on 1 gallon of gasoline. At
this rate, how far can the car travel on 7 gallons of gasoline?
In this problem, a unit rate (35 miles per gallon) is given, and
the solution is an equivalent rate obtained by multiplication: If
the car can travel 35 miles on 1 gallon, it can travel 7 times as far on
7 gallons. 7 ⴱ 35 245 miles per 7 gallons.
2. Jennifer received an allowance of $8 in 4 weeks. At this rate, how much
allowance did she receive per week?
In this problem, a rate that is not a unit rate ($8 in 4 weeks) is given,
and the solution is an equivalent unit rate obtained by division: If
1
Jennifer received $8 in 4 weeks, she received 4 of $8 in 1 week.
$8 / 4 $2 per week.
3. A gray whale’s heart beats 24 times in 3 minutes. At this rate, how
many times does a gray whale's heart beat in 2 minutes?
In this problem, a rate that is not a unit rate (24 beats in 3 minutes)
is given, and the solution is an equivalent rate that is not a unit rate.
The solution can be obtained by first finding the equivalent unit rate by
division and then using the unit rate to find the solution by multiplication:
1
If the gray whale’s heart beats 24 times in 3 minutes, it beats 3 of
24 times in 1 minute (24 / 3 8) and twice as many times in 2 minutes
(2 ⴱ 8 16 beats per 2 minutes).
To learn more about solving rate problems, see Section 18.4.3
in the Teacher’s Reference Manual.
Unit Organizer
905
The Unit-Rate Strategy
(Lessons 12◆ 2 and 12◆ 3)
The strategy illustrated in the last example—in which rate information
given for a number of things is converted to the equivalent unit rate,
from which other equivalent rates are calculated—is a powerful one.
This strategy is practiced throughout the unit in various pricing and
purchasing exercises, where students explore the “reasonableness” of
rate estimates involving very large numbers. While there are other, more
advanced strategies for dealing with rates, this basic strategy is perhaps
the most universally useful one and the one advocated by many science
and mathematics educators. (See Sci-Math, by Madeline P. Goodstein,
Addison-Wesley, 1983.)
The authors recommend frequent, brief practice sessions for solving such
problems. Use them as Mental Math and Reflexes routines. Even a single
problem, followed by a brief sharing of solutions, should help your
students improve their problem-solving skills.
Units Analysis in Rate Problems
(Lessons 12◆ 3 and following)
Scientists and science educators often complain that mathematics
teaching does not prepare students for the study of science. One of their
main criticisms is that too much teaching of arithmetic deals strictly with
numbers and ignores the role of units of measure that are nearly always
attached to those numbers. With this problem in mind, the authors of
Everyday Mathematics have insisted, starting in Kindergarten, that
nearly every number must come with a count or measure unit. You might
want to extend the work with units to a basic strategy used throughout
the natural sciences, called units analysis. This strategy involves
combining and canceling units in calculations involving measures.
Examples:
◆ There are 6 rows of chairs with 4 chairs per row. How many chairs
are there in all?
4 chairs
6 rows ⴱ 1 row
24 chairs
The “row” units cancel in much the same way as do numbers in the
numerators and denominators in the multiplication of fractions.
◆ Marsha earned $56 for 8 hours of work. How much did she earn
per hour?
56 dollars
8 hours
7 dollars per hour
Dollars divided by hours dollars per hour
You can begin to show the rate units as “word fractions” that come from
the rate fractions. As with many new emphases in Everyday Mathematics,
initial exposure involves simply calling these strategies to the attention
of students, without expecting them to use them, or even to fully
understand them.
More can be learned about units analysis in Section 9.3.3
of the Teacher’s Reference Manual.
906
Unit 12 Rates
Comparison Shopping
(Lessons 12◆ 4 and 12◆ 5)
Lessons 12-4 and 12-5 deal with a specific application of the unit-rate
strategy—calculating unit prices in order to compare the costs of similar
items. This application has become so commonplace that most
supermarkets display unit-price labels along with the price of the items
on their shelves. Students collect real data from newspaper ads, visits to
grocery stores, and other places of business; represent them symbolically,
usually in fraction form; and convert them to decimal form by doing the
indicated division. Performing these operations with a calculator will
usually result in an answer with more decimal places than are needed.
Hence, these activities also provide practice with simplifying complicated
results by rounding.
You may need to review problems involving dollars and cents. For
example, if a 3-pound bag of apples costs $1.49, the unit price is obtained
by dividing $1.49 by 3. On a calculator, this result may be displayed as
0.4966666667. The digit in the first place to the right of the decimal point
represents dimes, the second digit represents pennies, and the third digit
represents tenths of cents. In this problem then, the calculator display is
interpreted as 4 dimes, 9 pennies, 6 tenths of a cent, and so on. This is
equivalent to 49.7 cents when rounded to the nearest tenth of a cent.
See the Teacher’s Reference Manual, Section 9.3.3, for more about
comparison shopping.
World Tour and 50-Facts
Test Wrap-Ups
(Lesson 12◆ 6)
Students have participated in the World Tour Project and the 50-facts test
routine for most of the school year. Be sure to reserve time for at least one
sharing session, in which students describe and compare their experiences
from the World Tour and reflect on individual and class progress on the
50-facts tests.
To learn more about the World Tour Project and the 50-Facts Tests,
see Section 2.7 and Chapter 7, respectively, in the Teacher’s
Reference Manual.
Unit Organizer
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