Download Unit 10: Reflections and Symmetry

Overview
OverviewOverview_Txt
Unit
10 highlights a return to geometry from the point of view of
transformations
or “motions” of geometric figures. It focuses primarily
◆ Overview Text Bullet
on reflections and the related topic of symmetry. Unit 10 also introduces
formal operations with positive and negative numbers. Unit 10 has three
main areas of focus:
◆ To guide the discovery of basic properties of reflections, involving
2-dimensional figures and the connection with line symmetry,
◆ To guide the application of reflections, rotations, and translations, and
◆ To introduce addition involving negative integers.
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Unit 00 Unit Title
Unit 10 Reflections and Symmetry
Contents
Lesson
Objective
10◆ 1
Explorations with a Transparent Mirror
10◆ 2
Finding Lines of Reflection
10◆ 3
Properties of Reflections
10◆ 4
Line Symmetry
10◆ 5
Frieze Patterns
10◆ 6
Positive and Negative Numbers
10◆7
Progress Check 10
Page
794
To guide the exploration of reflections of 2-dimensional figures.
799
To guide the exploration of reflections; and to provide practice identifying
lines of reflection.
805
To guide the discovery of basic properties of reflections.
810
To guide exploration of the connection between reflections and
line symmetry.
816
To guide the application of reflections, rotations, and translations.
822
To introduce addition involving negative integers.
828
To assess students’ progress on mathematical content through the
end of Unit 10.
Unit Organizer
779
Learning In Perspective
780
Lesson Objectives
Links to the Past
Links to the Future
10◆1
To guide the exploration of reflections of
2-dimensional figures.
Grade 3: Explore symmetry with geoboards and
pattern blocks; complete symmetric figures; identify
symmetric shapes and draw lines of symmetry.
Grade 5: Applications and maintenance.
Grade 6: Study rotations, rotational symmetry,
and point symmetry.
10◆2
To guide the exploration of reflections;
and to provide practice identifying lines
of reflection.
Grade 3: Explore symmetry with geoboards and
pattern blocks; complete symmetric figures; identify
symmetric shapes and draw lines of symmetry.
Grade 5: Applications and maintenance.
Grade 6: Study rotations, rotational symmetry,
and point symmetry.
10◆3
To guide the discovery of basic
properties of reflections.
Grade 3: Explore symmetry with geoboards and
pattern blocks; complete symmetric figures; identify
symmetric shapes and draw lines of symmetry.
Grade 5: Applications and maintenance.
Grade 6: Study rotations, rotational symmetry,
and point symmetry.
10◆4
To guide exploration of the connection
between reflections and line symmetry.
Grade 3: Explore symmetry with geoboards and
pattern blocks; complete symmetric figures; identify
symmetric shapes and draw lines of symmetry.
Grade 5: Applications and maintenance.
Grade 6: Study rotations, rotational symmetry,
and point symmetry.
10◆5
To guide the application of reflections,
rotations, and translations.
Grades 1–3: Use straws, geoboards, and body
turns to demonstrate rotations. Grade 3: Model
polygons; change the shapes of constructed
polygons; perform polygon calisthenics to form
polygons and explore their properties.
Grade 5: Transform ordered number pairs and
explore the resulting transformations of geometric
figures. Explore regular tessellations.
Grade 6: Review regular tessellations; introduce
notation for tessellations; find semiregular
tessellations. Create translation tessellations.
Explore topological transformations.
10◆6
To introduce addition involving
negative integers.
Grade 3: Review uses of positive and negative
numbers to relate numbers to a zero point, as in
temperatures and elevations, and to record change.
Solve number stories about positive and negative
numbers.
Grade 4: Play Credits/Debits Game (Advanced
Version) to practice subtraction of positive and
negative integers. Grade 5: Solve addition/
subtraction stories with positive and negative
numbers. Grade 6: Develop a rule for adding and
subtracting positive and negative numbers;
practice adding and subtracting positive and
negative numbers.
Unit 10 Reflections and Symmetry
Key Concepts and Skills
10◆1
10◆2
10◆3
10◆4
10◆5
10◆6
Key Concepts and Skills
Grade 4 Goals*
Describe properties of congruent figures.
Identify, describe, and sketch reflections of two-dimensional figures.
Solve problems involving spatial visualization.
Geometry Goal 2
Geometry Goal 3
Geometry Goal 3
Describe properties of congruent figures, right angles, and perpendicular lines.
Explore lines of reflection and reflected images.
Solve problems involving spatial visualization.
Geometry Goal 2
Geometry Goal 3
Geometry Goal 3
Measure length.
Draw and describe congruent figures.
Explore basic properties of reflections.
Solve problems involving spatial visualization.
Measurement and Reference Frames Goal 1
Geometry Goal 2
Geometry Goal 3
Geometry Goal 3
Identify polygons and describe properties of regular polygons.
Identify and draw lines of symmetry.
Explore the connection between reflections and line symmetry.
Solve problems involving spatial visualization.
Describe rules for patterns and use them to solve problems.
Geometry Goal 2
Geometry Goal 3
Geometry Goal 3
Geometry Goal 3
Patterns, Functions, and Algebra Goal 1
Identify and draw congruent figures.
Identify, describe, and sketch reflections, rotations, and translations.
Extend, describe, and create geometric patterns.
Geometry Goal 2
Geometry Goal 3
Patterns, Functions, and Algebra Goal 1
Compare and order integers.
Add signed numbers.
Identify a line of reflection.
Number and Numeration Goal 6
Operations and Computation Goal 2
Geometry Goal 3
* See the Appendix for a complete list of Grade 4 Goals.
Unit Organizer
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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. Math Boxes also preview
content of the next unit.
Mixed practice
[10◆ 1, 10◆ 4 ], [10◆ 2, 10◆ 5], [10◆ 3, 10◆ 6]
Mixed practice with multiple choice
10◆ 3, 10◆ 4, 10◆ 6
Mixed practice with writing/reasoning opportunity
10◆ 1, 10◆ 3, 10◆ 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
10◆ 1
Over and Up Squares
Locating and plotting points on a
coordinate grid
Measurement and Reference Frames Goal 4
10◆ 2
Dart Game
Experimenting with transparent mirrors
and reflections
Geometry Goal 3
10◆ 2
Pocket-Billiards Game
Experimenting with transparent mirrors
and reflections
Geometry Goal 3
10◆ 2
Angle Tangle
Estimating and measuring angles
Measurement and Reference Frames Goal 1
10◆ 5
Polygon Pair-Up
Identifying properties of polygons
Geometry Goal 2
10◆ 6
Credits/Debits Game
Adding positive and negative numbers
Operations and Computation Goal 2
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 10 Family Letter provides families with an overview, Do-Anytime
Activities, Building Skills Through Games, and a list of vocabulary.
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Unit 10 Reflections and Symmetry
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:
◆ Drawing a picture
◆ Using a pattern
◆ Acting out the problem
◆ Using computation
◆ Using data in a table
Lesson
Activity
10 ◆ 1
Use the transparent mirror to “move” and draw reflected images of shapes.
10 ◆ 2
Play games that involve reflections.
10 ◆ 3
Fold paper to observe reflected images.
10 ◆ 4
Find lines of symmetry of polygons.
10 ◆ 5
Create and continue frieze patterns.
10 ◆ 6
Use credits and debits to practice addition of positive and negative numbers.
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 10, review your Content by Strand Poster to help you
set a monthly pace.
MOST CLASSROOMS
MARCH
APRIL
MAY
NCTM Standards
Unit 10
Lessons
10 ◆ 1
10 ◆ 2
NCTM
1, 3, 6–10 3, 5, 6–10
Standards
10 ◆ 3
10 ◆ 4
10 ◆ 5
10 ◆ 6
10 ◆ 7
1, 3–5,
6–10
3, 6–10
2, 3, 6–10
1, 2, 4,
6–10
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
783
Balanced Assessment
Ongoing Assessment
Recognizing Student Achievement
Opportunities to assess students’ progress toward Grade 4 Goals:
Lesson
10 ◆ 1
Content Assessed
Plot points in the first quadrant of a coordinate grid.
[Measurement and Reference Frames Goal 4]
10 ◆ 2
10 ◆ 3
Compare fractions with like numerators or like denominators; compare fractions to the
benchmark 12. [Number and Numeration Goal 6]
Use a transparent mirror to sketch and describe a reflection.
[Geometry Goal 3]
10 ◆ 4
Describe a pattern and use it to solve problems.
[Patterns, Functions, and Algebra Goal 1]
10 ◆ 5
Identify and sketch an example of a reflection and
identify examples of translations and rotations.
[Geometry Goal 3]
10 ◆ 6
Express the probability of an event as a fraction.
[Data and Chance Goal 4]
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 10 ◆2 Find a common factor to use as intervals when labeling
an axis
Lesson 10 ◆4 Understand the difference between line symmetry and
rotation symmetry
Lesson 10 ◆6 Encourage strategies for adding positive and negative
numbers to evolve over time
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Unit 10 Reflections and Symmetry
Periodic Assessment
10◆7 Progress Check 10
ASSESSMENT ITEMS
CONTENT ASSESSED
Self
Oral/Slate
Written
Name equivalent fractions, decimals, and percents.
[Number and Numeration Goal 5]
✔
✔
✔
Add signed numbers.
[Operations and Computation Goal 2]
✔
✔
✔
✔
✔
Add and subtract fractions.
[Operations and Computation Goal 5]
Make reasonable estimates for multiplication and
division problems. [Operations and Computation Goal 6]
Open Response
✔
Measure an angle.
[Measurement and Reference Frames Goal 1]
✔
✔
Locate multiple lines of symmetry in a two-dimensional
shape. [Geometry Goal 3]
✔
✔
Identify and sketch examples of reflections;
identify examples of translations and rotations.
[Geometry Goal 3]
✔
✔
✔
Portfolio Opportunities
Opportunities to gather samples of students’ mathematical writings,
drawings, and creations to add balance to the assessment process:
◆ Inserting decimal points, Lesson 10◆ 1
◆ Creating a paint reflection, Lesson 10◆ 2
◆ Interpreting a cartoon, Lesson 10◆ 4
◆ Calculating the mean, Lesson 10◆ 5
◆ Creating frieze patterns to practice reflections, rotations, and translations, Lesson 10◆ 5
◆ Solving problems involving pentominoes, Lesson 10◆ 7
Assessment Handbook
Unit 10 Assessment Support
◆ Grade 4 Goals, pp. 37–50
◆ Unit 10 Assessment Overview, pp. 126–133
◆ Unit 10 Open Response
• Detailed rubric, p. 130
• Sample student responses, pp. 131–133
Unit 10 Assessment Masters
◆ Unit 10 Self Assessment, p. 200
◆ Unit 10 Written Assessment, pp. 201–203
◆ Unit 10 Open Response, pp. 204 and 205
◆ Unit 10 Class Checklist, pp. 284, 285, and 303
◆ Unit 10 Individual Profile of Progress, pp. 282, 283, and 302
◆ Exit Slip, p. 311
◆ Math Logs, pp. 306–308
◆ Other Student Assessment Forms, pp. 304, 305, 309, and 310
Unit Organizer
785
Differentiated Instruction
Daily Lesson Support
ENGLISH LANGUAGE LEARNERS
EXTRA PRACTICE
10◆ 2 Building a Math Word Bank
10◆ 4 Creating a Line Symmetry Museum
10◆ 4 Creating shapes with line symmetry
10◆ 5 Creating frieze patterns
5-Minute Math 10◆ 3 Solving problems
involving reflections
READINESS
ENRICHMENT
10◆ 1 Exploring reflections with paper
folding
10◆ 2 Creating a paint reflection
10◆ 3 Creating reflections
10◆ 5 Exploring geometric patterns
10◆ 6 Using a calculator to skip count
10◆ 6 Using a number line to add positive
and negative integers
10◆ 1 Exploring shadows and reflections
10◆ 2 Solving paper-folding puzzles
10◆ 2 Using technology to investigate the
mirror as a virtual manipulative
10◆ 3 Constructing 3-dimensional buildings
and their reflections
10◆ 4 Interpreting a cartoon involving
line symmetry
10◆ 4 Exploring turn or rotation symmetry
10◆ 5 Exploring arrangements of four straws
10◆ 5 Exploring tessellations
Adjusting the Activity
10◆2 Devising a scoring system for
Pocket-Billiards Game
10◆3 Determining the latitude and longitude
of a capital city
10◆4 Summarizing the discussion of line
symmetry ELL
A U D I T O R Y
䉬
K I N E S T H E T I C
10◆4 Having a volunteer cut out shapes
ahead of time
10◆5 Sketching frieze patterns on
centimeter grid paper ELL
10◆6 Experimenting with a calculator
䉬
T A C T I L E
䉬
V I S U A L
Cross-Curricular Links
Language Arts
Technology
Lesson 10◆ 1 Students discuss the use of the prefix pre-.
Lesson 10◆ 2 Students investigate the
use of a mirror as a virtual
manipulative.
Lesson 10◆ 5 Students use technology
to explore tessellations.
Art
Lesson 10◆ 2
of reflection.
Lesson 10◆ 5
their own.
Lesson 10◆ 5
Lesson 10◆ 5
Students create a paint reflection to explore the concept
Students complete three frieze patterns and design one of
Social Studies
Students create and continue patterns using pattern blocks.
Students make frieze patterns by following directions.
Lesson 10◆ 3 Students visit a second
country in Region 4.
Using the Projects
Use Project 4, Making a Quilt, after Unit 10 to explore and apply ideas of pattern, symmetry,
rotation, and reflection in the context of quilts. See the Differentiation Handbook for
modifications to Project 4.
Differentiation Handbook
See the Differentiation Handbook for materials on Unit 10.
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Unit 10 Reflections and Symmetry
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
Lesson 10◆ 1 Shadows and Reflections, by Tana Hoban, Greenwillow, 1990
Lesson 10◆ 3 Count Your Way through Japan, by Jim Haskins, Carolrhoda
Books, 1988
Lesson 10◆ 3 Count Your Way through Russia, by Jim Haskins, Carolrhoda
Books, 1987
Lesson 10◆ 3 Count Your Way through India, by Jim Haskins, Carolrhoda
Books, 1992
Lesson 10◆ 3 Count Your Way through Israel, by Jim Haskins, Carolrhoda
Books, 1992
How the Second Grade Got $8,205.50 to Visit the Statue of Liberty,
by Nathan Zimelman, Albert Whitman & Company, 1992
Reflections, by Ann Jonas, Greenwillow, 1987
Unit 10 Vocabulary
congruent
credit
debit
frieze pattern
image
line of reflection
line of symmetry
opposite (of a number)
preimage
recessed
reflection
reflection (flip)
rotation (turn)
rotation (turn) symmetry
symmetric
translation (slide)
transparent mirror
Student Reference Book
pp. 60, 108, 109, 230, 238, 257, and 258
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
6◆ 3,
6◆ 9
6◆ 9
6◆ 3,
6◆ 9
6◆ 3,
6◆ 9
1◆ 12,
6◆ 3
Grade 4
10◆1
10◆ 2
10◆ 3
10◆ 4
10◆ 5
10◆ 6
3 ◆8
3 ◆8
7 ◆ 7–7 ◆ 9,
7 ◆ 11
Grade 5
Professional Development
Teacher’s Reference Manual Links
Lesson
Topic
Section
Lesson
Topic
Section
10 ◆ 1
Geometry Tools
3.2.4
10 ◆ 4
Line Symmetry
13.8.1
Transparent Mirrors
13.13.5
Polygons
13.4.2
Reflections, Rotations, and
Translations
13.7.1
Congruence and Similarity
13.6.2
10 ◆ 2
10 ◆ 3
See 10 ◆ 2.
10 ◆ 5
See 10 ◆ 2.
10 ◆ 6
Positive and Negative Numbers
9.4
Operations with Positive and
Negative Numbers
10.2.2
Unit Organizer
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Materials
Lesson
10◆ 1
10◆ 2
10◆ 3
10◆ 4
10◆ 5
10◆ 6
10◆ 7
Masters
Manipulative Kit Items
Other Items
Teaching Masters, pp. 304 and
305
Study Link Master, p. 306
Game Master, p. 494
Teaching Aid Master, p. 389
slate
2 six-sided dice
transparent mirror
sheets of paper; colored pencils;
crayons* ; scissors; Shadows and Reflections
Study Link 10 ◆ 1
Study Link Master, p. 307
Game Master, p. 457
Teaching Master, p. 308
Teaching Aid Master, p. 389
slate
Geometry Template
transparent mirror
ruler; large sheets of paper; paints,
brushes, and dark marker; computer with
Internet access (if available); scissors;
protractor
Study Link 10 ◆ 2
Teaching Master, p. 309
Study Link Master, p. 310
Teaching Aid Masters, pp. 389,
403, 419–421* , and 447
slate
centimeter cubes
pattern blocks
transparent mirror
ruler; blank paper
Study Link 10 ◆ 3
Teaching Masters, pp. 311–314,
316, and 317
Teaching Aid Master, p. 389
Study Link Master, p. 315
slate
pattern blocks
Geometry Template
transparent mirror
scissors; magazines and newspapers; tape
Study Link 10 ◆ 4
Teaching Aid Master, p. 389, 403* ,
and 437
Study Link Master, p. 318
Teaching Master, p. 319
Geometry Template
straws
pattern blocks
slate
transparent mirror
straightedge; computer with Internet access;
index cards; scissors; overhead or regular
pattern blocks* ; Polygon Pair-Up Polygon
Cards and Property Cards
Study Link 10 ◆ 5
Teaching Master, p. 320
Game Master, p. 468
transparencies of Math Masters,
pp. 318* and 321*
Study Link Master, p. 322
deck of number cards
calculator
number line
transparent mirror
masking tape
Study Link 10 ◆ 7
Assessment Masters, pp. 200–205
Study Link Masters, pp. 323–326
slate
transparent mirror
Geometry
Template
scissors
* Denotes optional materials
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Unit 10 Reflections and Symmetry
Technology
Assessment Management System, Unit 10
iTLG, Unit 10
Mathematical Background
The discussion below highlights the major content ideas presented
in Unit 10 and helps establish instructional priorities.
Types of Geometry
(Lessons 10◆ 1 and following)
You may remember your high school geometry course as dealing with
definitions, axioms, theorems (“Given ..., To Prove ...”), and perhaps
straightedge-and-compass constructions. This form of synthetic geometry
was first developed by Euclid about 300 B.C. and has been the model for
teaching geometry ever since.
10
(3,9)
9
8
7
6
However, there are two modern geometries that cover the same topics:
5
Analytic geometry The study of figures in a coordinate plane.
3
(8,5)
4
2
Transformation geometry The study of certain operations on figures.
These operations, or “transformations,” produce figures that are the same
shape as (similar to) the original figures, or the same size and shape as
(congruent to) the original figures.
1
0
0
(1,1)
1
2
3
4
5
6
7
8
9 10
Figure in a coordinate plane
These two geometries are probably more useful than synthetic geometry.
Both are featured in Grades 4–6 of Everyday Mathematics, with Unit 10
introducing the transformation approach.
Learn more about types of geometry in Chapter 13 of the Teacher’s
Reference Manual.
“Isometric” or “Congruence”
Transformations
(Lessons 10◆ 1 and following)
You may remember from your high school geometry course the
emphasis on congruent figures, especially on proving triangles
congruent by theorems called “side-angle-side” or “SAS,” and
so on.
This topic is handled in transformation geometry by rigid
motions or isometric transformations, which do not
change the size or shape of figures. These transformations—
translations (slides), reflections (flips), and rotations
(turns)—can duplicate any figure. (SAS and similar
theorems of synthetic geometry apply only to triangles.)
See the Teacher’s Reference Manual, Section 13.7,
for more information about isometric transformations.
Unit Organizer
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Reflections (Flips) and
Symmetry with Transparent
Mirrors (Lessons 10◆ 1–10◆ 4)
Everyone is familiar with mirrors and the exact—but reversed—images
one sees in them. The device used in this unit, the transparent mirror,
has an advantage over a regular mirror: It allows students to look through
a mirror and reach behind it to touch or trace the mirror image (almost
like Alice going “through the looking glass”).
Transparent mirror
As with any new tool, developing the skills for its use takes time, practice,
and patience. For accurate placement of images, have students practice
these skills:
◆ Lean down and look directly through the transparent mirror.
◆ Use the ends of the transparent mirror to keep it perpendicular
to the paper.
◆ Use the inner part of the recessed edge to place the transparent
mirror on points or lines or to draw mirror lines.
◆ Hold the transparent mirror firmly in position with one hand while
drawing behind it or along its recessed edge. (This is one of the main
skills to be learned.)
It is probably a good idea to acquire or practice these skills yourself
before teaching the lessons. Do the mirror exercises on journal pages and
masters until you feel comfortable using the transparent mirror.
To learn more about reflections and symmetry with a transparent mirror,
refer to Section 13.8 in the Teacher’s Reference Manual.
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Unit 10 Reflections and Symmetry
Transformations as Design Tools
(Lesson 10◆ 5)
Geometric patterns are part of many designs—in arts and crafts from
around the world, in architecture and engineering, and in paintings and
other works of art (sometimes in disguised forms). Lesson 10-5, on frieze
patterns, encourages students to explore reflections, symmetry, rotations,
and translations in order to analyze and create designs. The authors
believe that fourth graders will find these design tasks enjoyable. Some of
their creations may be quite elegant.
An interesting property of transformations is that two successive
reflections across parallel mirror lines are equivalent to one translation.
(The original image is reversed in the first reflection, but the mirror image
of the first reflection can be a translated image.) Hence one can make
friezes either by translating and tracing or by using transparent mirrors
twice for each frieze copy.
The authors hope that the principles learned here can be linked to
teaching visual arts in your school. Geometry that is useful in both art
and practical matters will be applied many times throughout Everyday
Mathematics.
For more about transformations as design tools, see Section 13.8
in the Teacher’s Reference Manual.
哬
哬
Unit Organizer
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Operations with Positive
and Negative Numbers
(Lesson 10◆ 6)
Since Kindergarten, Everyday Mathematics
students have been using positive and negative
numbers to identify locations on timelines, number
lines, number grids, and thermometers. Since first
grade, students have informally used addition and
subtraction in going from one place to another and
in finding distances. But Lesson 10-6 may be
students’ first exposure to operations with positive
and negative numbers. In this lesson, the numbers
are limited to integers—whole numbers and their
(negative) opposites.
“Credits and debits” number stories are used to
help make addition concrete. Single-digit numbers
ensure that most problems can be done mentally.
It is important in these “accounting” situations to
name both the operation and the number.
◆ “Add +$3” is read “Add positive 3 dollars”
(a credit transaction).
◆ “Add –$5” is read “Add negative 5 dollars”
(a debit transaction).
This distinguishes the addition operation from the
numbers involved.
Later lessons in Grade 4 introduce subtraction of
positive and negative numbers and teach the use
of the “change sign” key, which enables calculators
to work with negative as well as positive numbers.
Many practice and review exercises are included.
Other operations with positive and negative
numbers, as well as applications using them, are
featured in Grades 5 and 6.
See Section 10.2.2 of the Teacher’s
Reference Manual for more information
about operations with positive and negative
numbers.
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Unit 10 Reflections and Symmetry
Confusing Notation for Positive and
Negative Numbers (Lesson 10◆ 6)
The use of the same notation with several meanings can be confusing.
This is true of the symbol “”:
◆ The symbol “” attached to a numeral, as in 3, 0.5, or
37, is read “negative” and is used in naming numbers on
the number line (“negative three,” “negative five-tenths,”
“negative thirty-seven”).
◆ The symbol “” in a number model, preceding a positive
or negative number, as in (3) or (17), is read
“opposite of.” The opposite of a positive number is a
negative number; the opposite of a negative number is
a positive number. For example, the “opposite of
positive 3” is negative 3, and the “opposite of negative 17”
is positive 17.
9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9
Opposites
◆ The symbol “” in a number model, as in 17 3 14, is
read “minus,” “subtract,” or “take away” and indicates
the familiar subtraction operation.
Example 17 3 14 is read “17 minus 3 equals 14.”
17 (3) 14 is read “17 minus 3 equals 14.”
(17) 3 14 is read “the opposite of negative 17 minus
3 equals 14.”
17 (3) 14 is read “17 plus negative 3 equals 14.”
17 (3) 20 is read “17 minus negative 3 equals 20.”
The meanings of the symbol “” can get quite tangled in number models
like 17 3 = 20 (“Negative 17 minus 3 equals negative 20”) or
12 (4) = 8 (“12 take away the opposite of negative 4 equals 8”).
Some mathematics programs of the past tried to reduce confusion by using
“” only for subtraction. Positive and negative numbers were designated
with small raised symbols (for example, 3, 7, 17), and opposites were
indicated by “opp.” But everyday usage and nearly all algebra books
continued to use the traditional notation, so students eventually had to
reconcile the two notations.
Given the problems associated with both notations, the authors have
decided to use the traditional system. Help students sort it out when you
read expressions by consistently saying “plus” or “minus” for addition
or subtraction, and “positive,” “negative,” or “opposite” for numbers, as
indicated by the context. Encourage students to do likewise when they
eventually read expressions to each other and to themselves.
Note
Public or private
speech is very helpful
in dealing with
complexities of
meanings. It is verifiable
and observable
common sense for
students to use speech,
as well as sight, to sort
out complicated
symbolic expressions.
(This is a key tenet of
the learning theory of
Lev S. Vygotsky. An
excellent article on
Vygotskian learning
theory is “Why
Children Talk to
Themselves,” by
Laura E. Berk, in the
November 1994 issue
of Scientific
American.)
To further explore confusing notation for positive and negative numbers,
refer to Section 9.4.1 in the Teacher’s Reference Manual.
Unit Organizer
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